A Multidimensional Exponential Utility Indifference Pricing Model with Applications to Counterparty Risk
aa r X i v : . [ q -f i n . P R ] S e p A Multidimensional Exponential UtilityIndifference Pricing Model with Applications toCounterparty Risk
Vicky Henderson † , ♯ and Gechun Liang ‡ , ♯ † Department of Statistics, University of Warwick, Coventry, CV4 7AL, U.K. , [email protected] ‡ Department of Mathematics, King’s College London, London, WC2R 2LS, U.K. , [email protected] ♯ Oxford-Man Institute, University of Oxford, Oxford, OX2 6ED, U.K.
August 10, 2018
Abstract
This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model subject to inter-temporal default risk, and provides a semigroup approxi-mation for the utility indifference price. The key tool is the splitting method, whose convergenceis proved based on the Barles-Souganidis monotone scheme, and the convergence rate is derivedbased on Krylov’s shaking the coefficients technique. We apply our methodology to study thecounterparty risk of derivatives in incomplete markets.
Keywords : Utility indifference pricing, reaction-diffusion PDE with quadratic gradients, split-ting method, monotone scheme, shaking the coefficients technique, counterparty risk.
Mathematics Subject Classification (2010) : 91G40, 91G80, 60H30.
We thank the Oxford-Man Institute for partial support of this research. We thank participants at the 3rd AQFCconference (Hong Kong, July, 2015) and also Damiano Brigo, Rama Cont, David Hobson, Monique Jeanblanc, LishangJiang, Eva L¨utkebohmert, Andrea Macrina, Shige Peng, Martin Schweizer, Xingye Yue, and Thaleia Zariphopouloufor their helpful discussions. Introduction
The purpose of this article is to consider exponential utility indifference pricing in a multidi-mensional non-traded assets setting subject to intertemporal default risk, which is motivated byour study of counterparty risk of derivatives in incomplete markets. Our interest is in pricing andhedging derivatives written on assets which are not traded. The market is incomplete as the risksarising from having exposure to non-traded assets cannot be fully hedged. We take a utility in-difference approach whereby the utility indifference price for the derivative is the cash amount theinvestor is willing to pay such that she is no worse off in expected utility terms than she would havebeen without the derivative.There has been considerable research in the area of exponential utility indifference valuation,but despite the interest in this pricing and hedging approach, there have been relatively few explicitformulas derived. The well known one dimensional non-traded assets model is an exception and ina Markovian framework with a derivative written on a single non-traded asset, and partial hedgingin a financial asset, Henderson and Hobson [22], Henderson [20], and Musiela and Zariphopoulou[40] used the Cole-Hopf transformation ( or distortion power ) to linearize the non-linear PDE forthe value function. This trick results in an explicit formula for the exponential utility indifferenceprice. Subsequent generalizations of the model from Tehranchi [45], Frei and Schweizer [16] and [17]showed that the exponential utility indifference value can still be written in a closed-form expressionsimilar to that known for the Brownian setting, although the structure of the formula can be muchless explicit. On the other hand, Davis [14] used the duality to derive an explicit formula for theoptimal hedging strategy (see also Monoyios [39]), and Becherer [4] showed that the dual pricingformula exists even in a general semimartingale setting.As soon as one of the assumptions made in the one dimensional non-traded assets model breaksdown, explicit formulas are no longer available. For example, if the option payoff depends also onthe traded asset, Sircar and Zariphopoulou [42] developed bounds and asymptotic expansions forthe exponential utility indifference price. In an energy context, we may be interested in partiallyobserved models and need filtering techniques to numerically compute expectations (see Carmonaand Ludkovski [11] and Chapter 7 of [10]). If the utility function is not exponential, Henderson[20] and Kramkov and Sirbu [33] developed expansions in small quantity for the utility indifferenceprice under power utility.In this paper, we study exponential utility indifference valuation in a multidimensional settingsubject to intertemporal default risk with the aim of developing a pricing methodology. The maineconomic motivation for us to develop the multidimensional framework is to consider the coun-terparty default risk of options traded in over-the-counter (OTC) markets, often called vulnerableoptions . The credit crisis has brought to the forefront the importance of counterparty default riskas numerous high profile defaults lead to counterparty losses. In response, there have been manyrecent studies (see, for example, Bielecki et al [5] and Brigo et al [9]) addressing in particular thecounterparty risk of credit default swaps (CDS). In contrast, there is relatively little recent workon counterparty risk for other derivatives, despite OTC options being a sizable fraction of the OTCderivatives market. The option holder faces both price risk arising from the fluctuation of theassets underlying her option and counterparty default risk that the option writer does not honorher obligations. Default occurs either when the assets of the counterparty are below its liabilitiesat maturity ( the structural approach ) or when an exogenous random event occurs ( the reduced-form In fact, OTC options comprised about 10% of the $600 trillion (in terms of notional amounts) OTC derivativesmarket at the end of June 2010 whilst the CDS market was about half as large at around $30 trillion. pproach ), so intertemporal default is considered. In our setting, the assets of the counterparty andthe assets underlying the option may be non-traded and thus a multidimensional non-traded assetsmodel naturally arises.Our use of the utility indifference approach is motivated by its recent use in credit risk modelingwhere the concern is the default of the reference name rather than the default of the counterparty.Utility based pricing has also been utilized by Bielecki and Jeanblanc [6], Sircar and Zariphopolou[43] and recently Jiao et al [27] [28] in an intensity based setting. Several authors have applied it inmodeling of defaultable bonds where the problem remains one dimensional, see in particular Leunget al [35], Jaimungal and Sigloch [25], and Liang and Jiang [36]. In contrast, options subject tocounterparty risk are a natural situation where two or more dimensions arise.Our first contribution is the derivation of a reaction-diffusion partial differential equation (PDE)in Theorem 2.3 to characterize the utility indifference price in our multidimensional setting, wherewe do not rely on the dynamic programming principle. Instead, we consider the associated utilitymaximization problems for utility indifference valuation from a risk-sensitive control perspective.We first transform our utility maximization problems into risk-sensitive control problems, andemploy the comparison principle to derive a quadratic backward stochastic differential equation(BSDE) representation for the utility indifference price. See also Theorem 2.3 of Henderson andLiang [24] where we derived a quadratic BSDE representation for the utility indifference price in anon-Markovian setting with full recovery rate. The pricing PDE is then obtained by an applicationof nonlinear Feynman-Kac formula for quadratic BSDE (see Section 3 of Kobylanski [32]).Our main contribution is to develop a semigroup approximation for the pricing PDE by thesplitting method. In our multidimensional setting, the Cole-Hopf transformation (as in the onedimensional model) cannot be applied directly since the coefficients of the quadratic gradient termsdo not match, and due to the existence of the intertemporal payment term. Motivated by the ideaof the splitting method ( or fractional step, prediction and correction, etc ) in numerical analysis,we split the pricing equation into two semilinear PDEs with quadratic gradients and an ordinarydifferential equation (ODE) with Lipschitz coefficients, such that the Cole-Hopf transformation canbe applied to linearize both PDEs, and the Picard iteration can be used to linearly approximatethe ODE.The idea of splitting in our setting is as follows. The time derivative of the pricing equationdepends on the sum of semigroup operators corresponding to different factors. For each subproblemcorresponding to each semigroup there might be an effective way providing solutions, but for thesum of the semigroups, there may not be an accurate method. The application of the splittingmethod means that we treat the semigroup operators separately. We prove that when the meshof the time partition goes to zero, the approximate price will converge to the utility indifferenceprice in Theorem 2.9, relying on the monotone scheme introduced by Barles and Souganidis [3]:any monotone, stable and consistent numerical scheme converges to the correct solution providedthere exists a comparison principle for the limiting equation. Moreover, by employing the shakingthe coefficients technique introduced by Krylov [34] (see also Barles and Jacobsen [2, 26] for itsdevelopment), we are able to obtain the convergence rate of our splitting algorithm in Theorems2.11 and 2.12. The key to apply the monotone scheme and shaking the coefficients technique isto derive an consistent error estimate, which we prove by utilizing a coupled forward backwardstochastic differential equation (FBSDE) representation for the solution of PDEs in (2.19).Our third contribution is the application of the splitting method to compute prices of derivativeson a non-traded asset and where the derivative holder is subject to non-traded counterparty defaultrisk. In contrast to the complete market Black-Scholes style formulas obtained by Johnson and Stulz329], Klein [30] and Klein and Inglis [31], we show the significant impact that non-tradeable riskshave on the valuation of vulnerable options and the role played by partial hedging. In particular,our numerical illustrations quantify the effect of non-tradeable price and default risk on optionprices. The magnitude of the potential discounts to the complete market price depends upon thelikelihood of default and if default is likely, the discount can be extremely high. We also observethat put values can decrease with maturity (in absence of dividends) in situations where the riskof default is significant.Splitting methods have been used to construct numerical schemes for PDEs arising in mathe-matical finance (see the review of Barles [1], and Tourin [46] with the references therein). Recently,Nadtochiy and Zariphopoulou [41] applied splitting to the marginal Hamilton-Jacobi-Bellman (HJB)equation arising from optimal investment in a two-factor stochastic volatility model with generalutility functions. They show their scheme converges to the unique viscosity solution of the limitingequation. Whilst they also apply splitting to an incomplete market problem, their focus is to dealwith the lack of a verification theorem in their setting. In contrast, we propose a multidimensionalmodel subject to intertemporal default risk with time and space variable coefficients and exponentialutility. Our contributions include proposing a splitting approach for utility indifference pricing in amultidimensional non-traded assets model with intertemporal default risk, identifying how to splitthe resulting pricing PDE, and moreover, we prove the convergence rate of our splitting method byusing advanced techniques from the theory of viscosity solutions. Other recent works of Halperinand Itkin [18, 19] propose the use of splitting methods to price options on a single illiquid bond viamixed static-dynamic hedging. Our model is instead designed for multiple non-traded assets, whichis necessary for our treatment of counterparty risk in a hybrid structural-reduced form setting.Finally, Tan [44] proposes a splitting method for fully nonlinear degenerate parabolic PDEs andapplies it to Asian options and commodity trading.The paper is organized as follows: In Section 2, we present our multidimensional exponentialutility indifference pricing model, and propose a splitting method to solve the pricing equation.The convergence of the splitting algorithm and its convergence rate are proved in Section 2.3 and2.4, respectively. In Section 3, we apply the method to study counterparty risk. We conclude withthe verification theorem of the pricing equation in the Appendix.
Let W = ( W , . . . , W n +2 ) be an ( n + 2)-dimensional Brownian motion on a filtered probabilityspace (Ω , F , F = {F t } t ≥ , P ) satisfying the usual conditions , where F t is the augmented σ -algebragenerated by ( W u : 0 ≤ u ≤ t ). The market consists of a risk-free bank account with price 1, a setof observable but non-traded assets S = ( S , . . . , S n ), whose logarithm price processes are drivenby dS it = µ i ( S t , t ) dt + σ i ( S t , t ) dW it + ¯ σ i ( S t , t ) dW n +1 t (2.1)for i = 1 , . . . , n , and a traded financial index P , whose price process is driven by dP t P t = µ P dt + ¯ σ P dW n +1 t + σ P dW n +2 t . (2.2)The price of each non-traded asset S i reflects exposure to the traded or market risk W n +1 throughvolatility ¯ σ i ( S t , t ) and non-traded idiosyncratic risk W i through idiosyncratic or undiversifiable4olatility σ i ( S t , t ). We define the following parameters for the financial index P : θ P = ( µ P ) ( σ P ) ; ¯ θ P = ( µ P ) ( σ P ) + (¯ σ P ) ; ϑ P = µ P ¯ σ P ( σ P ) ; ¯ ϑ P = µ P ¯ σ P ( σ P ) + (¯ σ P ) ; κ P = (¯ σ P ) ( σ P ) ; ¯ κ P = (¯ σ P ) ( σ P ) + (¯ σ P ) . Our interest will be in pricing and hedging λ units of a contingent claim written on the non-traded assets S with maturity T . The payoff is delivered at either a random time τ or maturity T .The random time τ represents an inter-temporal default time, which is constructed in a canonicalway. Let e be an independent exponential random variable on the same probability space (Ω , F , P ).Then the random time τ is constructed as follows τ = inf (cid:26) s ≥ Z s a ( S u , u ) du ≥ e (cid:27) , where a ( · , · ) is an intensity function valued in R + . The original Brownian filtration F = {F t } t ≥ is enlarged by G t = F t ∨ H t for t ≥ H t = σ ( { τ ≤ u } : 0 ≤ u ≤ t ). Hence τ is the firstarrival time of a Cox process ( or doubly stochastic Poisson process ), which satisfies the followingenlargement of filtration property: The stochastic process W is an ( n + 2)-dimensional Brownianmotion under both filtrations {F t } t ≥ and {G t } t ≥ (see Chapter 8 of Bielecki and Rutkowski [7] fordetails).If such a random time τ happens before maturity T , i.e. τ ≤ T , the payoff at τ is R C λ ( S τ , τ ),where R ∈ [0 ,
1] represents the recovery rate, and C λ : R n × [0 , T ] → R + is the solution of the PDE(2.10). In Theorem 2.3 we shall show that C λ ( · , · ) is actually the utility indifference value functionof the contingent claim, so we consider the fractional recovery of market value. If τ > T , the payoffat T is λg ( S T ), where g ( · ) is a payoff function valued in R + . Therefore, the total payoff for λ unitsof this contingent claim is { τ ≤ T } R C λ ( S τ , τ ) + { τ>T } λg ( S T ) . Assumption 2.1 (i) The coefficients µ i , σ i , ¯ σ i , the intensity function a , and the payoff function g depend on the logarithm price of the non-traded assets: f = f ( S t , t ) ∈ C b ( R n × [0 , T ]) for f = µ, σ i , ¯ σ i , a, g , where C b ( R n × [0 , T ]) is the space of bounded and continuous functions which areLipschitz continuous in space.(ii) The volatilities of non-traded assets S satisfy the following uniformly elliptic condition: | σ i | ≥ ǫ > for i = 1 , . . . , n . The model is applicable in many situations. It might be that these assets are (i) not tradedat all, or (ii) that they are traded illiquidly, or (iii) that they are in fact liquidly traded but theinvestor concerned is not permitted to trade them for some reason. Our main application is tothe counterparty risk of derivatives where the final payoff depends upon both the value of thecounterparty’s assets and the assets underlying the derivative itself (see Section 3). A secondpotential area of application is to residual or basis risks arising when the assets used for hedgingdiffer from the assets underlying the contract in question (see Davis [14]). Typically this ariseswhen the assets underlying the derivative are illiquidly traded (case (ii) above) and standardized5utures contracts are used instead. Contracts may involve several assets, for example, a spreadoption with payoff ( K − e S T − e S T ) + or a basket option with payoff ( K − e S T − · · · − e S nT ) + . Suchcontracts frequently arise in applications to commodity, energy, and weather derivatives. Finally, aone dimensional example of the situation in (iii) is that of employee stock options (see [21]).On the other hand, the model also allows for intertemporal default risk, and the recovery atthe prepayment time is mark-to-market value. Hence, the model is well suited to counterpartycredit risk, where the intertemporal payment due to default usually depends on the market valueof the contract. Such an intertemporal default is modeled in a reduced form, so the correspondingcounterparty credit risk model is a hybrid between the structural and reduced form approaches.Other applications include optimal investment problems with uncertain time horizon (see Blanchet-Scalliet et al [8]), and modeling the prepayment risk of mortgage-backed securities (see Zhou [47]).Our approach is to consider the utility indifference valuation for such a contingent claim. Forthis we need to consider the optimization problem for the investor both with and without the option.The investor has initial wealth x ∈ R at any time t ∈ [0 , T ], and is able to trade the financial indexwith price P t (and riskless bond with price 1). This will enable the investor to partially hedge therisks she is exposed to via her position in the claim. Depending on the context, the financial indexmay be a stock, commodity or currency index, for example.The holder of the option has an exponential utility function with respect to her terminal wealth: U T ( x ) = − e − γx for γ ≥ . At time t ∈ [0 , T ], the investor holds λ units of the contingent claim, whose price is denoted as C λt and is to be determined, and invests her remaining wealth x − C λt in the financial index P . Theinvestor will follow an admissible trading strategy: π ∈ A F [ t, T ] = (cid:8) ( π s ) s ∈ [ t,T ] : π s = π ( S s , s ) is uniformly bounded (cid:9) , which results in the wealth on the event { τ > t } : X x − C λt s ( π ) = x − C λt + Z st π s P s dP s . (2.3)The investor will optimize over such strategies to choose an optimal π ∗ ,λ by maximizing herexpected terminal utility:ess sup π ∈A F [ t,T ] E P " − e − γ (cid:18) { t<τ ≤ T } (cid:18) X x − C λtτ ( π )+ R C λ ( S τ ,τ ) (cid:19) + { τ>T } (cid:18) X x − C λtT ( π )+ λg ( S T ) (cid:19)(cid:19) |G t . (2.4)To define the utility indifference price for the option, we also need to consider the optimizationproblem for the investor without the option. Her wealth equation is the same as (2.3) but startsfrom initial wealth x and she will choose an optimal π ∗ , by maximizingess sup π ∈A F [ t,T ] E P h − e − γX xT ( π ) |G t i . (2.5)Since the payoff of the optimal portfolio problem (2.5) is F -adapted, (2.5) is equivalent to thestandard Merton problem on the event { τ > t } :ess sup π ∈A F [ t,T ] E P h − e − γX xT ( π ) |F t i , {F t } t ≥ .The utility indifference price for the option is the cash amount that the investor is willing topay such that she is no worse off in expected utility terms than she would have been without theoption. For a general overview of utility indifference pricing, we refer to the monograph edited byCarmona [10] and the survey article by Henderson and Hobson [23] therein. Definition 2.2
The utility indifference price C λt of λ units of the derivative with the payoff { τ ≤ T } R C λ ( S τ , τ ) + { τ>T } λg ( S T ) at time t ∈ [0 , T ] is defined by the solution to ess sup π ∈A F [ t,T ] E P " − e − γ (cid:18) { t<τ ≤ T } (cid:18) X x − C λtτ ( π )+ R C λ ( S τ ,τ ) (cid:19) + { τ>T } (cid:18) X x − C λtT ( π )+ λg ( S T ) (cid:19)(cid:19) |G t = ess sup π ∈A F [ t,T ] E P h − e − γX xT ( π ) |G t i . (2.6) The hedging strategy for λ units of the derivative at time t on the event { τ > t } is defined by thedifference in the optimal trading strategies π ∗ ,λt − π ∗ , t . Our main result in this subsection is to show that the utility indifference price is given by C λt = { τ>t } C λ ( S t , t ) , where C λ ( S t , t ) is the solution of the reaction-diffusion PDE (2.10) withquadratic gradients, so it can be interpreted as the (pre-default) utility indifference value function.The function C λ ( S t , t ) is the main object that we are working on. Define the following operators: L = 12 n X i =1 σ i ( s , t ) ∂ s i s i + 12 n X i,j =1 ¯ σ i ( s , t )¯ σ j ( s , t ) ∂ s i s j + n X i =1 µ i ( s , t ) ∂ s i , (2.7) L = − n X i =1 ¯ ϑ P ¯ σ i ( s , t ) ∂ s i − γ n X i =1 σ i ( s , t )( ∂ s i ) − γ n X i,j =1 (1 − ¯ κ P )¯ σ i ( s , t )¯ σ j ( s , t )( ∂ s i )( ∂ s j ) , (2.8)and for s = ( s , · · · , s n ) and C λ ( s , t ), L C λ ( s , t ) = a ( s , t ) γ h − e γ (1 − R ) C λ ( s ,t ) i . (2.9)The operator L describes the infinitesimal behavior of the price processes of the non-traded assets S = ( S , · · · , S n ), the operator L reflects the investor’s risk aversion, and the operator L reflectsthe intertemporal payment which is distorted by the investor’s risk aversion. Theorem 2.3 (PDE representation for utility indifference price)Suppose that Assumption 2.1 is satisfied. Then the following reaction-diffusion PDE withquadratic gradients on the domain ( s , t ) ∈ R n × [0 , T ] : (cid:26) − ∂ t C λ ( s , t ) − ( L + L + L ) C λ ( s , t ) = 0 , C λ ( s , T ) = λg ( s ) (2.10)7 dmits a unique viscosity solution C λ ( s , t ) ∈ C b ( R n × [0 , T ]) . Moreover, the utility indifference priceof λ units of the derivative at time t ∈ [0 , T ] with the payoff { τ ≤ T } R C λ ( S τ , τ ) + { τ>T } λg ( S T ) isgiven by C λt = { τ>t } C λ ( S t , t ) , and the hedging strategy for λ units of the option at time t on the event { τ > t } is given by − ¯ κ P ¯ σ P n X i =1 ¯ σ i ( S t , t ) ∂ s i C λ ( S t , t ) . (2.11)The proof of Theorem 2.3 is provided in Appendix A, where we do not rely on the dynamicprogramming principle. Instead, we transform the optimal portfolio problems (2.4) and (2.5) intorisk-sensitive control problems, and derive a quadratic BSDE representation for the utility indif-ference price, inspired by Theorem 2.3 of Henderson and Liang [24]. Then PDE (2.10) is obtainedby an application of nonlinear Feynman-Kac formula for quadratic BSDE (see Kobylanski [32]).We note that the number of units λ only appears in the terminal condition. In the following, wepresent the case λ = 1, and the price at time t ∈ [0 , T ] is simply denoted by C t = { τ>t } C ( S t , t ).We first compare to the situation that the market is complete. If the underlying assets S =( S , · · · , S n ) could be traded, the market would become complete, and the pricing and hedging ofthe contingent claim with payoff: { τ ≤ T } R ¯ C ( S τ , τ ) + { τ>T } g ( S T )falls into the multidimensional Black-Scholes framework with intertemporal default risk. Corollary 2.4
Suppose that Assumption 2.1 is satisfied, and that S = ( S , · · · , S n ) are tradedassets. Let ¯ C ( s , t ) ∈ C b ( R n × [0 , T ]) be the unique viscosity solution of the reaction-diffusion PDEon the domain ( s , t ) ∈ R n × [0 , T ] : (cid:26) − ∂ t ¯ C ( s , t ) − (¯ L + ¯ L )¯ C ( s , t ) = 0 , ¯ C ( s , T ) = g ( s ) (2.12) where the operators ¯ L and ¯ L are given, respectively, by ¯ L = 12 n X i =1 σ i ( s , t ) ∂ s i s i + 12 n X i,j =1 ¯ σ i ( s , t )¯ σ j ( s , t ) ∂ s i s j ¯ L ¯ C ( s , t ) = − a ( s , t )(1 − R )¯ C ( s , t ) . Then the price of the option with payoff { τ ≤ T } R ¯ C ( S τ , τ ) + { τ>T } g ( S T ) at time t ∈ [0 , T ] is givenby ¯ C t = { τ>t } ¯ C ( S t , t ) . The pricing equation (2.10) has an additional nonlinear term L relative to the complete marketpricing PDE (2.12), and this L reflects the investor’s risk aversion. Moreover, the inter-temporalpayment term ¯ L in (2.12) is distorted to L by the investor’s risk aversion in (2.10). We have thefollowing asymptotic result relating the utility indifference price C t to the complete market price¯ C t at any time t ∈ [0 , T ]. 8 roposition 2.5 Assume that ¯ ϑ P = µ i ¯ σ i ( s , t ) for i = 1 , . . . , n. (2.13) Then the unit utility indifference value function C ( s , t ) uniformly converges to the complete marketvalue function ¯ C ( s , t ) as γ → on any compact subset of R n × [0 , T ] . Proof.
By the condition (2.13), the first-order linear terms in (2.10) become zero: n X i =1 µ i ( s , t ) ∂ s i − n X i =1 ¯ ϑ P ¯ σ i ( s , t ) ∂ s i = 0 . When γ →
0, the terms involving γ in L converge to zero, and L C ( s , t ) → ¯ L C ( s , t ). Therefore,by the stability of viscosity solutions, there exists a subsequence γ n → C ( s , t ; γ n ), uniformly converge to ¯ C ( s , t ) on any compact subset of R n × [0 , T ], where ¯ C ( s , t ) satisfies (2 . S and the financial index P . The Sharpe ratio of P is given by √ ¯ θ P . Similarly, wedefine the Sharpe ratio of S i to be √ ¯ θ i , where ¯ θ i = µ i σ i +¯ σ i ( s , t ). Then (2.13) is equivalent to therelation; p ¯ θ i = ¯ σ i ( s , t )¯ σ P p ¯ σ i ( s , t ) + σ i ( s , t ) p ¯ σ P + σ P ! p ¯ θ P = ρ iP p ¯ θ P , where ρ iP is the correlation between S i and P . This corresponds to the relation we expect fromthe capital asset pricing model (CAPM) when assets are traded. Since not all assets are tradedhere, we would not necessarily expect (2.13) to hold. The intuition is that when the idiosyncraticvolatilities disappear, and when assets are traded, there cannot be a difference in using the financialindex P or the assets themselves to hedge.Based on the pricing equation (2.10) and the PDE comparison principle, we present a numberof monotone properties of the utility indifference price. Their proofs are similar to Section A.2, sowe omit them. Proposition 2.6
The unit utility indifference value function C ( s , t ) is increasing with the recoveryrate R , the payoff g ( · ) and the intensity a ( · , · ) , and is decreasing with the risk aversion parameter γ .Moreover, if the condition (2.13) holds, then C ( s , t ) is also decreasing in the idiosyncratic volatilityof the traded asset σ P (or its proportion of total volatility, − ¯ κ P ). The last assertion of the above proposition tells us that the higher the idiosyncratic volatility σ P of the traded asset ( or as a proportion of total volatility ), the worse it is as a hedging instrument,and the lower the price one is willing to pay. This generalizes the monotonicity obtained in the onedimensional non-traded asset model (see, for example, Henderson [21] and Frei and Schweizer [16]in a non-Markovian model with stochastic correlation). For a reaction-diffusion PDE with quadratic gradients like (2.10), it is not possible to obtain anexplicit solution. A special case where an explicit solution does exist is the one dimensional version9ithout intertemporal default. Taking n = 1, σ P = 0 and R = 1 in (2.10) recovers the pricing PDEof [22], [20] and [40], which is solved by the Cole-Hopf transformation. However, this transformationdoes not apply directly to our multidimensional problem (2.10) because the coefficients of thequadratic gradient terms in L do not match, and the existence of the intertemporal payment term L . Instead, we will develop a splitting algorithm which will enable us to take advantage again ofthe Cole-Hopf transformation to linearize the PDEs.The splitting method ( or fractional step, prediction and correction, etc ) can be dated backto Marchuk [38] in the late 1960’s. The application of splitting to nonlinear PDEs such as HJBequations is difficult mainly because of the verification of the convergence for the approximatescheme. This was overcome by Barles and Souganidis [3], who employed the idea of viscositysolutions and proved that any monotone, stable and consistent numerical scheme converges providedthere exists a comparison principle for the limiting equation.The idea of splitting in our setting is the following. The time derivative of the pricing PDE (2.10)depends on the sum of semigroup operators ( or the associated infinitesimal operators ) correspondingto the different factors. These semigroups usually are of different nature. For each subproblemcorresponding to each semigroup there might be an effective way providing solutions. For the sumof these semigroups, however, we usually can not find an accurate method. Hence, application ofsplitting method means that instead of the sum, we treat the semigroup operators separately.The tricky part is how to split the equation ( or how to group factors ) effectively. In next lemma,we separate the pricing PDE (2.10) into three pricing factors by using the transformation (2.14),which is the key step to apply the splitting method to (2.10). Lemma 2.7
Define a new differential operator: ∂∂η = n X i =1 ¯ σ i ( s , t ) ∂∂s i . (2.14) Then (2.10) reduces to − ∂ t C − (cid:16) ˆ L + ˆ L + ˆ L (cid:17) C = 0 , (2.15) where ˆ L = 12 ∂ ηη − γ − ¯ κ P )( ∂ η ) , (2.16)ˆ L = 12 n X i =1 σ i ( s , t ) ∂ s i s i + n X i =1 A i ( s , t ) ∂ s i − γ n X i =1 σ i ( s , t )( ∂ s i ) (2.17)ˆ L = L (2.18) with A i ( s , t ) = µ i ( s , t ) − (cid:2) σ i ( s , t ) + ¯ σ i ( s , t ) (cid:3) − ¯ ϑ P ¯ σ i ( s , t ) . For any 0 ≤ t < t + ∆ ≤ T and any function φ ∈ C b ( R n +1 ), we define the following nonlinearbackward semigroup operators S i (∆) by φ ( · ) C i ( · , t ) where − ∂ t C i − ˆ L i C i = 0; C i ( · , t + ∆) = φ ( · ) (2.19)10n the domain R n +1 × [ t, t + ∆] for i = 1 , ,
3. That is, S i (∆) φ is the solution of (2.19) at time t with terminal data φ at time t + ∆.We observe that (2.19) for i = 1 , C = exp( − γ (1 − ¯ κ P ) C ), then we have ¯ C satisfying − ∂ t ¯ C − ∂ ηη ¯ C = 0 . (2.20)By letting ¯ C = exp( − γ C ), then we have ¯ C satisfying − ∂ t ¯ C − n X i =1 σ i ( s , t ) ∂ s i s i ¯ C − n X i =1 A i ( s , t ) ∂ s i ¯ C = 0 . (2.21)Moreover, (2.19) for i = 3 can be approximated by Picard iterations, since ˆ L = L is Lipschitzcontinuous (see Appendix A.2). Lemma 2.8
The operators S i (∆) for i = 1 , , have the following properties: • (i) For any function φ ∈ C b ( R n +1 ) , lim ∆ ↓ S i (∆) φ = φ uniformly on any compact subset of R n +1 . • (ii) S i (∆ ′ ) φ = S i (∆ ′ − ∆) S i (∆) φ for any ≤ t < t + ∆ < t + ∆ ′ ≤ T . • (iii) S i (0) φ = φ. ((i) (ii) and (iii) ensure that S i (∆) is indeed a strongly continuous semigroup operator.) • (iv) For any functions φ, ψ ∈ C b ( R n +1 ) such that φ ≥ ψ , S i (∆) φ ≥ S i (∆) ψ. • (v) S i (∆) φ is uniformly bounded, and moreover, | S i (∆) φ − S i (∆) ψ | ≤ C | φ − ψ | , where | · | represents the usual supremum norm. • (vi) For any φ ∈ C ∞ b ( R n +1 ) , the space of bounded and smooth functions, define the consistenterror: E i (∆ , φ ) = (cid:12)(cid:12)(cid:12)(cid:12) S i (∆) φ ( s ) − φ ( s )∆ − ˆ L i φ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . hen E (∆ , φ ) ≤ C ∆ | ∂ s φ | + X i + j =4 | ∂ i s φ | | ∂ j s φ | + | ∂ s φ | | ∂ s φ | + C ∆ (cid:0) | ∂ s φ | + | ∂ s φ | | ∂ s φ | (cid:1) , E (∆ , φ ) ≤ C ∆ X i =2 , , | ∂ i s φ | + X i + j =3 , | ∂ i s φ | | ∂ j s φ | + | ∂ s φ | | ∂ s φ | + C ∆ X i =2 , | ∂ i s φ | + | ∂ s φ | | ∂ s φ | , E (∆ , φ ) ≤ C ∆ , where s = ( η, s , . . . , s n ) with a slight abuse of notation. Proof. (i)-(v) are immediate. We only prove (vi) in the following. The key idea is to make useof the coupled FBSDE representation for C i ( · , t ) = S i (∆) φ ( · ). We first prove the case i = 1.Let (Ω , F , F = {F t } t ≥ , Q ) be a filtered probability space satisfying the usual conditions , onwhich supports a one-dimensional Brownian motion B . Then an application of nonlinear Feynman-Kac formula yields the following FBSDE representation of C ( η, t ): C ( η, t ) = φ ( X t +∆ ) − Z t +∆ t ∂ η C ( X s , s ) dB s = E Q [ φ ( X t +∆ ) | X t = η ] , (2.22)with X s = η − Z st γ − ¯ κ P ) ∂ η C ( X u , u ) du + Z st dW u . (2.23)By applying Itˆo’s formula to φ ( X t +∆ ), we obtain that S (∆) φ ( η ) − φ ( η ) − ∆ˆ L φ ( η )= E Q [ φ ( X t +∆ ) | X t = η ] − φ ( η ) − ∆ˆ L φ ( η )= E Q " φ ( η ) + Z t +∆ t ∂ η φ ( X s ) dX s + 12 ∂ η φ ( X s ) d h X i s | X t = η − φ ( η ) − ∆ˆ L φ ( η )= E Q "Z t +∆ t [ − γ − ¯ κ P ) ∂ η φ ( X s ) ∂ η C ( X s , s ) + 12 ∂ η φ ( X s )] ds | X t = η − ∆ˆ L φ ( η )Since ∂ η C ( X s , s ) = ∂ η E Q [ φ ( X t +∆ ) | X s ] = E Q [ ∂ η φ ( X t +∆ ) Y t +∆ | X s ] , with Y being the gradient flow of X : Y t +∆ = 1 − R t +∆ s γ (1 − ¯ κ P ) ∂ ηη C ( X u , u ) Y u du , we furtherhave S (∆) φ ( η ) − φ ( η ) − ∆ˆ L φ ( η )= E Q "Z t +∆ t − γ − ¯ κ P ) ∂ η φ ( X s ) ∂ η φ ( X t +∆ ) Y t +∆ + 12 ∂ η φ ( X s ) ds | X t = η − ∆ˆ L φ ( η ) .
12y applying Itˆo’s formula once again, we calculate the above conditional expectation as E Q − γ − ¯ κ P ) ∆ ∂ η φ ( η ) + Z t +∆ t Z st ( − γ − ¯ κ P ) X i + j =3 ∂ iη φ∂ jη C + 12 ∂ η φ ) duds · ∂ η φ ( η ) + Z t +∆ t ( − γ − ¯ κ P ) X i + j =3 ∂ iη φ∂ jη C + 12 ∂ η φ ) ds · (1 + O (∆))+ 12 ∆ ∂ η φ ( η ) + Z t +∆ t Z st ( − γ − ¯ κ P ) X i + j =4 ∂ iη φ∂ jη C + 12 ∂ η φ ) duds | X t = η where i, j ≥ φ ( · ) and C ( · , · ). Since φ ∈ C ∞ b and usingthe fact that ∂ iη C ( X s , s ) = E Q [ ∂ iη φ ( X t +∆ ) | X s ](1 + O (∆)), we obtain the consistent error estimate: E (∆ , φ ) ≤ C ∆ (cid:0) | ∂ η φ | + | ∂ η φ | | ∂ η φ | + | ∂ η φ | + | ∂ η φ | | ∂ η φ | (cid:1) + C ∆ (cid:0) | ∂ η φ | + | ∂ η φ | | ∂ η φ | (cid:1) . (2.24)The proof for the case i = 2 is similar, so we skip its proof and only provide the correspondingconsistent error estimate: E (∆ , φ ) ≤ C ∆ (cid:0) | ∂ s φ | + | ∂ s φ | | ∂ s φ | + | ∂ s φ | + | ∂ s φ | | ∂ s φ | + | ∂ s φ | + | ∂ s φ | | ∂ s φ | + | ∂ s φ | (cid:1) + C ∆ (cid:0) | ∂ s φ | + | ∂ s φ | | ∂ s φ | + | ∂ s φ | (cid:1) . (2.25)Finally, the proof for case i = 3 is a simple application of Taylor expansion, which yields E (∆ , φ ) ≤ C ∆ . (2.26)Next we use semigroup operators S i (∆) for i = 1 , , or PDE (2.10) with different coordinates ), which is the main result ofthis section. Theorem 2.9 (Semigroup approximation for utility indifference price)Suppose that Assumption 2.1 is satisfied. Let t i = i ∆ for i = 0 , , . . . , N with N ∆ = T . Thenthe unit utility indifference value function C ( · , · ) of the derivative with the payoff { τ ≤ T } R C ( S τ , τ )+ { τ>T } g ( S T ) at time t i is approximated by C ∆ ( · , t i ) = S (∆) C ∆ ( · , t i +1 ) = S (∆) S (∆) S (∆) C ∆ ( · , t i +1 ) (2.27) with C ∆ ( · , t N ) = g ( · ) . The values between any two adjacent partition points are obtained by usuallinear interpolation. Then, lim ∆ → C ∆ ( · , · ) = C ( · , · ) . uniformly on any compact subset of R n +1 × [0 , T ] . roof. The proof is based on the Barles-Souganidis monotone scheme [3], in which they provedthat any monotone, stable and consistent numerical scheme converges, provided there exists acomparison principle for the limiting equation.By the stability property (v) of Lemma 2.8, the following semi-relaxed limits of C ∆ are welldefined: C ( s , t ) = lim sup ( s ′ ,t ′ ) → ( s ,t ) , ∆ → C ∆ ( s ′ , t ′ ); C ( s , t ) = lim inf ( s ′ ,t ′ ) → ( s ,t ) , ∆ → C ∆ ( s ′ , t ′ ) . We show that C is a viscosity subsolution of (2.15). A symmetric argument will imply that C isa viscosity supersolution of (2.15), which proves that C = C = C , so C ∆ converges to C locallyuniformly.Let φ ∈ C ∞ b and ( s , t ) be such that0 = ( C − φ )( s , t ) = max ( s ′ ,t ′ ) ( C − φ )( s ′ , t ′ ) . By the definition of C , there exists a sequence ( s n , t n ) such that( s n , t n , ∆) → ( s , t , , and C ∆ ( s n , t n ) → C ( s , t ) . Moreover, by extracting a subsequence if necessary, ( s n , t n ) is also the maximum point of C ∆ − φ : δ ∆ = ( C ∆ − φ )( s n , t n ) = max ( s ′ ,t ′ ) ( C ∆ − φ )( s ′ , t ′ ) → . The monotone property (iv) of Lemma 2.8 then implies that φ ( s n , t n ) + δ ∆ − S (∆) (cid:0) φ ( s n , t n + ∆) + δ ∆ (cid:1) ∆ ≤ C ∆ ( s n , t n ) − S (∆) C ∆ ( s n , t n + ∆)∆ = 0From (v) of Lemma 2.8, S (∆) (cid:0) φ ( s n , t n + ∆) + δ ∆ (cid:1) ≤ S (∆) ( φ ( s n , t n + ∆)) + Cδ ∆ . In turn, usingthe consistent property (vi) of Lemma 2.8 together with the estimate (2.31) and letting ( s n , t n , ∆) → ( s , t , − ∂ t φ ( s , t ) − (ˆ L + ˆ L + ˆ L ) φ ( s , t ) ≤ . That is, C ( · , · ) is a viscosity subsolution of (2.15). The proof of the convergence rate of numerical viscosity solutions is much more difficult than theproof of the convergence itself. This open problem was first solved by Krylov [34], who developedthe so called shaking the coefficients technique , and was extensively studied by Barles and Jakobsenin a series of their papers, which also introduced an alternative optimal switching approximation method (see [2] and [26] with more references therein).The key step to apply Krylov’s technique in our setting is to obtain an estimate of consistenterror, and the other steps follow a similar line of argument as in [2] and [26]. Hence, we only derivethe consistent error estimate in the following, and outline the proofs of the other steps.14e first derive an upper bound of C − C ∆ . For any ǫ >
0, we may build a viscosity subsolution C ǫ ∈ C b ( R n +1 × [0 , T ]) of (2.15) by shaking its coefficients such that C − ǫ ≤ C ǫ ≤ C . (2.28)Next, we regularize C ǫ by convolution to obtain a smooth subsolution of (2.15). For this, let ρ be an R + -valued smooth function with support {| s | < } × { < t < } and mass 1. From thisfunction, we introduce a sequence of mollifiers ρ ǫ as follows: ρ ǫ ( s , t ) = 1 ǫ n +3 ρ (cid:18) s ǫ , tǫ (cid:19) . Define C ǫ ( s , t ) = C ǫ ∗ ρ ǫ ( s , t ) = Z − ǫ <τ< Z | e | <ǫ C ǫ ( s − e, t − τ ) ρ ǫ ( e, τ ) dedτ. A Riemann sum approximation shows that C ǫ ( s , t ) can be viewed as the limit of convex combi-nations of C ǫ ( s − e, t − τ ) for − ǫ ≤ τ ≤ | e | < ǫ . Since (2.15) is convex in ∂ t C and ∂ i s C for i = 0 , ,
2, the convex combinations of C ǫ ( s − e, t − τ ) are also the subsolutions of (2.15). In turn, C ǫ is itself a subsolution of (2.15) by the stability of viscosity solutions.On the other hand, standard properties of mollifiers imply that C ǫ ∈ C ∞ b , | C ǫ − C ǫ | ≤ Cǫ, (2.29)and moreover, | ∂ it C ǫ | ≤ Cǫ − i ; | ∂ j s C ǫ | ≤ Cǫ − j . (2.30)The key result to obtain the convergence rate of splitting is the following consistent error esti-mate. Lemma 2.10
Suppose that Assumption 2.1 is satisfied. Then, E (∆ , C ǫ ) = (cid:12)(cid:12)(cid:12)(cid:12) S (∆) C ǫ ( s , t ) − C ǫ ( s , t − ∆)∆ − ∂ t C ǫ ( s , t ) − (ˆ L + ˆ L + ˆ L ) C ǫ ( s , t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:0) ∆ + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − (cid:1) . (2.31) Proof.
Define C iǫ = C i − ǫ + ∆ˆ L i C i − ǫ for i = 1 , , C ǫ = C ǫ . We then split the consistenterror ∆ E (∆ , C ǫ ) into five parts as follows: | S (∆) C ǫ ( s , t ) − C ǫ ( s , t − ∆) − ∆ ∂ t C ǫ ( s , t ) − ∆(ˆ L + ˆ L + ˆ L ) C ǫ ( s , t ) | = | S (∆) C ǫ ( s , t ) − C ǫ ( s , t ) − ∆(ˆ L + ˆ L + ˆ L ) C ǫ ( s , t )+ C ǫ ( s , t ) − C ǫ ( s , t − ∆) − ∆ ∂ t C ǫ ( s , t ) | ≤ | S (∆) S (∆) S (∆) C ǫ ( s , t ) − S (∆) S (∆) C ǫ ( s , t ) | + | S (∆) S (∆) C ǫ ( s , t ) − S (∆) C ǫ ( s , t ) | + | S (∆) C ǫ ( s , t ) − C ǫ ( s , t ) | + | C ǫ ( s , t ) − C ǫ ( s , t ) − ∆(ˆ L + ˆ L + ˆ L ) C ǫ ( s , t ) | + | C ǫ ( s , t ) − C ǫ ( s , t − ∆) − ∆ ∂ t C ǫ ( s , t ) | = [1] + [2] + [3] + [4] + [5] . | S (∆) S (∆) S (∆) C ǫ ( s , t ) − S (∆) S (∆) C ǫ ( s , t ) | ≤ C | S (∆) C ǫ ( s , t ) − C ǫ ( s , t ) | ≤ C ∆(∆ ǫ − + ∆ ǫ − ) , (2.32)where the second inequality follows from the property (2.30) of the mollifier C ǫ , while keeping theworst ( highest order ) terms involving ǫ .For [2], we employ (v) of Lemma 2.8, the consistent error estimate (2.25), together with theproperty (2.30) of the mollifier C ǫ to derive that | S (∆) S (∆) C ǫ ( s , t ) − S (∆) C ǫ ( s , t ) | ≤ C ∆(∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − ) , (2.33)and similarly for [3], | S (∆) C ǫ ( s , t ) − C ǫ ( s , t ) | ≤ C ∆ . (2.34)Next, for [4], by the definition of C iǫ with i = 1 , ,
3, we have | C ǫ ( s , t ) − C ǫ ( s , t ) − ∆(ˆ L + ˆ L + ˆ L ) C ǫ ( s , t ) | = ∆ (cid:12)(cid:12)(cid:12) (ˆ L C ǫ ( s , t ) − ˆ L C ǫ ( s , t )) + (ˆ L C ǫ ( s , t ) − ˆ L C ǫ ( s , t )) (cid:12)(cid:12)(cid:12) ≤ C ∆ (cid:0) (∆ ǫ − + ∆ ǫ − ) + (∆ ǫ − + ∆ ǫ − + ∆ ǫ − ) (cid:1) ≤ C ∆(∆ ǫ − + ∆ ǫ − + ∆ ǫ − ) . (2.35)For [5], using (v) of Lemma 2.8 and Taylor expansion, we get | C ǫ ( s , t ) − C ǫ ( s , t − ∆) − ∆ ∂ t C ǫ ( s , t ) | ≤ | Z tt − ∆ (cid:18) ∂ t C ǫ ( s , t ) − Z tv ∂ t C ǫ ( s , u ) du (cid:19) dv − ∆ ∂ t C ǫ ( s , t ) | ≤ C ∆ | ∂ t C ǫ | ≤ C ∆ ǫ − . (2.36)Finally, the consistent error estimate (2.31) follows from (2.32)-(2.36) by keeping the worst termsinvolving ǫ . Theorem 2.11 (Upper bound of convergence rate)Suppose that Assumption 2.1 is satisfied. Then the difference between the unit utility indifferencevalue function C ( · , · ) and its approximation C ∆ ( · , · ) has an upper bound: C − C ∆ ≤ C ∆ . Proof.
Using the fact that C ǫ is a smooth subsolution of (2.15) together with the consistenterror estimate (2.31), we obtain that C ǫ ( s , t − ∆) − S (∆) C ǫ ( s , t )∆ ≤ C (cid:0) ∆ + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − (cid:1) . On the other hand, C ∆ ( s , t − ∆) = S (∆) C ∆ ( s , t ). Hence, C ǫ ( s , t − ∆) − C ∆ ( s , t − ∆) ≤ S (∆) C ǫ ( s , t ) − S (∆) C ∆ ( s , t )+ C (cid:0) ∆ + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − (cid:1) ≤ C | C ǫ ( s , t ) − C ∆ ( s , t ) | + C (cid:0) ∆ + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − (cid:1) , T / ∆times and get C ǫ − C ∆ ≤ C (cid:0) ∆ + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − (cid:1) . (2.37)In turn, (2.28), (2.29) and (2.37) imply that C − C ∆ = ( C − C ǫ ) + ( C ǫ − C ǫ ) + ( C ǫ − C ∆ ) ≤ C (cid:0) ǫ + ∆ + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − (cid:1) ≤ C ∆ by choosing ǫ = ∆ .To get a lower bound of C − C ∆ , we interchange the role of (2.15) and its splitting algorithm(2.27). For some technical reasons, we only consider the full recovery rate R = 1 when deriving thelower bound. Note that in this situation S (∆) is an identity operator. First, we need to rewritethe splitting algorithm (2.27) as the following equation so that the elliptic condition holds: C ∆ ( s , t − ∆) − S (∆) C ∆ ( s , t )∆ = 0 . (2.38)We shake the coefficients of (2.38) to construct its subsolution C ∆ ,ǫ , and regularize it by con-volution C ∆ ǫ = C ∆ ,ǫ ∗ ρ ǫ . Since S (∆) φ = S (∆) S (∆) φ is concave in φ , Jensen’s inequality impliesthat C ∆ ǫ ( s , t − ∆) − S (∆) C ∆ ǫ ( s , t )∆ ≤ Z − ǫ <τ< Z | e | <ǫ C ∆ ,ǫ ( s − e, t − ∆ − τ ) − S (∆) C ∆ ,ǫ ( s − e, t − τ )∆ ρ ǫ ( e, τ ) dedτ ≤ . Together with the consistent error estimate (2.31) with C ǫ replaced by C ∆ ǫ , we derive that − ∂ t C ∆ ǫ − (ˆ L + ˆ L ) C ∆ ǫ ≤ C (cid:0) ∆ + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − (cid:1) . Thus, C ∆ ǫ − ( T − t ) C (cid:0) ∆ + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − (cid:1) is a subsolution of (2.15).In turn, the comparison principle of (2.15) implies that C ∆ ǫ − C ≤ ( T − t ) C (cid:0) ∆ + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − + ∆ ǫ − (cid:1) . Combining with | C ∆ − C ∆ ǫ | ≤ Cǫ , we obtain the lower bound of the splitting algorithm (2.27): Theorem 2.12 (Lower bound of convergence rate)Suppose that Assumption 2.1 is satisfied, and that the recovery rate R = 1 . Then the differencebetween the unit utility indifference value function C ( · , · ) and its approximation C ∆ ( · , · ) has a lowerbound: C − C ∆ ≥ − C ∆ . Application to Counterparty Risk
In this section, we apply our multidimensional non-traded assets model to consider the coun-terparty risk of derivatives with possible default at maturity. Our concern as the buyer or holderof the option is that the writer or counterparty may default on the option with payoff h ( S T ) atmaturity T and we will not receive the full payoff. We have in mind several examples. A naturalexample is that of a commodity producer who is writing options as part of a hedging program (e.g.collars). Some of these options may be written on illiquidly traded assets and thus the option holderis subject to basis risk and in addition, is concerned with the default risk of the option writer. Asecond example is the default risk of a financial institution who has sold options on various un-derlying assets - stocks, foreign exchange or commodities. In addition to the possibility of basisrisk, the buyer of these options does not always have the ability to trade the underlying asset, orperhaps they choose not to (they may be using the derivative as part of a hedge already). A furtherexample may be that of a purchaser of insurance concerned with the default risk of the insurer.Typically the insured party does not trade at all, which motivates our consideration of this specialcase. Finally, the option holder may be an employee of a company who receives employee stockoptions if the company remains solvent. She is restricted from trading the stock of the company,but can trade other indices or stocks in the market. In contrast to the other examples, here theassets of the counterparty and the underlying stock are those of the same company.Consider an option written on an underlying asset with logarithm price S with payoff h ( S T ) atmaturity T . Counterparty default is modeled by comparing the value of the counterparty’s assetsexp( S ) to a default threshold D at maturity, which depends on the liabilities of the counterparty.Following Klein [30] we consider the situation D = L , where L refers to the option writer’s liabilities,assumed to be a constant. Generalizations to D = f ( S T ) are easily incorporated and allow for theoption liability itself to influence default, e.g. f ( x ) = h ( x )+ L was considered by Klein and Inglis [31]in a risk neutral setting. If the writer defaults, the holder will receive the proportion h ( S T ) /L of theassets exp( S ) that her option represents of the writer’s liabilities, scaled to reflect a proportionaldeadweight loss of α ∈ [0 , vulnerable option taking counterparty default intoaccount is g ( S T , S T ) = h ( S T ) { exp( S T ) ≥ L } + (1 − α ) h ( S T ) L exp( S T ) { exp( S T ) 0) of the vulnerable option. Following (2.14) we define a new operator: ∂∂η = ¯ σ ∂∂s + ¯ σ ∂∂s . • (i) Partition [0 , T ] into N equal intervals:0 = t < t < · · · < t N = T. • (ii) On [ t N − , t N ], predict the solution by solving the following PDE with the given terminaldata g ǫ : ∂ t C + ∂ ηη C − γ (1 − ¯ κ P )( ∂ η C ) = 0 , C | t = t N = g ǫ . The above equation can be linearized via the Cole-Hopf transformation:¯ C = exp( − γ (1 − ¯ κ P ) C ) . Thus, we obtain C | t = t N − by solving the corresponding linear PDE. • (iii) On [ t N − , t N ], correct the solution by solving the following PDE with the terminal data C | t = t N − : ∂ t C + σ ∂ s s C + σ ∂ s s C + A ∂ s C + A ∂ s C − γ σ ( ∂ s C ) − γ σ ( ∂ s C ) = 0 , C | t = t N = C | t = t N − where A , A are given in (2.7) to be A i = µ i − ( σ i + ¯ σ i ) − ¯ ϑ P ¯ σ i ; i = 1 , 2. The aboveequation can also be linearized by making the exponential transformation:¯ C = exp( − γ C ) . Thus, we obtain C | t = t N − by solving the corresponding linear PDE, which is used as theapproximation of C | t = t N − . • (iv) Repeat the above procedure on [ t N − , t N − ], and obtain C | t = t N − . . . .19 .2 Numerical Results We present results for the European put with payoff h ( S T ) = (cid:16) K − e S T (cid:17) + . If S and S arepositively correlated, this means when the put option is valuable (in-the-money), the firm’s assets S tend to be small, so there is a high risk of default. It is important to take counterparty risk intoaccount for puts in this case, as it will have a relatively large impact on the price. (This would beeven more significant when the default trigger involves the option liability). However, for a call,when the call is in-the-money, there is little default risk, and so counterparty risk is less important.Unless otherwise stated, the parameters are: K = 150; T = 1; exp( S ) = 50; exp( S ) = 100; L = 1000; α = 0 . γ = 1; µ P = 0 . σ P = 0 . 15; ¯ σ P = 0 . µ = 0 . σ = 0 . 25; ¯ σ = 0 . µ = 0 . σ = 0 . 3; ¯ σ = 0 . 2. These parameters result in correlation between the underlying assetand firm’s assets of ρ = 0 . 4; and correlations between each asset and the financial index P of ρ P = 0 . ρ P = 0 . . .In Figure 1 we show how the approximation converges as we increase the number of time steps N . For our parameter values, N = 11 steps is sufficient for the prices to converge and we use it in allsubsequent figures. We aim to compare the utility indifference price with hedging in the financialindex with the benchmark risk neutral price in a complete market (computed as in Corollary 2.4with n = 2, as studied in Klein [30]). We also compare to the situation where the financial index isindependent of the other assets and thus there is no hedging carried out. This price has an explicitformula: − γ ln E P h − e − γg ǫ ( S T ,S T ) i . Figure 2 provides a demonstration of the accuracy of the algorithm. We take ¯ σ P = 0 and comparethe splitting approximation to the above formula.Figure 3 shows the vulnerable option price(s) against the underlying asset price exp( S ). Thetwo panels of Figure 3 are intended to illustrate a “close or likely to default” scenario (the left panelwith exp( S ) = 500 relative to L = 1000) and a “far or unlikely to default” scenario (the right panelwith exp( S ) = 1400 relative to L = 1000). In both panels the risk neutral or complete marketprice is the highest. As the underlying asset price becomes very large, all option prices tend tozero, as the put is worthless, regardless of the default risk. At exp( S ) = 0, in the right panel, theoption price is equal to the option strike K = 150. In the left panel, the option price is lower dueto the risk of counterparty default. As S increases, all option prices decrease, as the moneyness ofthe put changes. When exp( S ) is close to zero, we see a dramatic drop in the utility indifferenceprices (relative to the risk neutral prices) due to the risk aversion towards unhedgeable price anddefault risks. Recall that since the underlying asset and firm’s assets are positively correlated,default risk becomes more important for low values of the underlying asset. The price drop is muchmore significant in the left panel (and in this extreme case, the option price drops down to zero ifno hedging can be carried out), where the likelihood of counterparty default is higher. The optionholder’s risk aversion causes the utility indifference prices to lie below the risk neutral price (ineach default scenario) with the relative discount to the risk neutral price being much greater in theleft panel where default is more likely. Assuming the holder can hedge in the financial index, thereis a drop of around 75% from the risk-neutral price to the utility indifference price. In the rightpanel, where the likelihood of default is relatively low, the difference between the utility indifferenceprice(s) and the risk-neutral price is not as dramatic, and is at most around 20% of the risk-neutralprice. We also see that the ability to hedge in the correlated financial index (versus no hedging atall) is more important when the default risk is higher (in the left panel).20igure 4 displays the impact of the option writer’s asset value exp( S ) on the option price fora fixed asset price exp( S ). We see a dramatic difference in the behavior of the risk neutral priceand the utility indifference prices. Under risk neutrality, the option price increases smoothly withexp( S ). However, under utility indifference, the prices are low and do not change much withvalues of exp( S ) below the default trigger of L = 1000. This is despite the put being in-the-money.As exp( S ) increases beyond the default level, the likelihood of default diminishes, and the utilityindifference prices start to increase with exp( S ). Note that the utility price is not always belowthe risk neutral price. Although Proposition 2.5 tells us that the risk neutral price is obtained as alimiting case of the utility indifference price, it requires condition (2.13) to hold.Figure 5 compares how the vulnerable option price changes with risk aversion parameter γ , andthe idiosyncratic volatilities σ , σ . The left panel plots vulnerable option prices against maturity T for various values of risk aversion γ . We see that the more risk averse the option holder is, the lessshe will pay for the option, consistent with Proposition 2.6. The other observation is that optionprices for a fixed γ are decreasing with maturity T . The risk neutral price is also decreasing with T , albeit very gradually. This is in contrast to risk neutral prices for non-default European putoptions which will increase in T provided there are no dividends. The reason is that there is atradeoff between price and default risk. If the maturity is longer, there is more chance for both S and S to fall - S falling means the put is more valuable, but S falling increases the default risk.For the parameters considered, the default risk is the dominant factor and thus the option pricedecreases with T . This is also in contrast to the call option, where Klein [30] reports that the riskneutral price increases with maturity.Recall that we do not expect price monotonicity in terms of the correlations, except in thesituation outlined in Proposition 2.6. Here we give an example of prices for various values of theidiosyncratic volatilities σ , σ . The left panel sets parameters to be µ = 0 . µ = 0 . 06 tosatisfy the CAPM restriction on Sharpe ratios. If σ = σ = 0, then we have ρ = 1, ρ P = 0 . ρ P = 0 . 8. Similarly if σ = 0 . 25 and σ = 0 . ρ = 0 . ρ P = 0 . ρ P = 0 . σ = σ = 1 then ρ = 0 . ρ P = 0 . ρ P = 0 . 16. We see that as σ , σ increase,the utility indifference price falls. Correspondingly, as the correlations ρ , ρ P , ρ P increase, theoption price rises. A Proof of Theorem 2.3 A.1 Derivation of Pricing PDE (2.10) The proof is an extension of Theorem 2.3 of Henderson and Liang [24], where we considered thespecial case R = 1 in a non-Markovian setting. We first transform the optimal portfolio problem(2.4) into a risk-sensitive control formulation, and derive a quadratic BSDE representation for itsvalue process and the associated optimal trading strategy. The problem (2.4) can be reformulatedas − e − γ ( x − C λt ) ess inf π ∈A F [ t,T ] E P h e − γ ( { t<τ ≤ T } ( X τ ( π )+ R C λ ( S τ ,τ ) ) + { τ>T } ( X T ( π )+ λg ( S T ) )) |G t i = − e − γ ( x − C λt ) exp ( − γ ess sup π ∈A F [ t,T ] − γ ln E P [ · · · |G t ] ) . For any π ∈ A F [ t, T ], by using the distribution property of the random time τ (see Chapter 8of Bielecki and Rutkowski [7] for example), we calculate the above conditional expectation, which21s { τ>t } ¯ Y t ( π ) with ¯ Y t ( π ) = E P "Z Tt a ( S s , s ) e − R st a ( S u ,u ) du − γX s ( π ) − γR C λ ( S s ,s ) ds + e − R Tt a ( S s ,s ) ds − γX T ( π ) − γλg ( S T ) |F t i . Therefore, in order to solve the optimal portfolio problem (2.4), we only need to solve thefollowing risk-sensitive control problem: Y t = ess sup π ∈A F [ t,T ] Y t ( π ) = ess sup π ∈A F [ t,T ] − γ ln ¯ Y t ( π ) , (A.1)and then the value process of (2.4) is given by − e − γ ( x − C λt ) × { τ>t } e − γY t . (A.2)In the following, we will first work out the BSDE representation for Y t ( π ), and then obtain theBSDE representation for Y t by the comparison principle. For any π ∈ A F [0 , T ], we change theprobability measure from P to Q ( π ) by defining d Q ( π ) d P = E ( N ( π )) = E (cid:18) − Z · γπ s (¯ σ P dW n +1 s + σ P dW n +2 s ) (cid:19) , where E ( · ) is the Dol´eans-Dade exponential. By Girsanov’s theorem, B ( π ) = W − hW , N ( π ) i is theBrownian motion under the new probability measure Q ( π ). With the new Brownian motion B ( π ) =( B ( π ) , . . . , B n +2 ( π )), the logarithm price processes of the non-traded assets S = ( S , · · · , S n )satisfy dS it = ( µ i ( S t , t ) − γ ¯ σ i ( S t , t )¯ σ P π t ) dt + σ i ( S t , t ) dB it ( π ) + ¯ σ i ( S t , t ) dB n +1 t ( π ) , (A.3)and ¯ Y t ( π ) becomes¯ Y t ( π ) = E P "Z Tt E ( N ( π )) s e − R st ρ u ( π ) du a ( S s , s ) e − γR C λ ( S s ,s ) ds + E ( N ( π )) T e − R Tt ρ u ( π ) du e − γλg ( S T ) |F t i = E Q ( π ) "Z Tt e − R st ρ u ( π ) du a ( S s , s ) e − γR C λ ( S s ,s ) ds + e − R Tt ρ u ( π ) du e − γλg ( S T ) |F t , with the stochastic discount factor ρ u ( π ): ρ u ( π ) = a ( S u , u ) + γµ P π u − γ (cid:0) σ P + ¯ σ P (cid:1) | π u | . By the martingale representation theorem, there exists an R n +1 -valued predictable process¯ Z ( π ) = ( ¯ Z ( π ) , . . . , ¯ Z n +1 ( π )) such that¯ Y t ( π ) = e − γλg ( S T ) + Z Tt h a ( S s , s ) e − γR C λ ( S s ,s ) − ρ s ( π ) ¯ Y s ( π ) i ds − n +1 X i =1 Z Tt ¯ Z is ( π ) dB is ( π ) . (A.4)22rom (A.1), Y t ( π ) = − γ ln ¯ Y t ( π ). Itˆo’s formula then implies that Y t ( π ) = λg ( S T ) + Z Tt (cid:20) ρ s ( π ) γ − a ( S s , s ) γ e γ (1 − R ) C λ ( S s ,s ) − γ | Z s ( π ) | (cid:21) ds − n +1 X i =1 Z Tt Z is ( π ) dB is ( π ) , (A.5)where Z it ( π ) = − γ ¯ Z it ( π ) / ¯ Y t ( π ). Equivalently, under the original probability measure P , we write Y t ( π ) = λg ( S T ) + Z Tt (cid:20) ρ s ( π ) γ − γ ¯ σ P Z n +1 s ( π ) π s − a ( S s , s ) γ e γ (1 − R ) C λ ( S s ,s ) − γ | Z s ( π ) | (cid:21) ds − n +1 X i =1 Z Tt Z is ( π ) dW is . (A.6)We notice that (A.6) is a quadratic BSDE with bounded terminal data and coefficients, whoseexistence and uniqueness is guaranteed by Theorems 2.3 and 2.6 of Kobylanski [32]. Moreover, thecomparison principle holds for (A.6). Let π , π ∈ A F [0 , T ] such that ρ s ( π ) γ − γ ¯ σ P zπ s ≥ ρ s ( π ) γ − γ ¯ σ P zπ s for z ∈ R n +1 . Then Y t ( π ) ≥ Y t ( π ).We claim that the value process of our risk-sensitive control problem (A.1) is given by thesolution of the following quadratic BSDE: Y t = λg ( S T ) + Z Tt " a ( S s , s ) γ (1 − e γ (1 − R ) C λ ( S s ,s ) ) + (cid:0) µ P − γ ¯ σ P Z n +1 s (cid:1) γ [( σ P ) + (¯ σ P ) ] − γ | Z s | ds − n +1 X i =1 Z Tt Z is dW is , (A.7)with the associated optimal portfolio: π ∗ ,λt = ¯ ϑ P γ ¯ σ P − ¯ κ P ¯ σ P Z n +1 t . Indeed, by Theorems 2.3 and 2.6 of Kobylanski [32], the quadratic BSDE (A.7) admits a uniquebounded solution Y with its martingale representation part Z = ( Z , . . . , Z n +1 ). To prove theclaim, we notice that for any π ∈ A F [ t, T ], ρ s ( π ) γ − γ ¯ σ P Z n +1 s π s ≤ a ( S s , s ) γ + (cid:0) µ P − γ ¯ σ P Z n +1 s (cid:1) γ [( σ P ) + (¯ σ P ) ] , and for π = π ∗ ,λ , the equality holds: ρ s ( π ∗ ,λ ) γ − γ ¯ σ P Z n +1 s π ∗ ,λs = a ( S s , s ) γ + (cid:0) µ P − γ ¯ σ P Z n +1 s (cid:1) γ [( σ P ) + (¯ σ P ) ] . Then the claim follows from the comparison principle for (A.6).23he optimization problem (2.5) is a special case of (2.4) with λ = 0, whose value process is givenby − e − γx × { τ>t } e − ¯ θP ( T − t ) , and the optimal control on the event { τ > t } is given by π ∗ , t = ¯ ϑ P γ ¯ σ P .By Definition 2.2, the utility indifference price C λt is such that − e − γ ( x − C λt ) × { τ>t } e − γY t = − e − γx × { τ>t } e − ¯ θP ( T − t ) , from which we obtain C λt = { τ>t } (cid:18) Y t − ¯ θ P γ ( T − t ) (cid:19) . The hedging strategy π ∗ ,λt − π ∗ , t on the event { τ > t } is given as − ¯ κ P ¯ σ P Z n +1 t .Finally, the pricing PDE (2.10) is obtained by an application of nonlinear Feynman-Kac repre-sentation (see Section 3 of [32]) by noticing that Y t − ¯ θ P γ ( T − t ) = C λ ( S t , t ) and Z it = σ i ( S t , t ) ∂ s i C λ ( S t , t ) for i = 1 , . . . , n ; Z n +1 t = n X i =1 ¯ σ i ( S t , t ) ∂ s i C λ ( S t , t ) . A.2 Well posedness of the Pricing PDE (2.10) The existence and uniqueness of bounded solutions to (2.10) are easily obtained from [32]. Wenext derive explicit bounds for the solution C λ ( s , t ), which are used in Proposition 2.6.It is obvious that the solution is nonnegative: C λ ( s , t ) ≥ 0. Let C λ ( s , t ; I ) be the solution of thepricing PDE with the full recovery R = 1, so it satisfies the following PDE with quadratic gradients: (cid:26) − ∂ t C λ ( s , t ; I ) − ( L + L ) C λ ( s , t ; I ) = 0 , C λ ( s , T ; I ) = λg ( s ) . (A.8)Note that − ( ∂ t + L + L + L ) C λ ( s , t ; I ) ≥ 0, since L C λ ( s , t ; I ) ≤ a ( s ) γ h − e γ (1 − R ) C λ ( s ,t ; I ) i = 0 , so that C λ ( s , t ; I ) is a supersolution to (2.10). On the other hand, C λ ( s , t ) is the (sub)solutionto (2.10), and C λ ( s , T ; I ) = C λ ( s , T ). By the comparison principle, we conclude that C λ ( s , t ) ≤ C λ ( s , t ; I ) on R n × [0 , T ].The solution C λ ( s , t ; I ) to PDE (A.8) is interpreted as the utility indifference price withoutintertemporal default, and we claim that C λ ( s , t ; I ) is bounded from above by ¯ C λ ( s , t ; I ), which isthe solution to PDE (A.8) with γ = 0, that is, ¯ C λ ( s , t ; I ) satisfies the following linear PDE: − ∂ t ¯ C λ ( s , t ; I ) − L ¯ C λ ( s , t ; I ) + n X i =1 ¯ ϑ P ¯ σ i ∂ s i ¯ C λ ( s , t ; I ) = 0 , ¯ C λ ( s , T ; I ) = λg ( s ) . (A.9)Indeed, note that − ( ∂ t + L + L )¯ C λ ( s , t ; I ) ≥ 0, since the terms involving γ in L can be regroupedas (omitting the arguments in σ i and ¯ σ i ) − γ n X i =1 σ i (cid:0) ∂ s i ¯ C λ ( s , t ; I ) (cid:1) − γ − ¯ κ P ) n X i =1 ¯ σ i ∂ s i ¯ C λ ( s , t ; I ) ! ≤ C λ ( s , t ; I ) is a supersolution to (A.8). On the other hand, C λ ( s , t ; I ) is the (sub)solution to(A.8), and ¯ C λ ( s , T ; I ) = C λ ( s , T ; I ). The comparison principle implies that C λ ( s , t ; I ) ≤ ¯ C λ ( s , t ; I )on R n × [0 , T ].It is well known that the linear PDE (A.9) admits a unique bounded solution since the terminaldata g ( · ) is bounded. Hence, there exists a constant K such that0 ≤ C λ ( s , t ) ≤ C λ ( s , t ; I ) ≤ ¯ C λ ( s , t ; I ) ≤ K. Finally, we show that C λ ( s , t ) is Lipschitz continuous in s . This in turn gives us the uniformboundedness of π ∗ ,λ and therefore the hedging strategy π ∗ ,λ − π ∗ , . 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The left panel takes T ∈ [0 , T ∈ [1 , t he v u l ne r ab l e op t i on p r i c e explicit formula splitting approximation Figure 2: Comparison of the explicit price and the approximated price via splitting when ¯ σ P = 0,exp( S ) = 1400, and N = 11. t he v u l ne r ab l e op t i on p r i c e financial indexno hedgingrisk−neutral 0 20 40 60 80 100 120 140 160 180 200050100150 the underlying asset price S t he v u l ne r ab l e op t i on p r i c e financial indexno hedgingrisk−neutral Figure 3: Vulnerable option price against the underlying asset price exp( S ). The left panel takesexp( S ) = 500; the right panel takes exp( S ) = 1400.28 500 1000 150005101520253035404550 the writer’s asset value S t he v u l ne r ab l e op t i on p r i c e financial indexno hedgingrisk−neutral Figure 4: Vulnerable option price against the writer’s asset value exp( S ). t he v u l ne r ab l e op t i on p r i c e risk−neutral γ = 0.5 γ = 1 γ = 1.5 t he v u l ne r ab l e op t i on p r i c e σ =0; σ =0 σ =0.25; σ =0.3 σ =1; σ =1 Figure 5: Impact of risk aversion and correlation. The left panel gives the option price againstmaturity for various risk aversion parameters γ . The right panel gives the price against variouscorrelation parameters. We set µ = 0 . µ = 0 ..