Forward indifference valuation and hedging of basis risk under partial information
aa r X i v : . [ q -f i n . M F ] J a n Forward indifference valuation andhedging of basis risk underpartial information
Mahan TahvildariLincoln CollegeMathematical InstituteUniversity of OxfordA thesis submitted in partial fulfilment of the
Master of Science in Mathematical Finance
December 16, 2018 dedicate this work to my parents. “Be happy for this moment. This moment is your life.”
Omar Khayy´am, Persian mathematician, astronomer, and poet cknowledgments
I would like to thank my supervisor Professor Dr Michael Monoyios for his guidance,constructive suggestions and encouragement for this research work. I would also liketo express my great appreciation to my employer d-fine, that made it possible forme to study at Oxford University by granting me a full scholarship. Finally, I wishto thank my parents for their restless support and encouragement throughout mystudies. bstract
We study the hedging and valuation of European and American claims ona non-traded asset Y , when a traded stock S is available for hedging, with S and Y following correlated geometric Brownian motions. This is an incompletemarket , often called a basis risk model . The market agent’s risk preferences aremodelled using a so-called forward performance process (forward utility) , whichis a time-decreasing utility of exponential type. Moreover, the market agent(investor) does not know with certainty the values of the asset price drifts.This market setting with drift parameter uncertainty is the partial informationscenario . We discuss the stochastic control problem obtained by setting upthe hedging portfolio and derive the optimal hedging strategy . Furthermore, a (dual) forward indifference price representation of the claim and its PDE areobtained. With these results, the residual risk process representing the basisrisk (hedging error), pay-off decompositions and asymptotic expansions of theindifference price in the European case are derived. We develop the analogous stochastic control and stopping problem with an American claim and obtain thecorresponding forward indifference price valuation formula. ontents References 52
Introduction
A fundamental theory of mathematical finance is the problem of a market agentwho invests in a financial market in order to maximise trxivehe expected utility ofhis terminal wealth under his individual preferences. Problems of expected utilitymaximisation go back at least to the two seminal articles of Merton [90], [91] (see alsoMerton [92]), who studied the framework of time-continuous models, and the seminalarticle of Samuelson [119] treating the time-discrete case. Merton derived a non-linearpartial differential equation (Hamilton-Jacobi-Bellman (HJB) equation) for the valuefunction of the maximisation problem using methods from stochastic control theory.The modern approach for solving such problems uses dual characterisations ofportfolios through defining an appropriate set of martingale measures. Harrison andPliska [49] developed a general stochastic model of a continuous, multi-dimensional, complete market and obtained the corresponding general Black-Scholes pricing for-mula. The setting of a complete market, where the martingale measure is unique,was also studied by Pliska [112], Cox and Huang [27], [28] and by Karatzas, Lehoczkyand Shreve [70]. One of the main results is, that the marginal utility of the optimalterminal wealth is equal to the density of the martingale measure modulo a constant.The setting of an incomplete market, where perfect hedging is not possible, is amore difficult case and was studied via time-discrete models by He and Pearson [50]and by Karatzas, Lehoczky, Shreve and Xu [71], who realised that the use of dualmethods from convex analysis provided comprehensive solutions to stochastic controlproblems. The dual variational problem of the primal problem is formulated andsolved with convex dual relationship as Bismut [17] demonstrated. Kramkov andSchachermayer [77] studied the classical utility maximisation problem under weakerassumptions on the model and on the utility function. Rogers [115] delved deeperinto the theory by applying methods from functional analysis and presented variousexamples solved with duality methods (see also ˇZitkovi´c [138] and Berrier, Rogersand Tehranchi [16]). Davis [29], [30], Rouge and Karoui [118], Henderson and Hob-son [51], [53], Musiela and Zariphopoulou [104] investigated utility-based hedging inan incomplete market case, where hedging becomes imperfect and a hedging error ,the basis risk , remains. It is the risk associated with the trading of a derivative se-curity on a non-traded underlying asset, hedged with a imperfectly correlated tradedasset. Examples are weather derivatives or options on illiquid securities. Ankirch-ner and Imkeller [6] introduced a typical example for a cross hedge, where an airlinecompany wants to manage kerosene price risk. Ankirchner et al. [5], [7], [8] dealt1lso with applied basis risk models. Monoyios [95] derived perturbation series givingaccurate analytic approximations for the price and hedging strategy of the claim us-ing an exponential utility and carried out an numerical performance analysis betweenthe improved optimal hedge and the naive hedge with the traded asset. Kallsen andRheinl¨ander dealt with classical utility-based pricing and hedging using an quadratichedging approach and extended the results obtained by Mania and Schweizer [87],Becherer [13] and Kramkov and Sˆırbu [78]. Zariphopoulou [136] studied optimisationmodels with power utility and produced reduced form solutions of the indifferenceprice by applying a distortion method to the indifference price PDE. The settingwith exponential utility in a multi-dimensional model was treated by Musiela andZariphopoulou [104]. Monoyios [96] derived representations for the optimal martin-gale measures in a two-factor Markovian model by using the distortion power solutionfor the primal problem to obtain a dual entropic representation of the stochastic con-trol problem. We refer to the introductions of the aforementioned papers for morereferences in the field of classical utility-based optimisation problems.Monoyios [97] explored the impact of drift parameter uncertainty in an incompletemarket model having an European option on a non-traded asset hedging a correlatedtraded stock. He developed analytic expansions for the indifference price and hedgingstrategies. The key approach is the development of a filtering approach , the
Kalman-Bucy filter , in which the investor updates the market price of risk parameter from theobservations of the asset prices. Applications of filtering can be found in Kallianpur[65], Rogers and Williams [116] and Fujisaki et al. [47]. Filtering originates fromsignal processing by Wiener [135] and Kolmogorov [60] during the 1940s. In the1960s, it was further development by Kalman and Bucy [67], [68]. The setting, inwhich the investor does not observe the assets’ Brownian motions is called the partialinformation scenario . Problems under partial information scenarios were also studiedby Rogers [114], Lakner [82] and Brendle [21]. Monoyios [98] used a two-dimensionalKalman-Bucy filter with Gaussian prior distribution in a partial information modeland derived the optimal hedging strategy and indifference price representations usingdual methods. Dependent on the prior estimations of the asset price drifts, the pricerepresentations formulas uses the minimal entropy martingale measure or the minimalmartingale measure .Musiela and Zariphopoulou [105], [106], [107] introduced a new class of forwardutilities (forward performances) that are generated forward in time. They discussedassociated value functions, optimal investment strategies and indifference price rep-resentations. They defined the concept of forward performance processes in order to2uantify the dynamically changing preferences of an investor. Independently, Hen-derson defined in [52] and Henderson and Hobson [54] analysed the same class ofdynamic utilities, but called them horizon-unbiased . Forward utilities are defined bythe dynamic programming principle and ensure more flexibility as they are specifiedfor today and not for a fixed future time. Berrier and Theranchi [15] broadened thedefinitions by adding a process for the investor’s consumption.In this work, we investigate the utility-based valuation of European and Americanclaims on a non-traded asset Y , when a correlated traded stock S is available forhedging, with S and Y following correlated geometric Brownian motions, and whenthe agent’s risk preferences are modelled using the forward performance process fromMusiela and Zariphopoulou [106], and when the agent does not know with certaintythe values of the asset price drifts. Since the market becomes incomplete, we retainan unhedgeable (basis) risk. The basis risk model will first be constructed undera standard full information hypothesis, where the drifts of both assets are knownconstants. In this setting, the utility-based valuation of European claims on Y hasbeen well studied, using classical (as opposed to forward) exponential utility. Thepartial information case, where the asset drifts are taken as unknown constants, whosevalues are filtered from price observations, has also been studied for European claimvaluation using classical utility (see, for example, Monoyios [98]). The thesis willinvestigate the valuation and hedging of European and then American claims on Y with a exponential forward utility under partial information. We apply the partialinformation model with the Kalman-Bucy filter from Monoyios [98] to get analogousresults for valuation and hedging with forward instead of classical utility. The novelapproach is the embedding of the specific partial information model, making themarket prices of risks depending on both asset prices, into the aforementioned forwardperformance framework. We compare the optimal hedging strategies and indifferenceprice representations for European and American claims associated with forward andclassical utility under the partial information scenario. One of the key results isthe change of optimal measure from the minimal entropy martingale measure Q E to the minimal martingale measure Q M . In the European option’s case, we obtainthe dual representation of the forward indifference price with its semi-linear PDE ofsecond order, the residual risk, a pay-off decomposition of the European claim and anasymptotic expansion of the forward indifference price. In the case of an Americanclaim, we define the control and stopping problem and derive the dual representationof the forward indifference price under the partial information model.Oberman and Zariphopoulou [108] and Leung and Sircar [83] studied the valua-3ion and hedging of American options in a basis risk model under full informationusing classical utility. Leung, Sircar and Zariphopoulou [84] investigated the full in-formation model using forward utility to price executive stock options (ESOs) . ESOsare American calls issued by a company to its employees (mostly executives) as aform of variable payments as instruments for motivation (cf. Kraizberg et al. [76],Chen et al. [23], Brandes et al. [20]). We extend the framework of Leung, Sircarand Zariphopoulou [84] to the partial information model, derive the indifference pricevaluation and prepare the groundwork for future applications.The remainder of the dissertation is organised as follows. In Section 2 the basisrisk market model in the full and partial information scenarios, the concept of filteringand forward utilities defined via a certain class of risk tolerance functions are treated.The forward utility-based valuation and hedging problem with an European option isdealt in Section 3. It begins with the setting of perfect hedging in a complete marketand continues with the incomplete market case, followed by the formulation of theperformance maximisation of the investor’s hedging portfolio. The problem is solvedwith dual methods and results in the optimal hedging strategy and the dual repre-sentation formula for the forward indifference price. Furthermore, the residual riskof the strategy, option’s pay-off decompositions and an asymptotic expansion for theforward indifference price are derived. In Section 4 we set up the partial informationmodel with an American option, which can be early exercised and develop the (dual)optimal control and stopping optimisation problem and obtain the entropic represen-tation of the forward indifference price. We conclude in Section 5 by performing ananalysis of essential model assumptions and results obtained in this work, and discussalternatives from present topics as well as future directions for research. In this section the financial market is modelled by a basis risk model premised on thegeometric Brownian motion. A distinction is made in the assumption of the assetprice
Sharpe ratios , which leads to the full information scenario for certain Sharperatios and partial information scenario for uncertain Sharpe ratios. The
Kalman-Bucy filtering approach is developed and applied to the partial information scenarioto transform it into the case of full information. Lastly, the concept of forward utility is introduced as a dynamic extension of the classical utility theory and used withinthe basis risk model. Herein, a useful function called local risk tolerance serves theclassification of forward utility functions . 4 .1 Full information scenario
The classical basis risk model defined in this subsection was initially explored byDavis [30]. Consider a filtered probability space (Ω , F , F := ( F t ) ≤ t ≤ T , P ) as thesetting of a financial market, where the terminal filtration F is generated by the two-dimensional standard P -Brownian motion ( W S , W ⊥ ) with correlation between theWiener processes W S := ( W St ) ≤ t ≤ T and W ⊥ := ( W ⊥ t ) ≤ t ≤ T . A traded stock price S := ( S t ) ≤ t ≤ T follows a geometric Brownian motion process given byd S t = σ S S t ( λ S d t + d W St ) , (1)in which the stock’s volatility σ S > market price of risk (MPR) or Sharperatio λ S = µ S − r m σ S with drift µ S are known constants. For simplicity, the risk-freemarket interest rate r m is taken to be zero. A non-traded asset Y := ( Y t ) ≤ t ≤ T follows the correlated geometric Brownian motiond Y t = σ Y Y t ( λ Y d t + d W Yt ) , (2)with σ Y > λ Y known constants. The Brownian motion W Y := ( W t ) ≤ t ≤ T fromthe non-traded asset dynamics is correlated with the stock’s Brownian motion W S according to W Yt = ρW St + p − ρ W ⊥ t with a known constant ρ ∈ [ − ,
1] as thecorrelation coefficient. In the case | ρ | = 1, the market is called complete and perfecthedging is possible; see Subsection 3.1. If | ρ | 6 = 1, the market is called incomplete .An investor with initial wealth x > F -predictable (portfolio or trading) strategy θ := ( θ t ) ≤ t ≤ T ( π := ( π t ) ≤ t ≤ T ), that isan S -integrable process representing the number of shares held in the portfolio (re-spectively the cash amount π t := θ t S t invested in the stock). Contextually, both θ and π are called strategy. Under self-financing trading condition, the investor’s portfoliowealth is denoted by the positive process X := ( X t ) ≤ t ≤ T and satisfiesd X t = θ t d S t = σ S π t ( λ S d t + d W St ) , X = x . (3)The process ( θ · S ) = (( θ · S ) t ) ≤ t ≤ T given by the stochastic integral( θ · S ) t := Z t θ u d S u = Z t d X u = X t − x , represents the profit and loss from trading up to time t ∈ [0 , T ]. The next definitiongives the space of admissible trading strategies to make the market model suitable formeasure changes. 5 efinition 2.1.1 (Relative entropy and admissible strategies) . The set of equivalentlocal martingale measures M e := { Q ∼ P | S is a local ( Q , F )-martingale } and itssubset M e,f := { Q ∈ M e | H ( Q , P ) < ∞} of measures with finite relative entropy H ( Q , P ) := E (cid:20) d Q d P log d Q d P (cid:21) (4)between Q and P are assumed to be non-empty. The set of admissible strategies isΘ := { θ ∈ Θ p | ( θ · S ) is a ( Q , F )-martingale for all Q ∈ M e,f } , (5)where Θ p is the superset of ( P , F )-predictable and S -integrable strategies. An admis-sible strategy satisfies R t π u d u < ∞ almost surely for each t ∈ [0 , T ]. △ Condition (5) for admissible strategies is taken from Becherer [12, pp. 28–29] (seealso Mania and Schweizer [87, p. 2116]) and appears as one of the candidate sets(Θ ) examined in Delbaen et al. [33, p. 104]. The latter paper prove that for threedifferent choices of Θ the resulting primal and dual problem have the same value andthus establish in particular a robustness result for the duality of classical exponentialutility-based hedging.The relative entropy was introduced in information theory by Kullback and Leibler[80] and developed by Kullback in his book [79]. It is H ( Q , P ) ≥ Q = P . The profit and loss process ( θ · S ) is identical to ( X t − x ) ≤ t ≤ T and hencethe martingale property in (5) holds also for the wealth process X . With the choice ofadmissible strategies, arbitrage opportunities for the investor are excluded. More onarbitrage and self-financing strategies can be found in Jeanblanc et al. [61, pp. 81–84].Since the MPRs λ S , λ Y are assumed to be constants, the investor has access to theso-called background filtration F and hence is able to observe the Brownian motionprocess ( W S , W ⊥ ), as well as the stock price process S . This set-up is referred to asa full information scenario . Remark 2.1.2 (Solution to the stock price SDE) . To solve (1), firstly apply Itˆo’slemma on the logarithmic stock prices,d(log S t ) = d S t S t −
12 d h S i t S t = σ S (cid:18)(cid:18) λ S − σ S (cid:19) d t + d W St (cid:19) , and the integrate over [0 , t ] to obtain S t = S exp (cid:18) σ S (cid:18) λ S − σ S (cid:19) t + W St (cid:19) . If the MPR or volatility were not constant, then an integral would remain in theexpressed solution. ♦ .2 Partial information scenario Based on historical data analyses from Ang and Bekaert [4], Clarke et al. [24] andFrench et al. [45], one might assume that the parameter values for an annual stockreturn (drift) and volatility are µ S = 8% and σ S = 16% respectively, so that theSharpe ratio is λ S = 0 . λ S , λ Y in the asset price dynamics (1), (2) is practicallyimpossible, due to the lack of long-term historical market data. The subsequentargument for the MPR parameter uncertainty is taken from Monoyios [97, pp. 342–343]. The normalised stock returns σ S d S t S t = λ S d t + d W St can be observed by theinvestor over a time interval [0 , t ], to make the best estimate λ S ( t ) = 1 t Z t σ S d S u S u = λ S + W St t ∼ N (cid:18) λ S , t (cid:19) , which leads to a 95% confidence interval h λ S ( t ) − . √ t , λ S ( t ) + . √ t i for λ S . In orderto determine with 95% confidence the observation time t for the estimated value λ S ( t )being 5% to within of its true value λ S , meaning (cid:12)(cid:12)(cid:12) λ S ( t ) − λ S (cid:12)(cid:12)(cid:12) ≤ .
05, the equality . √ t = 0 .
05 needs to be solved, which gives t ≈ σ S , σ Y and the correlation ρ are assumed to be known con-stants, because they can be inferred from quadratic and co-variationsd h S i t = ( σ S ) S t d t, d h Y i t = ( σ Y ) Y t d t, d h S, Y i t = ρσ S σ Y S t Y t d t, through the best estimators σ S = s t Z t d h S i u S u , σ Y = s t Z t d h Y i u Y u , ρ = 1 σ S σ Y t Z t d h S, Y i u S u Y u . when price observations are taken to be approximately continuous. The problemof estimating quadratic variation using realised variance is discussed in Barndorff-Nielsen and Shephard [10]. Chakraborti et al. [22] analysed asset correlations onan empirical basis. If the requirement of constant MPRs is omitted, the agent willhave no access to the background filtration F , but instead, only to the so-called observation filtration b F := ( b F t ) ≤ t ≤ T , which is generated by the asset price processes7 and Y . Hence, only the observation of ( S, Y ) but not the Brownian motion process( W S , W ⊥ ) is possible. The values of the parameters λ S , λ Y become uncertain, sothey can be modelled as random variables. This set-up is referred to as a partialinformation scenario . An agent with full information (partial information) is called outsider (insider) (see Henderson, Klad´ıvko and Monoyios [55]). General filtering theory deals with the estimation of an unobservable stochastic pro-cess given a related observable process. Treatments of filtering theory can be found inKallianpur [65, Chapter 10], Rogers and Williams [116, pp. 322–331] and Fujisaki etal. [47]. Wiener [135] and Kolmogorov [60] paved the way for filtering problems in thefrequency domain in signal processing theory during the 1940s. In the 1960s linearfiltering theory was developed further by Kalman [67] and Kalman and Bucy [68],where filtering problems were considered in the time rather than frequency domainwith state space representations.The partial information scenario can be converted into a full information scenarioby changing from the background to the observation filtration using the so-called
Bayesian approach in a
Kalman-Bucy filtering framework . Following Monoyios [98],the asset MPRs are modelled as F -random variables with a given initial distributionconditional on b F . Definition 2.3.1 (Observation and signal process) . Define the two-dimensional observation process
Ξ := (Ξ t ) ≤ t ≤ T byΞ t := ξ St ξ Yt ! := σ S R t S u S u σ Y R t Y u Y u ! = λ S t + W St λ Y t + W Yt ! , (6)given the dynamics (1) and (2), generating the observation filtration b F := ( b F t ) ≤ t ≤ T , b F t := ( ξ Su , ξ Yu | ≤ u ≤ t ). The corresponding unobservable signal process is given byΛ := λ S λ Y ! , (7)which is an unknown two-dimensional constant in this market model. Moreover,assume a Gaussian prior distribution Λ | b F ∼ N (Λ , Σ ) , Λ := λ S λ Y ! , Σ := z S c c z Y ! , c := ρ min { z S , z Y } , (8)for given constant parameters λ S , λ Y , z S , z Y . △ S t = S exp (cid:0) σ S (( λ S − σ S ) t + W St ) (cid:1) ,Y t = Y exp (cid:0) σ Y (( λ Y − σ Y ) t + W Yt ) (cid:1) , (9)from which the observation process may be expressed as deterministic functions ofthe asset prices and time, ξ St = ξ S ( t, S t ) = 1 σ S log (cid:18) S t S (cid:19) + 12 σ S t, ξ Yt = ξ Y ( t, Y t ) = 1 σ Y log (cid:18) Y t Y (cid:19) + 12 σ Y t. For any process ζ expressed by a function of time and current asset prices, the abbre-viation ζ t := ζ ( t, S t , Y t ) may be used. The SDEs of the observation and signal process(6), (7) are dΞ t = Λ d t + ρ p − ρ ! W St W ⊥ t ! , dΛ = ! . According to (8), an unbiased estimator of Λ is Gaussian with initial estimations for λ S , λ Y , z S , z Y .The idea behind the Kalman-Bucy filter is to choose a prior distribution withspecific parameter values for the MPR process Λ and continuously update it overtime. The prior distribution initialises the probability law of Λ conditional on b F ,and through filtering done in the next definition, this is updated with the evolutionof the asset prices under the observation filtration b F . An in-depth discussion of thisfiltering procedure is made in Section 5. We describe in Remark 2.3.4 a partialinformation model apart from the Kalman-Bucy filter. Definition 2.3.2 (Kalman-Bucy filter) . The optimal filter process b Λ := ( b Λ t ) ≤ t ≤ T defined by b Λ t := E [Λ | b F t ], is the two-dimensional MPR process under conditionalexpectation, b λ it := E [ λ i | b F t ] for 0 ≤ t ≤ T , i ∈ { S, Y } . The conditional covariancematrix process Σ = (Σ t ) ≤ t ≤ T is given byΣ = z St c t c t z Yt ! , z it := E [( λ i − b λ it ) | b F t ] , c t := E [( λ S − b λ St )( λ Y − b λ Yt ) | b F t ] , (10)for 0 ≤ t ≤ T , i ∈ { S, Y } . △ The Kalman-Bucy filter transforms the partial into a full information scenario byreplacing the constant parameters λ S , λ Y by stochastic processes b λ S , b λ Y and changingthe filtration of the probability space from F to the observation filtration b F . Theupcoming result of the partial information model under b F come from Monoyios [98].9 roposition 2.3.3 (Model under partial information) . The Kalman-Bucy filter fromDefinition 2.3.2 converts the model from the partial to the full information scenariowith asset price SDEs d S t = σ S S t ( b λ St d t + d c W St ) , d Y t = σ Y Y t ( b λ Yt d t + d c W Yt ) , (11) on the filtered probability space (Ω , b F , b F , P ) , where c W S , c W Y are ( P , b F ) -Brownian mo-tions with correlation ρ according to c W Yt = ρ c W St + p − ρ c W ⊥ t and b λ S , b λ Y are b F -adapted processes. For | ρ | 6 = 1 , z i ≤ z j with i, j ∈ { S, Y } , the drift processes are b λ it = λ i + z i ξ it z i t , b λ jt = λ j + w ξ jt w t − ρ (cid:18) λ i + w ξ it w t − b λ it (cid:19) , ≤ t ≤ T where w := z j − ρ z i − ρ for z i < z j and w := z i for z S = z Y . They satisfy the SDEs d b λ it = z it d c W it = z it (d ξ it − b λ it d t ) , b λ i = λ i , d b λ jt − ρ d b λ it = w t (d c W jt − ρ d c W it ) = w t (d( ξ jt − ρξ jt ) − ( b λ jt − ρ b λ it ) d t ) , b λ j = λ j , (12) with the entries of the covariance matrix in (10), given by z it = z i z i t , z jt = ρ z it + (1 − ρ ) w t , w t := w w t , c t = ρz it , ≤ t ≤ T. Proof.
A proof can be found in Monoyios [98, Proposition 1].Thus, under partial information, the investor’s portfolio wealth dynamics from (3)is transformed into d X t = σ S π t (cid:16)b λ St d t + d c W St (cid:17) . (13)If the prior variances z S , z Y are identical, then z t := z St = z Yt = w t and henced b λ Yt = z t d c W Yt holds for all t ∈ [0 , T ]. This case of filtering is similar to the twoone-dimensional Kalman-Bucy filters on each asset as developed in [97]. Accordingto Proposition 2.3.3, the MPR of the asset prices have the dependencies b λ St = b λ S ( t, S t ) , b λ Yt = b λ Y ( t, S t , Y t ) , if z S < z Y , b λ St = b λ S ( t, S t ) , b λ Yt = b λ Y ( t, Y t ) , if z S = z Y , b λ St = b λ S ( t, S t , Y t ) , b λ Yt = b λ Y ( t, Y t ) , if z S > z Y , (14)solving the SDEsd b λ St = z St d c W St , d b λ Yt − ρ d b λ St = w t (d c W Yt − ρ d c W St ) , if z S < z Y , d b λ St = w t d c W St , d b λ Yt = w t d c W Yt , if z S = z Y , d b λ Yt = z Yt d c W Yt , d b λ St − ρ d b λ Yt = w t (d c W St − ρ d c W Yt ) , if z S > z Y . In the remainder of this thesis, except where otherwise stated, we are working withthe partial information model of Proposition 2.3.3.10 emark 2.3.4 (Ornstein-Uhlenbeck model for MPRs) . A more complicated methodof modelling the MPRs would implicate an unknown stochastic process for each un-known MPR. For instance, the MPR dynamics could be expressed as processes of
Ornstein-Uhlenbeck type ,d λ it = η i ( ν i − λ it ) d t + δ i d B it , i = S, Y, (15)with Brownian motions B S , B Y and constant mean reversion rates η S , η Y , mean re-version levels ν S , ν Y and volatilities δ S , δ Y . The mean reversion level represents theequilibrium or long-term mean of the MPR variable and the mean reversion rate thevelocity by which the MPR variable reverts to its equilibrium. The volatility definesthe impact of stochastic shocks on the MPR change. The resulting issue would con-tain the estimations of these unknown parameters. Brendle [21] modelled the driftsby a multidimensional Ornstein-Uhlenbeck model in the context of a power-utilitybased optimal portfolio problem under partial information. However, it is not clearhow the model parameters from above can be estimated using real market data, be-cause they are assumed as known constants. This model is not pursued here to seekmaximally explicit formulas for the valuation and optimal hedge. More applicationsof stochastic differential equations of Ornstein-Uhlenbeck type in financial economicswere treated by, for instance, Barndorff-Nielsen and Shephard [9]. ♦ In the classical utility framework , the expected utility criteria is typically formulatedthrough a deterministic, concave and increasing function of terminal wealth, whereboth the investment time horizon T and the associated risk preferences are chosena priori. The value function as the optimal solution in the relevant market modelhas the fundamental property of supermartingality for arbitrary investment strategiesand martingality at an optimum, which is a consequence of the dynamic programmingprinciple (see, for example, Merton [90, p. 249]). Since the classical utility U ( x ) isfixed at a time T and its value function v ( t, x ) generated at previous times t ∈ [0 , T ]with the wealth argument x , it is also called the backward utility by Musiela andZariphopoulou [105, pp. 304, 315]. As depicted therein, backward utilities does notaccurately capture future changes in the risk preference as the market environmentevolves. Therefore, they introduce a new class of dynamic utilities that are con-structed forward in time, which offers flexibility with regards to the a priori choicesmentioned above while the natural optimality properties of the value function process11s preserved. Contrary to the classical utility framework, the forward utility U t ( x ) isnormalised at present time t but not for a fixed investment horizon T , and generatedfor all future times via a self-financing criterion. The forward measurement crite-rion is defined by Musiela and Zariphopoulou [106] in terms of a family of stochasticprocesses on [0 , ∞ ) indexed by a wealth argument.In this section, only proofs are outlined in cases that they are instructive, otherwisethe reference to the original source is given. Definition 2.4.1 (Forward performance process) . Let Θ t ⊆ Θ be the subset ofstrategies starting at t . An b F t -adapted stochastic process U := ( U t ( x )) t ≥ , where(i) for each t ≥ U t : x U t ( x ) is concave and increasing in x ∈ R ,(ii) for each t ≥ θ ∈ Θ t , E [ U t ( X t )] + < ∞ , E [ U s ( X s ) | b F t ] ≤ U t ( X t ) , s ≥ t, (iii) there exists a self-financing (optimal) strategy θ ∗ ∈ Θ t , for which E [ U s ( X ∗ s ) | b F t ] = U t ( X ∗ t ) , s ≥ t, (iv) it satisfies the initial datum U ( x ) = u ( x ) at t = 0 for all x ∈ R , with a concaveand increasing function u : R → R of wealth,is called a forward performance process . △ The function U t in (i) is the (forward) performance function . The conditions (ii)and (iii) represent the supermartingality and martingality properties, respectively.Among others, forward formulations of optimal control problems were proposedand studied in the past by Seinfeld and Lapidus [122] and Vit [133] for the deter-ministic case and Kurtz [81] for the stochastic case. As in [106], we will considera special class of time-decreasing and time-monotone forward performance processesexpressed by a deterministic function u ( x, t ) of wealth and time, where the time ar-gument is replaced by an increasing process A := ( A t ) t ≥ depending on the marketcoefficients and not the investor’s preferences. On the contrary, the function u isindependent of market changes and only depends on the initial datum u satisfyinga market independent differential constraint for t ≥ Definition 2.4.2 (Mean-variance trade-off process) . For the stock’s market price ofrisk process b λ S , define by A t := Z t (cid:16)b λ Su (cid:17) d u (16)the (mean-variance) trade-off process A := ( A t ) t ≥ . △ λ S ( t, S t , Y t ) and likewise the trade-off process A ( t, S t , Y t ) are, in general,dependent of the asset prices S, Y . In the full information scenario, when b F = F , thetrade-off process simplifies to A t = (cid:0) λ S (cid:1) t as the MPR b λ St = λ S becomes constant. Proposition 2.4.3 (Forward performance process and general optimal strategy) . Let u : R × [0 , ∞ ) → R be a concave and increasing function of the wealth argument with u ∈ C , , satisfying the non-linear partial differential equation u t u xx = 12 u x (17) and the initial condition u ( x,
0) = u ( x ) , where u ∈ C ( R ) . Then, the time-decreasingprocess U := ( U t ( x )) t ≥ defined by U t ( x ) := u ( x, A t ) , (18) is a forward performance process with the time argument replaced by the mean-variance process (16) of Definition 2.4.2. Moreover, the optimal trading strategy isgiven by π ∗ t = − b λ St σ S u x ( X ∗ t , A t ) u xx ( X ∗ t , A t ) , (19) where X ∗ is the associated wealth process following (3) with π ∗ t = θ ∗ t S t .Proof. We refer to Musiela and Zariphopoulou [106, Proposition 3].The monotonicity of u ( x, A t ) follows from the related assumptions on u and thetime-monotonicity is obtained from the definition of A and from the fact that thetime-derivative of u is negative, u t <
0. The process U will be simply also referred toas forward utility or dynamic utility . The mean-variance trade-off process A behavesas a stochastic time change of the deterministic utility function u ( x, t ). Optimalportfolio choice problems under space-time monotonicity was studied in detail byMusiela and Zariphopoulou [107]. As the representation (19) shows, the optimalstrategy does not directly depend on u but on the differential quantity − u x u xx , whichwas separately analysed by Zariphopoulou and Zhou [137]. Definition 2.4.4 (Local risk tolerance) . The local risk tolerance function is r : R × [0 , ∞ ) −→ [0 , ∞ ) , ( x, t ) u x ( x, t ) u xx ( x, t ) , (20)with initial function r ( x,
0) = r ( x ) = − u x ( x, u xx ( x, = − u ′ ( x ) u ′′ ( x ) and u satisfying (17). The local risk tolerance process R := ( R t ) t ≥ is defined by R t := r ( X t , A t ). △ X ∗ can beexpressed as d X ∗ t = R ∗ t b λ St (cid:16)b λ St d t + d c W St (cid:17) (21)with R ∗ t = r ( X ∗ t , A t ) as the local risk tolerance process benchmarked at optimal wealth .This brings up the question whether u can be indirectly derived from r . Corollary 2.4.5 (Transport equation) . The utility function u satisfies the transportequation u t + 12 r ( x, t ) u x = 0 . (22) With the knowledge of r the first-order partial differential equation (22) can be solvedto yield u .Proof. The transport equation (22) follows from (17) and (20). It can be solved usingthe method of characteristics. Consider dd t u (˜ x ( t ) , t ) = ˜ x ′ ( t ) u x (˜ x ( t ) , t )+ u t (˜ x ( t ) , t ) withthe characteristic curves ˜ x ( t ). The solution are the curves whose slope is equal to halfof the risk tolerance, i. e. ˜ x ′ ( t ) = r (˜ x ( t ) , t ) with initial value ˜ x (0) = x . Then, thefunction u can be successively constructed through the initial condition u computedfrom Definition 2.4.4 and its evaluation along the characteristic curves.This means that for an infinitesimal time interval (0 , ǫ ), the performance level u ( x + r ( x ) ǫ, ǫ ) at time ǫ is identical to u ( x,
0) at t = 0, when the wealth is moved from x to a higher level x + r ( x ) ǫ . The infinitesimal amount r ( x ) ǫ can be interpretedas the compensation required by the investor in order to satisfy his impatience in thetime interval (0 , ǫ ). More about the theory of investor’s impatience can be found inFisher [41, Chapter IV], Koopmans [75, p. 296] and Diamond et al. [34].Apart from the transport equation (22), the local risk tolerance function r solvesan autonomous non-linear heat equation , which gives an alternative approach to con-struct u from r . Corollary 2.4.6 (Fast diffusion equation) . Let u ∈ C ( R × [0 , ∞ )) satisfy the condi-tions from Proposition 2.4.3. Then, the associated local risk tolerance r is the solutionof an equation of fast diffusion type, namely r t + 12 r r xx = 0 and r ( x,
0) = − u ′ ( x ) u ′′ ( x ) . (23) Proof.
We will quote the proof from Musiela and Zariphopoulou [106, Proposition 6].Differentiating the non-linear partial differential equation u t = u x u xx from (17) withrespect to t yields u tx = u x − u x (cid:18) u x u xxx u xx (cid:19) , (24)14nd a second derivation with respect to x gives u txx = u xx − u xx (cid:18) u x u xxx u xx (cid:19) − u x (cid:18) u x u xxx u xx (cid:19) x . (25)The spatial derivatives of the local risk tolerance (20) are r x = − u xx − u x u xxx u xx = − u x u xxx u xx , r xx = (cid:18) u x u xxx u xx (cid:19) x . (26)The preceding identities (26), (25) and (24) imply r t + 12 r r xx = − u tx u xx + u x u txx u xx + 12 u x u xx (cid:18) u x u xxx u xx (cid:19) x = − u tx u xx + u x u xx − u x u xx (cid:18) u x u xxx u xx (cid:19) = 0 , which proves the assertion.After choosing an initial condition r ( x ) = r ( x,
0) = − u ′ ( x ) u ′′ ( x ) , the initial datum u ( x,
0) = u ( x ) and furthermore, with (23) and r , the values of r ( x, t ) for t > u ( x, t ), t > Corollary 2.4.7 (Autonomous SDE system for ( X ∗ , R ∗ )) . Let r satisfy (23) and let A be as in (16). Then, the processes X ∗ and R ∗ solve the system d X ∗ t = R ∗ t b λ St (cid:16)b λ St d t + d c W St (cid:17) , d R ∗ t = r x ( X ∗ t , A t ) d X ∗ t , for t > .Proof. The first equation of the optimal wealth dynamics is taken from (21). Itsquadratic variation is d h X ∗ i t = R t (cid:16)b λ St (cid:17) d t ( ) ==== R t d A t . (27)Using Itˆo’s lemma, the dynamics of the risk tolerance process at optimum wealth canbe deduced byd R ∗ t = d r ( X ∗ t , A t ) = r x ( X ∗ t , A t ) d X ∗ t + r t ( X ∗ t , A t ) d A t + 12 r xx ( X ∗ t , A t ) d h X ∗ i t = r x ( X ∗ t , A t ) d X ∗ t + (cid:18) r t ( X ∗ t , A t ) + 12 r xx ( X ∗ t , A t ) R t (cid:19) d A t ( ) ==== r x ( X ∗ t , A t ) d X ∗ t + (cid:18) r t ( X ∗ t , A t ) + 12 r ( X ∗ t , A t ) r xx ( X ∗ t , A t ) (cid:19) d A t ( ) ==== r x ( X ∗ t , A t ) d X ∗ t , because r solves the fast diffusion equation.15he reciprocal of the local risk tolerance is called local risk aversion , which solves asimilar partial differential equation of second order. The risk aversion is a well-knownparameter in utility theory to express the investor’s risk preference . Corollary 2.4.8 (Local risk aversion) . The local risk aversion function, defined as γ : R × [0 , ∞ ) −→ (0 , ∞ ) , ( x, t ) r ( x, t ) , (28) satisfies the partial differential equation γ t = 12 (cid:18) γ (cid:19) xx , γ ( x,
0) = − u ′′ ( x ) u ′ ( x ) , (29) where u is the local risk tolerance function from Definition 2.4.4.Proof. By (28), insert r = γ into the fast diffusion equation (23) solved by r to get0 (23) ==== r t + 12 r r xx (28) ==== − γ t γ + 12 γ (cid:18) γ (cid:19) xx = − γ (cid:18) γ t − (cid:18) γ (cid:19) xx (cid:19) , which directly implies the partial differential equation (29) for γ .The partial differential equation (29) is of porous medium type ; see for exampleVasquez [131]. This and the fast diffusion equation (23) may not have well-definedglobal solutions for arbitrary initial conditions. Zariphopoulou and Zhou [137] intro-duced a two-parameter family of so-called asymptotically linear local risk tolerancefunctions solving (23), which includes the most common cases that lead to exponen-tial , power , and logarithmic utilities . Proposition 2.4.9 (Asymptotically linear local risk tolerance) . Let α, β > beconstant parameters, then the function r ( x, t ) = p αx + βe − αt , ( x, t ) ∈ R × [0 , ∞ ) , (30) solves (23) with initial datum r ( x ) = p αx + β . The limiting cases lead to local risktolerance functions with corresponding utilities for t ≥ as follows: lim α → r ( x, t ) = p β =: r e , u ( x, t ) = − e − x √ β + t , x ∈ R , (exponential) ; (31)lim β → r ( x, t ) = √ αx, u ( x, t ) = x δ δ e − δ − δ t , x ≥ , α = 1 , (power) ; (32)lim β → r ( x, t ) = x, u ( x, t ) = log( x ) − t , x > , α = 1 , (logarithmic) , (33) where δ := √ α − √ α . roof. The first partial derivatives of r are r t = − αβe − αt r − and r x = αxr − . Thesecond derivative with respect to x is r xx = αr − − ( αx ) r − . As a result, (30) solvesthe fast diffusion equation r t + 12 r r xx = − αβe − αt r − + 12 (cid:0) αr − ( αx ) r − (cid:1) = − α (cid:0) αx + βe − αt (cid:1)| {z } = r r − + 12 αr = −
12 ( r − r ) = 0 . To construct the utilities in the limiting cases, Definition 2.4.4 and the transportequation from Corollary 2.4.5 can be applied. For the exponential case (31), consider √ β = u x ( x,t ) u xx ( x,t ) from (20) and make the exponential ansatz u ( x ) = e − x √ β for t = 0 asthe left side is independent of time. By (22), it is u t + √ βu x = 0, which yields theproduct solution u ( x, t ) = u ( x ) e t = e − xβ + t .In the power case (32), the risk tolerance is expressed by √ αx = − u ′ ( x ) u ′′ ( x ) . Similarto the exponential case, make a multiplicative ansatz u ( x, t ) = u ( x )˜ u ( t ), but with amonomial initial function u = x δ δ to solve the problem, because of − u ′ u ′′ = − x δ − ( δ − x δ − = − xδ − √ αx. Further, the transport equation gives the homogeneous ordinary differential equationof first order x δ δ ˜ u ′ + √ αx δ ˜ u = 0. Excluding the trivial solution u = ˜ u = 0 for x = 0 simplifies the equation to ˜ u ′ + δ − δ ˜ u = 0. After a rearrangement, we getthe logarithmic derivative dd t log(˜ u ( t )) = ˜ u ′ ( t )˜ u ( t ) = − δ − δ , which is solved by simpleintegration and taking the inverse function, i. e. ˜ u ( t ) = e − δ − δ t . The restriction x ≥ u ( x ) = log( x ) solves x = − u ′ u ′′ and the transport equation becomes u t + xu x = 0.Since an attempt to solve the problem through a multiplicative separation fails, we tryan additive approach to get an appropriate solution u ( x, t ) = u ( x ) + ˜ u ( t ). With this,the transport equation is apparently solved by ˜ u ( t ) = − t . Obviously, the domain ofthe utility is defined only for x > asymptotically linear due to its limiting behaviourlim | x |→∞ r ( x, t ) | x | = √ α, t ≥ . The local risk tolerances in (31), (32) and (33) are referred to as exponential , power and logarithmic risk tolerance , respectively. The form of u only depends on the rangeof the parameter α , specifically, one the cases α = 1 and α = 1.17 roposition 2.4.10 (Class of forward utility functions) . Let r be an asymptoticallylinear local risk tolerance function as defined in (30) with α, β > . The correspondingutility function is given by u ( x, t ) = M ( √ α )
1+ 1 √ α α − e −√ α t (cid:16) β √ α e − αt +(1+ √ α ) x (cid:16) √ αx + √ αx + βe − αt (cid:17)(cid:17)(cid:16) √ αx + √ αx + βe − αt (cid:17)
1+ 1 √ α + N, α = 1 M (cid:16) log (cid:16) x + p x + βe − t (cid:17) − e t β x (cid:16) x − p x + βe − t (cid:17) − t (cid:17) + N, α = 1 , for ( x, t ) ∈ R × [0 , ∞ ) , where M > , N ∈ R are constants derived from integration.Proof. We refer to Zariphopoulou and Zhou [137, Proposition 3.2].To preserve the monotonicity of u , the constraint M > x ∈ R with exception of the situation β →
0, the non-negativity limitation on the investor’s wealth is omitted. This property is useful forindifference valuation.
The previous section prepared for the option’s indifference pricing and optimal hedging of basis risk in an incomplete market model with partial information using a forwardexponential utility approach. In this section, we derive the optimal hedging strategy ,the dual representation of the forward indifference price with a PDE, the residual risk , pay-off decompositions and asymptotic expansions of the indifference price as results. Suppose the market is complete, this means that the Brownian motions c W S , c W Y of the assets S, Y are perfectly negatively or positively correlated with correlationcoefficient | ρ | = 1. In this case, Y effectively becomes a traded asset and perfecthedging of the stock S by an European contingent claim (European option) C on Y is possible due to the no-arbitrage requirement of the market. More about thearbitrage theory of capital asset pricing can be found in Delbaen and Schachermayer[32, p. 473] and Ross [117]. The complete market case under full information wastreated by Monoyios [97, p. 334]. An important result is that the perfect hedge doesnot require the knowledge of the MPR processes b λ S , b λ Y , making the hedging strategy in the full and the partial information scenario identical.18 roposition 3.1.1 (Pricing in a complete market) . In a complete market, that meansa correlation of | ρ | = 1 , the claim price process C := ( C ( t, Y t )) ≤ t ≤ T is given by theBlack-Scholes pricing formula.Proof. We will give a proof based on Davis [30] and Monoyios [95]. Apply thescenario under partial information with its notation from Subsection 2.2, becausethe calculations and results under full information are exactly the same. With-out loss of generality let the correlation coefficient be ρ = 1, implying the identity c W Y = ρ c W S + p − ρ c W ⊥ = c W S . The no-arbitrage theory requires an unique mar-ket price of risk, since the random process W S is the only existing risk factor in thebasis risk model. Therefore, the MPRs are related by b λ S = b µ S − r m σ S = b µ Y − r m σ Y = b λ Y with r = 0 according to Subsection 2.1. Like (9), the solutions of the asset pricedynamics (11) are S t = S exp (cid:18) σ S (cid:18)Z t b λ Su d u − σ S t + c W St ) (cid:19)(cid:19) ,Y t = Y exp (cid:18) σ Y (cid:18)Z t b λ Su d u − σ Y t + c W St ) (cid:19)(cid:19) . (34)Thus, the asset Y is a function of the stock S , given by Y t Y ) ==== (cid:18) S t S (cid:19) σYσS exp (cid:18) σ Y ( σ S − σ Y ) t (cid:19) . Apply the Itˆo lemma on the (contingent) claim price process (value process of theEuropean option on Y ) C := ( C ( t, Y t )) ≤ t ≤ T , so thatd C = C t d t + C y d Y t + 12 C yy d h Y i t = (cid:18) C t + C y σ Y Y t b λ St + 12 C yy (cid:0) σ Y (cid:1) Y t (cid:19) d t + C y σ Y Y t d c W St . (35)The replication conditions are X ∗ t = C ( t, Y t ) , d X ∗ t = d C ( t, Y t ) , ≤ t ≤ T, for theinvestor’s optimal wealth X ∗ t . A comparison of the random terms between the wealthdynamics (13) and (35) provides the perfect hedging strategy θ ∗ t = σ Y σ S Y t S t C y ( t, Y t ) , (36)which is independent of the MPRs and so is conform with both the full and partialinformation scenarios. The claim price process solves the Black-Scholes SDE C t ( t, Y t ) + 12 (cid:0) σ Y (cid:1) Y t C y ( t, Y t ) = 0 , with a bounded continuous process C ( T, y ) and the non-negative random variable C ( Y T ) := C ( T, Y T ) as the pay-off at expiry T of the European contingent claim.19 .2 Forward performance problem in an incomplete market Now, suppose the market is incomplete, meaning that the correlation of c W S , c W Y isnot perfect, | ρ | 6 = 1. Then the claim is not perfectly replicable in general. The ensuing indifference valuation and hedging problem of the claim is embedded in a exponentialforward performance maximisation framework . Firstly, we define essential terms ofthe valuation and hedging theory regardless of the specific forward utility and theinformation scenario. Definition 3.2.1 (Value process, indifference price and optimal hedging strategy) . Presume, the investor holds a long position in the stock S and a short position in theclaim C on the non-traded asset Y to hedge the stock. The maximal b F t -conditionalexpected forward performance of terminal portfolio wealth X T − C ( Y T ) from trading, v C ( t, X t , S t , Y t ) := ess sup θ ∈ Θ t E h U T ( X T − C ( Y T )) (cid:12)(cid:12)(cid:12) b F t i , ≤ t ≤ T, (37)is called (primal forward) value process . When no claim is sold, the value process is v ( t, X t , S t , Y t ) := ess sup θ ∈ Θ t E h U T ( X T ) (cid:12)(cid:12)(cid:12) b F t i , ≤ t ≤ T. (38)The terminal values (maximal expected performances) are v C ( T, X T , S T , Y T ) = U T ( X T − C ( Y T )) , v ( T, X T , S T , Y T ) = U T ( X T ) . (39)The (forward performance) indifference price process p is defined by (Hodges andNeuberger [56, p. 226]) v C ( t, X t + p ( t, S t , Y t ) , S t , Y t ) = v ( t, X t , S t , Y t ) , ≤ t ≤ T. (40)Evaluating the value and indifference price processes given the deterministic point( X t , S t , Y t ) = ( x, s, y ) delivers the value and indifference price functions v ( t, x, s, y )and p ( t, s, y ), respectively. We abbreviate with E t,x,s,y [ · ] the conditional expectation E [ · | ( t, X t , S t , Y t ) = ( t, x, s, y )]. As in Becherer [11, p. 7] defined, the optimal hedgingstrategy θ H := ( θ Ht ) ≤ t ≤ T , θ Ht := θ Ct − θ t , (41)is the difference between the optimal strategy θ C := ( θ Ct ) ≤ t ≤ T for the problem withthe claim (37) and the optimal strategy θ := ( θ t ) ≤ t ≤ T without the claim (38). △ The notion of essential supremum ess sup (likewise essential infimum ess inf) istaken from Karatzas and Shreve [72, p. 323]. For a real-valued function f , it isess sup f := inf { a ∈ R | µ R ( f − ( a, ∞ )) = 0 } with the Lebesgue measure µ R .20y Definition 2.4.1, it is v ( t, X t , S t , Y t ) = U t ( X t ) with the associated optimalwealth X t in absence of the claim. In terms of the stock-weighted trading strategy π = θS , the optimal hedging strategy is π H = π C − π . The portfolio strategies are denotedas θ t = θ ( t, S t , Y t ) ( π t = π ( t, S t , Y t )) to express them as functions of the asset prices.The indifference price p implicitly defined in (40) is also called the writer’s indifferenceprice , since the option C in the portfolio is sold. The solution to the optimisationproblem (37) in classical utility theory is well-studied in Zariphopoulou [136] andMonoyios [95, p. 248] using the so-called distortion transformation to linearise the Hamilton-Jacobi-Bellman (HJB) equation for the value function. References for theHJB equation are, for example, Pham [109, pp. 42–46] and [72, p. 130].
Theorem 3.2.2 (Optimal strategy in terms of the value process) . The general solu-tion to the maximisation problems (37), (38) with terminal performance values (39)in terms of the value process is the optimal strategy process θ ∗ ( t, S t , Y t ) = − b λ St v x + σ S S t v xs + ρσ Y Y t v xy σ S S t v xx ! , ≤ t ≤ T, (42) for θ ∗ = θ C , θ and v = v C , v .Proof. The value function v ( t, x, s, y ) solves the non-linear HJB equation v t + sup θ ∈ Θ t (cid:16) σ S s b λ St ( θ t v x + v s ) + σ Y y b λ Yt v y + θ t (cid:0) σ S s (cid:1) v xs + ρσ S σ Y sy ( θ t v xy + v sy )+ 12 (cid:0) σ S s (cid:1) ( θ t v xx + v ss ) + 12 (cid:0) σ Y y (cid:1) v yy (cid:17) = 0 , (43)where the supremum is derived through differentiation with respect to θ t , σ S s b λ St v x + (cid:0) σ S s (cid:1) v xs + ρσ S σ Y syv xy + (cid:0) σ S s (cid:1) θ ∗ t v xx = 0 , which gives the optimal strategy function θ ∗ ( t, s, y ). Evaluating the optimal strategyfunction at the random point ( t, S t , Y t ) provides the optimal strategy process (42).In terms of the cash value, the optimal strategy is π ∗ t = θ ∗ t S t = − b λ St v x σ S v xx − S t v xs v xx − ρσ Y Y t v xy σ S v xx . (44)The first term π Mt := − b λ St v x σ S v xx of (44) is called the Merton strategy , because it madeits first appearance in [90, p. 250] as Merton’s optimal solution in the setting of asimplified market with only one asset. Since in the full information scenario the value21rocess does not directly depend on S regarding the known MPR λ S , the mixedpartial derivative v xs is zero and thus the partial information component π St := S t v xs v xx of the strategy vanishes. The last term π Yt := ρσ Y Y t v xy v xx is induced by the claim onthe non-traded asset Y and reflects the hedging component of the strategy. Thesensitivity of the marginal utility of wealth with respect to changes of the option’sprice is measured by v xy . For the uncorrelated case ρ = 0, the hedging component π Yt becomes zero and the stock cannot be hedged by the option. Therefore, the optimalstrategy in the uncorrelated full information scenario is identical to the one of Merton.By Proposition 2.4.9, the limiting case (31) of the exponential linear local risktolerance lim α → r ( x, t ) = r e = √ β corresponds to the exponential utility func-tion u ( x, t ) = − e − x √ β + t . By Corollary 2.4.8, the utility has the more familiar form u ( x, t ) = − e − γx + t with the local risk aversion γ = 1 /r e = 1 / √ β >
0. ApplyingProposition 2.4.3, the exponential forward performance process is U t ( x ) = u ( x, A t ) = − exp (cid:18) − γx + 12 Z t (cid:16)b λ Su (cid:17) d u (cid:19) , ( x, t ) ∈ R × [0 , ∞ ) . (45)In comparison to the classical exponential utility function u ( x ) = − e − γx the dynamicutility decreases in time, valuing less future utility. The primal value process (37) isthe maximal expected forward performance v C ( t, X t , S t , Y t ) = ess sup θ ∈ Θ t E (cid:20) − exp (cid:18) − γ ( X T − C ( Y T )) + 12 Z T (cid:16)b λ Su (cid:17) d u (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) . (46) Remark 3.2.3 (Wealth independence of indifference price) . As with classical utility,the forward indifference price p defined in (40) is independent of initial wealth X t under b F t . This becomes clear, when regarding at v C ( t, X t , S t , Y t ) = e − γX t + R t ( b λ Su ) d u ess sup θ ∈ Θ t E h − e − γ ( R Tt θ u d S u − C ( Y T ) ) + R Tt ( b λ Su ) d u (cid:12)(cid:12)(cid:12) b F t i , since it is X T = X t + R Tt θ u d S u by (3), where X t factors out of the problem. ♦ Since under full information, the stock’s MPR λ S is observable and therefore aknown constant with respect to the background filtration F , the trade-off process A t = R t (cid:0) λ S (cid:1) d s = (cid:0) λ S (cid:1) t becomes a deterministic linear function of time. Hence,the investor’s risk preference simplifies to the exponential performance process U t ( x ) = − e − γx + ( λ S ) t , ( x, t ) ∈ R × [0 , ∞ ) . Factoring out the MPR term, one gets the classical primal problem v C ( t, X t , S t , Y t ) = e ( λ S ) T ess sup θ ∈ Θ t E (cid:2) − e − γ ( X T − C ( Y T )) | F t (cid:3) , ≤ t ≤ T. .3 Dual representation of the stochastic control problem In the 1970s and 80s, Bismut [17], Karatzas et al. [70], [71] and Cox & Huang [27]realised that the use dual methods from convex analysis provided valuable comprehen-sion of solutions to optimal stochastic control problems, which are more general thanthe original problem from Merton [90]. Kramkov and Schachermayer [77] studied thedual approach for solving maximisation problems under classical utility. Rogers [115]delved deeper into the theory by applying methods from functional analysis and pre-senting various examples solved with duality methods. We follow ˇZitkovi´c [138] andBerrier et al. [16] to briefly introduce the dual approach for solving forward perfor-mance maximisation problems. Firstly, recall the notion of relative entropy H ( Q , P )for equivalent local martingale measures Q ∼ P from Definition 2.1.1. For the subset M e,f of these measures with finite relative entropy, we introduce the Radon-Nikodymderivative (see Shreve [126, pp. 65–79] or Platen and Heath [111, pp. 338–339]), al-lowing us to perform measure changes.
Definition 3.3.1 (Radon-Nikodym derivative process) . For measures Q ∈ M e,f , thepositive likelihood ratio ( P , b F )-martingale process Z Q = ( Z Q t ) ≤ t ≤ T defined by Z Q t = d Q d P (cid:12)(cid:12)(cid:12)(cid:12) b F t , (47)is called the Radon-Nikodym derivative process . It is the density process of Q withrespect to P . △ For admissible portfolio strategies (5), Z Q X := ( Z Q t X t ) ≤ t ≤ T is a non-negative( P , b F )-local martingale, hence a supermartingale satisfying Z Q = 1 , E [ Z Q T ] = 1 , E [ Z Q T X T | b F t ] ≤ Z Q t X t almost surely(see [138, pp. 2180–2181] and [16, p. 1]).The map Q Z Q induced by (47) creates an one-to-one correspondence betweenthe class M e,f of equivalent locale martingale measures with finite relative entropyand the set of density processes Z := (cid:8) Z Q | Q ∈ M e,f (cid:9) . If t = T , then the Radon-Nikodym derivative is Z Q T = d Q d P and the relative entropy (4) can be expressed by H ( Q , P ) = E (cid:2) Z Q T log Z Q T (cid:3) = E Q (cid:2) log Z Q T (cid:3) , where E Q denotes the expectation with respect to Q , whereas E is the P -expectation.The relative entropy can be interpreted as a measure of distance, even though it is nota metric. The density process and relative entropy will be generalised to conditionalversions in order to formulate the dual problem to Definition 3.2.1.23 efinition 3.3.2 (Conditional density and conditional relative entropy) . The ratio Z Q t,T := Z Q T Z Q t , ≤ t ≤ T, (48)is called conditional density process and motivates the conditional relative entropy H t ( Q , P ) := E Q h log Z Q t,T (cid:12)(cid:12)(cid:12) b F t i (49)of Q with respect to P over the interval [ t, T ]. △ At t = 0, the conditionality becomes trivial with density Z Q ,T = Z Q T and relativeentropy H ( Q , P ) = H ( Q , P ). Frittelli [46, p. 42] showed the existence and unique-ness of a minimal entropy martingale measure (MEMM) Q E , that minimises H ( Q , P )over all Q ∈ M e,f . According to Kabanov and Stricker [64, pp. 131–132], Q E alsominimises the H t ( Q , P ) for an arbitrary t ∈ [0 , T ], so that we can write Q E := arg min Q ∈M e,f H t ( Q , P ) . (50)We say, the minimal conditional density process ( Z Q E t,T ) ≤ t ≤ T minimises the conditionalrelative entropy process ( H t ( Q , P )) ≤ t ≤ T .Our aim is to give the optimal strategy from Theorem 3.2.2 in terms of derivativesof the indifference price from Definition 3.2.1, which we will approach through framingthe dual problem . Definition 3.3.3 (Convex conjugate (dual) performance and its inverse marginal) . The (convex) conjugate (or dual ) ˜ U t : (0 , ∞ ) → R of the performance U t is˜ U t (˜ x ) = ess sup x> [ U t ( x ) − x ˜ x ] = U t ( I t (˜ x )) − ˜ xI (˜ x ) , t ≥ , ˜ x > , (51)where I t := ( U ′ t ) − denotes its inverse of the marginal U ′ t := dd x U t satisfying U ′ t ( I t (˜ x )) = I t ( U ′ t (˜ x )) = ˜ x, t ≥ , ˜ x > . (52)The conjugate function ˜ U t solves the bidual relation U t ( x ) = ess inf ˜ x> h ˜ U t (˜ x ) + x ˜ x i = ˜ U t ( U ′ t ( x )) + xU ′ t ( x ) , t ≥ , x > , as well as ˜ U t (˜ x ) ≥ U t ( x ) − x ˜ x with equality if and only if x = I t (˜ x ). The marginaldual performance ˜ U ′ t satisfies the identity ˜ U ′ t (˜ x ) = − I t (˜ x ). △ U ′ t and I t are continuous, strictly decreasing and map (0 , ∞ ) onto itselfsatisfying the Inada conditions (see F¨are and Primont [40], Inada [59]) I t (0 + ) = U ′ t (0 + ) = ∞ , I t ( ∞ ) = U ′ t ( ∞ ) = 0 , where we have abbreviated I t (0 + ) = ∞ for the limit lim ˜ x → + I t (˜ x ) = ∞ (analogousthe other limits). The conjugate function ˜ U t is convex, decreasing, continuouslydifferentiable with the limits˜ U ′ t (0+) = −∞ , ˜ U ′ t ( ∞ ) = 0 + , ˜ U t (0 + ) = U t ( ∞ ) , ˜ U t ( ∞ ) = U t (0 + ) . The dual function ˜ U t (˜ x ) is the Legendre-transform of − U t ( − x ) (cf. Rockafellar [113,p. 251]). Pliska [112] showed some useful applicatios for computing value functionsand optimal strategies. The methods and the exposition of the results given thereare similar to the corresponding methods used by [113]. Lemma 3.3.4 (Dual value process and dual problem) . For the primal value function v = v C from Definition 3.2.1 the dual value process is ˜ v ( t, ˜ X t , S t , Y t ) := ess inf Z Q ∈Z E h ˜ U T ( ˜ X t Z Q t,T ) − ˜ X t Z Q t,T C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i , ≤ t ≤ T. (53) The primal and dual value functions are conjugate with respect to the wealth space, ˜ v ( t, ˜ x, s, y ) = sup x> [ v ( t, x, s, y ) − x ˜ x ] , ˜ x > ,v ( t, x, s, y ) = inf ˜ x> [˜ v ( t, ˜ x, s, y ) + x ˜ x ] , x > . (54) The partial derivatives of he primal and dual value functions at the optimum arerelated by v x ( t, x ∗ , s, y ) = ˜ x ∗ , ˜ v ˜ x ( t, ˜ x ∗ , s, y ) = − x ∗ . (55) Proof.
We refer to the theorems in Kramkov and Schachermayer [77, pp. 908–911],Rogers [115, pp. 107–113] and ˇZitkovi´c [138, pp. 2184–2188].As noticed in Mania and Schweizer [87, p. 2116], the terminology “primal” corre-sponds for any problem optimising over the portfolio strategy and “dual” when theoptimiser is the density Z Q resp. measure Q .Since the main results of duality theory for solving stochastic control problemsare worked out, they will be applied to the primal performance maximisation prob-lem (46) of exponential forward type U t ( X t ) = − exp (cid:0) − γX t + A t (cid:1) to obtain thecorresponding dual problem. The next theorem covers the valuation of the dualperformance process and the dual entropic representation of the problem.25 heorem 3.3.5 (Dual forward performance problem) . The dual representation ofthe primal optimisation problem (46) is given by v C ( t, X t , S t , Y t ) = − exp (cid:18) − γX t − ess inf Z Q ∈Z (cid:18) H t ( Q , P ) − γ E Q (cid:20) C ( Y T ) + 12 γ A T (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21)(cid:19)(cid:19) . (56) Proof.
Definition 3.3.3 is considered to calculate the dual performance process. In-serting the derivative U ′ t ( x ) = γ exp (cid:0) − γx + A t (cid:1) into (52), leads to its inverse I t (˜ x ) = − γ (cid:18) log (cid:18) ˜ xγ (cid:19) − A t (cid:19) . Putting the inverse into (51), provides the dual performance˜ U t (˜ x ) = U t ( I t (˜ x )) − ˜ xI (˜ x ) = ˜ xγ (cid:18) log (cid:18) ˜ xγ (cid:19) − − A t (cid:19) . (57)Before moving to the dual value function, consider the conditional relative entropy H t ( Q , P ) ( ) ==== E Q h log Z Q t,T (cid:12)(cid:12)(cid:12) b F t i = 1 Z Q t E h Z Q T log Z Q t,T (cid:12)(cid:12)(cid:12) b F t i = E " Z Q T Z Q t log Z Q t,T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b F t ( ) ==== E h Z Q t,T log Z Q t,T (cid:12)(cid:12)(cid:12) b F t i , (58)where in the second equation we have applied Lemma 5.2.2 from Shreve [125, p. 212],concerning the expression of conditional expectations of random variables under mea-sure change. Obviously, the conditional density has the expectation E h Z Q t,T (cid:12)(cid:12)(cid:12) b F t i = 1 Z Q t E h Z Q T (cid:12)(cid:12)(cid:12) b F t i = E Q h (cid:12)(cid:12)(cid:12) b F t i = 1 . (59)Then, by Lemma 3.3.4, the dual value function is given by˜ v ( t, ˜ x, s, y ) ( ) ==== ess inf Z Q ∈Z E t,x,s,y h ˜ U T (˜ xZ Q t,T ) − ˜ xZ Q t,T C ( Y T ) i ( ) ==== ess inf Z Q ∈Z E t,x,s,y (cid:20) ˜ xγ Z Q t,T (cid:18) log (cid:18) ˜ xγ Z Q t,T (cid:19) − − A T (cid:19) − ˜ xZ Q t,T C ( Y T ) (cid:21) = ˜ xγ (cid:18) log (cid:18) ˜ xγ (cid:19) − (cid:19) ess inf Z Q ∈Z E t,x,s,y (cid:2) Z Q t,T (cid:3) + ˜ xγ ess inf Z Q ∈Z E t,x,s,y (cid:20) Z Q t,T log Z Q t,T − γZ Q t,T (cid:18) C ( Y T ) + 12 γ A T (cid:19)(cid:21) ( ) , ( ) ======= ˜ U (˜ x ) + ˜ xγ ess inf Z Q ∈Z E t,x,s,y (cid:20) Z Q t,T log Z Q t,T − γZ Q t,T (cid:18) C ( Y T ) + 12 γ A T (cid:19)(cid:21) ( ) ==== ˜ U (˜ x ) + ˜ xγ ess inf Z Q ∈Z (cid:18) H t ( Q , P ) − γ E Q t,x,s,y (cid:20) C ( Y T ) + 12 γ A T (cid:21)(cid:19) . (60)26hus, the dual forward performance problem amounts to the minimisation of H C ( t, S t , Y t ) := ess inf Z Q ∈Z (cid:18) H t ( Q , P ) − γ E Q (cid:20) C ( Y T ) + 12 γ A T (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21)(cid:19) , ≤ t ≤ T, (61)which will be referred to as the minimal entropy process . Given (55), the derivativeof the dual value function at the optimum ˜ x = ˜ x ∗ , x = x ∗ satisfies˜ v ˜ x ( t, ˜ x ∗ , s, y ) = ˜ U ′ (˜ x ∗ ) + 1 γ H C ( t, s, y ) = 1 γ (cid:18) log (cid:18) ˜ x ∗ γ (cid:19) + H C ( t, s, y ) (cid:19) = − x ∗ . As the latter equation defines the functional expression of x ∗ by ˜ x ∗ , rearrange it toget the inverse expression ˜ x ∗ = γ exp (cid:0) − γx ∗ − H C ( t, s, y ) (cid:1) . Using this in the bidual relation (54) delivers v ( t, x ∗ , s, y ) = ˜ v ( t, ˜ x ∗ , s, y ) + x ∗ ˜ x ∗ = ˜ x ∗ γ (cid:18) log (cid:18) ˜ x ∗ γ (cid:19) − (cid:19) + ˜ x ∗ γ H C ( t, s, y ) + x ∗ ˜ x ∗ = ˜ x ∗ γ (cid:18) log (cid:18) ˜ x ∗ γ (cid:19) − H C ( t, s, y ) + γx ∗ (cid:19) = − exp (cid:0) − γx ∗ − H C ( t, s, y ) (cid:1) , which proves (56).When writing the primal forward performance problem (46) as v C ( t, X t , S t , Y t ) = ess sup θ ∈ Θ t E (cid:20) − exp (cid:18) − γ (cid:18) X T − (cid:20) C ( Y T ) + 12 γ A T (cid:21)(cid:19)(cid:19) (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) , and comparing both this and its dual representation (56) to the classical utility casefrom [98], the claim pay-off term in the forward model is C ( Y T ) + γ A T instead of C ( Y T ) in the classical model. Thus, the forward case adds value to the terminal valueof the option. Corollary 3.3.6 (Dual representation of the indifference price) . The indifferenceprice process has the entropic representation p ( t, S t , Y t ) = − γ (cid:0) H C ( t, S t , Y t ) − H ( t, S t , Y t ) (cid:1) , ≤ t ≤ T. (62) Proof.
Denote by H the minimal entropy process (61) with no claim present, orequivalently, C = 0. The expression (62) follows directly from Theorem 3.3.5 and thedefinition of the indifference price (40). 27ext, we give the optimal hedging strategy in terms indifference price derivatives,which is derived analogously to the classical case from Monoyios [98, Theorem 1]. Theorem 3.3.7 (Optimal hedging strategy in terms of the indifference price) . Sup-pose the forward indifference price function p is of class C , , ([0 , T ] × [0 , ∞ ) ) . Thenthe optimal hedging strategy for a short position in the claim is given by θ H ( t, S t , Y t ) = p s ( t, S t , Y t ) + ρ σ Y Y t σ S S t p y ( t, S t , Y t ) , ≤ t ≤ T. (63) Proof.
By differentiating the entropic representation of the value function given inTheorem 3.3.5, we obtain the partial derivatives v x = − γv, v xx = γ v, v xy = γH y v, v xs = γH s v, (64)for the case v = v C , H = H C with the claim and the case v = v , H = H withoutthe claim. Apply them to Theorem 3.2.2 to obtain the optimal strategy in terms ofderivatives of the minimal entropy process, θ ∗ ( t, S t , Y t ) ( ) ==== − b λ St v x + σ S S t v xs + ρσ Y Y t v xy σ S S t v xx ! ( ) ==== b λ St γσ S S t − γ (cid:18) H s ( t, S t , Y t ) + ρ σ Y Y t σ S S t H y ( t, S t , Y t ) (cid:19) for θ = θ C , θ . Finally, consider the optimal hedging strategy formula θ H = θ C − θ from (41) and use Corollary 3.3.6 to eliminate the Merton strategy term and obtainthe optimal hedging strategy expressed by the indifference price (63). The requiredregularity of the indifference price is shown in [98, Subsection 3.3]. After we have defined the dual stochastic control problem, we are going to give a moreexplicit representation formula for the forward indifference price from Corollary3.3.6following Monoyios [98, Subsection 3.2] and Leung et al. [84, Subsection 3.2]. For this,we will characterise the martingale measure Q by giving the corresponding densityprocess Z Q and then perform a measure change to the basis risk model. The mea-sures Q ∈ M e,f characterised by their densities Z Q , are parametrised via b F -adaptedprocesses ψ := ( ψ t ) ≤ t ≤ T satisfying R T ψ u d u < ∞ P -a.s. and E [ Z Q T ] = 1, according tothe stochastic exponential Z Q t := E (cid:16) − b λ S · c W S − ψ · c W ⊥ (cid:17) t = exp (cid:18) − Z t b λ Su d c W Su − Z t ψ u d c W ⊥ u − Z t (cid:20)(cid:16)b λ Su (cid:17) + ψ u (cid:21) d u (cid:19) . (65)28ince R t (( b λ Su ) + ψ u ) d u = A t + R t ψ u d u is a strictly positive square-integrable con-tinuous process, Novikov’s condition E (cid:20) exp (cid:18) Z t (cid:20)(cid:16)b λ Su (cid:17) + ψ u (cid:21) d u (cid:19)(cid:21) < ∞ (66)is fulfilled. Denote with Ψ the set of integrands ψ such that (66) is satisfied. Hence,the density process Z Q is indeed a ( P , b F )-martingale. Applying Girsanov’s theorem from [125, pp. 224–225] for a measure change to Q , provides the two-dimensionalBrownian motion ( c W S, Q , c W ⊥ , Q ) defined by c W S, Q t = c W St + Z t b λ Su d u, c W ⊥ , Q t := c W ⊥ t + Z t ψ u d u, ≤ t ≤ T. (67)The integrand process ψ is commonly referred to as the volatility risk premium for thesecond Brownian motion c W ⊥ . The so-called minimal martingale measure (MMM) Q M corresponds to the case ψ = 0. It was originally introduced by F¨ollmer andSchweizer [42] for the risk-minimised (optimal) quadratic hedging strategy in an in-complete market. It alters the MPR of the stock’s Brownian motion, but does notchange the MPR of Brownian motions orthogonal to those driving the stock. By (65),it has the Radon-Nikodym derivative process Z Q M t = E (cid:16) − b λ S · c W S (cid:17) t = exp (cid:18) − Z t b λ Su d c W Su − Z t (cid:16)b λ Su (cid:17) d u (cid:19) . The second equation of (67) implies that c W ⊥ t is also a ( Q M , b F )-Brownian motion. Proposition 3.4.1 (Representation of conditional relative entropy) . The conditionalrelative entropy between Q and P satisfies H t ( Q , P ) = 12 E Q (cid:20)Z Tt (cid:20)(cid:16)b λ Su (cid:17) + ψ u (cid:21) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) < ∞ . (68) In the full information scenario, the conditional relative entropy simplifies to H t ( Q , P ) = 12 (cid:0) λ S (cid:1) ( T − t ) + 12 E Q (cid:20)Z Tt ψ u d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) , (69) which is minimised by ψ = 0 . Hence, in the full information scenario, the MEMM Q E coincides with the MMM Q M . Using Q M as the reference measure, the conditionalrelative entropy above can be additively decomposed to H t ( Q , P ) = H t ( Q , Q M ) + H t ( Q M , P ) with H t ( Q , Q M ) = 12 E Q (cid:20)Z Tt ψ u d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) , H t ( Q M , P ) = 12 E Q (cid:20)Z Tt (cid:16)b λ Su (cid:17) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) . (70)29 roof. The conditional relative entropy from Definition (3.3.2) is H t ( Q , P ) ( ) ==== E Q h log Z Q t,T (cid:12)(cid:12)(cid:12) b F t i ( ) ==== E Q " log Z Q T Z Q t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b F t ( ) ==== E Q (cid:20) − Z Tt b λ Su d c W Su − Z Tt ψ u d c W ⊥ u − Z Tt (cid:20)(cid:16)b λ Su (cid:17) + ψ u (cid:21) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) ( ) ==== E Q (cid:20) − Z Tt b λ Su d c W S, Q u − Z Tt ψ u d c W ⊥ , Q u + 12 Z Tt (cid:20)(cid:16)b λ Su (cid:17) + ψ u (cid:21) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) = 12 E Q (cid:20)Z Tt (cid:20)(cid:16)b λ Su (cid:17) + ψ u (cid:21) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) , where the integrability on the right hand side implied by Novikov’s condition (66) isassociated with the finite conditional relative entropy condition. The second assertion(69) directly follows from (68) for b λ St = λ S . To show (70), consider the Radon-Nikodym derivative under Q M ,d Q d Q M = d Q d P (cid:18) d Q M d P (cid:19) − = Z Q T Z Q M T ( ) ==== exp (cid:18) − Z T ψ u d c W ⊥ u − Z T ψ u d u (cid:19) . (71)Then again, apply Lemma 5.2.2 from [125, p. 212], giving the conditional expectationunder measure change to compute the density process of Q with respect to Q M , E Q M " Z Q T Z Q M T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b F t = 1 Z Q M t E h Z Q T (cid:12)(cid:12)(cid:12) b F t i = Z Q t Z Q M t =: Z Q , Q M t , ≤ t ≤ T. (72)As in Definition 3.3.2, for any measure Q ∈ M e,f , the conditional density is given by Z Q , Q M t,T := Z Q , Q M T Z Q , Q M t ( ) ==== Z Q T Z Q M T Z Q M t Z Q t ( ) ==== exp (cid:18) − Z Tt ψ u d c W ⊥ u − Z Tt ψ u d u (cid:19) . (73)Then, directly compute the conditional relative entropy over the interval [ t, T ], H t ( Q , Q M ) = E Q h log Z Q , Q M t,T (cid:12)(cid:12)(cid:12) b F t i ( ) ==== 12 E Q (cid:20)Z Tt ψ u d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) , by using the ( Q M , b F )-martingale property of R Tt ψ u d c W ⊥ u . The conditional relativeentropy H t ( Q M , P ) is analogously determined given c W S, Q = c W S, Q M .Proposition 3.4.1 implies the generalised additivity formula H t ( Q , P ) = H t ( Q , e Q ) + H t ( e Q , P ) , (74)for any martingale measure e Q ∈ M e,f (see also Monoyios [99, p. 902]).30 heorem 3.4.2 (Forward indifference price valuation) . The forward indifferenceprice is the solution of the stochastic control problem p ( t, S t , Y t ) = − γ ess inf ψ ∈ Ψ (cid:16) H t ( Q , Q M ) − γ E Q h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i(cid:17) , (75) with optimal control ψ H ( t, S t , Y t ) = − γ p − ρ σ Y Y t p y ( t, S t , Y t ) , (76) and solves the semi-linear partial differential equation of second order p t + A Q M S,Y p + 12 γ (1 − ρ ) (cid:0) σ Y yp y (cid:1) = 0 , p ( T, s, y ) = C ( y ) . (77) The marginal performance-based price (marginal forward indifference price) is p M ( t, S t , Y t ) := lim γ → p ( t, S t , Y t ) = E Q M h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i . (78) Proof.
Denote with Ψ the set of volatility risk premia ψ such that (68) is satisfied.Then Ψ parametrises all the Q ∈ M e,f through the well-defined map induced by(65). Therefore in control theory Ψ is called the control set and ψ a control . UsingProposition 3.4.1, the minimal entropy process (61) can be represented as H C ( t, S t , Y t ) = ess inf ψ ∈ Ψ E Q (cid:20) Z Tt (cid:20)(cid:16)b λ Su (cid:17) + ψ u (cid:21) d u − γ (cid:20) C ( Y T ) + 12 γ Z T (cid:16)b λ Su (cid:17) d u (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) = ess inf ψ ∈ Ψ E Q (cid:20) Z Tt (cid:20)(cid:16)b λ Su (cid:17) + ψ u (cid:21) d u − Z Tt (cid:16)b λ Su (cid:17) d u − γC ( Y T ) − A t (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) = ess inf ψ ∈ Ψ E Q (cid:20) Z Tt ψ u d u − γC ( Y T ) − A t (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) ( ) ==== − A t + ess inf ψ ∈ Ψ (cid:16) H t ( Q , Q M ) − γ E Q h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i(cid:17) . (79)Hence, the dual value process (56) becomes v C ( t, X t , S t , Y t ) = − exp (cid:18) − γX t + 12 A t − ess inf ψ ∈ Ψ (cid:16) H t ( Q , Q M ) − γ E Q h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i(cid:17)(cid:19) . (80)Denote by ψ C the optimal control in (79) with the claim. Analogous, let ψ be theoptimal control in absence of the claim. In the latter case, the relative entropy isminimised by the control ψ = ψ M = 0 and gives the minimal entropy process H ( t, S t , Y t ) = − A t + ess inf Q ∈M e,f H t ( Q , Q M ) = − A t + H t ( Q M , Q M ) = − A t , (81)31o that the value function has the optimal wealth X t = X t and simplifies to v ( t, X t , S t , Y t ) = − exp (cid:18) − γX t + 12 A t (cid:19) = U t ( X t ) = U ( X t ) . (82)With (79), (81), the forward indifference price from Corollary 3.3.6 has the expression p ( t, S t , Y t ) = − γ ess inf ψ ∈ Ψ (cid:16) H t ( Q , Q M ) − γ E Q h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i(cid:17) = ess sup ψ ∈ Ψ (cid:18) E Q h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i − γ Z Tt ψ u ( u, S u , Y u ) d u (cid:19) . (83)By (11), (67), the asset prices have the Q -dynamicsd S t = σ S S t d c W S, Q t , d Y t = σ Y Y t h ( b λ Yt − ρ b λ St − p − ρ ψ t ) d t + d c W Y, Q t i , (84)with the ( Q , b F )-Brownian motion c W Y, Q = ρ c W S, Q + p − ρ c W ⊥ , Q . (85)Then, with A Q M S,Y as the generator of (
S, Y ) under Q M (cf. (43)), the HJB equationfor p , by (83), is given by p t + A Q M S,Y p + max ψ ∈ Ψ (cid:20) − γ ψ − p − ρ σ Y yψp y (cid:21) = 0 , (86)with terminal value p ( T, s, y ) = C ( Y T ) and ψ = ψ ( t, s, y ). Consider the function f ( ψ ) = − γ ψ − p − ρ σ Y yψp y and determine its maximum by solving the equation f ′ ( ψ C ) = 0, which gives the optimal control (76), because of ψ H = ψ C − ψ = ψ C .Substituting this into the HJB equation (86) yields the PDE (77) for the forwardindifference price. The marginal performance-based indifference price (78) followsfrom the Feynman-Kac theorem (cf. Theorem 6.4.1 from [125, p. 268]), when thenon-linear term in the PDE (77) vanishes for γ → H C ( t, S t , Y t ) := ess inf ψ ∈ Ψ (cid:16) H t ( Q , P ) − γ E Q h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i(cid:17) , (87)with H t ( Q , P ) as in (68). Without the claim, the formula turns into H ( t, S t , Y t ) := ess inf ψ ∈ Ψ H t ( Q , P ) = ess inf ψ ∈ Ψ E Q (cid:20) Z Tt (cid:20)(cid:16)b λ Su (cid:17) + ψ u (cid:21) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) = 12 E Q (cid:20)Z Tt (cid:20)(cid:16)b λ Su (cid:17) + (cid:0) ψ Eu (cid:1) (cid:21) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) = H t ( Q E , P ) , (88)32here Q E is the MEMM, H t ( Q E , P ) the minimal conditional relative entropy and ψ E = ψ the minimal entropy control process . The value processes with and withoutthe claim are v C ( t, X t , S t , Y t ) = − exp (cid:0) − γX t − H C ( t, S t , Y t ) (cid:1) ,v ( t, X t , S t , Y t ) = − exp (cid:0) − γX t − H t ( Q E , P ) (cid:1) . (89)Because of the relative entropy additivity H t ( Q , P ) = H t ( Q , Q E ) + H t ( Q E , P ) from(74), the classical indifference price is the solution of the dual control problem p ( t, S t , Y t ) = − γ ess inf ψ ∈ Ψ (cid:18) E Q (cid:20) Z Tt h ψ u − (cid:0) ψ Eu (cid:1) i d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) − γ E Q h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i(cid:19) = − γ ess inf ψ ∈ Ψ (cid:16) H t ( Q , Q E ) − γ E Q h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i(cid:17) = ess sup ψ ∈ Ψ (cid:18) E Q h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i − γ H t ( Q , Q E ) (cid:19) , (90)which was shown by Monoyios [99, p. 903]. The HJB equation for (87) is H Ct + A Q M S,Y H C + 12 (cid:16)c λ S (cid:17) +min ψ ∈ Ψ (cid:20) ψ − p − ρ σ Y yψp y (cid:21) = 0 , H C ( T, s, y ) = − γC ( y ) . Solving yields the optimal control ψ C = p − ρ σ Y yH Cy and further the PDE H Ct + A Q M S,Y H C + 12 (cid:16)c λ S (cid:17) −
12 (1 − ρ ) (cid:0) σ Y yH Cy (cid:1) = 0 , H C ( T, s, y ) = − γC ( y ) . (91)Without the claim, the same approach returns ψ E = p − ρ σ Y yH y and an analo-gous PDE for H with H ( T, s, y ) = 0. Hence, the optimal (hedging) control is ψ H = ψ C − ψ E = − γ p − ρ σ Y yp y . (92)Subtract the PDEs (91) for H C and H according to (62) and apply the identities − γp y = H Cy − H y and γp y − p y H y = γ (cid:16)(cid:0) H Cy (cid:1) − (cid:0) H y (cid:1) (cid:17) to obtain the PDE for p , p t + A Q M S,Y p + 12 γ (1 − ρ ) (cid:0) σ Y yp y (cid:1) − p − ρ σ Y yp y ψ E = 0 , p ( T, s, y ) = C ( y ) . Expressed by the differential operator A Q E S,Y , the indifference price PDE has the form p t + A Q E S,Y p + 12 γ (1 − ρ ) (cid:0) σ Y yp y (cid:1) = 0 , p ( T, s, y ) = C ( y ) , (93)and the marginal utility-based price process is p M ( t, S t , Y t ) = E Q E h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i .In comparison to the well known classical case (90), (93), the relative entropyterm in the forward indifference problem of Theorem 3.4.2 is computed with respect33o Q M instead of Q E . The reason is that through the suitable choice of the mean-variance trade-off process A t = R t ( b λ Su ) d u , the conditional relative entropy H t ( Q , P )under the physical measure P in the minimal entropy function H C is transformedinto the relative entropy H t ( Q , Q M ) under the MMM by eliminating the entropyterm H t ( Q M , P ). These representations for American versions of the indifferenceprices were derived by Leung and Sircar [83] for classical utility and by Leung, Sircarand Zariphopoulou [84] for forward utility. The optimal hedging control ψ H in (76)and (92) have the same representation formula. The classical value processes (89)have, in comparison to the value processes (80), (82) of the forward model, no trade-off term, which only shows up in the forward performance process. In the forwardproblem, the optimal control without the claim vanishes, i. e. ψ = ψ M = 0, butin the classical model ψ = ψ E is not in general zero. This difference only occursin the partial information scenario when z S > z Y from (14). In the case z S ≤ z Y ,the stock’s MPR b λ S loses the dependence on the non-traded asset price Y , so that,after (84), Y is directly affected by ψ and therefore b λ S becomes independent of ψ .Thus, the drift term is excluded from the minimal entropy process (88). If the fullinformation scenario is applied, then the classical problem takes the MMM Q E = Q M because of ψ E = 0 and the trade-off term with the drift λ S under the backgroundfiltration F is again excluded from (88). In conclusion, an appropriate selection ofthe initial variance estimations z S , z Y with z S ≤ z Y in the Kalman-Bucy filter underpartial information from Proposition 2.3.3, ensures the same pricing in the forwardand classical model. Remark 3.4.3 (Distortion solution of the indifference price) . Monoyios [96] proved,that if the asset prices follow SDEs with stochastic volatilities of the formd S t = σ ( Y t ) S t ( λ ( Y t ) d t + d W t ) , d Y t = a ( Y t ) d t + b ( Y t ) (cid:16) ρ d W t + p − ρ d W ⊥ t (cid:17) , then the distortion transformation from Musiela and Zariphopoulou [104, pp. 222–223] leads to the solution of the classical indifference price PDE (93), p ( t, y ) = 1 γ (1 − ρ ) log E Q E t,y (cid:2) exp (cid:0) γ (1 − ρ ) C ( Y T ) (cid:1)(cid:3) , (94)given by Oberman and Zariphopoulou [108]. Leung et al. [84, pp. 16–17] gave thesolution in the forward performance model using the appropriate measure Q M . Theindifference price solution under full information looks like (94) with Q M and can befound in Henderson and Hobson [53, p. 344], Musiela and Zariphopoulou [104, p. 233]and Monoyios [95, p. 248]. ♦ .5 Residual risk In Subsection 3.1 we have discussed the complete market case, where perfect hedgingof the stock S by the derivative C ( Y ) is possible, when the underlying non-tradedasset price Y perfectly correlates with the stock price. One says, that the stock is replicated by the derivative on the non-tradeable underlying. If the asset prices arenot perfectly correlated, as in Subsection 3.2, then the hedge becomes imperfect and anon-hedgeable basis risk (hedging error) remains. The basis is the difference betweenthe price of the asset to be hedged and the price of the hedging instrument, whichis why residual risk is commonly also referred to as basis risk. Hedging a financialinstrument by another correlated instrument is called cross hedging .The reasons why an instrument is practically non-tradeable are diverse. For in-stance, its liquidity (trading volume) in the market could be very low or the spreadsand commission fees very high, so that trading is not economical. Or it is simplynot tradeable, because the instrument is an abstract synthetic product, like an in-dex. There are many examples in the commodities and OTC (over-the-counter) andderivatives markets with exotic products like weather and insurance indices or creditdefault derivatives. Ankirchner and Imkeller [6] introduced a typical example for across hedge, where an airline company wants to manage kerosene price risk. Sincethere is no liquid kerosene futures market, the airline company may fall back on fu-tures on less refined oil, such as crude oil futures, for hedging its kerosene risk. Thisis a reasonable approach, if the price evolutions of kerosene and crude oil are highlycorrelated. Ankirchner, Imkeller and Popier [8] dealt with optimal cross hedgingstrategies for insurance related derivatives. Other papers dealing with cross hedgingincluding practical examples are Ankirchner et al. [5], [7].
Definition 3.5.1 (Residual risk process) . Suppose, the investor shorts the claim C ( Y ) at time t = 0 for the price p (0 , S , Y ). To hedge this position over [0 , T ],the optimal hedging strategy θ H is used. His overall portfolio value is given by the residual risk process ̺ := ( ̺ t ) ≤ t ≤ T , defined by ̺ t = X t − p ( t, S t , Y t ) , (95)with initial and terminal values ̺ = 0 , ̺ T = X T − C ( Y T ) and the forward indifferenceprice p . The terminal residual risk is the terminal portfolio value that appeared inthe forward performance problem (37). The stock’s position value process is given byd X t = θ Ht d S t + r ( X t − θ t S t ) d t = θ Ht d S t , X = p (0 , S , Y ) (96)and the riskless interest rate r = 0. △ roposition 3.5.2 (Residual risk process) . The residual risk process of the forwardperformance-based model under the partial information scenario solves the SDE d ̺ t = 12 γ (1 − ρ ) (cid:0) σ Y Y t p y ( t, S t , Y t ) (cid:1) d t − p − ρ σ Y Y t p y ( t, S t , Y t ) d c W ⊥ t . (97) Proof.
By Definition 3.5.1, the residual risk has the differential expressiond ̺ t ( ) ==== d X t − d p ( t, S t , Y t ) ( ) ==== θ Ht d S t − d p ( t, S t , Y t ) . Using Theorem 3.3.7, Theorem 3.4.2 and Itˆo’s lemma, we obtain the SDEd ̺ t ( ) ==== (cid:18) p s + ρ σ Y Y t σ S S t p y (cid:19) d S t − (cid:18) p t d t + p s d S t + p y d Y t + 12 (cid:16) p ss d h S i t + p yy d h Y i t + p sy d h S, Y i t (cid:17)(cid:19) ( ) ==== ρσ Y Y t p y d c W S, Q t − (cid:16) p t + A Q M S,Y p (cid:17) d t − σ Y Y t p y d c W Y, Q M t ( ) ==== − (cid:16) p t + A Q M S,Y p (cid:17) d t − p − ρ σ Y Y t p y d c W ⊥ t ( ) ==== 12 γ (1 − ρ ) (cid:0) σ Y Y t p y (cid:1) d t − p − ρ σ Y Y t p y d c W ⊥ t , for the residual risk process ̺ , where c W Y, Q M = ρ c W S, Q + p − ρ c W ⊥ .The version of the residual risk SDE (97) under full information and classicalutility is in [104] and [97]. The residual risk evolution is expressed by a forwardindifference price-based drift term containing the coefficient 1 − ρ together with astochastic term including the orthogonal Brownian motion c W ⊥ and the scale parame-ter p − ρ . In the complete market scenario | ρ | = 1, the residual risk ̺ vanishes andno hedging error remains. But even if the absolute correlation is very high, meaningclose to 1, then a considerably high residual risk remains. If the correlation was highas ρ = 98%, the scale parameter of the drift term would be 1 − ρ ≈
4% and of thestochastic term even p − ρ ≈ Exchange Traded Funds (ETF) or derivatives try to replicate another (untrade-able) instrument like an index, then the measured correlation is not always perfectas desired and a so-called tracking-error arises. This was shown by Jorion [62], Aber,Can and Li [1] and Lobe, R¨oder and Schmidhammer [86] in various settings. Mod-els for dynamic conditional correlation were studied by Engle [39] and Franses andHafner [44].Figure 1: Effect of the correlation coefficient on the residual risk correlation s t d e v b a s i s / s t d e v t o t a l r i s k Remark 3.5.3 (Effect of correlation on diversification) . As the instability of theresidual risk ̺ ( ρ ) for absolute correlations close to 1 makes hedging more difficult,a similar effect can be found in classical portfolio theory from Markowitz [88]. Fordiversification purpose, consider a portfolio P = ϑS + (1 − ϑ ) Y containing two assets S, Y with relative weights ϑ, − ϑ ∈ [0 , P is then σ P = q ϑ ( σ S ) + (1 − ϑ ) ( σ Y ) + 2 ρϑ (1 − ϑ ) σ S σ Y ≤ ϑσ S + (1 − ϑ ) σ Y . The inequation follows from the evaluation of the binomial ( ϑσ S + (1 − ϑ ) σ Y ) anddelivers an equation when the assets are perfectly correlated. Since √ ρ has a lowslope when ρ is close to 1, a decrease of σ P and therefore a diversification effect onlyoccurs, when ρ rapidly falls towards 0. In the case of negative correlation this effectreverses. A small negative correlation may significantly lower the portfolio volatility.Sharpe [123], [124] and Lintner [85] deal also with classical portfolio theory. ♦ .6 Pay-off decompositions and asymptotic expansions In this subsection, we shall obtain pay-off decompositions of the claim followed by anasymptotic representation for the forward indifference price valid for small values ofrisk aversion. We pursue an approach as for classical utility from Monoyios [98].Recall from (67) and (84) the asset price dynamics under Q M ,d S t = σ S S t d c W S, Q t , d Y t = σ Y Y t h ( b λ Yt − ρ b λ St ) d t + d c W Y, Q M t i , with c W S, Q M = c W S, Q , c W ⊥ , Q M = c W ⊥ and c W Y, Q M = ρ c W S, Q + p − ρ c W ⊥ . Definition 3.6.1 (Preference-adjusted exponential of the residual risk) . The process L := ( L t ) ≤ t ≤ T , L t := − exp ( − γ̺ t ) , L = − , (98)is called preference-adjusted exponential of the residual risk (PAERR) . △ Corollary 3.6.2 (Preference-adjusted exponential of the residual risk) . The PAERRprocess L from Definition 3.6.1 is a ( P , b F ) -martingale with dynamics d L t = p − ρ σ Y Y t p y ( t, S t , Y t ) d c W ⊥ t . (99) Proof.
By Proposition 3.5.2 and Itˆo’s lemma it isd L t ( ) ==== − γL t d ̺ t + 12 γ L t d h ̺ i t ( ) ==== p − ρ σ Y Y t p y ( t, S t , Y t ) d c W ⊥ t . The martingale property follows, because the orthogonal Brownian motion c W ⊥ is amartingale under both measures Q M and P .Corollary 3.6.2 is similar to Proposition 6 of [104, p. 237] under full informationand classical utility, but with the forward indifference price depending on ( S, Y ) ratherthan the single variable Y due to the partial information scenario. Under classicalutility and partial information as in [98, Subsection 4.1], the dynamics (99) is ingeneral a ( Q E , b F )-martingale. Remark, that therein the process L starts with L = 0rather than L = −
1. Since L is a martingale, the classical exponential utility of theresidual risk is E [ U ( ̺ t )] = E [ − exp( γ̺ t )] = E [ L t ] = L = − E [ U t ( ̺ t )] = E (cid:20) − exp (cid:18) − γ̺ t + 12 Z t (cid:16)b λ Su (cid:17) d u (cid:19)(cid:21) = − E (cid:20) exp (cid:18) Z t (cid:16)b λ Su (cid:17) d u (cid:19)(cid:21) , decreases over time. 38 orollary 3.6.3 (Pay-off decomposition) . The claim pay-off decomposes into C ( Y T ) = p ( t, S t , Y t ) + Z Tt θ Hu d S u + L T − L t + 12 γ ( h L i T − h L i t ) , ≤ t ≤ T, (100) where θ H is the optimal hedging strategy for the claim, given in Theorem 3.3.7.Proof. By Proposition 3.5.2 and Corollary 3.6.2, the differential of the forward indif-ference price isd p ( t, S t , Y t ) ( ) ==== − γ (1 − ρ ) (cid:0) σ Y Y t p y (cid:1) d t + p − ρ σ Y Y t p y d c W ⊥ t + θ Ht d S t ( ) ==== − γ d h L i t + d L t + θ Ht d S t . Integration from t to T delivers the pay-off decomposition (100).The classical version under the full information scenario of Corollary 3.6.3 is Theo-rem 7 of [104, p. 238]. Under the partial information classical model of [98, Lemma 1],the pay-off decomposition (100) is measured under Q E , whereas the forward versionalways takes Q M . Pay-off decomposition is the suitable term, because L is a Q M -martingale with respect to c W ⊥ , which is strongly orthogonal to the Q M -martingale X T − X t = R Tt θ Hu d S u , that in turn, is defined as a stochastic integral with respect tothe Q M -Brownian motion c W S, Q induced by S . Mania and Schweizer [87, pp. 2129–2130] obtained an analogous pay-off decomposition in a more general backward SDEmodel under the classical framework. Definition 3.6.4 (Marginal preference-adjusted exponential of the residual risk) . Define the marginal preference-adjusted exponential of the residual risk (MPAERR) by the process L M := ( L t ) ≤ t ≤ T , L Mt := lim γ → L t . With the marginal performance-based price p M from (78) the evolution is d L Mt = p − ρ σ Y Y t p My ( t, S t , Y t ) d c W ⊥ t . TheMPAERR is also a Q M -martingale. △ Corollary 3.6.5 (F¨ollmer-Schweizer-Sondermann pay-off decomposition) . The claimpay-off admits the decomposition C ( Y T ) = p M ( t, S t , Y t ) + Z Tt θ Mu d S u + L MT − L Mt , ≤ t ≤ T, (101) where p M is the marginal performance-based price (78) and θ M the optimal hedgingstrategy (63) with p M in place of p .Proof. Equation (101) is the
F¨ollmer-Schweizer-Sondermann pay-off decomposition [42], [43] under Q M in our model and is immediately implied by Corollary 3.6.3 andDefinition 3.6.4 as γ → orollary 3.6.6 (Forward indifference price representation) . The forward indiffer-ence price admits the representation p ( t, S t , Y t ) = p M ( t, S t , Y t ) + 12 γ E Q M h h L i T − h L i t (cid:12)(cid:12)(cid:12) b F t i . (102) Proof.
Applying the conditional Q M -expectation given b F t on the pay-off decomposi-tion (100) eliminates R Tt θ Hu d S u + L T − L t , due to the martingale property. By themarginal performance-based price formula (78) the representation (102) follows.The classical version of Corollary 3.6.6 under Q E is dealt in [98, Corollary 2].Again, in the forward performance framework, the measure Q M is used. Proposition 3.6.7 (Asymptotic expansion of the forward indifference price) . Theforward indifference price has the asymptotic representation p ( t, S t , Y t ) = p M + 12 γ (cid:16) Var Q M h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i − E Q M h h X M i t,T (cid:12)(cid:12)(cid:12) b F t i(cid:17) + O ( γ ) , (103) where X Mt,T := X MT − X Mt := R Tt θ Mu d S u denotes the profit and loss of the wealth from t to T under the marginal hedging strategy and h X M i t,T its covariation.Proof. We make the same ansatz as in the classical version from [98, Theorem 2] andwrite the asymptotic expansion p ( t, S t , Y t ) = p M ( t, S t , Y t ) + γg ( t, S t , Y t ) + O ( γ ) , (104)with an appropriate process g := ( g t ) ≤ t ≤ T . By Corollary 3.6.6 and Corollary 3.6.2 itfollows γg ( t, S t , Y t ) + O (cid:0) γ (cid:1) ( ) ===== 12 γ E Q M h h L i T − h L i t (cid:12)(cid:12)(cid:12) b F t i ( ) ==== 12 γ (1 − ρ ) (cid:0) σ Y (cid:1) E Q M (cid:20)Z Tt Y u p y ( u, S u , Y u ) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) ( ) ===== 12 γ (1 − ρ ) (cid:0) σ Y (cid:1) E Q M (cid:20)Z Tt Y u (cid:0) p My + γg y + O ( γ ) (cid:1) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) = 12 γ (1 − ρ ) (cid:0) σ Y (cid:1) E Q M (cid:20)Z Tt (cid:0) Y u p My ( u, S u , Y u ) (cid:1) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) + O ( γ )and further leads to the solution g ( t, S t , Y t ) = 12 E Q M h h L M i T − h L M i t (cid:12)(cid:12)(cid:12) b F t i . (105)Inserting (105) into (104) gives the asymptotic expansion of the indifference price p ( t, S t , Y t ) = p M ( t, S t , Y t ) + 12 γ E Q M h h L M i T − h L M i t (cid:12)(cid:12)(cid:12) b F t i + O ( γ ) . (106)40otice, that by switching from the PAERR L in (102) to the MPAERR L M in (106),an expansion term of order O ( γ ) is added to the indifference price representation.The F¨ollmer-Schweizer-Sondermann decomposition (101) implies the pay-off varianceVar Q M h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i = E Q M h(cid:0) C ( Y T ) − p M ( t, S t , Y t ) (cid:1) (cid:12)(cid:12)(cid:12) b F t i = E Q M "(cid:18)Z Tt θ Mu d S u + L MT − L Mt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b F t = E Q M h h X M i T − h X M i t + h L M i T − h L M i t (cid:12)(cid:12)(cid:12) b F t i , (107)because L and X are orthogonal Q M -martingales. Inserting (107) after a rearrange-ment into (106) gives the asymptotic expansion (103). Early exercise claims arise often in situations in which a certain project is undertakenor abandoned (Smith and Nau [130], Smith and McCardle [129]), executives exercisetheir employee stock options (Aboody [2], Huddart [58]), household owners prepaytheir mortgages or sell their property (Hall [48], Kau and Keenan [74], Schwartz andTorous [121]). Allowing early exercise gives rise to stochastic control problems withstopping times. Early exercise options were priced for the first time by Davis and Za-riphopoulou [31] in the setting, where the option’s underlying asset is traded but withproportional transaction costs. Karatzas and Wang [73] studied utility maximisationproblems of mixed optimal stopping and control type in complete markets, whichcan be solved by reduction to a family of related pure optimal stopping problems.Oberman and Zariphopoulou [108] introduced a utility-based methodology for the val-uation of early exercise contracts in incomplete markets. Henderson and Hobson [54]considered the case of infinite time horizon, where the problem is expressed with re-spect to horizon-unbiased utility functions , a class of utility functions satisfying certainconsistency conditions over time, which are nothing less than forward utilitie. Leungand Sircar [83] studied problems of hedging American options with exponential utilitywithin a general incomplete market model. In Leung, Sircar and Zariphopoulou [84]this theory was expanded to the forward performance framework.In this section, we apply the forward performance model under the partial infor-mation scenario from Proposition 2.3.3 to American options to derive hedging andvaluation results comparable to the European counterparts of Section 3.41 .1 Optimal control and stopping problem
Suppose C is now an early exercise claim (American option) written on the non-traded asset Y . The investor sets up a hedging portfolio consisting of a long positionin the stock S and a short position in the option C as in the European scenario ofSection 3. Definition 4.1.1 (Admissible exercise times) . The collection of admissible exercisetimes is the set T of stopping times τ with respect to the observation filtration b F = ( b F t ) ≤ t ≤ T that take values in [0 , T ]. For 0 ≤ t ≤ u ≤ T , define the subset T t,u := { τ ∈ T | t ≤ τ ≤ u } of stopping times taking values in [ t, u ]. △ In addition to the dynamic trading strategy θ ∈ Θ, the investor chooses an exercisetime τ ∈ T , in order to maximise his expected forward performance of his hedgingportfolio X t − C t = θ t S t − C t . Therefore, let Θ t,τ denote the subset of strategies startingat t and terminating at τ . The claim pay-off becomes C ( Y τ ) := C ( τ, Y τ ) = C τ withthe exercise time τ as the terminal date instead of the fixed date T . Definition 4.1.2 (Optimal control and stopping problem) . The value process of theinvestor’s portfolio is the combined stochastic control and optimal stopping problem v C ( t, X t , S t , Y t ) := ess sup τ ∈T t,T ess sup θ ∈ Θ t,τ E h U τ ( X τ − C ( Y τ )) (cid:12)(cid:12)(cid:12) b F t i , ≤ t ≤ T. (108)The double essential supremum notation will be shortened to ess sup τ ∈T t,T ,θ ∈ Θ t,τ . △ In comparison to the European case (37), the optimisation is additionally per-formed under the stopping time. The forward indifference price is defined as inDefinition 3.2.1 and is useful to characterise the optimal exercise time τ ∗ . Corollary 4.1.3 (Optimal stopping time) . By (40) and (108), the optimal stoppingtime τ ∗ is the first time the value process reaches the forward performance process, i. e. τ ∗ t = inf (cid:8) u ∈ [ t, T ] (cid:12)(cid:12) v C ( u, X u , S u , Y u ) = U u ( X u − C ( Y u )) (cid:9) = inf (cid:8) u ∈ [ t, T ] (cid:12)(cid:12) v ( u, X u − p ( u, S u , Y u ) , S u , Y u ) = U u ( X u − C ( Y u )) (cid:9) = inf { u ∈ [ t, T ] | U u ( X u − p ( u, S u , Y u )) = U u ( X u − C ( Y u )) } = inf { u ∈ [ t, T ] | p ( u, S u , Y u ) = C ( Y u ) } , under appropriate integrability conditions (see [72, Theorem D.12]). Corollary 4.1.3 implies, that the investor exercises the American option as soonas the forward indifference price reaches from above the option pay-off and allowsanalysing the optimal exercise time through the forward indifference price.42 orollary 4.1.4 (Primal forward performance problem with American claim) . Underthe exponential forward performance (45), the primal problem (108) becomes v C ( t, X t , S t , Y t ) = ess sup τ ∈T t,T ,θ ∈ Θ t,τ E (cid:20) − exp (cid:18) − γ ( X τ − C ( Y τ )) + 12 Z τt (cid:16)b λ Su (cid:17) d u (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) , = e − γX t + R t ( b λ Su ) d u | {z } = U t ( X t ) ess sup τ ∈T t,T ,θ ∈ Θ t,τ E h − e − γ ( R τt θ u d S u − C ( Y τ ) ) + R τt ( b λ Su ) d u (cid:12)(cid:12)(cid:12) b F t i . To obtain the dual optimal control and stopping problem, some preparation isrequired. Firstly, a reconsideration and extension of the conditional relative entropyfrom Definition 3.3.2 is needed, to include the case of stopping times. Secondly, arelation between the conditional relative entropies up to time τ and T is derived.Lastly, a particular dynamic programming property of the classical Merton problemis recalled and applied to the American option case. Definition 4.1.5 (Stopped conditional relative entropy) . Define by H t,τ ( Q , P ) := E Q h log Z Q t,τ (cid:12)(cid:12)(cid:12) b F t i , ≤ t ≤ τ ∈ T , (109)the right stopped (conditional) relative entropy over the stochastic interval [ t, τ ] and by H τ,T ( Q , P ) := E Q h log Z Q τ,T (cid:12)(cid:12)(cid:12) b F τ i , T ∋ τ ≤ t ≤ T, (110)the left stopped (conditional) relative entropy over the stochastic interval [ τ, T ]. △ By Proposition 3.4.1, the right stopped relative entropy (109) is given by H t,τ ( Q , P ) = 12 E Q (cid:20)Z τt (cid:20)(cid:16)b λ Su (cid:17) + ψ u (cid:21) d u (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21) . The only difference to the European case is, that T is replaced by τ . From here, thenew notation H t,T ( Q , P ) is used for H t ( Q , P ). Remark, that the left stopped relativeentropy (110) is b F τ -conditional. According to Definition 4.1.5, the conditional relativeentropy over [ t, T ] splits into H t,T ( Q , P ) = H t,τ ( Q , P ) + E Q [ H τ,T ( Q , P ) | b F t ]. Lemma 4.1.6 (Decomposition of the relative entropy under stopping times) . Theconditional relative entropy H t,T ( Q , P ) decomposes into the right and left entropies ess inf ψ ∈ Ψ H t,T ( Q , P ) = ess inf ψ ∈ Ψ (cid:18) H t,τ ( Q , P ) + E Q (cid:20) ess inf ψ ∈ Ψ H τ,T ( Q , P ) (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21)(cid:19) , under the stopping time τ .Proof. A proof is given by Leung and Sircar [83, Lemma 2.7].43 roposition 4.1.7 (Primal and dual classical Merton problem with stopping time) . For an investor with starting wealth X τ at τ ∈ T , the classical Merton value process v ( τ, X τ ) = ess sup θ ∈ Θ τ,T E h U ( X T ) (cid:12)(cid:12)(cid:12) b F t i has the dual separable representation v ( τ, X τ , S τ ) = U ( X τ ) exp (cid:18) − ess inf ψ ∈ Ψ H τ,T ( Q , P ) (cid:19) . With starting wealth X t at t ∈ [0 , T ] , the classical value process can be written as v ( t, X t , S t ) = ess sup θ ∈ Θ t,τ E h v ( τ, X τ , S τ ) (cid:12)(cid:12)(cid:12) b F t i , τ ∈ T . (111) Proof.
We refer to [83, Propositions 2.5 and 2.6].The dynamic programming property (111) is called the self-generating condition by Musiela and Zariphopoulou [105], and horizon-unbiased condition by Hendersonand Hobson [54].
Proposition 4.1.8 (Dual classical problem with American option) . The dual classicalvalue process in the American option case is given by v C ( t, X t , S t , Y t )= U ( X t ) exp − ess sup τ ∈T t,T ess inf ψ ∈ Ψ (cid:16) H t,τ ( Q , P ) + E Q h H τ,T ( Q E , P ) − γC ( Y τ ) (cid:12)(cid:12)(cid:12) b F t i(cid:17)! . The classical exponential indifference price is given by p ( t, S t , Y t ) = − γ ess sup τ ∈T t,T ess inf ψ ∈ Ψ (cid:16) H t,τ ( Q , Q E ) − γ E Q h C ( Y τ ) (cid:12)(cid:12)(cid:12) b F t i(cid:17) . Proof.
A detailed proof is given in [83, Propositions 2.4 and 2.8]. Therein, the claimis additionally dependent on the stock S , i. e. C τ = C ( τ, S τ , Y τ ). Theorem 4.1.9 (Forward indifference price valuation with American option) . Thedual forward performance problem with the American option has the representation v C ( t, X t , S t , Y t )= U t ( X t ) exp − ess sup τ ∈T t,T ess inf ψ ∈ Ψ (cid:16) H t,τ ( Q , Q M ) − γ E Q h C ( Y τ ) (cid:12)(cid:12)(cid:12) b F t i(cid:17)! (112) with the entropic representation of the forward indifference price p ( t, S t , Y t ) = − γ ess sup τ ∈T t,T ess inf ψ ∈ Ψ (cid:16) H t,τ ( Q , Q M ) − γ E Q h C ( Y τ ) (cid:12)(cid:12)(cid:12) b F t i(cid:17) . (113)44 roof. Follow the approach from [84, Proposition 2.7] by transforming the pay-off into e C ( τ, S τ , Y τ ) = C ( τ, Y τ ) + 12 γ Z τt (cid:16)b λ S ( u, S u , Y u ) (cid:17) d u + 1 γ H τ,T ( Q E , P ) . (114)Then recall the primal forward performance problem from Corollary 4.1.4, v C ( t, X t , S t , Y t ) = U ( X t ) ess sup τ ∈T t,T ,θ ∈ Θ t,τ E h − e − γ ( R τt θ u d S u − C ( Y τ ) ) + R τt ( b λ Su ) d u (cid:12)(cid:12)(cid:12) b F t i = U ( X t ) ess sup τ ∈T t,T ,θ ∈ Θ t,τ E h − e − γ ( R τt θ u d S u − e C ( τ,S τ ,Y τ ) ) −H τ,T ( Q E , P ) (cid:12)(cid:12)(cid:12) b F t i , through a substitution of the claim pay-off C by the transform e C . Now, the applica-tion of Proposition 4.1.8 yields v C ( t, X t , S t , Y t )= U t ( X t ) exp − ess sup τ ∈T t,T ess inf ψ ∈ Ψ (cid:16) H t,τ ( Q , P ) + E Q h H τ,T ( Q E , P ) − γ e C τ (cid:12)(cid:12)(cid:12) b F t i(cid:17)! = U t ( X t ) exp − ess sup τ ∈T t,T ess inf ψ ∈ Ψ (cid:18) H t,τ ( Q , P ) − E Q (cid:20) Z τt (cid:16)b λ Su (cid:17) d u + γC τ (cid:12)(cid:12)(cid:12)(cid:12) b F t (cid:21)(cid:19)! = U t ( X t ) exp − ess sup τ ∈T t,T ess inf ψ ∈ Ψ (cid:16) H t,τ ( Q , P ) − H t,τ ( Q M , P ) − γ E Q h C ( Y τ ) (cid:12)(cid:12)(cid:12) b F t i(cid:17)! = U t ( X t ) exp − ess sup τ ∈T t,T ess inf ψ ∈ Ψ (cid:16) H t,τ ( Q , Q M ) − γ E Q h C ( Y τ ) (cid:12)(cid:12)(cid:12) b F t i(cid:17)! , which proves (112). The forward indifference price representation (113) is then im-plied by (40). In this thesis we applied the forward performance framework, defined by Musiela andZariphopoulou [106] to the basis risk model with partial information from Monoyios[98] to solve the forward utility maximisation problem of exponential type of an in-vestor with a hedging portfolio consisting of a long position in the traded stock S and a short position of a claim written on the non-traded asset Y . We obtained theoptimal hedging strategy, value function and indifference price representation usingmethods from duality theory. In the case of an European option, we discussed themain result containing the change of the MEMM Q E to the MMM Q M after Theo-rem 3.4.2. We derived the residual risk, pay-off decompositions and an asymptotic45xpansion of the indifference price. Then we changed to the market model withan American option having a random exercise time inspired by Leung, Sircar andZariphopoulou [84]. We formulated the optimal control problem with stopping timeand obtained the representations for the value function and forward indifference price.Hereinafter, we take up some points of the thesis to discuss future research topics.We carry out a comprehensive review of the parameter uncertainty in the Kalman-Bucy filter used in our partial information model and discuss alternatives from recentpublications. Furthermore, we present the semi-martingale framework for utlity max-imisation problems allowing to use weaker assumptions on the model. Moreover, weoutline the approach of solving the forward indifference price PDE in the Europeanoption’s case with numerical methods, by applying the asymptotic expansion of theindifference price as an approximation. For the case with an American option, wedescribe the variational inequality for the forward indifference price to be expectedand a suggestion for solving it numerically. In addition, we propose a larger mar-ket model by making more claims available for the market agent, and discuss otherlarge markets in utility maximisation theory. Lastly, we present a generalisation ofstochastic utilities used in forward utility-based optimisation theory. In Subsection 2.2 and Subsection 2.3 we developed our partial information modelthrough a Kalman-Bucy filter with known Gaussian prior distribution based onMonoyios [97], [98]. We assumed in Definition 2.3.1 the signal process Λ = λ S λ Y ! with unknown MPR constants λ S , λ Y of the asset prices S, Y to have a Gaussian priordistributionΛ | b F ∼ N (Λ , Σ ) , Λ := λ S λ Y ! , Σ := z S c c z Y ! , c := ρ min { z S , z Y } , for given constants λ S , λ Y , z S , z Y . This is the underlying distribution of the Kalman-Bucy Filter introduced in Definition 2.3.2. The first assumption is made by choos-ing Gaussian random variables, the second is the knowledge of the prior distribu-tion parameters. If the second assumption is omitted, the problem of uncertainMPRs is shifted to the problem of unknown parameters of the Gaussian prior dis-tribution. One could specify intervals for the parameters, e. g. for λ S , using thebest estimate approach from Subsection 2.2. The single standard deviation interval[ λ S ( t ) − √ t , λ S ( t ) + √ t ] with confidence 66 .
27% leads to approximately t ≈
10 years46f empirical data. With 90% confidence t ≈
271 years of market data is required. Ahigher confidence of 95% needs historical data collected since the Early Middle Ages.However, this example shows the statistical error by adopting the estimated interval,which affects the accuracy of the filter. Decision making as utility optimisation basedon the filter gets an additional inherent risk.Monoyios [98, Section 6] carried out extensive numerical simulations with empiri-cal examples of hedging under the partial information model with classical utility. Hedemonstrated, that the filtering procedure can improve the performance of the hedge,provided that the prior is not extremely poor. The rate of learning by the filter on theasset price MPRs is too slow to counteract parameter uncertainty without the extrainsurance of an increased option premium. Monoyios concluded, that considering thecombined valuation and hedging program, taking parameter uncertainty into accountvia an increased option premium and using a filtering approach is of benefit.
Robustness with respect to model uncertainty in stochastic filtering has beenconsidered for diverse linear and non-linear systems. Miller and Pankov [94] andSiemenikhin [127], for instance, studied linear dynamics with parameter uncertaintyin the noise covariance matrices using a so-called minimax filter , which is basicallyan estimator minimising the maximal expected loss over a range of possible models.This idea emerged in Wald [134] in 1945, in which the problem is to find a distribu-tion minimising the maximum risk, which is a general statistical inference problem .Therein, the risk is defined as an integral function of the unknown parameters andweighted statistical decision functions . Martin and Mintz [89] examined the existenceand behaviour of game-theoretic solutions for robust linear filters and predictors inthe context of time-discrete models. They discovered that robust Kalman-Bucy filterscan be realised when the least favourable prior distribution is either independent, of,or only weakly dependent upon the specific decision interval. Moreover, they con-cluded, based on practical experience with times series data, that uncertain dynamics(drifts) can have far greater effect on filter and predictor performance than typical un-certainties in either the signal or observation noise covariances. Verd´u and Poor [132]noted that minimax estimators are criticised as being too pessimistic and having apoor performance in the most statistically probable model, since they are dependenton the specification of an often arbitrary uncertainty class, likely taking implausiblemodels into consideration.Allan and Cohen [3] have recently discussed some other filter techniques and pro-posed a new approach to parameter uncertainty in stochastic filtering, specificallywhen working with the time-continuous Kalman-Bucy filter by making evaluations47ia a non-linear expectation, represented in terms of a penalty function . The penalty is a measure for the error evolving in time caused by the uncertainty, and is calculatedby propagating the a priori uncertainty forward through time using filter dynamics.An idea, that has been taken from Cohen [25] and [26], in which the investigationconcerned time-discrete models in a binomial and Markov chain framework, respec-tively.We proposed in Remark 2.3.4 an Ornstein-Uhlenbeck model for the signal process(MPRs) and mentioned the parameter uncertainty issue. This model is more com-plicated than the constant signal Λ filtered with the determined Gaussian prior inthe sense, that it has multiple parameter uncertainties. An alternative model is thelinear equation d λ it = α t λ it d t + β t d W it , i = S, Y, with Gaussian prior Λ | b F ∼ N (Λ , Σ ) as defined in (8), and measurable, locallybounded, deterministic functions α and β of time on appropriate real intervals asdefined in [3]. The parameter functions α and β are assumed to be uncertain. Throughfollowing the methods from Allan and Cohen [3], its feasible to tackle this issue byformulating the penalty problem and measuring penalties dependent of different trueand estimated parameters. Robust upper and lower expectations of the signal canprovide error bounds for the Kalman-Bucy filter. If the signal Λ follows the Ornstein-Uhlenbeck process (15), then one could try to apply the theory of [3] with the aim toanalyse penalties and calculate robust bounds for the Kalman-Bucy filter.In addition, one may consider a broader class of prior distributions for the Kalman-Bucy filter. For instance, Beneˇs and Karatzas [14] analysed filtering with non-Gaussian prior distribution and showed that the conditional distribution is a mixtureof Gaussians, which is propagated by two sets of sufficient statistics . These statis-tics obey usually non-linear SDEs implementable of a filter. For a Gaussian initialdistribution, there is only one random sufficient statistic propagating the conditionaldensity, in accordance with the classical theory.Mostovyi and Sˆırbu [102] have recently studied the sensitivity of an expected util-ity maximisation problem in a continuous semi-martingale market with respect tosmall changes in the MPR. They analyse the stochastic control problem under theperturbation and give an explicit form of the correction terms for an example withpower utility. Eventually, this discussion brings up the question, whether and how pa-rameter uncertainty of the Kalman-Bucy filter affects the forward utilities, (optimal)hedging strategies, residual risk, forward indifference price and claim representationstreated in this dissertation, which is a good topic for future research.48 .2 Utility maximisation in semi-martingale financial models For our dual performance maximisation problems, we expressed in Definition 3.3.1the equivalent local martingale measures (ELMMs) Q ∈ M e,f by the Radon-Nikodymderivative processes Z Q under b F . Since Karatzas and Kardaras [69], it has been ac-knowledged that one does not need ELMMs. Karatzas and Kardaras studied optimalutility-based hedging strategies in a general semi-martingale model with a weakerassumption, the “No Unbounded Profit with Bounded Risk” (NUPBR) instead of thestronger “No Free Lunch with Vanishing Risk” (NFLVR) condition. They proved,that the optimal portfolio even exists, when the NFLVR assumption is replaced byNUPBR and filled the gap between the “No Arbitrage” (NA) and NFLVR condi-tions. The NUPBR rule involves the boundedness in probability of the terminalvalues of wealth processes and is the minimal a priori assumption required in orderto proceed with utility optimisation (cf. [69, p. 449]). Using semi-martingale modelswith NUPBR is a topic of current research, for example, treated by Mostovyi andSˆırbu [102], [103] and Mostovyi [101]. In Theorem 3.4.2, we gave the forward indifference price for the European option interms of a control problem (75), solving the semi-linear PDE (77), which is p t + A Q M S,Y p + 12 γ (1 − ρ ) (cid:0) σ Y yp y (cid:1) = 0 , p ( T, s, y ) = C ( y ) . In our partial information model from Subsection 2.3, it was not possible to derivea closed probabilistic representation with the distortion method similar to the fullinformation scenario (94) in Remark 3.4.3. The same issue was present under classicalutility in Monoyios [98].A potential further action is the derivation of a numerical solution to the aforemen-tioned PDE for the forward indifference price and investigate valuation and hedgingperformances inspired by the classical case from [98, Section 6]. Comparable to [98,Section 5], one can try to obtain an analytic formula for the conditional variance ofthe claim Var Q M h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i in the asymptotic expansion of the indifference price(103) from Proposition 3.6.7, and to specify the distribution parameters of log Y T in terms of the partial information model parameters from Proposition 2.3.3. Thenext step is the attempt to give the Black-Scholes representations of the marginalforward indifference price p M and marginal hedging strategy θ M , and approximatethe indifference price by the asymptotic expansion to obtain the derivatives of p M Q M h C ( Y T ) (cid:12)(cid:12)(cid:12) b F t i . Using the explicit formulas for themarginal indifference price and hedging strategy, the final step is to give an integralrepresentation of the expected covariation of the profit and loss E Q M h h X M i t,T (cid:12)(cid:12)(cid:12) b F t i in (103). If this approach succeeds, then one can try to numerically evaluate thisexpression using a Monte-Carlo simulation . One expects to find that the forwardutility approach is like a low risk aversion limit of the classical approach, since theminimal martingale measure Q M is used.In the case of an American option from Section 4, a further approach is to derivethe variational inequality for the forward indifference price p ( t, s, y ) under partialinformation similar to the one given by Leung et al. [84, Subsection 3.1] under fullinformation. One expects, that the indifference price solves the free boundary problem p t + A Q M S,Y p + γ (1 − ρ ) (cid:0) σ Y yp y (cid:1) ≤ ,p ( t, s, y ) ≥ C ( t, y ) , (cid:16) p t + A Q M S,Y p + γ (1 − ρ ) (cid:0) σ Y yp y (cid:1) (cid:17) ( C ( t, y ) − p ( t, s, y )) = 0 ,p ( T, s, y ) = C ( T, y ) , (115)for ( t, s, y ) ∈ [0 , ∞ ) × R × [0 , T ]. The crucial difference to [84] is that the indifferenceprice under our partial information model depends additionally on S and Y , ratherthan only on Y . Nevertheless, [84, equation (32)] displays the variational inequalityin the general case, when p as well as C depend on S and Y . To derive a numeri-cal solution for the indifference price, one needs to solve the free boundary problem(115) in three dimensions, which is a non-trivial task. In [84, Section 4] early exerciseproblems of employee stock options (ESOs) are modelled under the full informationscenario with constant MPRs λ S , λ Y , and solved numerically using a fully explicitfinite-difference scheme for the exponential forward performance case. Full and par-tial information models of ESOs are analysed, for instance, by Henderson et al. [55]and Monoyios and NG [100]. In our basis risk market model, we considered a single European (American) option C with fixed expiry T (early exercise time τ ) on the non-traded asset Y . This marketmodel can be enlarged by offering n ( n >
1) European (American) claims C , . . . , C n written on Y with expiries T , . . . , T n (early exercise times τ , . . . , τ n ) and pay-offs C ( Y T ) , . . . , C n ( Y T n ) ( C ( Y τ ) , . . . , C n ( Y τ n )). Furthermore, a setting of mixed Euro-pean and American claims may be considered. When creating the hedging portfolio,50he market agent must decide, how many and which claims he is going to include.One may simply value each single option C j , j = 1 , . . . , n separately by setting up n different hedging portfolios with a long position in the stock S and a short positionwith a unit of the option C j , but it is not clear, if one misses on possible effects in therisk management strategies. Specifically, with forward utility and different, flexibleexercise times, the number of options held in the portfolio at the same time can vary.Generally, one may enlarge the financial market by adding more assets. The con-cept of a large security market was described by Kabanov and Kramkov [63] as a se-quence of probability spaces (general models), whereas Bj¨ork and N¨aslund [18] defineda large market to be one probability space with countably many assets (( S it ) ≤ t ≤ T ) ∞ i =1 .Donno, Guasoni and Pratelli [35] applied the classical utility maximisation theory on alarge market, studied them with duality methods and characterised replicable claims.Mostovyi [101] considered the model from [35] with stochastic utility. He concluded,that the value function with countably many assets is the limit of the value functionsof the finite-dimensional models, but the optimal strategy with infinite assets is nota limit of the trading strategies of the finite-dimensional markets, in general. We used the forward utility U t ( x ) = − e − γx + R t ( b λ Su ) d u of exponential type, defined inSubsection 3.2, for the utility optimisation problems. It is a specific forward utilityderived by the class of asymptotically linear local risk tolerance functions dealt inSubsection 2.4. Musiela and Zariphopoulou [105], [107] suggested this model to givemore flexibility to the individual risk preferences of an investor adapting the marketdevelopment. El Karoui and M’Rad [36], [38] studied the consistency of dynamicutilities. They introduced the general notion of progressive utility , which is a collectionof Itˆo semi-martingales with dynamicsd U ( t, X t ) = β ( t, X t ) d t + γ ( t, X t ) d W t , (116)including drift and volatility processes β, γ . The stochastic utilities are often referredto as utility random fields . Utility random fields of investment and consumption wereconsidered at first by Berrier and Tehranchi [15] and Berrier et al. [16]. El Kaouri etal. [37] extended the forward utility setting through market-consistent utility randomfields that are calibrated to a given learning σ -algebra. They provided differentialregularity conditions on stochastic utility properties ensuring the existence of consis-tency and optimal strategies. Defining utility random fields through SDEs of the form(116) offers an opportunity for future research in utiliy-based valuation and hedging.51 eferences [1] Aber, J.W., Li, D. and Can, L. , Price volatility and tracking ability of ETFs ,Journal of Asset Management 10(4) (2009), pp. 210–221.[2]
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