Malliavin Calculus in Finance

Transcription

1 Malliavin Calculus in Finance Peter K. Friz 1 Greeks and the logarithmic derivative trick Model an underlying assent by a Markov process with values in R m with dynamics described by the SDE dx t = b(x t )dt + (X t )dw t. (1) This includes BS and local vol models with m = 1. Heston-model or BS for Asians options require m = 2. To prize general path-dependent (but non-american type) options one must compute integrals of the form u(x) = E[φ(X t1,..., X tn ) X() = x] = E x [φ(x t1,..., X tn )] Greeks are the sensitivities of u with respect to some parameter. Assume you know the joint law of (X t1,..., X tn ) given by a density f(x 1,..., x n ) = f λ (x 1,..., x n ) depending on some parameter λ. For instance λ could be the spot-price X() = x. Or it could be the volatility in the BS-model, where (X t ) = X t for some constant >. Then E λ [φ(x t1,..., X tn )] = φ(x t1,..., x tn )f λ (x 1,..., x n )dx 1...dx n Under mild condition we can interchange integration and derivation w.r.t. λ and obtain λ E λ[φ(x t1,..., X tn )] = φ(x t1,..., x tn ) λ f λ(x 1,..., x n )dx 1...dx n (2) [ ] = φ(x t1,..., x tn ) λ ln f λ(x 1,..., x n ) f λ (x 1,..., x n )dx 1...dx n = E λ [φ(x t1,..., X tn ) π ] where π = π (X t1,..., X tn ) = λ ln f λ(x t1,..., X tn ). We view π as a random-weight and observe it is universal for all payofffunctions φ.however, π is not unique. We can replace it by any r.v. π such that its conditional expectation given (X t1,..., X tn ) equals π, i.e. such that E[π X t1,..., X tn ] = π. 1

2 Clearly we should pick π such as to minimize the variance of φ(x t1,..., X tn ) π since this will make the Monte-Carlo simulation for (2) converge faster. As shown in [F2] π (or any π which depends only on X t1,..., X tn ) is the variance-minimizing choice. All that looks nice but in most case we simply don t know the density f λ.malliavin calculus helps to get around this problem. In the end, we will be able to compute Greeks as weighted averages over the payoff-functions. 2 Basic Malliavin Calculus Consider r.v.s on the Wiener-space Ω of the form F = F (ω) = h t dw t for some deterministic h(.) in L 2 [, T ]. These are just the Gaussians with zero-mean and standard-deviation h L2 [,T ]. We define the Malliavin derivative by DF = h, D t F = h t. Let ψ : R R be a smooth function. Then G = ψ(f ) is another r.v., in general non-gaussian. Inspired by the chain-rule for classical derivatives we set In more detail this reads DG = D[ψ(F )] = ψ (F )DF = ψ (F )h D t G(ω) = ψ (F (ω))h t. For fixed ω the r.h.s. is in L 2 [, T ] H. We can view DG as H-valued random-variable. Alternatively, it is a function of (t, ω) hence can be seen as stochastic process. An assumption of finite second moments translates to DG = D t G(ω) L 2 ([, T ] Ω). Example: Let G(W.) = (W 2 s ) for some s (, T ]. Then D t G = 2W s 1 [,s] (t) i.e. D t G = 2W s when t s and zero otherwise. The path-perturbation picture. Take any deterministic h(.) in L 2 [, T ]. and perturb the Brownian path W.(ω), d W ε t = dw t + εh(t)dt. or equivalently, W ε t = W t + ε t h(τ)dτ = W t + ε h t. 2

3 Now consider G( W ε ) G(W ) ε = (W s + ε h s ) 2 (W 2 s ) ε = 2W s hs + O(ε) After ε we see that the directional derivative of G in direction h is given by 2W s hs = 2W s h(τ)1 [,s] (τ)dτ = 2W s 1[,s], h (standard inner product in H = L 2 [, T ]) = 2W s 1 [,s], h = DG, h This is no coincidence. It is the true meaning of the Malliavin derivative and true for general G: bracket DG with h and obtain a directional derivative of G in direction of h.formally, D t G is obtained by h = δ t (.), the Diracdelta. In the path-perturbation picture we perturb a Brownian path W. (ω) by a Heaviside-function jumping for to 1 at time t. Integration by Parts In the preceding example, D t G depends on the Brownian path at time s. Hence, viewed as stochastic process, it can not be adapted. Now, take any process U t = U(t, ω) L 2 ([, T ] Ω).Both U and DG are in L 2 [, T ] = H for fixed ω.write U, DG = U t D t Gdt for the standard inner product in L 2 [, T ] and note that the it is still stochastic: U, DG = U, DG (ω) Integrating over ω yields the expression E U, DG.Then define δ as adjoint to D by requiring that E( U, DG ) = E(δ[U]G). (3) In operator-language, defined on appropriate subspaces D : L 2 (Ω) L 2 ([, T ] Ω) while δ : L 2 ([, T ] Ω) L 2 (Ω). Relation (3) is called Integration by Parts on the Wiener-space. Finite-dimensional analogue. Replace E (i.e. integration over Ω) by integration over some O R n.let g be a nice functions with compact support 3

4 on O and g its gradient-vectorfield. Let u be any vectorfield on O. Then, by integration by parts, u, g R n dx = ( div u)gdx O This motivates to call D resp. δ the gradient- resp. divergence operator on the Wiener-space. Skorohod integral. We introduced δ as abstract divergence-operator which takes (not necessarily adapted) processes and spits out a random-variable. It is one of the striking facts of Malliavin Calculus that, for adapted processes U the divergence coincides with the standard Ito-integral. O U adapted = δ[u] = U t dw t (4) Hence δ can be viewed as generalization of the standard Ito-integral to nonadapted processes (known as Skorohod integral). For numerical purposes, we note that there is a Riemann-sum expression for the Skorohod integral, see [N] What is it good for? It allows us to get ride of derivatives under the expectation. Here is a simple example - the later application to Greeks may look different since we stick with a special example but it is similar in spirit. By the chain-rule for Malliavin-derivatives, Dψ(F ) = ψ (F )DF, (5) for all F sufficiently nice in the sense of Malliavin. Then ( ) E (ψ 1 (F )) = E Dψ(F ), DF, DF DF ( [ ]) 1 = E ψ(f )δ DF, DF DF = E (ψ(f )π) (6) where the r.v. π is defined as [ π = δ Observe that π is the same for all ψ. 1 DF, DF DF ]. 4

5 3 Application to Greeks in the BS-model Consider a European option with European payoff φ(s T ),where ds t /S t = r t dt + t dw t, S = x. with deterministic, possibly time-dependent risk-free rate and volatility. We assume inf t >. Let us focus on λ = x, the spot-price. To indicate S = x we will also write S t = St x. Set Y t := x Sx t, called tangent- or first-variation process. Since S x t it follows that = x exp t t s dw s (ω) + Y t = 1 x S t, t T. ( ) r s 2 s 2 ds On the other hand, the Malliavin derivative of S t (t fixed!) is easily computed via the chain-rule { } D s St x = St x S x (ω) s 1 [,t] (s) = t (ω) s if s t else It follows that the Malliavin derivative and the first-variation process are related, Y t = D ss x t x s whenever s t. (7) Let s compute delta. Since we assume deterministic interest-rates it suffices to consider the undiscounted payoff 5

6 x E[φ(Sx T )] = E[φ (ST x )Y T ] (8) [ φ (ST x = E )D sst x ] (any s [, T ]) x s [ Ds φ(st x = E ) ] (by Malliavin s chain-rule) x s = E 1 D s φ(st x) ds (average over s [, T ]) T x s = 1 xt E Dφ(ST x ), 1 (inner product on L 2 [, T ]) = 1 ( [ ]) 1 xt E φ(st x )δ (by Malliavin s IBP) = 1 xt E φ(st x dw t ) (by (4)) = E φ(s x T ) 1 xt dw t In particular, for constant r and we have x E[e rt φ(s x T )] = E [ e rt φ(s x T ) W T xt ] (9) Note that the weight π := W T xt is a function of Sx T (check) and hence the variance-minimzing choice among all weights, as mentioned earlier. Exercise: Derive (9) via the logarithmic derivative - trick. Exercise: By iterating the preceding argument (you can assume r, constant) find the following expression for gamma, 2 [ ( x 2 E[e rt φ(st x )] = E e rt φ(st x 1 W 2 ) T x 2 T T W T 1 )] (1) Exercise: In the end of the day, we want to use these expressions in Monte- Carlo simulations. Hence the variance of the expression inside the bracket [...] must be small to obtain good numerical results. Even the optimal weight-choice π does not guarantee that the variance is small - it just says it s smaller than for other weights which do the same job. Discuss delta and gamma for a European calls, digitals and a doubledigitals with payoff φ(x) = 1 [a,b] (x). 6

7 Which payoffs are good, which are bad? How can you fix the bad ones? (Hint: localize around the singularity or consult [F1] to see how to do it). Exercise: From the preceding it is may not be clear how to obtain other Greeks as vega, rho. A study of drift- resp. volatility- perturbation via Malliavin Calculus, Girsanov Theorem etc. is possible (and successfully done in [F1]). However, we prefer to use a trick by Peter Carr. As shown in [C, (29),(3)], one has vega = x 2 T gamma (11) rho = xt delta T price from which [ ( W E[e rt φ(st x )] = E e rt φ(st x 2 ) T T W T 1 )]. [ ( )] r E[e rt φ(st x )] = E e rt φ(st x WT ) T What are Carr s assumptions? Is (11) OK for the Heston model? (Yes, you should look up his paper to find out!) 4 Beyond European options The preceding methods carry through for more complex instruments. To be concrete, consider an Asian barrier in with payoff φ S T, S t dt where φ(x, y) = 1 {y B} (x K) +. In [F1] it is shown that its delta is given by E e rt φ S T, S t dt δ(g) for a non-adapted process G given explicitly in terms of S. and its tangentprocess Y. On the other hand the logarithmic derivative trick won t work - since we don t know the joint law of S T, S t dt. 7

8 5 When Malliavin Calculus fails... Consider a simple Barrier option with payoff { if τ(ω) < T Φ(ω) = φ (S T (ω) else } where τ is the first hitting time of the knock-out region {x : x B}. For Malliavin Calculus to work, we need Φ(ω) smooth in the sense of Malliavin such that DΦ exists. We won t go into the maths, but the intuition of Malliavin s smoothness is that small perturbations in the paths ω must not change Φ(ω) too much. But there are paths extremely close to each other, one of which hits the barrier while the other doesn t. We see that Φ(ω) is not smooth and Malliavin Calculus doesn t work. On the other hand, there is a smooth joint law for (S T (ω), τ(ω)) i.e. a smooth probability distribution, say f(.,.), on R 2 for the 2-dimensional r.v. (S T, τ). In general, we don t know f but by the logarithmic derivative trick, the existence of some optimal weight π for computing Greeks is ensured. In [F2, p29] a solution to this serious difficulty is promised, but hasn t (to my knowledge) appeared yet. 6 Beyond B&S In the preceding section, for the sake of clarity, we restricted ourself to lognormal dynamics of the underlying S. In particular, we saw an immediate relation between the Malliavin derivative and the tangent-process Y which in turn was the main-ingredient for the derivation of (8). A similar relation holds in the general case. First, assume that the underlying asset follows a diffusion dx x t = b(x x t )dt + (X x t )dw t, X x = x. (12) The tangent-process Y t satisfies the linear SDE obtained by formal derivation w.r.t. x dy t = b(x x t )Y t dt + (X x t )Y t dw t, Y t = 1. (13) Then it is known that, for t fixed, D s X t = Y t Ys 1 (Xs x )1 {s t}. 8

9 With this result computations like the one in (8) extends to general dynamics of the underlying. References [F1] Fournie et al. Applications of Malliavin Calculus to Monte Carlo Methods in Finance (1999) [F2] Fournie et al. same title, Part II (21) [C] Carr, Deriving Derivatives of Der. Sec. (2) [N] Nualart, The Malliavin Calculus (1995) 9