Calibration of Barrier Options

Transcription

1 Calibration of Barrier Options Olivier Pironneau On Yuri Kuznetsov s 6 th birthday Abstract A barrier option is a financial contract which is used to secure deals on volatile assets. The calibration of the volatility is an interesting mathematical and numerical problem which we analyze. In the process we will recall the modern tools of numerical analysis for fast solutions. We report also on the behavior of some optimization algorithms for this problem. Introduction A European option on a financial asset S is the right to buy (Call) or sell (Put) S at the future date T (maturity) and at the price K (strike). For a given asset, there exists on the market several options with different maturities and strikes which we may write as C t := C T,K (S t, t) where S t is the price of the asset at time t. The model of Black & Scholes assumes a stochastic ODE for S t. where ds t = S t (µdt + σ t dw t ), S known (1) µ is the tendency of the of the asset price, (W t ) is a standard Brownian motion, σ t is the volatility of the asset price at time t. No arbitrage requires that the price of the option should be the expected value of the profit at maturity discounted by the interest rate r. For the Call option (E denotes the expected value): C t = e r(t t) E ( (S T K) +) () Laboratoire Jacques-Louis Lions, Université Paris VI, 175 rue du Chevaleret, F Olivier.Pironneau@upmc.fr Y.K. is a long standing friend and collaborator, way back from Siberia times. I am most grateful to be able to express my friendly gratitude by contributing to this volume. 1

2 Barrier options provide a safety mechanism in case the fluctuations of S are too large. It is part of the contract of a barrier option that it loses its value if S t becomes smaller than S m e rt or larger than S M e rt. Ito Calculus on (1) reframed with the adapted probability law which brings µ = r gives The Put-Call parity gives t C + σ S SSC + rs S C rc = C(T, S) = (S K) +, (3) P (S, t) = Ke r(t t) S + C(S, t) (4) and naturally P satisfies the same PDE but with P (t, ) = Ke rt if S m =. 1 Variational and Numerical Methods As S m = or S M = + is often the case (one barrier only) it is preferable to work in the variational setting of weighted Sobolev spaces (6) where the problem is always well posed. Numerically the problem is easier with the Put because it decays to zero for large S; one can use (4) to recover C from P. Existence, Unicity and Regularity have been studied at length in [4]. For an easier reading we change t T t, x S, u(x, t) = P (T t, S) Ω = (x m, x M ), Q = (x m, x M ) (, T ) (5) Then one seeks for u in V = {v H 1 p : v(x m ) = v(x M ) = } where H 1 p = {u L (Ω) : x x u L (Ω)} (6) such that for all w V w t u + x (w x σ Q ) xu wxr x u + rwu)dxdt = u(x, ) = (K x) + x Ω u(x m, t) = or u(, t) = Ke rt if x m =, u(x M, t) =, t (, T ) (7) There is no singularity at the origin x =. But there is a weak singularity at t = due to the initial condition. Numerical Methods are necessary in the case of barrier option because there are no analytical solutions to (7). The Crank-Nicolson scheme is second order

3 accurate and so is the finite element method of order 1. In this case the scheme is equivalent to the centered finite difference scheme u n+1 i u n i (x iσ n+ 1 i ) ( u n+ 1 i+1 1 un+ i u n+ 1 i u n+ 1 i 1 ) δt n δx i x i+1 x i x i x i 1 rx i (u n+ 1 i+1 δx 1 un+ i 1 ) + 1 run+ i = (8) i with δx i = 1 ((x i x i 1 ) 1 + (x i+1 x i ) 1 ) 1. Mesh adaptivity is an essential ingredient for speed and accuracy and is best investigated in the context of Finite Element Methods, even with the simplest method of order 1. Traders and banks need.1% precision and without adaptivity it is expensive. Numerical experiments (Achdou in [4]) show that a fine mesh near x = K and a small δt near t = growing larger with what the aposteriori estimates indicate. In one dimension Gauss LU factorizations are hard to beat as the computing time depends linearly on the number of mesh points. Calibration The Black & Scholes model does not predict perfectly the price of options and discrepencies can be observed daily in the market; then one may either look for a better model (the trend is with stochastic volatility models) or pass from a constant volatility to a local volatility σ(s, t). The situation is as follows: each day (time t ) the underlying asset S is known and the past of some options on the same asset c i := C Ki,T i (S t, t), t < t for i I are known too; the options differ only by their strike and maturity. The problem is to use these observations (c i ) i I to predict correctly C K,T (S t, t) for all K, T, t > t..1 Dupire s Equation One difficulty is that the Black-Scholes equation has to be solved as many times as the number of different strikes in the observations, a number which can be as big as I. Dupire s argument brings this number down to one in the case of standard European option. By making use of the adjoint equation instead of Kolmogorov equation we prove here that the argument works also on barrier option. Consider (3) in Q = {S, t : S (S m e rt, S M e rt ), t (, T ) ; assume that < S m < K < S M. Consider the following change: c(x, t) = e r(t t) C(xe rt, t). 3

4 Then So c verifies: t c = e r(t t) ( t C + rs S C rc) at S = xe rt t c + σ x xx c = in (S m, S M ) (, T ), c(x, T ) = (x K) +, c(s m, t) = c(s M, t) =, x (S m, S M ), t (, T ). (9) In other words, this change of variable bring us to the case r = in the variable S, t. Proposition 1 Let v be solution in Q of Then t v 1 σ S SS v + rs S v =, v(s, t ) = (S S) + v(s m e rt, t) = (S S m e rt ) +, v(s M e rt, t) = t (1) C(S, t ) = v(k, T ) v(s M e rt, T ) + (S M e rt K) S v(s M e rt, T ) (11) Proof. We operate in the variable x, t or, equivalently, we assume r = in the Black-Scholes model. Consider the Call C and its adjoint p: t C + 1 σ S SS C, C(T ) = (S K) + (1) t p SS ( 1 σ S p) =, p(t ) = δ(s S ) (13) with natural boundary conditions or zero conditions at the barriers for both C and p. An integration by parts in time and Green s formula in space applied to (1) multiplied by p and integrated over Q = (S m, S M ) (, T ) yields C(S, t ) = + T t SM S m C(S, t )δ(s S ) = SM S m C T p(t )ds [p 1 σ S S C C S ( 1 σ S p)] S M Sm (14) Now all terms vanish in the last integral because C = p = at S M or, if S M = +, C S and p at infinity, C = p = at S m or, if S m =, pσ S, pc are zero at S =. Let v be a double primitive of p, i.e. SS v = p; then (13) becomes t v 1 σ S SS v + rs S v = as + b, v(s, t ) = c + ds + (S S ) + in Q (15) 4

5 Now C T = (S K) +, so SS C T = δ(s K); hence by a double integration by parts, (14) becomes C(S, t ) = C T SS vds = v SS C T ds + [C T S v v S C T ] S M Sm = v(k, T ) v(s M, T ) + (S M K) S v(s M, T ) (16) because C T (S m ) = S C T (S m ) =, C T (S M ) = S M K, S C T (S M ) = 1. By choosing a = b = and c = S, d = 1, this proves the result because (S S ) + (S S ) = (S S ) = (S S) + so v vanishes at infinity and the initial condition in (1) is the desired one. This calculation is subject to the condition that SS v = at S m, S M ; the reader will check that it is verified by using (1) with r =. Dupire s formula (1) allows the computation of any option at S, t for any K, T at the cost of a single integration of the PDE of v, provided that S m and S M are the same for all. Numerical Simulations We have used the Euler implicit scheme in time and centered finite differences in S for C and v. The equation for C is integrated once only but to compare it with (1) the later must be integrated at each different S. The parameters are K = 1, r =., σ =.4, time steps and 5 mesh points for S. The accuracy (figure.1) is quite good.. Least squares Observe that whatever the value of σ in the region S small, C is always extremely small there, so data observed in that region will be useless. Similarly for large S, C is exponentially close to S Ke r(t t), so C is very insensitive to the value of σ at large S. This leads to restrict the search for η = σ in K = {η : η(s, t) [η m, η M ] S (S 1, S ) η(s, t) = η S / (S 1, S ), t} As the problem is not well posed the easiest is to Find η K minimizing J(η)) = i I w i C(t i, K i ) c i + J r (η), where C solves (1) with σ = η and with J r (η) = [α( τ η) + β( K η) + γ( τ,kη) ] dτdk (17) Q This choice is dictated by the constraints on η for the continuity of the application η C; these are sufficient conditions; their necessity is an open problem. 5

6 "u.txt" using 1: "u.txt" using 1:3 "u.txt"using 1:4 6 5 "u.txt" using 1: "u.txt" using 1:3 "u.txt"using 1: "u.txt" using 1: "u.txt" using 1:3 "u.txt" using 1: Figure 1: European option with K = 1, r = one barrier at 9 (last) or one at 15 (middle) and two barriers at 9 and 13 (top); the 3 curves are C by solving (3), C by the Dupire formula (11) and the unconstrained C (no barrier) computed by the Black-Scholes exact formula. The third plot is at T = 1, while the two others are at T =.1 because the curves are too flat at T = 1. 6

7 .3 Parametric Volatility Surface To reduce the number of unknown one may try η = σ + i<n a i e (K Ki) sin(t T i ) (18) and use automatic differentiation to compute ai J. Then the problem is solved with a Conjugate Gradient algorithm and by using operator overloading (e.g. C++) the programming effort is reduced to nothing. Figure shows one such "sig.dat" "sigd.dat" "sigd.dat" "sig.dat" Figure : Volatility surface in two cases for which the exact solution is known. In one case the exact surface is in the space of parametrized approximations and the result is very good; in the second case it is not and the results are not so precise. There N = 5 4, in (18) i.e. 4 time levels and 5 splines at each time level. It is small but increasing these numbers just make the answer more oscillatory. simulation..4 Quasi-Newton Method of Levenberg - Marquardt Let C be the solution of Dupire s equation with a local volatility σ(a) and a R na the spline parameters. Following [5][6] we observe that Therefore min σ J(a) = 1 J a i = j is an approximate Newton step. n E j=1 E j a i E j E j E j = C(K j, t j ) C j J = E E + E T E E T E + αi (19) (E T E + αi)(a m+1 a m ) = E E 7

8 "s.txt" "s.txt" Figure 3: Volatility surface computed with the quasi-newton method (left) and with the CG method (right). "od.txt" "ud.txt" "od.txt" "ud.txt" Figure 4: Predicted (red) and exact (green) option prices versus price of underlying asset and time computed with the quasi-newton method (left) and with the CG method (right). 8

9 The numerical results are good when the cost function can be driven to zero exactly and not so otherwise. So this method should be used only when the control space is large enough to reach zero cost. Here figures 3 and 4 show that the solution obtained depend on the method for optimization, another sign of instability of the problem, even though the cost is reduced by 6 orders of magnitude..5 Physicist s Approach Many inverse problems are solved for geology (petrol) and meteorology (data assimilation). The 4D-var is universally adopted for the formulation of the least square problem. Following Tarantola[8] we need an a priori solution σ d and correlation matrices N, M and then solve min σ {(C C d ) T N(C C d ) + (σ σ d ) T M(σ σ d )} () We did not observe any significant improvement in our numerical test over the "u.txt" "u.txt" Figure 5: On the same inverse problem for the volatility surface, the call option price surfaces versus strike and maturity computed after optimization are displayed. On the left by the quasi-newton method of Levenberg and Marquardt and on the right with a conjugate gradient algorithm on the physicist s formulation. No perceptive difference can be seen. previous methods (see figure 5). We have used the implied volatility for σ d and M = N = I, which, according to physicists [9] is not ideal, but a better choice would require a field study with a bank, something which we project to do in the future. Perspectives The real numerical challenge comes with more complex problems like basket options or with more complex models like the Orstein-Uhlenbeck s stochastic volatility model (see [3] or [7] and [4] ): t C + 1 y S SS C + ρβs y Sy C + 1 β yy C 9

10 +rs S C + (α(m y) βλy ) y C rc = (1) for all S >, all y R and t (, T ) and boundary conditions like C(S, y, T ) = (S K) +, lim yc =. y However even for the one dimensional case we do not have a robust strategy, so these calibration problems need to be studied further. References [1] Paul Wilmott, Sam Howison, and Jeff Dewynne. The mathematics of financial derivatives. Cambridge University Press, Cambridge, A student introduction. [] B. Dupire. Pricing with a smile. Risk, pages 18, [3] E. Stein and J. Stein. Stock price distributions with stochastic volatility : an analytic approach. The review of financial studies, 4(4):77 75, [4] Yves Achdou and O. Pironneau Numerical Methods for Option Pricing SIAM series, Philadelphia, USA, 5. [5] K. Levenberg, A Method for the Solution of Certain Problems in Least Squares. Quart. Appl. Math., , [6] D. Marquardt, An Algorithm for Least-Squares Estimation of Nonlinear Parameters. SIAM J. Appl. Math. 11, , [7] R. Cont and P. Tankov. Financial modelling with jump processes. Chapman and Hall, 3. [8] Albert Tarantola Inverse Problem: Theory and Methods for Model Parameter Estimation. SIAM series, Philadelphia, USA, 5. [9] Albert Tarantola Private Conversation April 6. 1