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Georgian Mathematical Journal Volume 8 (2001), Number 1, 189-199 OPTIMAL MEAN-VARIANCE ROBUST HEDGING UNDER ASSET PRICE MODEL MISSPECIFICATION T. TORONJADZE Abstract. The problem of constructing robust optimal in the mean-variance sense trading strategies is considered. The approach based on the notion of sensitivity of a risk functional of the problem w.r.t. small perturbation of asset price model parameters is suggested. The optimal mean-variance robust trading strategies are constructed for one-dimensional diffusion models with misspecified volatility. 2000 Mathematics Subject Classification: 60G22, 62F35, 91B28. Key words and phrases: Robust mean-variance hedging, misspecified asset price models. 1. Introduction and statement of the problem. Let (Ω, F, F, P ) be a filtered probability space with a fitration F = (Ft )0≤t≤T satisfying the usual conditions, where T ∈ (0, ∞] is a fixed time horizon. Let for each ε > 0 n o Λε := λ : λ = λ0 + εh, h ∈ H , (1) where λ0 is a fixed F -predictable vector or matrix valued process satisfying some additional integrability conditions (see, e.g., (7), (9), (18) below), n H := h : h bounded, F -predictable, vector or matrix valued process, o h ∈ BallL (0, R), 0 < R < +∞ . (2) Here BallL (0, R) denotes the closed R-radius ball of processes h in an appropriate metric space L with center at the origin. The class H is called the class of alternatives. Let X λ , λ ∈ Λε , be a continuous Rd -valued semimartingale describing the misspecified discounted price of a risky asset (stock) in a frictionless financial market. A contingent claim is an FT -measurable square-integrable random variable (r.v.) H, and a trading strategy θ is Ra F -predictable Rd -valued process such that the stochastic integral G(λ, θ) := θ dX λ , λ ∈ Λε , is a well-defined real-valued square-integrable semimartingale. Intuitively, H models the payoff from a financial product one is interested in and for each λ, G(λ, θ) describes the trading gains induced by the self-financing c Heldermann Verlag www.heldermann.de ISSN 1072-947X / $8.00 / ° 190 T. TORONJADZE portfolio strategy associated with θ when the asset price process follows the semimartingale X λ . For each λ ∈ Λε , the total loss of a hedger, who srarts with the initial capital x, uses the strategy θ, believes that the stock price process follows X λ and has to pay a random amount H at the date T , is thus H − x − GT (λ, θ). The robust mean-variance hedging means solving the optimization problem ³ ´2 minimize sup E H − x − GT (λ, θ) over all strategies θ. (3) λ∈Λε Denote by J(λ, θ) the risk functional of problem (3), ³ ´2 J(λ, θ) := E H − x − GT (λ, θ) , and consider the following approximation (which is common in the robust statistics theory, see, e.g., [1], [2]): n o sup J(λ, θ) = exp sup ln J(λ0 + εh; θ) λ∈Λε ( h∈H · DJ(λ0 , h; θ) ' exp sup ln J(λ , θ) + ε J(λ0 , θ) h∈H ( sup DJ(λ0 , h; θ) ) , = J(λ0 , θ) exp ε h∈H J(λ0 , θ) ¸) 0 where DJ(λ0 , h; θ) := ¯ d J(λ0 + εh; θ) − J(λ0 , θ) ¯ J(λ0 + εh; θ)¯ , = lim ε=0 ε→0 dε ε (4) is the Gateaux differential of the functional J at the point λ0 in the direction h. Our approach consists in approximating (in the leading order ε) the optimization problem (3) by the problem ( sup DJ(λ0 , h; θ) ) minimize J(λ , θ) exp ε h∈H 0 J(λ0 , θ) over all strategies θ, (5) and note that every solution θ∗ of problem (5) minimizes J(λ0 , θ) under the constraint sup DJ(λ0 , h; θ) sup DJ(λ0 , h; θ∗ ) h∈H h∈H ≤ k := . J(λ0 , θ) J(λ0 , θ∗ ) This gives a characterization of an optimal strategy θ∗ of problem (5), and thus leads to OPTIMAL MEAN-VARIANCE ROBUST HEDGING 191 Definition 1. The trading strategy θ∗ is called optimal mean-variance robust against the class of alternatives H if it is a solution of the optimization problem minimize J(λ0 , θ) over all strategies θ, subject to constraint sup DJ(λ0 , h; θ) h∈H J(λ0 , θ) ≤c (6) (c is some general constant). In the present paper we consider first a simple diffusion model with zero drift and show (see the Proposition in Section 2) that the solutions of problems (3) and (6) coincide. Then we pass to a more complicated diffusion model with nonzero drift and a deterministic mean-variance tradeoff process and solve the optimization problem (6) which will be at the same time an approximation (in leading order ε) solution of problem (3) (see the Theorem in Section 3). The consideration of misspecified asset price models was initiated by Avellaneda et al. [3], Avelaneda and Paras [4], and Avellaneda and Lewicki [5]. They obtained pricing and hedging bounds in markets with bounds on uncertain volatility. El Karoui et al. [6] investigate the robustness of the Black– Scholes formula, C. Gallus [7] give an estimate of the variance of additional costs at maturity if the hedger uses the classical Black–Scholes strategy, but the volatility is uncertain. H. Ahn et al. [8] consider the Black–Scholes model with misspecified volatility of the form σe 2 = σ 2 + δS(t, x), |S(t, x)| ≤ 1. The trading strategies are also the Black–Scholes ones, and the risk functional is an expected exponencial utility. Based on Feynman–Kac formula, they write the partial differential equation for the corresponding optimization problem whose solution cannot be obtained in explicit form. Instead, they find an approximate solution in the functional form. 2. Diffusion model with zero drift. Let a standard Wiener process w = (wt )0≤t≤T be given on the complete probability space (Ω, F, P ). Denote by F w = (Ftw , 0 ≤ t ≤ T ) the P -augmentation of the natural filtration Ftw = σ(ws , 0 ≤ s ≤ t), 0 ≤ t ≤ T , generated by w. Let the stock price process be modeled by the equation dXtλ = Xtλ · λt dwt , X0λ > 0, 0 ≤ t ≤ T, (7) where λ ∈ Λε , see (1), (2), with ZT (λ0t )2 dt < ∞, 0 P -a.s., and h ∈ BallL∞ (dt×dP ) (0, R), 0 < R < ∞. All considered processes are real-valued. Denote by Rλ the yield process, i.e., dRtλ = λt dwt , R0 = 0, 0 ≤ t ≤ T, (8) 192 T. TORONJADZE and let θ = (θt )0≤t≤T be the dollar amount invested in the stock X λ . Define the class of admissible strategies Θ = Θ(Λε ) = Θ(λ0 , H). Definition 2. A class of admissible strategies Θ = Θ(λ0 , H) is a class of F w -predictable real-valued processes θ = (θt )0≤t≤T such that ZT ZT θt2 (λ0t )2 E dt < ∞, θt2 h2t dt < ∞, ∀h ∈ H, E 0 0 or, equivalently, ZT ZT θt2 (λ0t )2 E θt2 dt < ∞. dt < ∞, E 0 (9) 0 The corresponding gain process has the form Zt θs dRsλ , 0 ≤ t ≤ T, Gt (λ, θ) = (10) 0 (recall that θ is the dollar amount invested in the risky asset rather than the number of shares). Evidently, GT (λ, θ) ∈ L2 (P ) for each λ ∈ Λε . The wealth at maturity T , with the initial endowment x, is equal to VTx,θ (λ) ZT θt dRtλ . =x+ 0 Let, further, the contingent claim H be FTw -measurable P -square-integrable r.v. For simplicity, we suppose that a risk-free interest rate r ≡ 0; hence the corresponding bond price Bt ≡ 1, 0 ≤ t ≤ T . Consider the optimization problem (3). It is easy to see that if λ ∈ Λε ; then λ0t − εR ≤ λt ≤ λ0t + εR, 0 ≤ t ≤ T, P -a.s. By the martingale representation theorem ZT ϕH t dwt , P -a.s., H = EH + (11) 0 where ϕ H w is the F -predictable process with ZT 2 (ϕH t ) dt < ∞. E (12) 0 Hence ³ E H− ´2 VTx,θ (λ) ZT 2 (ϕH t − λt θt ) dt. 2 = (EH − x) + E 0 OPTIMAL MEAN-VARIANCE ROBUST HEDGING 193 From this it directly follows that the process λ∗t (θ) = (λ0t − εR)I + (λ0t + εR)I ϕH t θt ≥λ0t } ϕH t θt <λ0t } { { I{θt 6=0} I{θt 6=0} , 0 ≤ t ≤ T, (13) is a solution of the optimization problem ³ ´2 maximize E H − VTx,θ (λ) over all λ ∈ Λε , with a given θ ∈ Θ. It remains to minimize (w.r.t. θ) the expression ZT ³ E ∗ ϕH t − λt (θ)θt ´2 dt. 0 From (13) it easily follows that the equation (w.r.t. θ) ∗ ϕH t − λt (θ)θt = 0, has no solution, but θt∗ = ϕH t I 0 , 0 ≤ t ≤ T, λ0t {λt 6=0} (14) solves problem (3). We assume that 0/0 := 0. Consider now the optimization problem (6). For each fixed h µ ZT θt dRtλ J(λ, θ) = E H − x − ¶2 0 Ã ZT =E H −x− dwt − ε 0 θt ht dwt 0 "µ 0 = J(λ , θ) − 2εE !2 ZT θt λ0t EH − x + ZT ³ ϕH t − θt λ0t ´ # ¶ ZT dwt 0 θt ht dwt 0 ZT θt2 h2t dt, 2 +ε E 0 and hence 0 DJ(λ , h; θ) = 2E ZT ³ 0 ´ θt λ0t − ϕH t θt ht dt, (15) 194 T. TORONJADZE as follows from (9), (12), the definition of the class H and the estimation µ E ZT ³ θt λ0t − ϕH t ´ ¶2 θt ht dt 0 ≤E ZT ³ θt λ0t − ´2 θtH ZT 0 Ã ZT ≤ const ·R2 E 0 ! ZT θt2 (λ0t )2 dt + E 0 θt2 h2t dt dt E ZT 2 (ϕH t ) dt E 0 θt2 dt < ∞. (16) 0 Since, further, DJ(λ0 , h; θ) = 0 for h ≡ 0, using (16) we get 0 ≤ sup DJ(λ0 , h; θ) < ∞. h∈H Hence we can take 0 ≤ c < ∞ in problem (6). Now if we substitute θ∗ from (14) into (15), we get DJ(λ0 , h; θ∗ ) = 0 for each h, and thus sup DJ(λ0 , h; θ∗ ) h∈H J(λ0 , θ∗ ) = 0. If we recall that θ∗ = arg min J(λ0 , θ), we get that θ∗ defined by (14) is a θ∈ΘΛε solution of the optimization problem (6) as well. Thus we prove the following Proposition. In scheme (7), (8) under assumptions (9): (a) the optimal mean-variance robust trading strategy θ∗ = (θt∗ )0≤t≤T for the optimization problem (6) is given by the formula θt∗ = ϕH t I 0 ; λ0t {λt 6=0} (b) this strategy is an approximation (in leading order ε) strategy for the optimization problem (3) and coincides with the exact optimal strategy of this problem. 3. Diffusion model with nonzero drift. Let us consider the filtered probability space (Ω, F, F w = (Ftw )0≤t≤T , P ) with a given standard Wiener process w = (wt , Ftw ), 0 ≤ t ≤ T , and a given P -square-integrable FTw -measurable r.v. H. Let the stock price process be defined by the equation ³ ´ dXtλ = Xtλ µt dt + λt dwt , X0λ > 0, (17) µt = kt λt , 0 ≤ t ≤ T, (18) where and k = (kt )0≤t≤T is a bounded deterministic function, λ = (λt )0≤t≤T ∈ Λε , i.e., λt = λ0t + εht , OPTIMAL MEAN-VARIANCE ROBUST HEDGING 195 RT where λ0 is an F w -predictable process with (λ0t )2 dt < ∞, P -a.s. h ∈ H, with 0 L =L∞ (dt×dP ), i.e., h is a bounded F -pedictable process, h ∈ BallL∞ (dt×dP ) (0,R), 0 < R < +∞. All processes in (17), (18) are real-valued. Consider the optimization problem (6). Denote, as in the previous section, by Rλ the yield process defined by the equation dRtλ = λt (kt dt + dwt ), R0λ = 0, 0 ≤ t ≤ T, (19) and for each θ = (θt )0≤t≤T ∈ Θ(λ0 , H) (see Definition 2) introduce the risk functional of the problem µ ZT θt dRtλ J(λ, θ) = E H − x − ¶2 . 0 Rt Note that since the mean-variance tradeoff process ( ks2 ds)0≤t≤T is continuous 0 and bounded, the space {GT (λ, θ) : θ ∈ Θ} is closed in L2 (P ) for each λ ∈ Λε , see, e.g., Corollary 4 of [9]. Further, there exists a unique equivalent martingale measure (which does not depend on the parameter λ) given by the relation dPe = zeT dP, where zeT = ET (−k.w), zeT > 0, E zeT = 1, Et (−k.w) is the Dolean exponential of Rt the martingale −k.wt = − ks dws , 0 ≤ t ≤ T . Moreover, the process G(λ, θ) 0 belongs to S 2 , the set of square integrable semimartingales, for each λ ∈ Λε . Now, following [10] and [11], introduce the objects: zet = E Pe(zeT /Ftw ), 0 ≤ 2 zT zT e (and thus dQ e = e e = e d P dP ). Since ze > 0 is a strictly positive t ≤ T , dQ e z0 e z0 e is a probability measure with Q e ≈ P. Pe -martingale, Q Introduce, further, the process Zt wt0 = wt + ks ds, 0 ≤ t ≤ T. 0 0 Then Ftw = Ftw , 0 ≤ t ≤ T , because k is deterministic, and hence the process w0 = (wt0 )0≤t≤T is a standard (Pe , F w )-Wiener process. Consider now the new filtered probability space (Ω, F, F w , Pe ), rewrite the process Rλ in the form dRtλ = λt dwt0 , R0λ = 0, 0 ≤ t ≤ T, (20) and decompose the r.v. zeT w.r.t. w0 : ZT ζt dwt0 . zeT = ze0 + 0 (21) 196 T. TORONJADZE In this notation, based on Proposition 5.1 of [10], we can write µ T Z ze2 ze2 J(λ, θ) = E T2 20 H − x − θt dRtλ ze0 zeT ¶2 0 = 2 e ze ze0−1 E Q 20 zeT Ã = e ze0−1 E Q µ ZT ¶2 0 H ze0 − x − zeT 0 θt λt dwt0 H −x− ZT z0 ψt0 (λ) d zet e 0 ZT − 0 w0 ψt1 (λ) d t zet !2 ze0 1 := J(λ, ψ , ψ ). (22) Here Zt ψt1 (λ) ψt0 (λ) = θt λt , θs λs dws0 − θt λt wt0 , 0 ≤ t ≤ T. =x+ 0 Thus 0 ψt1 (λ) = ψt1 (λ0 ) + εψt1 (h), ψt0 (λ) = ψt0 (λ0 ) + εψ t (h), (23) 0 where ψ t (h) = ψt0 (h) − x. If now µ ¶ T T Z Z e0 w0 H e H 0,H z Q ze0 = E ze0 + ψt d + ψt1,H d t ze0 zeT zeT zet zet (24) 0 0 is the Galtchouk–Kunita–Watanabe decomposition of the r.v. H e z w.r.t. e zT 0 w0 z0 e F w )-local martingales e (Q, and eztt ze0 , then, using (22), (23) and (24), we e zt get for each fixed h 0 1 ("µ µ ¶¶ e H ze0 zeT # ZT ³ ZT ³ ´ w0 ´ ze 0 1,H 0,H t 0 1 0 0 0 d + ψt (λ ) − ψt (λ d ze0 + ψt (λ ) − ψt zet zet ³ 0 0 0 1 0 ´ J(λ, ψ , ψ ) = J λ , ψ (λ ), ψ (λ ) + e ε2ze0−1 E Q 0 0 Ã ZT × 0 + x − EQ ze0 0 ψ t (h) d zet e ε2 ze0−1 E Q Ã ZT 0 ZT + 0 w0 ψt1 (h) d t zet ze0 0 ψ t (h) d zet ZT + 0 !) ze0 w0 ψt1 (h) d t zet !2 ze0 . (25) OPTIMAL MEAN-VARIANCE ROBUST HEDGING 197 Consequently, ( 0 0 1 2ze0−1 DJ(λ , h; ψ , ψ ) = E ZT ³ e Q ¿ ´ 0 ψt0 (λ0 ) − ψt0,H ψ t (h) d 0 +E e Q ZT ·³ ψt1 (λ0 ) − ´ 0 ³ ψt1 (λ0 ) ψt1,H ψt1,H ψ t (h) + ψt0 (λ0 ) − ψt0,H ´ +E Zt ³ − ´ ¿ 0 w ψt1 (h) d 0 t zet À ¸ ¿ 0 w ze0 d ze0 , zet zet À t ψt1 (h) 0 e Q ze0 zet À) ze0 . (26) 0 From the definition of ψ 1 (h) and ψ (h) (see (23)) it follows that for h ≡ 0, 0 ψ 1 (h) = 0 and ψ (h) = 0. Hence DJ(λ0 , 0; ψ 0 , ψ 1 ) = 0, and thus sup DJ(λ0 , h; ψ 0 , ψ 1 ) ≥ 0. (27) h∈H We show now that sup DJ(λ0 , h; ψ 0 , ψ 1 ) < ∞. (28) h∈H For this it is sufficient to estimate the expression (as it easily follows from (25)) I=E e Q Ã ZT 0 ze0 0 ψ t (h) d zet ZT w0 ψ (h) d t ze0 zet 1 + 0 !2 . But using Proposition 8 of [11], we have for each h ( ZT = 0 ³ ´ ze0 GT h, Θ(0, H) zet ze0 0 ψ t (h) d zet ZT + 0 ¯ µ w0 ¯ ze0 w0 0 e ψ 1 (h) d t ze0 ¯¯ ψ (h), ψ 1 (h) ∈ L2 , ze0 , Q zet ze ze ¶) , e is the space of F w -predictable processes (ψ 0 , ψ 1 ) such where L2 ( ezez0 , ez ze0 , Q) R 0 e R 0 e F w ) of martingales. that ψ d zez0 + ψ 1 d wez ze0 is in the space M2 (Q, Hence, using notation (10), we have w0 I= 2 e ze E Q 20 zeT µ ZT G2T (h, θ) = ze0 EG2T (h, θ) θt dRth = ze0 E 0 µ ZT = ze0 E ¶2 θt ht (kt dt + dwt ) 0 ¶2 198 T. TORONJADZE Ã θt2 h2t kt2 ≤ ze0 const · E µ ≤ ze0 const ·R θt2 h2t dt + E 0 2 ! ZT ZT dt 0 ¶ ZT θt2 dt < ∞, E 0 by the definitions of the classes Θ, H, and the boundedness of the function k = (kt )0≤t≤T . From (27) and (28), as in the previous section, it follows that we can take 0 ≤ c < ∞ in problem (6). Now if we substitute ψ 1,∗ (λ0 ) := ψ 1,H and ψ 0,∗ (λ0 ) := ψ 0,H into J(λ0 , ψ 0 , ψ 1 ) and DJ(λ0 , h, ψ 0 , ψ 1 ), we get ³ ´ J λ0 , ψ 0,∗ , ψ 1,∗ = min J(λ0 , ψ 0 , ψ 1 ) 0 1 ψ ,ψ (see Lemma 5.1 of [10]), and DJ(λ0 , h; ψ 0,∗ , ψ 1,∗ ) =0 J(λ0 , ψ 0,∗ , ψ 1,∗ ) h∈H sup (hence the constraint of problem (6) is satisfied). Consequently, using Proposition 8 of [11], we arrive at the following Theorem. In model (17)–(19) the optimal mean-variance robust trading strategy (in the sense of Definition 1) is given by the formula " 1,H θt∗ µ 0 e0 ψt ζt 0,H z 1,H wt ∗ = + V − ψ − ψ ze0 t t t λ0t λ0t zet zet where Ã Vt∗ t ψ and ψ 1,H I{λ0t 6=0} , 0 ≤ t ≤ T, ! Z ze0 Z w0 ze0 x + ψs0,H d + ψs1,H d s ze0 , = zet zes zes 0 0,H t ¶# 0 are given by relation (24), ζt is defined in (21). Acknowledgement This work was supported by INTAS Grants Nos. 97-30204 and 99-00559. References 1. F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw, and W. A. Stahel, Robust statistics. The approach based on influence functionals. John Wiley and Sons, N.Y. etc., 1986. 2. H. Rieder, Robust asymptotic statistics. Springer-Verlag, N.Y., 1994. 3. M. Avellaneda, A. Levy, and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2(1995), 73–88. OPTIMAL MEAN-VARIANCE ROBUST HEDGING 199 4. M. Avellaneda and A. Paras, Managing the volatility of risk of portfolios of derivative securities: the Lagrangian uncertain volatility model. Appl. Math. Finance 3(1996), 21– 52. 5. M. Avellaneda and P. Lewicki, Pricing interest rate contingent claims in markets with uncertain volatilities. Preprint, Courant Institute of Mathematical Sciences, New York University. 6. N. El Karoui, M. Jeanblanc-Picque, and S. E. Shreve, Robustness of the Black and Scholes formula. Math. Finance 8(1998), No. 2, 93–126. 7. C. Gallus, Robustness of hedging strategies for European options. Engelbert, HansJuergen (ed.) et al., Stochastic processes and related topics. Proceedings of the 10th winter school, Siegmundsberg, Germany, March 13–19, 1994. Gordon and Breach Publishers. Stochastics Monogr., Amsterdam 10(1996), 23–31. 8. H. Ahn, A. Muni, and G. Swindle, Misspecified asset price models and robust hedging strategies. Appl. Math. Finance 4(1997), 21–36. 9. H. Pham, T. Rheinlaender, and M. Schweizer, Mean-Variance hedging for continuous processes: new proofs and examples. Finance Stochs. 2(1998), 173–198. 10. C. Gourieroux, J. P. Laurent, and H. Pham, Mean-variance hedging and numeraire. Math. Finance 8(1998), No. 3, 179–200. 11. T. Rheinlaender and M. Schweizer, On L2 -projections on a space of stochastic integrals. Ann. Probab. 25(1997), No. 4, 1810–1831. (Received 20.11.2000) Author’s Address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: toro@imath.acnet.ge