Quantification of Risk in Classical Models of Finance
QQuantification of Risk in Classical Models of Finance
Alois Pichler ∗ Ruben Schlotter † April 15, 2020
Abstract
This paper enhances the pricing of derivatives as well as optimal control problems to a levelcomprising risk. We employ nested risk measures to quantify risk, investigate the limiting behavior ofnested risk measures within the classical models in finance and characterize existence of the risk-averselimit. As a result we demonstrate that the nested limit is unique, irrespective of the initially chosen riskmeasure. Within the classical models risk aversion gives rise to a stream of risk premiums, comparableto dividend payments. In this context, we connect coherent risk measures with the Sharpe ratio frommodern portfolio theory and extract the Z-spread—a widely accepted quantity in economics to hedgerisk. By involving the Z-spread we demonstrate that risk-averse problems are conceptually equivalent tothe risk-neutral problem.The results for European option pricing are then extended to risk-averse American options, where westudy the impact of risk on the price as well as the optimal time to exercise the option.We also extend Merton’s optimal consumption problem to the risk-averse setting.
Keywords:
Risk measures, Optimal control, Black–Scholes
Classification:
This paper studies discrete classical models in finance under risk aversion and their behavior in ahigh-frequency setting. Using nested risk measures we first study risk aversion in the multiperiod model.We develop risk aversion in a discrete time and discrete space setting and find an important consistencyproperty of nested risk measures. This consistency property, termed divisibility , is crucial in high-frequencytrading environments. For this, our study of risk-averse models extends to continuous time processesas well. This very property allows consistent decision making, i.e., decisions, which are independentof individually chosen discretizations or trading frequencies. Our results also give rise to a generalizedBlack–Scholes framework, which incorporates risk aversion in addition.Riedel (2004) has introduced risk measures in a dynamic setting. Later, Cheridito et al. (2004) studyrisk measures for bounded càdlàg processes and Cheridito et al. (2006) also discuss risk measures in adiscrete time setting. Ruszczyński and Shapiro (2006) introduce nested risk measures, for which Philpottet al. (2013) provide an economic interpretation as an insurance premium on a rolling horizon basis. Fora recent discussion on risk measures and dynamic optimization we refer to De Lara and Leclère (2016).Applications can be found in Philpott and de Matos (2012) or Maggioni et al. (2012), e.g., where stochasticdual dynamic programming methods are addressed, see also Guigues and Römisch (2012).Divisibility is an indispensable prerequisite in defining an infinitesimal generator based on discretizations.This generator, called risk generator , constitutes the risk averse assessment of the dynamics of the underlying ∗ Funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 416228727 – SFB 1410 † Both authors: Technische Universität Chemnitz, 09126 Chemnitz, Germany. Contact: [email protected] a r X i v : . [ q -f i n . M F ] A p r tochastic process. Using the risk generator we characterize the existence of the risk-averse limit of discretepricing models. For coherent risk measures the risk generator constitutes a nonlinear operator, comparableto the classical infinitesimal generator but with an additional term, accounting for risk, which takes the form s ρ | σ ∂ x ·| . Here, s ρ is a scalar expressing the risk aversion and σ is the volatility of the diffusion process describing theasset price. It turns out that the risk generator does not dependent on the risk measure, which is employedto set up the generator. This surprising feature has important conceptual implications, as evaluating a riskmeasure is often an optimization problem itself. As well we derive that the scaling quantity s ρ allows theeconomic interpretation of a Sharpe ratio and s ρ · σ is the Z-spread.Using the risk generator we derive a nonlinear Black–Scholes equation, which we relate to theBlack–Scholes formula for dividend paying stocks proposed by Merton (1973). Moreover we relaterisk-averse pricing models to foreign exchange options models as in Garman and Kohlhagen (1983).Nonlinear Black Scholes equations have been discussed previously in Barles and Soner (1998) and Ševčovičand Žitňanská (2016) in the context of modeling transaction costs. There, the nonlinearity is in the secondderivative. In contrast, risk aversion leads to drift uncertainty and causes nonlinearity in the first derivative.For coherent risk measures we derive an explicit solution for the European option pricing problem. Weshow that risk aversion expressed via coherent risk measures can be interpreted either as an extra dividendpayment or capital injection. Furthermore we relate risk-aversion with a change of currency as in theforeign exchange option model. The amount of the dividend payment or, equivalently, the interest rate inthe risk-averse currency, is given by a multiple of the Sharpe ratio and the volatility of the underlying stock.This ratio, which expresses risk aversion, arises for any coherent risk measure and does not depend on aspecific market model such as the Black–Scholes model.Using a free boundary formulation we extend the analysis from European to American option pricing.For the Black–Scholes option pricing of European and American options, risk-aversion naturally leads to abid-ask spread, which we quantify explicitly.Similarly we extend the Merton optimal consumption problem to a risk-averse setting. We elaborateon the optimal controls and show that risk-aversion reduces the investment in risky assets and increasesconsumption. We observe the same pattern as for European and American options, i.e., risk-aversioncorrects the drift of the underlying market model. For all classical models discussed here, the risk averseassessment still allows explicit pricing and control formulae. Recall the definition of law invariant, coherent risk measures ρ : L → R defined on some vector space L of R -valued random variables first. They satisfy the following axioms introduced by Artzner et al. (1999).A1. Monotonicity: ρ ( Y ) ≤ ρ ( Y (cid:48) ) , provided that Y ≤ Y (cid:48) almost surely;A2. Translation equivariance: ρ ( Y + c ) = ρ ( Y ) + c for c ∈ R ;A3. Convexity: ρ (cid:0) ( − λ ) Y + λ Y (cid:48) (cid:1) ≤ ( − λ ) ρ ( Y ) + λ ρ ( Y (cid:48) ) for 0 ≤ λ ≤ ρ ( λ Y ) = λ ρ ( Y ) for λ ≥ ρ ( Y ) = ρ ( Y (cid:48) ) , whenever Y and Y (cid:48) have the same law, i.e., P ( Y ≤ y ) = P ( Y (cid:48) ≤ y ) for all y ∈ R .The expectation ( ρ ( Y ) = E Y ) is the risk-neutral risk measure, satisfying all axioms above. In contrast tothe risk-neutral setting, the risk averse buyer and seller have an opposite perception of risk. We shall referto ρ ( Y ) as the seller’s ask price and to − ρ (− Y ) as the buyer’s bid price. Note, that − ρ (− Y ) ≤ ρ ( Y ) . The inequality 0 = ρ ( Y − Y ) ≤ ρ ( Y ) + ρ (− Y ) implies that − ρ (− Y ) ≤ ρ ( Y ) . .1 Nested risk measures We consider a filtered probability space ( Ω , F , (F t ) t ∈T , P ) and associate t ∈ T with stage or time . For thediscussion of risk in a dynamic setting we introduce nested risk measures corresponding to the evolutionof risk over time. Nested risk measures are compositions of conditional risk measures (cf. Pflug andRömisch (2007)). Following Ruszczyński and Shapiro (2006), we introduce conditional risk measures ρ t ,conditioned on the sigma algebra F t , as ρ t ( Y |F t ) : = ess sup Q ∈Q E Q [ Y | F t ] , (1)where Q is a convex set of probability measures absolutely continuous with respect to P (cf. also Delbaen(2002)). The conditional risk measures ρ t satisfy conditional versions of the Axioms A1–A5 above. Forfurther details we refer the interested reader also to Shapiro et al. (2014, Section 6.8.2). For the essentialsupremum of a set of random variables as in (1) we refer to Karatzas and Shreve (1998, Appendix A).We now introduce nested risk measures in discrete time. Definition 1 (Nested risk measures) . The nested risk measure , nested at times t < · · · < t n , is ρ t : t n ( Y ) : = ρ t (cid:0) . . . ρ t n ( Y | F t n ) · · · | F t (cid:1) , (2)where ( ρ t i ) ni = is a family of conditional risk measures. For a partition P = ( t , t , . . . , t n ) we denote thenested risk measure also by ρ P ( Y ) .Similar as above, we distinguish the buyer and seller perspective and consider the bid price − ρ t : t n (− Y ) : = − ρ t (cid:0) . . . ρ t n (− Y | F t n ) · · · | F t (cid:1) , as well as the ask price in (2). To illustrate key properties of nested risk measures as defined in (2) we discuss the binomial model,well-known from finance by employing the mean semi-deviation, a coherent risk measure satisfying allAxioms A1–A5 above.
Definition 2 (Semi-deviation) . The mean semi-deviation risk measure of order p ≥ Y ∈ L p at level β ∈ [ , ] is SD p ,β ( Y ) : = E Y + β (cid:107)( Y − E Y ) + (cid:107) p . Consider the stochastic process S = ( S , . . . , S T ) with initial state S . The process S models a stock instochastic finance over time. The discrete stock price changes according to P (cid:16) S t + ∆ t = S t · e ± σ √ ∆ t (cid:17) = p ± ,where p : = p + : = e r ∆ t − e − σ √ ∆ t e σ √ ∆ t − e − σ √ ∆ t and p − : = − p + . This setting describes the risk free risk measure, because E S t + ∆ t = pS t e σ √ ∆ t + ( − p ) S t e − σ √ ∆ t = S t e r ∆ t ,where r is the risk free interest rate.We can evaluate various classical coherent risk measures for this binomial model explicitly. Thefollowing remark considers the mean semi-deviation for the binomial model as well as the nested meansemi-deviation for the n -period model. 3 S · e σ √ ∆ t S · e − σ √ ∆ t p − p (a) single stage S · e σ n √ ∆ t S · e − σ n √ ∆ t p − p . . . p . . . .... . . − p (b) multistage Figure 1: Binomial option pricing model
Remark . Consider the one stage setting in Figure 1afirst. The risk-averse bid price for the stock S ∆ t employing the mean semi-deviation SD ,β of order 1 withrisk level β in the Bernoulli model is − SD ,β (− S ∆ t ) = E S ∆ t − β E (− S ∆ t + E S ∆ t ) + = pS e σ √ ∆ t + ( − p ) Se − σ √ ∆ t − β p ( − p ) (cid:16) Se σ √ ∆ t − Se − σ √ ∆ t (cid:17) . We introduce the new probability weights (cid:101) p : = p (cid:0) − β ( − p ) (cid:1) (3)so that − SD ,β (− S ∆ t ) = (cid:101) E S ∆ t . We now repeat this observation in n stages and consider an n -period binomial model with step size ∆ t = Tn ,cf. Figure 1b. The nested risk measure is − SD T ,β (− S T ) = − SD ,β (cid:0) . . . SD ,β ( S T | S T − ∆ t ) . . . (cid:1) = (cid:101) E S T , where the last expectation is with respect to the probability measure (cid:101) P (cid:18) S T = S e σ (cid:16) k √ ∆ t − n √ ∆ t (cid:17) (cid:19) = (cid:18) nk (cid:19) (cid:101) p k ( − (cid:101) p ) n − k , k = , . . . , n . It follows from inversion and the central limit theorem that σ log S T S + n √ ∆ t √ ∆ t → N ( n (cid:101) p , n (cid:101) p ( − (cid:101) p )) , (4)the limit is normally distributed. For this model to be non-degenerate it is inevitable that ˜ p → as n → ∞ and it is important to note that this forces specific choices of β in (3).In what follows we develop the setting in Remark 3 from an economic perspective and elaborate arigorous mathematical solution to the question of convergence in (4). We characterize the risk measuresfor which the risk-averse model converges by involving a new consistency property. This consistencyproperty is naturally formulated in continuous time and related to a risk-averse analogue of the infinitesimalgenerator in dynamic optimization. 4 The risk-averse limit of discrete option pricing models
Most well-known coherent risk measures in the literature as the Average Value-at-Risk, the EntropicValue-at-Risk as well as the mean semi-deviation involve a parameter which accounts for the degree of riskaversion. As Remark 3 elaborates, the nested risk-averse binomial model does not necessarily lead to awell-defined limit. It is essential to relate the coefficient of risk aversion of the conditional risk measures toits time period. We therefore introduce re-parameterized families of coherent risk measures which we call divisible . The divisibility property is central in discussing the limiting behavior of risk-averse economicmodels.
Definition 4 (Divisible families of risk measures) . Let Y ∼ N ( , ) be a standard normal random variable.A family of coherent measures of risk ρ = { ρ ∆ t : ∆ t > } , parameterized by ∆ t , is called divisible if thelimit lim ∆ t ↓ ρ ∆ t (√ ∆ t · Y ) ∆ t = s ρ (5)exists for some s ρ ≥ s E =
0. We provide some further examples of divisiblefamilies.
Example 5 (Entropic Value-at-Risk family) . The Entropic Value-at-risk at level κ ≥ EV @ R κ ( Y ) : = inf x > x ( κ + log E e xY ) . The family (cid:8) EV @ R β · ∆ t : ∆ t > (cid:9) is divisible with s EV @ R β = √ β . For a comprehensive discussion on theEntropic Value-at-Risk in this context we refer the reader to Pichler and Schlotter (2019, Proposition 9).For completeness we provide an additional family of divisible risk measures and remark that any convexcombination of them is divisible too. Lemma 6.
The family (cid:110) SD p ,β ; ∆ t : = SD p ,β ·√ ∆ t (cid:111) , ∆ t > , of mean semi-deviations is divisible with limit s SD p ,β = β ( π ) − p − p · Γ (cid:18) p + (cid:19) p . Proof.
Let Y ∼ N ( , ∆ t ) , then E Y p + = ∫ R max ( y , ) · y p · √ π ∆ t e − y ∆ t d y = √ π ∆ t ∫ ∞ y p · e − y ∆ t d y . Employing the Gamma function, the latter integral is1 √ π ∆ t ∫ ∞ y p · e − y ∆ t d y = √ π (cid:18) p − Γ (cid:18) p + (cid:19) ∆ t p (cid:19) . Taking the p -th root and multiplying by β √ ∆ t we obtain SD p ,β √ ∆ t ( Y ) ∆ t = β ( π ) − p − p · Γ (cid:18) p + (cid:19) p , showing the assertion. (cid:3)
5e now extend nested risk measures to continuous time and demonstrate that the extension is well-defined for divisible families of risk measures. As a result we show that the risk-averse binomial optionpricing model converges exactly for divisible families of risk measures.
Definition 7 (Nested risk measures) . Let T > t ∈ [ , T ) and let ρ P be divisible for every partition P ⊂ [ t , T ] , cf. Definition 1. The nested risk measure ρ t : T in continuous time for a random variable Y is ρ t : T ( Y | F t ) : = lim P ⊂[ t , T ] ρ P ( Y | F t ) almost surely , (6)where the almost sure limit is among all partitions P ⊂ [ t , T ] with mesh size (cid:107)P (cid:107) tending to zero for thoserandom variables Y , for which the limit exists.The following proposition evaluates the nested mean semi-deviation for the Gaussian random walk, thebasic building block of diffusion processes. Proposition 8 (Nested mean semi-deviation for the Gaussian random walk) . Let W = ( W t ) t ∈P be a Wienerprocess evaluated on the partition P . For the family of conditional risk measures (cid:18) SD t i p ,β ti ·√ ∆ t i (cid:19) t i ∈P , thenested mean semi-deviation is SD Tp ,β ( W T ) = n − (cid:213) i = β t i ∆ t i · ( π ) − p − Γ (cid:18) p + (cid:19) p . (7) Proof.
Note that Z : = W t i + − W t i ∼ N ( , t i + − t i ) and the conditional mean semi-deviation is (usingconditional translation equivariance A2) SD t i p ,β ti ·√ ∆ t i (cid:0) W t i + (cid:12)(cid:12) W t i (cid:1) = W t i + SD t i p ,β ti ; √ ∆ t i (cid:0) W t i + − W t i (cid:12)(cid:12) W t i (cid:1) . Furthermore the conditional expectation is zero as Brownian motion has independent and stationaryincrements with mean zero and thus, with Lemma 6, SD t i p ,β ti ·√ ∆ t i (cid:0) W t i + (cid:12)(cid:12) W t i (cid:1) = W t i + β t i ∆ t i · ( π ) − p − Γ (cid:18) p + (cid:19) p . Iterating as in Definition 1 shows SD Tp ,β ( W T ) = n − (cid:213) i = β t i ∆ t i · ( π ) − p − Γ (cid:18) p + (cid:19) p , the assertion. (cid:3) Remark . For constant risk level β we obtain SD Tp ,β ( W T ) = n − (cid:213) i = ∆ t i · β · ( π ) − p − Γ (cid:18) p + (cid:19) p = T · s SD p ,β , so that the accumulated risk grows linearly in time. 6 .1 The risk generator This section addresses nested risk measures for Itô processes. Furthermore, we characterize convergenceunder risk using a natural condition involving normal random variables and introduce a nonlinear operator,the risk generator, which also allows discussing risk-averse optimal control problems.It is well-known that the binomial model in Figure 1b converges to the geometric Brownian motion.We therefore discuss Itô process ( X s ) s ∈T solving the stochastic differential equationd X s = b ( s , X s ) d s + σ ( s , X s ) d W s , s ∈ T , (8) X t = x for T = [ t , T ] . We assume that X following (8) is well-defined and refer to Øksendal (2003, Theorem 5.2.1)for sufficient conditions. We further assume that s (cid:55)→ σ ( s , X s ) is Hölder continuous for some γ ∈ ( , / ) .We introduce the risk generator for families of divisible coherent risk measures. The risk generatordescribes the momentary evolution of the risk of the stochastic process. Definition 10 (Risk generator) . Let X = ( X t ) t be a continuous time process and ( ρ ∆ t ) ∆ t be a family ofdivisible risk measures. The risk generator based on ( ρ ∆ t ) ∆ t is R ρ Φ ( t , x ) : = lim ∆ t ↓ ∆ t (cid:16) ρ ∆ t (cid:0) Φ ( t + ∆ t , X t + ∆ t ) | X t = x (cid:1) − Φ ( t , x ) (cid:17) (9)for those functions Φ : T × R → R , for which the limit exists.Using the ideas from Proposition 8 we obtain explicit expressions for the risk generator for Itô diffusionprocesses. Proposition 11 (Risk generator) . Let X be the solution of (8) and let the family ( ρ ∆ t ) ∆ t be divisible. For Φ ∈ C (T × R ) such that Φ ( t , X t ) ∈ L p the risk generator based on ( ρ ∆ t ) ∆ t is given by the nonlineardifferential operator R ρ Φ ( t , x ) = (cid:18) Φ t + b Φ x + σ Φ xx + s ρ · | σ Φ x | (cid:19) ( t , x ) . (10) Proof.
By assumption, Φ ∈ C (T × R ) and hence we may apply Itô’s formula. For convenience and easeof notation we set f ( t , x ) : = (cid:16) Φ t + b Φ x + σ Φ xx (cid:17) ( t , x ) and f ( t , x ) : = ( σ Φ x ) ( t , x ) . In this setting, Eq. (9)rewrites as R ρ Φ ( t , x ) = lim ∆ t ↓ ∆ t ρ ∆ t (cid:20) ∫ t + ∆ tt f ( s , X s ) d s + ∫ t + ∆ tt f ( s , X s ) d W s (cid:12)(cid:12)(cid:12)(cid:12) X t = x (cid:21) . To show (10) for each fixed ( t , x ) it is enough to show that (cid:12)(cid:12) R ρ Φ ( t , x ) − f ( t , x ) − s ρ | f ( t , x )| (cid:12)(cid:12) ≤ . Using convexity of the risk measure together with the triangle inequality we have0 ≤ lim ∆ t ↓ (cid:12)(cid:12)(cid:12)(cid:12) ρ ∆ t (cid:20) ∆ t ∫ t + ∆ tt f ( s , X s ) d s − f ( t , x ) (cid:12)(cid:12)(cid:12)(cid:12) X t = x (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) + lim ∆ t ↓ (cid:12)(cid:12)(cid:12)(cid:12) ρ ∆ t (cid:20) h ∫ t + ∆ tt f ( s , X s ) d W s − s ρ | f ( t , x )| (cid:12)(cid:12)(cid:12)(cid:12) X t = x (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . (11)7e continue by looking at each term separately. Note that s (cid:55)→ f ( s , X s ) − f ( t , x ) is continuous almostsurely and hence the mean value theorem for definite integrals implies that there exists a ξ ∈ [ t , t + ∆ t ] such that 1 ∆ t ∫ t + ∆ tt f ( s , X s ) d s − f ( t , x ) = f ( ξ, X ξ ) − f ( t , x ) , almost surely . From continuity of ρ in the L p norm we may concludelim ∆ t ↓ h ρ ∆ t (cid:18) (cid:12)(cid:12)(cid:12)(cid:12)∫ t + ∆ tt f ( s , X s ) − f ( t , x ) ds (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) X t = x (cid:19) = . Note that the stochastic integral term in (11) can be bounded by ρ ∆ t (cid:20) ∆ t ∫ t + ∆ tt f ( s , X s ) d W s (cid:12)(cid:12)(cid:12)(cid:12) X t = x (cid:21) ≤ ρ ∆ t (cid:20) ∆ t ∫ t + ∆ tt f ( s , X s ) − f ( t , x ) d W s (cid:12)(cid:12)(cid:12)(cid:12) X t = x (cid:21) + ρ ∆ t (cid:20) ∆ t ∫ t + ∆ tt f ( t , x ) d W s (cid:12)(cid:12)(cid:12)(cid:12) X t = x (cid:21) , where ρ ∆ t (cid:104) ∆ t ∫ t + ∆ tt f ( t , x ) d W s (cid:12)(cid:12)(cid:12) X t = x (cid:105) converges to s ρ | f ( t , x )| and hence(11) ≤ lim ∆ t ↓ (cid:12)(cid:12)(cid:12)(cid:12) ρ ∆ t (cid:20) ∆ t ∫ t + ∆ tt f ( s , X s ) − f ( t , x ) d W s (cid:12)(cid:12)(cid:12)(cid:12) X t = x (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) . Furthermore, the stochastic integral M ∆ t : = ∫ t + ∆ tt f ( s , X s ) − f ( t , x ) d W s is a continuous martingale with M = ρ shows that there exists a constant C independent of ∆ t such that ρ ∆ t ( M ∆ t ) ≤ C √ ∆ t · (cid:107) M ∆ t (cid:107) p . Applying the Burkholder–Davis–Gundy inequality implies the upper bound ρ ∆ t ( M ∆ t ) ≤ (cid:101) C √ ∆ t (cid:34) E (cid:12)(cid:12)(cid:12)(cid:12)∫ t + ∆ tt ( f ( s , X s ) − f ( t , x )) d s (cid:12)(cid:12)(cid:12)(cid:12) p (cid:35) p for some constant (cid:101) C depending on p . Because the diffusion coefficient σ is Hölder continuous withexponent γ and Φ x is continuous it follows that f is locally Hölder continuous in t , i.e., there exists a C > γ < such that | f ( s , X s ) − f ( t , X t )| ≤ C | t − s | γ almost surely and hence E (cid:12)(cid:12)(cid:12)(cid:12)∫ t + ∆ tt ( f ( s , X s ) − f ( t , x )) d s (cid:12)(cid:12)(cid:12)(cid:12) p ≤ (cid:18)∫ t + ∆ tt C | s − t | γ d s (cid:19) p = (cid:18) C ∆ t γ + γ + (cid:19) p . We conclude ρ ∆ t ( M ∆ t ) ≤ (cid:101) C √ ∆ t (cid:18) C ∆ t γ + γ + (cid:19) = (cid:101) C ∆ t + γ (cid:115) C γ + , such that h ρ ∆ t ( M ∆ t ) vanishes, which concludes the proof. (cid:3) emark . For random variables Y of the form Y = ∫ Tt c ( s , X s ) d s + Ψ ( X T ) , where X is a Itô diffusion process based on Brownian motion, the limit (6) exists as a consequence ofDefinition 4 as well as the arguments in the proof of Proposition 11 above.The next proposition relates the convergence of the binomial model under risk to the risk generator. Proposition 13.
Denote by S n the n -period binomial tree model and suppose that the sequence of binomialprocesses ( S n ) n converges to an Itô process X in distribution. The risk-averse discrete binomial model inRemark 3 converges if and only if the family of nested risk measures is divisible.Proof. Let ( ρ ∆ t ) ∆ t be a divisible family of risk measures, then Proposition 11 shows that the risk generatorexists for the diffusion process X and hence ρ ∆ t ( X t + ∆ t | X t ) − X t = c ρ · ∆ t + o ( ∆ t ) (12)for some constant c ρ ∈ R . Moreover, applying Itô’s lemma to (12) implies divisibility. In addition,existence of the limit of risk-averse binomial models as in Remark 3 is equivalent to ρ ∆ t (cid:0) S nt + ∆ t | S nt (cid:1) − S nt = c ρ · ∆ t + o ( ∆ t ) , for n large enough. It follows from Fatou’s lemma that ρ T ( X T ) ≤ lim n →∞ ρ T ( S nT ) and hence the convergence of the risk-averse binomial model implies the existence of the risk generator.For the converse, note that X = S and hencelim n →∞ ρ ∆ t (cid:0) S n ∆ t − S (cid:1) = lim n →∞ ρ ∆ t (cid:0) S n ∆ t + X ∆ t − X ∆ t − X (cid:1) ≤ lim n →∞ ρ ∆ t (cid:0) S n ∆ t − X ∆ t (cid:1) + ρ ∆ t ( X ∆ t − X ) . The latter term satisfies ρ ∆ t ( X ∆ t − X ) = c ρ · ∆ t + o ( ∆ t ) by assumption. For the first term notice that ( S n ∆ t − X ∆ t ) n tends to zero in distribution and hence alsoconverges in probability. Moreover, (cid:16) ρ ∆ ( S n ∆ t − X ∆ t ) (cid:17) n is uniformly bounded and hencelim n →∞ ρ ∆ t (cid:0) S n ∆ t − X ∆ t (cid:1) = , showing the assertion. (cid:3) This section introduces risk-averse dynamic equations using nested risk measures. In what follows weconsider the value function involving nested risk measures defined by V ( t , x ) : = ρ t : T (cid:16) e − r ( T − t ) Ψ ( X T ) | X t = x (cid:17) . (13)Here, r is a discount factor and Ψ a given terminal payoff function. The structure of nested risk measuresallows extending the dynamic programming principle to the risk-averse setting.9 emma 14 (Dynamic programming principle) . Let ( t , x ) ∈ [ , T ) × R and ∆ t > , then it holds that V ( t , x ) = ρ t : T (cid:16) e − r ∆ t V ( t + ∆ t , X t + ∆ t ) (cid:12)(cid:12) X t = x (cid:17) . (14) Proof.
By definition of the risk-averse value function (13) it holds V ( t + ∆ t , X t + ∆ t ) = ρ t + ∆ t : T (cid:16) e − r ( T − t − ∆ t ) Ψ ( X T ) | X t + ∆ t (cid:17) and hence the definition of nested risk measure gives ρ t : t + ∆ t (cid:16) e − r ∆ t V ( t + ∆ t , X t + ∆ t ) (cid:12)(cid:12) X t = x (cid:17) = ρ t : T (cid:16) e − r ( T − t ) Ψ ( X T ) | X t = x (cid:17) , which shows the assertion. (cid:3) To derive the dynamic equations for V we consider (14) in the form0 = ∆ t ρ t : t + ∆ t (cid:16) e − r ∆ t V ( t + ∆ t , X t + ∆ t ) − V ( t , x ) (cid:12)(cid:12) X t = x (cid:17) (15)for ∆ t →
0. The following theorem employs the risk generator to obtain dynamic equations for therisk-averse value function (13).
Theorem 15.
The value function (13) solves the terminal value problem V t ( t , x ) + b ( t , x ) V x ( t , x ) + σ ( t , x ) V xx ( t , x ) + s ρ | σ ( t , x ) · V x ( t , x )| − rV ( t , x ) = , (16) V ( T , x ) = Ψ ( x ) , provided that V ∈ C in a neighborhood of ( t , x ) .Proof. Let ( t , x ) ∈ [ , T ] × R be fixed. Similarly to the risk neutral case we define Y s : = e − r ( s − t ) V ( s , X s ) , s ≥ t . The process Y s satisfies the Itô formula Y t + ∆ t = Y t + ∫ t + ∆ tt e − r ( s − t ) (cid:18) V t + b · V x + σ V xx − rV (cid:19) ( s , X s ) d s + ∫ t + ∆ tt e − r ( s − t ) σ ( s , X s ) · V x ( s , X s ) d W s . As σ is Hölder continuous it follows thatlim ∆ t ↓ ∆ t ρ t : t + ∆ t (cid:18) ∫ t + ∆ tt e − r ( s − t ) σ · V x d W s (cid:12)(cid:12)(cid:12)(cid:12) X t = x (cid:19) = s ρ · | σ · ∂ x V | . Following the lines of the proof of Proposition 11 implies0 = lim ∆ t ↓ ∆ t ρ t : t + ∆ t (cid:16) e − r ∆ t Y t + ∆ t − Y t (cid:12)(cid:12) X t = x (cid:17) = lim ∆ t ↓ ∆ t ρ t : t + ∆ t (cid:20)∫ t + ∆ tt e − r ( s − t ) (cid:18) V t + b · V x + σ V xx − rV (cid:19) d s + ∫ t + ∆ tt e − r ( s − t ) σ · V x d W s (cid:21) = V t ( t , x ) + b ( t , x ) · V x ( t , x ) + σ ( t , x ) V xx ( t , x ) + s ρ | σ ( t , x ) · V x ( t , x )| − rV ( t , x ) , which demonstrates the assertion. (cid:3) emark
16 (Optimal controls) . The dynamic programming principle and Theorem 15 are usually consideredin an environment involving controls u . This extends to the risk-averse setting as well. The proofs aresimilar and we thus state the result only. The value function V ( t , x ) : = inf u ρ t : T (cid:18)∫ Tt c ( s , X us , u s ) d s + Ψ ( X uT ) (cid:19) with diffusion process X uT governed by an adapted control process u satisfies the Hamilton–Jacobi–Bellmanequation inf u (cid:26) V t (·) + b (· , u ) V x (·) + σ (· , u ) V xx (·) + s ρ | σ (· , u ) · V x (·)| − rV (·) + c (· , u ) (cid:27) = , V ( T , ·) = Ψ (·) . This is (16), but with an extra infimum among all controls u . We resume this discussion in Section 5 below.For a general overview on stochastic optimal control and controlled processes in a risk neutral context werefer the interested reader to Fleming and Soner (2006). The previous section discusses a discrete, risk-averse binomial option pricing problem and subsequentlyderives characterizations for the risk-averse limit to exist. In this section we study the risk-averse valuefunctions of the limiting process of the binomial tree process, i.e., the geometric Brownian motion. Inthe risk-averse setting we find again explicit formulae. The resulting explicit pricing formulae lead us tointerpret risk aversion as dividend payments and to relate the risk level s ρ to the Sharpe ratio.Consider a market with one riskless asset, e.g., a bond and a risky asset, usually a stock. The returnof the riskless asset is constant and denoted by r . As usual in the classical Black–Scholes framework,the underlying stock S is modeled by a geometric Brownian motion following the stochastic differentialequation d S t = r S t d t + σ S t d W t (17)with initial condition S = s . Similarly as above we distinguish the risk-averse value function V ( t , x ) : = − ρ t : T (cid:104) − e − r ( T − t ) Ψ ( S T ) | S t = x (cid:105) (18)for the bid price and the corresponding value function for the ask price given by (cid:101) V ( t , x ) : = ρ t : T (cid:104) e − r ( T − t ) Ψ ( S T ) | S t = x (cid:105) . (19)Notice that the discount rate r is the same as in the dynamics (17) of the stock S = ( S t ) t . In the risk-neutralsetting the bid and ask prices coincide.Theorem 15 shows that the risk-averse value function (18) of the bid price satisfies the PDE withterminal condition V t ( t , x ) + r x V x ( t , x ) + σ x V xx ( t , x ) − s ρ · | σ x · V x ( t , x )| − r V ( t , x ) = , (20) V ( T , x ) = Ψ ( x ) , Ψ ( x ) is the payoff function. Similarly, the following PDE representation the value function (cid:101) V describing the ask price derives as (cid:101) V t ( t , x ) + r x (cid:101) V x ( t , x ) + σ x (cid:101) V xx ( t , x ) + s ρ · (cid:12)(cid:12)(cid:12) σ x · (cid:101) V x ( t , x ) (cid:12)(cid:12)(cid:12) − r (cid:101) V ( t , x ) = , (21) (cid:101) V ( T , x ) = Ψ ( x ) . Notice that (21) and (20) differ only in the sign of the nonlinear term, showing again that in the risk-neutralsetting (i.e., s ρ = ) the bid and ask prices coincide. We have the following explicit solution of (21)and (20) for the price of the call option. Proposition 17 (Call option) . Let Ψ ( x ) : = max ( x − K , ) , define the auxiliary functions (cf. Delbaen andSchachermayer (2006, Section 4.4)) d ± (cid:66) σ √ T − t · (cid:20) log (cid:16) xK (cid:17) + (cid:18) r ± s ρ σ + σ (cid:19) ( T − t ) (cid:21) , d ± (cid:66) d ± − σ √ T − t (22) and the value functions V ± ( t , x ) : = xe ± s ρ σ ( T − t ) Φ ( d ± ) − Ke − r ( T − t ) · Φ ( d ± ) . (23) Then V + solves the risk-averse Black–Scholes PDE (21) for the ask price, while V − solves (20) , thecorresponding PDE for the bid price; further, we have that V − ≤ V + . We can solve the problem for the European put option similarly.
Proposition 18 (Put option) . Let Ψ ( x ) : = max ( K − x , ) and define the value functions V ∓ ( t , x ) : = Ke − r ( T − t ) · Φ (− d ∓ ) − xe ∓ s ρ σ ( T − t ) Φ (− d ∓ ) , (24) with d ± and d ± as in Proposition 17. Then V − solves the risk-averse Black–Scholes PDE (21) and V + solves (20) , respectively. Note that V + ≤ V − .Proof. Plugging the derivatives into the PDE (21) and (20) shows the assertion. (cid:3)
Nature of the risk level s ρ . Proposition 17 and 18 show that the value function for the risk-averseEuropean option pricing problem can be identified with the risk neutral problem, where the stock paysdividends at rate s ρ σ . In case of the bid price of a European call option the dividend payments are given by s ρ σ . Similarly, the dividend payments for the bid price for a European put option are − s ρ σ , thus negative.For an increasing risk level s ρ , the bid price for the put and the call price decrease. This monotonicityreverses for the ask price.The value functions (23) and (24) can also be interpreted within the framework of the Garman–Kohlhagenmodel on foreign exchange options. In this sense the terms ± s ρ σ represent the interest rate in the risk-aversecurrency. Illustration of the risk level s ρ . Figure 2 shows the put and call prices for different values of s ρ . Forthis illustration we choose T = S = K = .
2, the interest rate is r = σ =
15 %. 12 .0 0.5 1.0 1.5 2.0 2.5 3.0stock price S p r i c e Call prices for different values of s s = 1 s = 0 s = 1 (a) Call prices S p r i c e Put prices for different values of s s = 1 s = 0 s = 1 (b) Put prices Figure 2: European option prices for different risk levelsFigure 3 exhibits the bid-ask spread, which is present in the risk-averse situation.
Discussion of the risk level s ρ . The introduction outlines that s ρ is related to the Sharpe ratio, a specificreward-to-variability ratio introduced by William F. Sharpe. The Sharpe ratio is r − r free σ , where r is the mean return of an asset with volatility σ and r free is the risk free interest rate. Comparingunits in (22) we see that s ρ σ is an interest rate and hence s ρ has unitinterestvolatility , the same unit as the Sharpe ratio. For the risk free return r of the market (see Equation (17)) and the riskaverse interest r averse the investor expects, we equate s ρ = r − r averse σ with s ρ as in (5) above. Notice that r averse should not exceed r and may be negative so that s ρ is alwayspositive. It follows that the risk-aversion coefficient s ρ has the structure of a Sharpe ratio. Furthermore, s ρ σ is the Z-spread for the risk-averse investor. Regarding the sign of s ρ σ , notice that the bid price of theEuropean call option is increasing in the interest rate and hence the risk neutral interest rate decreases to r − s ρ σ . The bid price for the European put option is decreasing with respect to the interest rate and hencethe interest rate increases to r + s ρ σ . The sign changes again when considering the respective ask prices.13 .0 0.5 1.0 1.5 2.0 2.5risk level s P r i c e Price of European call option askbid (a) Call options s P r i c e Price of European put option askbid (b) Put options
Figure 3: The bid-ask spread for varying risk level s ρ We return to the binomial model with risk-averse probabilities. The preceding sections show that the risklevel β for the mean semi-deviation risk measure needs to be proportional to √ ∆ t . In this case we obtain the risk-averse probabilities (cid:101) p = p ( − β √ ∆ t p ) = + r − βσ − σ σ √ ∆ t + β (− r + σ ) σ ∆ t + O( ∆ t ) and following the standard arguments we obtain the distribution for the stock S T in the limit as S T = S exp (cid:26) T (cid:18) r − βσ − σ (cid:19) + σ W T (cid:27) . Recall from Lemma 6 that s ρ , for the mean semi deviation, is β √ π . However the binomial model convergesto a process with dividends β . The discrepancy in the scaling factor is in line with the discontinuity of riskmeasures with respect to convergence in distribution, described in Bäuerle and Müller (2006, Theorem 4.1). In the risk-averse setting explicit formulae for European option prices in the Black–Scholes model areavailable. This is surprising given the initial nonlinear PDE formulation (18). Similarly we may reformulatethe risk-averse American option pricing problem and in what follows we introduce the risk-averse optimalstopping problem for American put options and introduce the value functions.Again we assume that the stock S follows the geometric Brownian motion (17). Here, the risk-aversebid price of an American option is given by sup τ ∈[ , T ] − ρ τ [− e − r τ Ψ ( S τ )] , where Ψ (·) denotes the payofffunction and the supremum is among all stopping times with τ ∈ [ , T ] . The ask price is given bysup τ ∈[ , T ] ρ τ [ e − r τ Ψ ( S τ )] . We can further define the value functions V ( t , x ) : = sup τ ∈[ t , T ] − ρ t : τ (cid:104) − e − r ( τ − t ) Ψ ( S τ ) | S t = x (cid:105) (cid:101) V ( t , x ) : = sup τ ∈[ t , T ] ρ t : τ (cid:104) e − r ( τ − t ) Ψ ( S τ ) | S t = x (cid:105) for the ask price. For ease of notation we only discuss the bid price for American put options, the argumentsfor the ask price are analogous. Analogously to the risk-neutral setting we obtain the free boundary problemfor the optimal exercise boundary t (cid:55)→ L ( t ) . V t ( t , x ) + r xV x ( t , x ) + σ x V xx ( t , x ) − s ρ σ x | V x | = rV ( t , x ) for x ≥ L ( t ) , (25) V ( t , x ) = ( K − x ) + for 0 ≤ x < L ( t ) , (26) V x ( t , x ) = − x = L ( t ) , (27) V ( T , x ) = ( K − x ) + L ( T ) = K lim x →∞ V ( t , x ) = ≤ t ≤ T . (28)For an overview on American options and free boundary problems in general we refer to Peskir andShiryaev (2006). The following result follows with standard arguments for American options. Theorem 19.
The value function V ( t , x ) = sup τ ∈[ t , T ] − ρ t : τ (cid:104) − e − r ( τ − t ) ( K − S τ ) + | S t = x (cid:105) (29) solves the free boundary problem (25) – (28) . Similarly to European options, risk-aversion reduces to a modification of the drift term and the standardAmerican put option problem is recovered where the underlying stock pays dividends. First we notice that − s ρ σ x | V x | ≤ − s ρ σ x y V x , y ∈ [− , ] , and hence for each ( t , x ) ∈ C V t ( t , x ) + r xV x ( t , x ) + σ x V xx ( t , x ) − s ρ σ x | V x | = inf y ∈[− , ] (cid:26) V t ( t , x ) + (cid:0) r − s ρ σ y (cid:1) xV x ( t , x ) + σ x V xx ( t , x ) (cid:27) . For x ≥ L ( t ) , the American option is not exercised and the same arguments as for the European optionsshow that the infimum over all constraints is attained at y ≡ −
1. The first line of the free boundaryformulation (25) is thus equal to V t ( t , x ) + (cid:0) r + s ρ σ (cid:1) xV x ( t , x ) + σ x V xx ( t , x ) = rV ( t , x ) for x ≥ L ( t ) . Consequently, we deduce that the value function V ( t , x ) : = sup τ ∈[ t , T ] E (cid:104) e − r ( τ − t ) Ψ ( S τ ) | S t = x (cid:105) solves the free boundary problem (25)–(28), where the state process is given byd S s = (cid:0) r + s ρ σ (cid:1) S s d s + σ S s d W s , S t = x . umerical illustration Consider the geometric Brownian motiond S t = . S t d t + . S t d W t , < t ≤ , S = . The strike price in the next Figure 4 is K =
1. We consider the optimal stopping region for different risklevels s ρ . A risk-averse option buyer (bid price) would generally exercise earlier, he accepts less profitsdue to his risk aversion. Compared with the risk neutral investor, the accumulating construction of nestedrisk measures ensures that the risk aware option buyer prefers exercising prematurely rather than delayedexercise.The reverse is true for the option holder (ask price), where the investor waits longer. time to expiry s t o c k p r i c e continuation regionstopping regionoptimal stopping for American put options s = 0.0 s = 0.1 s = 0.5 (a) bid price s t o c k p r i c e continuation regionstopping region optimal stopping for American put options s = 0.0 s = 0.1 s = 0.3 (b) ask price Figure 4: optimal stopping regions for put optionsIn the risk neutral case it is never optimal to exercise an American call option before expiry. However,this is only the case if the underlying asset does not pay dividends (see, for instance, Shreve (2010, Chapter8.5) for details). As risk-aversion expressed with coherent risk measures can be represented by dividendpaying stocks we conclude that it is generally optimal to exercise the call option early. Figure 5 shows theoptimal exercise boundary for the risk-averse call option with strike K = S = .20.40.60.8 time to expiry s t o c k p r i c e continuation regionstopping regionoptimal stopping for American call options s = 0.5 s = 1.0 s = 1.2 Figure 5: optimal stopping regions for different risk-levels (call option)
The preceding sections demonstrate that classical option pricing models generalize naturally to a risk-aversesetting by employing nested risk measures. In what follows we demonstrate that the classical Mertonproblem, which allows an explicit solution in specific situations, as well allows extending to the risk-aversesituation.Consider a risk-less bond B satisfying the ordinary differential equation d B t = r B t d t and a risky asset S driven by the stochastic differential equationd S t = µ S t d t + σ S t d W t . We are interested in the optimal fraction π t of the wealth w t one should invest in the risky asset. Considerthe wealth process d w t = [( π t µ + ( − π t ) r ) w t − c t ] d t + π t σ w t d W t , where c t is the rate of consumption. Merton employs the power utility function u ( x ) = x − γ − γ . We considerthe risk-averse objective function R ( t , x ) : = sup π, c − ρ t : T (cid:18) − ∫ Tt e − (cid:37) ( s − t ) u ( c s ) d s − (cid:15) γ e − (cid:37) ( T − t ) u ( w T ) | w t = x (cid:19) . Surprisingly, R has a closed form solution and, moreover, the optimal portfolio allocation is π ∗ = max (cid:18) µ − r − s ρ σσ γ , (cid:19) . We observe again that risk aversion leads to a modified drift term r + s ρ σ in place of r . The optimalportfolio allocation π ∗ is a decreasing function of s ρ . This is in line with the usual economic perception, asincreasing risk-aversion corresponds to less investments into the risky asset. The optimal consumption isgiven by c ∗ t ( x ) = x ν + ( ν (cid:15) − ) e − ν ( T − t ) , where ν is a constant depending on the model parameters. Consumption generally increases with riskaversion as the value of immediate consumption offsets the present value of uncertain wealth in the future.17n what follows we derive the optimal value function R and verify the optimal portfolio allocation π ∗ and optimal consumption c ∗ given above.The following result is a consequence of Proposition 11. Proposition 20.
The optimal discounted value function R satisfies the following Hamilton–Jacobi–Bellmanequation = max π, c (cid:20) R t + [( π t µ + ( − π t ) r ) x − c t ] R x + σ π x R xx + e − (cid:37) t u ( c t ) − s ρ | σπ t xR x | (cid:21) , (30) R ( T , x ) = (cid:15) γ e − (cid:37) T − γ x − γ . The Hamilton–Jacobi–Bellman equation allows for explicit optimal controls, the following propositionoutlines them.
Proposition 21.
In the risk-averse setting, the optimal controls are given by π ∗ t ( x ) = − ( µ − r ) R x σ xR xx + s ρ σ | R x | σ xR xx , c ∗ t ( x ) = (cid:0) e (cid:37) t R x (cid:1) − γ . The Hamilton-Jacobi-Bellman equation (30) rewrites as = R t − (cid:16) ( µ − r ) + s ρ σ (cid:17) R x σ R xx + s ρ R x | R x | σ R xx + r xR x + γ e − (cid:37) t γ − γ R γ − γ x , (31) R ( T , x ) = (cid:15) γ e − (cid:37) T − γ x − γ . The preceding proposition derives first order conditions for the fraction π ∗ t and consumption rate c ∗ t .Employing the Hamilton–Jacobi–Bellman equations we obtain nonlinear second order partial differentialequations for the optimally controlled value function. Surprisingly, this nonlinear equation has an explicitsolution too. Theorem 22 (Solution of the risk-averse Merton problem) . The PDE (31) has the explicit solution R ( t , x ) = e − (cid:37) t (cid:18) + ( ν (cid:15) − ) e − ν ( T − t ) ν (cid:19) γ x − γ − γ , where ν : = (cid:37)γ − r − γγ − − γγ (cid:18) ( ( µ − r ) + s ρ σ ) σ − s ρ σ (cid:19) . Moreover, the optimal controls are π ∗ = µ − s ρ σ − r σ γ and c ∗ t ( x ) = x ν + ( ν (cid:15) − ) e − ν ( T − t ) . Proof.
We recall the PDE (31),0 = R t − (cid:16) ( µ − r ) + s ρ σ (cid:17) R x σ R xx + s ρ R x | R x | σ R xx + r xR x + γ e − (cid:37) t γ − γ ( R x ) γ − γ , R ( T , x ) = e − (cid:37) T (cid:15) γ x − γ − γ , R ( t , x ) = e − (cid:37) t f ( t ) γ x − γ − γ . In this case the partial derivatives are given by R t = e − (cid:37) t (cid:16) − (cid:37) f ( t ) γ + γ f ( t ) γ − f (cid:48) ( t ) (cid:17) x − γ − γ , R x = e − (cid:37) t f ( t ) γ x − γ , R xx = − γ e − (cid:37) t f ( t ) γ x − γ − . The terminal condition for our Merton problem is v ( T , x ) = (cid:15) γ e − (cid:37) T x − γ − γ hence f ( T ) = (cid:15) >
0. Setting C : = − ( ( µ − r ) + s ρ σ ) σ and C : = s ρ σ for ease of notation we substitute the derivatives in the PDE (31) andobtain the following ordinary differential equation for f ; f (cid:48) ( t ) = f ( t ) (cid:18) (cid:37)γ − r − γγ + − γγ ( C + C f γ ) (cid:19) − . (32)For ν as defined in Theorem 22, the general solution of the ordinary differential equation (32) is f ( t ) = + ( ν(cid:15) − ) e − ν ( T − t ) ν , which is positive. The optimal value function thus is R ( t , x ) = e − (cid:37) t (cid:18) + ( ν(cid:15) − ) e − ν ( T − t ) ν (cid:19) γ x − γ − γ . We assumed that π ≥ π ∗ t = max (cid:16) ( µ − r )− s ρ σσ γ , (cid:17) . The optimal consumptionprocess is c ∗ t = x ν + ( ν(cid:15) − ) e − ν ( T − t ) , which concludes the proof. (cid:3) The following Figure 6 illustrates the optimal consumption c ∗ as a function of the risk level s ρ for (cid:37) = . γ = . r = . µ = . σ = . (cid:15) = .
1. The time horizon is T = w =
1. Note that s ρ can take only values smaller than µ − r σ as otherwise π ∗ < s o p t i m a l c o n s u m p t i o n consumption vs. risk aversion Figure 6: optimal consumption19
Summary
This paper introduces risk awareness in classical financial models by introducing nested risk measures.We demonstrate that classical formulae, which are of outstanding importance in economics, are explicitlyavailable in the risk-averse setting as well. This includes the binomial option pricing model, theBlack–Scholes model as well as the Merton optimal consumption problem.We also give an explicit Z-spread, which reflects risk awareness. The Z-spread involves the volatilityof the risky asset and a constant s ρ , which derives from nesting risk measures. The results thus provide aneconomic verification of the Z-spread by thorough risk management employing coherent risk measures.To aid the discussion on risk-averse value functions we extend nested risk measures from a discrete timesetting to continuous time. This allows us to derive a risk generator expressing the momentary dynamicsof our model under risk aversion. We show that for every coherent risk measure the risk generator is of thesame form, implying that in continuous time there is only one nested risk measure. Moreover, a constant s ρ expresses risk aversion which we associate with the Sharpe ratio. References
P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent Measures of Risk.
Mathematical Finance , 9:203–228, 1999. doi:10.1111/1467-9965.00068.G. Barles and H. Soner. Option pricing with transaction costs and a nonlinear black-scholes equation.
Finance and Stochastics , 2, 1998. doi:10.1007/s007800050046.N. Bäuerle and A. Müller. Stochastic orders and risk measures: Consistency and bounds.
Insurance:Mathematics and Economics , 38(1):132–148, 2006. doi:10.1016/j.insmatheco.2005.08.003.P. Cheridito, F. Delbaen, and M. Kupper. Coherent and convex monetary risk measures forbounded càdlàg processes.
Stochastic Processes and their Applications , 112(1):1–22, jul 2004.doi:10.1016/j.spa.2004.01.009.P. Cheridito, F. Delbaen, and M. Kupper. Dynamic monetary risk measures for bounded discrete-timeprocesses.
Electronic Journal of Probability , (3):57–106, 2006. ISSN 1083-6489. doi:10.1214/EJP.v11-302.M. De Lara and V. Leclère. Building up time-consistency for risk measures and dynamic optimization.
European Journal of Operational Research , 249:177–187, 2016. doi:10.1016/j.ejor.2015.03.046.F. Delbaen. Coherent risk measures on general probability spaces. In
Essays in Honour of DieterSondermann , pages 1–37. Springer-Verlag, Berlin, 2002.F. Delbaen and W. Schachermayer.
The Mathematics of Arbitrage . Springer, 2006.W. H. Fleming and H. Soner.
Controlled Markov Processes and Viscosity Solutions . 2006. ISBN0-387-26045-5.M. B. Garman and S. W. Kohlhagen. Foreign currency option values.
Journal of International Money andFinance , 2:231–237, 1983. doi:10.1016/s0261-5606(83)80001-1.V. Guigues and W. Römisch. Sampling-based decomposition methods for multistage stochastic programsbased on extended polyhedral risk measures.
SIAM Journal on Optimization , 22(2):286–312, jan 2012.doi:10.1137/100811696. 20. Karatzas and S. E. Shreve.
Methods of Mathematical Finance . Springer New York, 1998. doi:10.1007/978-1-4939-6845-9.F. Maggioni, E. Allevi, and M. Bertocchi. Bounds in Multistage Linear Stochastic Programming.
Journalof Optimization Theory and Applications , 163(1):200–229, 2012. doi:10.1007/s10957-013-0450-1.R. C. Merton. Theory of rational option pricing.
The Bell Journal of Economics and Management Science ,4(1):141–183, 1973.B. Øksendal.
Stochastic Differential Equations . Springer Berlin Heidelberg, 2003. doi:10.1007/978-3-642-14394-6.G. Peskir and A. Shiryaev.
Optimal Stopping and Free-Boundary Problems . Birkhäuser Basel, 2006.doi:10.1007/978-3-7643-7390-0.G. Ch. Pflug and W. Römisch.
Modeling, Measuring and Managing Risk . World Scientific, 2007.doi:10.1142/9789812708724.A. Philpott, V. de Matos, and E. Finardi. On solving multistage stochastic programs with coherent riskmeasures.
Operations Research , 61(4):957–970, aug 2013. doi:10.1287/opre.2013.1175.A. B. Philpott and V. L. de Matos. Dynamic sampling algorithms for multi-stage stochastic pro-grams with risk aversion.
European Journal of Operational Research , 218(2):470–483, 2012.doi:10.1016/j.ejor.2011.10.056.A. Pichler and R. Schlotter. Risk-averse optimal control, 2019. URL .F. Riedel. Dynamic coherent risk measures.
Stochastic Processes and their Applications , 112(2):185–200,2004. doi:10.1016/j.spa.2004.03.004.A. Ruszczyński and A. Shapiro. Conditional risk mappings.
Mathematics of Operations Research , 31(3):544–561, 2006. doi:10.1287/moor.1060.0204.D. Ševčovič and M. Žitňanská. Analysis of the nonlinear option pricing model under variable transactioncosts.
Asia-Pacific Financial Markets , 23(2):153–174, mar 2016. doi:10.1007/s10690-016-9213-y.A. Shapiro, D. Dentcheva, and A. Ruszczyński.
Lectures on Stochastic Programming: Modeling andTheory . 2nd edition, 2014. doi:10.1137/1.9780898718751.S. E. Shreve.
Stochastic Calculus for Finance II . Springer New York, 2010.ISBN 0387401016. URL