Srdjan D. Stojanovic

Comptes Rendus Mathématique

1/y:givenneara Black–er-ime deaire. Dene EDPC. R. Acad. Sci. Paris, Ser. I 340 (2005) 551–556http://france.elsevier.com/direct/CRASSMathematical EconomicsRisk premium and fair option prices under stochastic volatilitthe HARA solutionSrdjan D. StojanovicDepartment of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USAReceived 5 April 2004; accepted after revision 29 October 2004Available online 5 March 2005Presented by Pierre-Louis LionsAbstractWe have solved the problem of finding (HARA) fair option price under a general stochastic volatility model. For aHARA utility, the ‘risk premium’, i.e., the ‘market price of volatility risk’ is determined via a solution of a certain nonliPDE. Equivalently, the fair option price is determined as a solution of an uncoupled system of a non-linear PDE andScholes type PDE.To cite this article: S.D. Stojanovic, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméPrime de risque et prix de l’option sous l’hypothèse de volatilité stochastique.Nous résolvons le problème de la détmination du prix de l’option sous un modèle général de volatilité stochastique. Pour une fonction d’utilité HARA, « la prrisque », i.e. le prix du marché associé au risque de volatilité, est déterminé en utilisant la solution d’une EDP non linéla même façon, le prix de l’option est déterminé comme solution d’un système découplé d’une EDP non linéaire et d’ude type Black–Scholes.Pour citer cet article : S.D. Stojanovic, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégéeConsidérons un « marché boursier à volatilité stochastique », consistant d’un titre unique au prixY(t) à la datet et vérifiant le système suivant d’équations différentielles stochastiques de type ItôdY(t) = Y(t)(a(t, Y (t),v(t)) − D1(t, Y (t),v(t)))dt + Y(t)p(t, Y (t),v(t))dB1(t) (1)E-mail address: srdjan@math.uc.edu (S.D. Stojanovic).URL: http://math.uc.edu/~srdjan/ (S.D. Stojanovic).1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2004.11.002552 S.D. Stojanovic / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 551–556ARA :anteedv(t) = q(t, Y (t),v(t))dt + w(t, Y (t),v(t))(ρ(t, Y (t),v(t))dB1(t) +√1− ρ(t, Y (t),v(t))2 dB2(t))où B1 et B2 représentent des mouvements browniens indépendants,a est le taux d’appréciation du titre,D1 estle dividende,p est la volatilité, et v est un facteur scalaire arbitraire (par exemple, la volatilité :p(t, Y,v) = v)à directionq, à diffusionw et où−1  ρ  1 est le coefficient de corrélation entre prix et facteur. Soitr le tauxd’intérêt.Soit également un marché à options constitué d’une option unique sur le titre ci-dessus, de gain fixeυ(Y ) à ladate de maturitéT , et au prixV (t, Y (t),v(t)) à la datet < T , et oùV (t, Y,v) est une fonction inconnue a priori.Soit X > 0, la richesse. L’utilité associée à la richesse est mesurée par une fonction d’utilité de type Hψγ (X) = X1−γ /(1− γ ) pourγ ∈ (0,∞), γ = 1, etψ1(X) = log(X).Considérons une stratégie de couverture du portefeuille par autofinancementΠ(t,X,Y,v) = {Π1,Π2}, où Π1est la valeur monétaire de l’investissement dans l’action etΠ2 est la valeur de l’investissement dans l’option. Etdonné (1) et une stratégie fixeΠ , il existe une EDS qui caractérise l’évolution de la richesseX(t) = XΠ(t). Lastratégie de portefeuilleγ -optimale (le problème de Merton), notéeΠγ ={Πγ 1,Πγ 2}est telle que :supΠEt,X,Y,vψγ(XΠ(T )) = Et,X,Y,vψγ (XΠγ (T )). (2)Définition 0.1.Pour toutγ ∈ (0,∞), la fonction du prix de l’optionVγ (t, Y,v) est définie comme la solution d’unéquation Black–Scholes (abstraite)Πγ 2 = 0.Theorem 0.2.Pour γ ∈ (0,∞), γ = 1, le prix de l’option correspondant Vγ (t, Y,v) est solution de l’EDP Black–Scholes généraliséeVγ(1,0,0) + 12Y 2Vγ(0,2,0)p2 + YwρVγ (0,1,1)p + 12w2Vγ(0,0,2) + Y(r − D1)Vγ (0,1,0) − rVγ+(q − (a − r)wρp+ w2(1− ρ2)fγ(0,0,1)fγ)Vγ(0,0,1) = 0 (3)de condition terminale Vγ (T ,Y,v) = υ(Y ), et où fγ est une solution appropriée de l’EDP de type Monge–Ampère−((a − r)22p2+ rγ)fγ2 +(qγγ − 1 −w(a − r)ρp)fγ(0,0,1)fγ + w2γ2(γ − 1)fγ(0,0,2)fγ+ Y(a − r + γ (r − D1))γ − 1 fγ(0,1,0)fγ + pwYγργ − 1 fγ(0,1,1)fγ + p2Y 2γ2(γ − 1)fγ(0,2,0)fγ+ γγ − 1fγ(1,0,0)fγ − 12w2ρ2(fγ(0,0,1))2 − 12p2Y 2(fγ(0,1,0))2 − pwYρfγ (0,0,1)fγ (0,1,0) = 0 (4)de condition terminale fγ (T ,Y,v) = 1.Pour γ = 1, le prix de l’option correspondant V1 = V1(t, Y,v) est solution de l’EDP Black–Scholes généralisée(3) où f1 = 1, soit pour f1(0,0,1)/f1 = 0 ((4) n’est pas nécessaire dans ce cas).1. Statement of problem and it’s solutionConsider a ‘stochastic volatility stock market’ (cf. [1,3,4,6–8,11]), consisting of a single security with pricY(t)at timet , obeying Itô SDE systemdY(t) = Y(t)(a(t, Y (t),v(t)) − D1(t, Y (t),v(t)))dt + Y(t)p(t, Y (t),v(t))dB1(t)( ) ( )( ( ) √ ( )2 ) (5)dv(t) = q t, Y (t),v(t) dt + w t,Y (t),v(t) ρ t, Y (t),v(t) dB1(t) + 1− ρ t,Y (t),v(t) dB2(t)S.D. Stojanovic / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 551–556 553.a fixedhee-]))short,ir price.g., [11],ns forwhereB1 andB2 are independent standard Brownian motions, and wherea is the appreciation rate,D1 is thedividend rate,p is the volatility, andv is an arbitrary scalar factor (for example, volatility:p(t, Y,v) = v), with adrift q, diffusionw, and where−1 ρ  1 is the instantaneous price/factor correlation. Letr be the interest rateConsider also an associated option market, consisting of a single option on the above underlying, withpayoff υ(Y ) (for example,υcall(Y ) = max[0, Y − k], for some strike pricek) at a fixed expiration timeT , with apriceV (t, Y (t),v(t)) at timet < T , where functionV (t, Y,v) is a priori unknown.The problem is to characterize, and be able to compute, ‘a fair option price function’V (t, Y,v).Let X > 0 denote the wealth. We measure utility of wealth via HARA class of utility functionsψγ (X) =X1−γ /(1− γ ), for γ ∈ (0,∞), γ = 1, andψ1(X) = log(X). Parameterγ is calledrisk-aversion.Consider a self-financing portfolio hedging strategyΠ(t,X,Y,v) = {Π1(t,X,Y,v),Π2(t,X,Y,v)}, whereΠ1is the cash value of the investment into the underlying stock, andΠ2 is the cash value of the investment into toption. Assuming (5), and for a fixed strategyΠ , it is not difficult to write down an Itô SDE characterizing corrsponding evolution of the wealthX(t) = XΠ(t). Theγ -optimal portfolio strategy (Merton’s problem; see [9,10is a strategyΠγ = {Πγ 1,Πγ 2}, such thatsupΠEt,X,Y,vψγ(XΠ(T )) = Et,X,Y,vψγ (XΠγ (T )). (6)Definition 1.1.For any risk-aversionγ ∈ (0,∞), the fair option price functionVγ (t, Y,v) is defined as a (smoothsolution of the (abstract) Black–Scholes equationΠγ 2 = 0.‘Smooth’ means once continuously differentiable int , and twice continuously differentiable in variablesYandv.Intuitively, the fair option price is such a price for which it is not rational to speculate by investing, long orinto options. Variants of this definition have been present in the literature already (see, e.g., [8]; cf. Davis’ fadefinition – see, e.g., (7.3) and (7.4) in [7]). As it turns out, Definition 1.1, for eachγ ∈ (0,∞) implies a selectionof the ‘risk premium’ process (see (9) below). For some other issues in regard to the risk premium see, ewhere a notion ofviability of the risk premium process is studied in details. We shall use alternative notatiopartial derivatives: for example,∂2V (t, Y,v)/∂v2 = V (0,0,2)(t, Y,v).Theorem 1.2. For a risk-aversion γ ∈ (0,∞), γ = 1, the corresponding (smooth) fair option price Vγ =Vγ (t, Y,v) is a solution of a generalized Black–Scholes PDE:Vγ(1,0,0) + 12Y 2Vγ(0,2,0)p2 + YwρVγ (0,1,1)p + 12w2Vγ(0,0,2) + Y(r − D1)Vγ (0,1,0) − rVγ+(q − (a − r)wρp+ w√1− ρ2Mγ)Vγ(0,0,1) = 0 (7)with the terminal conditionVγ (T ,Y,v) = υ(Y ) (8)and withMγ = w√1− ρ2 fγ(0,0,1)fγ(9)where fγ = fγ (t, Y,v) is an appropriate solution of a Monge–Ampère type PDE (see, e.g. [2])−((a − r)2 + rγ)fγ2 +(qγ − w(a − r)ρ)fγ(0,0,1)fγ + w2γfγ(0,0,2)fγ2p2 γ − 1 p 2(γ − 1)554 S.D. Stojanovic / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 551–556s that,,mple,case ofif, aftereliedetiolityiontilityalso+ Y(a − r + γ (r − D1))γ − 1 fγ(0,1,0)fγ + pwYγργ − 1 fγ(0,1,1)fγ + p2Y 2γ2(γ − 1)fγ(0,2,0)fγ+ γγ − 1fγ(1,0,0)fγ − 12w2ρ2(fγ(0,0,1))2 − 12p2Y 2(fγ(0,1,0))2 − pwYρfγ (0,0,1)fγ (0,1,0) = 0 (10)with the terminal condition:fγ (T ,Y,v) = 1. (11)For γ = 1, i.e., in the case of log-utility, the corresponding fair option price V1 = V1(t, Y,v) is characterized asa solution of the generalized Black–Scholes PDE (7) with M1 = 0.Remark 1. Comparing (7) with the literature (see, e.g., Eq. (15) in [4]) we can see that Theorem 1.2 impliein the case of HARA utility, for a risk-aversionγ ∈ (0,∞), the ‘market price of volatility risk’, or ‘risk premium’is equal toM = Mγ , above characterized by (9)–(11).Remark 2. It is probably quite useful to have an intuitive understanding of what the risk premium is. For exaafter mimicking the classical derivation of the Black–Scholes PDE (see, e.g., Section 3.2.1 of [12]), in thethe underlying price dynamics (5), one can see that pricing options under stochastic volatility is done asthe randomness is eliminated as much as possible (by means of hedgingΠ∞ = {−wρV (0,0,1)/p − YV (0,1,0), V },which is a stochastic volatility analogue of the Black–Scholes hedging{−YV (0,1,0), V }), the remaining white noisdB2(t)/dt is thought of as, or substituted by, the risk premiumM. This suggests that arbitrage arguments, if appin the stochastic volatility setting, would require extreme considerations, such asM → ±∞ (cf. the notion oflowerand upper arbitrage prices and Theorem 6.1 in [7]). We continue this discussion in Remark 7 below.Remark 3. Analyzing (7), one can see that fair option price is unique, i.e., it is same for allγ ∈ (0,∞), iffMγ = M1 = 0, and assuming furthermorew2(1 − ρ2) > 0, iff fγ (0,0,1) = 0. Furthermore, analyzing (10), oncan conclude that (say,r > 0) for fγ (0,0,1) = 0 to hold it is not sufficient that the square of the Sharpe ra((a − r)/p)2 does not depend on the factorv, unless, additionally, which is trivial, the same holds for the volatip(t, Y,v) = p(t, Y ).Remark 4. Dividing by fγ , (10) can be written also as a quasi-linear PDE. Further, multiplying by(γ − 1), andsendingγ → 1, one arrives (at least formally) atqf1(0,0,1) + w22f1(0,0,2) + Y(a − D1)f1(0,1,0) + pwYρf1(0,1,1) + p2Y 22f1(0,2,0) + f1(1,0,0) = 0 (12)which (is a linear equation, and) together with (10), is solved, obviously, byf1 = 1.Remark 5. System (7)–(11) isuncoupled, which is very useful. Furthermore,fγ and therefore alsoMγ do not de-pend on a particular option payoffυ. Assuming thatMγ = 0 is continuous, since (11) implies that at the expirattime Mγ (T ,Y,v) = 0, we conclude that even if data is time-independent, the (HARA) market price of volarisk Mγ is not.Remark 6. If w = 0, or if ρ = ±1, thenMγ = 0, and the same fair option priceV = Vγ = Vγ (t, Y,v) holds forall γ ∈ (0,∞). For example, ifw = 0, the price is characterized as a solution of a (hypoelliptic (see [5], and[12,13])) Black–Scholes PDEV (1,0,0) + 12Y 2V (0,2,0)p2 + Y(r − D1)V (0,1,0) − rV + qV (0,0,1) = 0. (13)If alsoq = 0, then (13) simplifies further to the (usual) Black–Scholes PDE.S.D. Stojanovic / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 551–556 555ypaper,all withte thellFig. 1.fγ1, the correspondingfγ1(0,0,1)/fγ1, and the market price of volatility riskMγ1, for γ1 = 1/10.Fig. 2. As Fig. 1, but forγ2 = 100.Fig. 3. Approximations of upper arbitrage, upper fair, lower fair, and lower arbitrage prices.2. Computational exampleLet a = (3√v + 0.1)√v, D1 = 0, p = √v, q = 16(0.12− v), ρ = 1/2, w = √v, which amounts to a vervolatile market, and letr = 0.025,T = 0.5.For γ1 = 1/10, and some timet < T , fγ1, the correspondingfγ1(0,0,1)/fγ1, and the market price of volatilityrisk (the risk premium)Mγ1 are shown in Fig. 1, while forγ2 = 100, and same timet < T , fγ2, the correspondingfγ2(0,0,1)/fγ2, and the market price of volatility riskMγ2 are shown in Fig. 2.Remark 7. We continue Remark 2 above, and compare the methodology of HARA fair option prices of thiswith the concept of lower and upper arbitrage prices (see [7]). We compute the fair option prices (for a cstrikek = 60) corresponding toγ1 andγ2, and refer to them as upper and lower fair prices. Furthermore, denosolution of (7)–(8) corresponding to a particular risk premiumM asVM (for example,Vγ = VMγ ). We approximatethe lower and upper arbitrage prices byV±m, wherem is sufficiently large. For example, ifm = 100,200, in theabove example we compute and plot togetherV±100,V±200,V1/10,V100 (V100 andV200 are almost same, as weasV−100 andV−200), shown in Fig. 3.AcknowledgementsThe author wishes to thank very much Dr. Christelle Viauroux for the French translation.556 S.D. Stojanovic / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 551–556s, Cam-view ofmaticalon,43 (4)References[1] J.P. Fouque, G. Papanicolaou, K.R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Presbridge, 2000.[2] C.E. Gutiérrez, The Monge–Ampère Equations, Birkhäuser, Boston, 2001.[3] S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The ReFinancial Studies 6 (2) (1993) 327–343.[4] D.G. Hobson, Stochastic volatility, in: D. Hand, S. Jacka (Eds.), Applications of Statistics Series, Arnold, London, 1998.[5] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967) 147–171.[6] J. Hull, A. White, The pricing of options on assets with stochastic volatility, J. Finance 42 (2) (1987) 281–300.[7] I. Karatzas, S.G. Kou, On the pricing of contingent claims under constraints, Ann. Appl. Probab. 6 (2) (1996) 321–369.[8] J. Kallsen, Utility-based derivative pricing in incomplete markets, in: H. Geman, D. Madan, S.R. Pliska, T. Vorst (Eds.), MatheFinance—Bachelier Congress 2000, Springer, Berlin, 2002.[9] R.C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Economic Theory 3 (1971) 373–413.[10] R.C. Merton, Continuous-Time Finance, Blackwell, Cambridge, 1992.[11] H. Pham, N. Touzi, Equilibrium state prices in a stochastic volatility model, Mathematical Finance 6 (2) (1996) 215–236.[12] S.D. Stojanovic, Computational Financial Mathematics using Mathematica: Optimal Trading in Stocks and Options, Birkhäuser, Bost2003.[13] S.D. 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Risk premium and fair option prices under stochastic volatility: the HARA solution

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Risk premium and fair option prices under stochastic volatility: the HARA solution

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