Application of large deviation methods to the pricing of index options in finance. Méthodes de grandes déviations et pricing d options sur indice

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1 Application of large deviation methods to the pricing of index options in finance Méthodes de grandes déviations et pricing d options sur indice Marco Avellaneda 1, Dash Boyer-Olson 1, Jérôme Busca 2, Peter Friz 1 Abstract: We develop an asymptotic formula for calculating the implied volatility of European index options based on the volatility skews of the options on the underlying stocks and on a given correlation matrix for the basket. The derivation uses the steepest-descent approximation for evaluating the multivariate probability distribution function for stock prices, which is based on large-deviation estimates of diffusion processes densities by Varadhan [V]. A detailed version of these results can be found in [ABBF]. Keywords: Large deviations, mathematical finance, index options. Résumé : Nous montrons une formule asymptotique donnant la volatilité implicite d une option sur indice à partir des volatilités des actifs sous-jacents. La démonstration repose sur les estimations de densités de diffusion en temps petit de type grandes déviation de Varadhan [V]. On pourra trouver une version détaillée de ces résultats dans l article [ABBF]. 1 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York, 112. We are grateful to Andrew Lesniewski and Paul Malliavin for interesting conversations and insights on the steepest descent approximation and the topics of this paper. 2 CNRS, Ceremade, Université Paris Dauphine, Pl. du Maréchal de Lattre de Tassigny, Paris Cedex 16, France. 1

2 Mots-clé : Grandes déviations, finance mathématique, options sur indice. I. INTRODUCTION We consider a basket of n stocks described by their price processes S i (t), i = 1,..., n and an index on these stocks B(t) = w i S i (t), with the w i s constant. For simplicity we assume that each stock follows a one-factor risk-neutral process ds i S i = σ i (S i, t) dz i + µ i dt, where σ i (S i, t) is the so-called local volatility function, µ i is the drift associated with the cost of carry, and Z i = Z i (t) are standard Brownian motions which satisfy E (dz i dz j ) = ρ ij dt, ρ ij constant 1. Note that the function σ i is, in theory, uniquely determined from (an infinite set of) market prices of European options written on S i [D] [Ru]. It is known [DKK] that the multi-dimensional problem of determining option prices on the basket B is equivalent to computing options prices written on a one-dimensional diffusion process db(t) B(t) = σ B,loc (B, t) dw + µ B dt (1) governed by a suitable effective local volatility σ B,loc. Specifically, the price of an European call option with maturity t and strike B is given by E (e µ Bt (B(t) B) + ), where B(t) follows (1). The volatility function in (1) is found to be the expectation of the stochastic volatility σb(b, 2 t) = 1 ρ B 2 ij σ i σ j w i w j S i S j 1 The results presented here apply to more general correlation/volatility structures, including for instance the case of multivariate stochastic volatility/stochastic correlation models. 2

3 conditional on the value of the index B at time t, that is, σb,loc(b, 2 t) = E { σb 2 B (t) = B } { } 1 = E ρ B 2 ij σ i σ j w i w j S i (t) S j (t) w i S i (t) = B. ij=1 (2) Hence σ B,loc can be thought of as a way of representing option prices written on B and similarly for σ i and S i. A natural question is thus to relate those quantities. Another important way of representing option prices in practice is the so-called Black-Scholes implied volatility: one replaces σ B,loc in (1) by a (uniquely determined) constant σb I so as to leave the value of the option maturing at t with strike B unchanged. Note that this constant depends on strike and maturity as parameters. One proceeds in a similar way to define the σi I s. Our main result establishes an asymptotic relation between the local and implied volatilities of the index and the underlying assets in the limit σ 2 t 1, if σ denotes a typical level of the volatilities involved. In practice an usual order of magnitude is σ 2 t.2, leading to very good agreement with market quotes [ABBF]. II. MAIN RESULTS It is convenient to introduce the forward spots F i = S i ()e µ it, F = B()e µ Bt, the forward log-moneyness x i fraction p i (x) = = ln S i F i, x = ln (B/F ) and the F ie x i w i. (3) F k e x k w k k=1 Slightly abusing the notations, we simply write σ i (x i ) = σ i (F i e xi, ) and σ B,loc (x) = σ B,loc (F e x, ) (and similarly for σ I i, σ I B ). Theorem 1 Link between local volatilities by In the limit σ 2 t 1 the local volatility of the index is given, to first order, σ 2 B,loc(x) = ( ) ρ ij σ i (x i ) σ j x j pi (x ) p j (x ), (4) 3

4 where x = (x 1,..., x n) is the solution of the nonlinear system x i du σ i (u) = λ p i (x)e x i xi = 1. ρ ij p j (x )σ j (x j), j=1 i = 1,..., n (5) Theorem 2 Link between Black-Scholes implied volatilities 1) In the limit σ 2 t 1 the implied volatility functions of the index and the underlying stocks are related by σ I B(x) = ( 1 x ) 1 ( ( ) ) 1 du, σ i (x i d y ) = x σ B,loc (u) dy σi I(y), (6) y=x i (i = 1,..., n) together with (4), (5). 2) In the at-the-(forward)money region { x 1, x i 1} this relation reduces, to first order, to 2σ B I (x) σi B () = ρ ij p i (x ) p j (x ) (2σi I (x i ) σi i ()) ( 2σj I ( x j ) σ I j () ) (7) II. SKETCH OF THE PROOF Proof of Theorem 1. We formally rewrite (2) as σb,loc 2 = E {σ2 B δ (B (t) B)} E {δ (B (t) B)} (8) so that σb,loc 2 appears as an average of σ2 B. When σ 2 t is small, we shall prove that a concentration phenomenon appears, reducing (8) to an evaluation at some point. For this purpose, we introduce the transition probability, or Green function, π (, ; x, t) of the diffusion process x = (x 1,..., x n ) with matrix a ij = σ i σ j ρ ij and we analyze it thanks to a classical formula by 4

5 Varadhan [V] that we now recall. Introducing the inverse matrix (g ij ) = (a ij ) 1 and the associated Riemmanian metric we have where ds 2 = g ij dx i dx j, (9) π (, ; x, t) e d2 (,x) 2t = e (σ)2 d 2 (,x) 2(σ) 2 t (1) d 2 (, x) = 1 inf x()=,x(1)=x g ij (x (τ), ) ẋ i ẋ j dτ. (11) Here ẋ is the time-derivative of x. The asymptotic in (1) are understood in the sense that the ratio of the logarithms of the two terms tends to 1 as σ 2 t. Hence π (, ; x, t) is strongly peaked near the points x where d 2 (, x) is minimal. The method of steepest descent thus implies σ B,loc (x, t) σ B (x, t), (12) where x is the (generically unique) point realizing { the distance (in} the sense of (11)) of the origin to the manifold Γ B = x : w i F i e x i = B. Introducing the change of variable y i = shown to be equivalent to minimizing 1 x i du, determining σ i (u) x is easily (ρ 1 ) ij ẏ i ẏ j dτ under the con- straint n w i F i () e xi (y i ) = B. Writing the first-order Euler-Lagrange condi- tion results in j=1 ( ρ 1 ) ij ẏj = λ B w if i () e xi (y i ) xi (y i ) y i = λp i (x i (y i ))σ i (x i (y i )), where for convenience the Lagrange multiplier has been written as λ/b. Looking for a solution y(τ) linear in τ (so that ẏ j = y j ) and multiplying by (ρ) 1 easily yields (5). Proof of Theorem 2. 5

6 Part 1) follows readily from the harmonic-mean relation between implied and local volatilities in the limit σ 2 t 1 which can be found in [BBF1] [BBF2]. Part 2) follows from a first-order expansion of that relation near x = (resp. x i = ). References [ABBF] Avellaneda M., Boyer-Olson D., Busca J., Friz P. Reconstruction of volatility; Pricing index options using the steepest-descent approximation. Submitted. [BBF1] Berestycki H, Busca J. & Florent I. An inverse parabolic problem arising in finance. Comptes Rendus de l Académie des Sciences de Paris - Series I - Mathematics, 331 (12 part 1) (2), [BBF2] Berestycki H., Busca J. & Florent I. Asymptotics and Calibration of Local Volatility Models. Quantitative Finance 2, No 1 (22), [BN] Britten Jones & Neuberger M. A. Option prices, implied prices processes, and stochastic volatility, Journal of Finance, 55, 2 (2), [DK] Derman E. & Kani I. Riding on a smile, RISK, 7 (2) (1994). [DKK] Derman E., Kani I. & Kamal Trading and hedging of local volatility, Journal of Financial Engineering, Vol 6, No 3 ( 1997), [D] Dupire, B. Pricing with a smile, RISK, 7 (1) (1994). [G] Gatheral, J. Stochastic Volatility and Local Volatility, lecture notes for Case Studies in Financial Modeling, M.S. Program in Math Finance, N.Y.U., mathematics (21). [L] Lim, Juyoung. Pricing and Hedging Options on Baskets of Stocks, Ph.D. Thesis, NYU (22). [Ro] Roelfsema, M.R. Non-Linear Index Arbitrage, WBBM Report Series 46, Delft University Press (2). [Ru] Rubinstein M. Implied binomial trees. Journal of Finance, 49 (3)(1994),

7 [V] Varadhan S.R.S. On the behaviour of the fundamental solution of the heat equation with variable coefficients, Commun. Pure Appl. Math. 2 (1967). avellane@cims.nyu.edu dolson@gargoylestrategic.com busca@ceremade.dauphine.fr Peter.Friz@cims.nyu.edu 7