Monte Carlo Methods and Black Scholes model

Transcription

1 Monte Carlo Methods and Black Scholes model Christophe Chorro MASTER MMMEF 22 Janvier 2008 hristophe Chorro (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

2 Bibliography The slides of this lecture (and other documents) are available on the following website http : //christophe.chorro.fr/enseignements.html J. JACOD, P. PROTTER : Probability essentials, Springer, Course of ANNIE MILLET in Paris 1: ftp : //samos.univ paris1.fr/pub/samos/cours/millet/master1/polym1_english.pdf D. LAMBERTON, B. LAPEYRE : Introduction to stochastic calculus applied to finance MARK BROADIE AND PAUL GLASSERMAN, Estimating Security Price Derivatives Using Simulation, Management Science, 1996, Vol. 42, No. 2, Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

3 Study Plan of the Lecture Introduction to Monte Carlo Methods Presentation of the problem An historical example Strong law of large numbers Control of the error (the central limit theorem) Conclusion Black scholes model Simulations of Gaussian random variables Simulation of the Brownian motion Reminder on the Black Scholes model The greeks Finite difference method for Greeks Integration by parts method for Greeks hristophe Chorro (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

4 Plan 1 Introduction to Monte Carlo Methods Presentation of the problem An historical example Strong law of large numbers (SLLN) Control of the error: the CLT Conclusion 2 Black Scholes model Simulations of Gaussian random variables Simulation of Brownian motion Reminder on the Black Scholes model The greeks Finite difference method for Greeks Integration by parts method for Greeks Numerical results Concluding remarks Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

5 Presentation of the problem The basic problem is to estimate a multi-dimensional integral R d f (x)dµ(x) for which an analytic answer is not known. More precisely we look for a stochastic algorithm that gives: A numerical estimate of this integral, An estimate of the error, A good accuracy with an interesting computational cost. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

6 Presentation of the problem This kind of problem appears in a wide field of areas: Von Neuman, Ulam: Neutron diffusion in fissionable material (Manhattan project 1947) Biology Mathematical finance (pricing and hedging of contingents claims) hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

8 The Buffon needle We consider a floor with equally spaced lines, a distance δ apart and a needle of length 0 < l < δ dropped on it Question: What is the probability that a needle dropped randomly intersects one of the lines? Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

9 The Buffon needle If θ = θ 0 is fixed, P(Intersection) = l sinθ 0. δ If the needle is dropped randomly, θ is uniformly distributed on [0, π[ and P(Intersection) = π 0 l sinθ 0 δ dθ 0 π = 2l πδ. Now, if we drop N needles and denote by X the number of them crossing a line, one has X N 2l πδ. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

10 The Buffon needle Laplace (1812) suggested to use this experiment to approximate π: π 2lN Xδ. Lazzarini (1901) made the experiment with l = 2.5cm, δ = 3cm and N = He obtained X = 1808 thus π Problem: Time consuming and bad accuracy... Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

11 The Buffon needle See for computer simulation of this experiment. Christophe Chorro (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

12 The Buffon needle in modern language We consider a random variable Z such that We have Z = 1 if the randomly dropped needle crosses a line Z = 0 otherwise. E[Z ] = P(Intersection) = 2l πδ and if we denote by Z 1,...Z N a N-sample of Z we previously use the following approximation Z Z N N E[Z ]. Aim: Prove the validity of this approximation. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

14 SLLN We consider a probability space (Ω, A, P) and we denote by E the expectation under P. Theorem Let (X n ) n N be i.i.d random variables with values in R such that E[ X 1 ] <. Then, denoting S n = X X n, one has S n n E[X 1 ]. a.s and L 1 hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

15 SLLN Numerical illustration (Buffon needle) Illustration of the SLLN when X 1 B( 1 2 ) and n = 500 Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

16 SLLN Remark 1: The hypothesis E[ X 1 ] < is necessary !5!10!15!20! SLLN is not fulfilled when X 1 C(1) (here n = 10000) Remark 2: Possible extension for random variables with values in R d hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

17 SLLN: Sketch of the proof We suppose that E[ X 1 4 ] < and without loss of generality we assume that E[X 1 ] = 0. Using that the random variables are independent and centered we obtain [ ( ) ] ( ) 4 S E n n n E[Xi 2 ]E[Xj 2 ] = 1 n 4 k=1 E[X 4 k ] + 3 i j = 1 n 4 ( ne[x 4 1 ] + 3n(n 1)E[X 2 1 ]2) CS 3E[X 4 1 ] n 2. Thus [ (Sn ) ] 4 [ ] S n E 0 E n n 0. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

18 SLLN: Sketch of the proof Moreover from the preceding inequality and the monotone convergence theorem: [ ( ) ] [ 4 Sn (Sn ) ] 4 E = E < n n thus This implies that n=1 n=1 n=1 ( ) 4 Sn < a.s. n S n n a.s 0. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

20 Control of the error An asymptotic bound via the Central Limit Theorem (CLT) Definition We say that a sequence of random variables (X n ) n>0 converges toward X in distribution ( X n D X) if f C b (R, R), E[f (X n ] E[f (X)]. n This convergence extends when f is an indicator function. Theorem Let (X n ) n N be i.i.d random variables with values in R such that E[ X 1 2 ] <. Then where σ 2 = Var(X 1 ). S n ne[x 1 ] nσ D N (0, 1) hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

21 Control of the error Numerical illustration: $%'& $%'$ $%!& $%!$ $%"& $%"$ $%#& $%#$ $%$& $%$$!!!"!# $ # "! Illustration of the CLT when X 1 U([0, 1]) and n = 500 We have an obvious extension of this result in any finite dimension Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

22 Control of the error From the preceding theorem we obtain that ( aσ P S n n n E[X 1] aσ ) n a n a In practice we know from tables (cf next slide that) 1 2π e x2 2 dx. P( N (0, 1) 1.96) = 0.95 thus when n is large enough, with a confidence of 95%, E[X 1 ] [ Sn n 1.96σ n, S n n σ n ]. The magnitude of the error is given by 1.96σ n : the size of σ is fundamental for the speed of convergence. hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

23 Christophe Chorro (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

24 Control of the error When σ is unknown it may be estimated easily: Theorem Let (X n ) n N be i.i.d random variables with values in R such that E[ X 1 2 ] <. Then if we define one has a) ˆσ n 2 = ( n 1 n 1 n n i=1 X 2 i E[ ˆσ n 2 ] = σ 2 ( 1 n ) n X i ) 2 i=1 b) ˆσ n 2 a.s σ 2. hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

25 Control of the error We have the following result that allows the building of confidence intervals even if σ is unknown. Theorem Let (X n ) n N be i.i.d random variables with values in R such that E[ X 1 2 ] <, then S n ne[x 1 ] n ˆσn D N (0, 1) Proof: It is just a consequence of the classical CLT and of the Slutsky lemma: Lemma If X n X and Y n a (a being a constant) then D D X n Y n Xa. D hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

27 Reminder of the method To basically compute E[f (X)] by Monte Carlo Methods we have to Generate a n-sample of the distribution of X Compute 1 n n f (X k ) for large n k=1 Precise the confidence interval coming from the CLT Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

28 Advantages Easy to implement on any software No regularity on f The control of the error ( σ n ) is independent of the dimension of the problem Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

29 Limits The error is a random variable (we only have confidence intervals) This method may be slow if we don t use extra-techniques Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

30 Limits Consider that we want to approximate by Monte Carlo simulations I = E[e βn (0,1) ] (even if the exact value is given by e β2 2 ). In this way we generate (see part) a n-sample (G 1,..., G n ) of a N (0, 1) and use I n = eβg e βg n E[e βn (0,1) ]. n By the CLT the order of magnitude of the relative error is given by I n I I σ (n)i = e β 2 1 n. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

31 Limits When β = 1, we need n 1000 to obtain a relative error 10% When β = 5, we need n to obtain a relative error 100% (THIS IS NUMERICALLY IMPOSSIBLE) Moreover, in this case Exact value= Approximated value (for n=100000)= Confidence interval (level 95%)= ]20188, [ hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

33 Simulations of Gaussian random variables Starting from i.i.d uniform random variables on [0, 1] (U n ) n N ( See: we want to generate random samples of a Gaussian distribution. hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

34 Gaussian random variables from Inversion Method Let X be a real random variable, the distribution function of X is defined as Properties: non-decreasing, right C 0, F X (x) = P(X x); x R. lim F (x) = 1 and lim F (x) = 0. x + x Definition We define the generalized inverse of F X denoted by F X where u ]0, 1[, F X (u) = inf {x F X (x) u}. Remark: When F X is strictly increasing and continuous, F X = F 1 X. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

35 Gaussian random variables from Inversion Method Proposition If U U([0, 1]) then F X (U) has the same distribution then X. Proof: We just have to remark that u ]0, 1[, x R, F X (u) x u F X (x). Figure: Illustration of the definition of F X hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

36 Gaussian random variables from Inversion Method Example 1: Particular continuous distributions If X E(λ), λ > 0 then F X (x) = (1 e λx )1 x 0 and 1 Log(1 U) E(λ). λ If X C(a), a > 0, then F X (x) = 1 π [Arctan( x a ) + π 2 ] and a tan (π(u 12 ) ) C(a). Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

37 Gaussian random variables from Inversion Method Example 2: When X N (0, 1), F X is unknown but we have the following approximation: If u > 0; 5, let t = 2log(1 u) F X (u) t If u 0; 5, let t = 2log(u) c 0 + t(c 1 + tc2) 1 + t(d 1 + t(d 2 + td 3 )). Where F X (u) c 0 + t(c 1 + tc2) 1 + t(d 1 + t(d 2 + td 3 )) t. c 0 = , c 1 = , c 3 = , d 1 = , d 2 = , d 3 = Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

38 Gaussian random variables from Inversion Method Figure: Empirical density of standard normal distribution obtained by Inversion method hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

39 Gaussian random variables from Transformation method Here we try to express X as a function of another random variable Y easy to generate. One of the main tool is the following result Proposition Let D and be two open sets of R d and Φ = (Φ 1,...Φ d ) : D a C 1 -diffeomorphism. If g : R is measurable and bounded then g(v)dv = g(φ(u)) J Φ (u)) du where J Φ (u) = det [ ( ) Φ i u j (u) 1 i,j d D ]. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

40 Gaussian random variables from Transformation method Example 1: Box-Muller method Proposition If U 1 and U 2 are two independent uniform random variables on [0, 1] then G 1 = 2log(U 1 )cos(2πu 2 ) and G 2 = 2log(U 1 )sin(2πu 2 ) are two independent N (0, 1). Proof: Let us define the following C 1 -diffeomorphism Ψ : (x, y) ]0, 1[ 2 (u = 2log(x)cos(2πy), v = 2log(x)sin(2πy)) fulfilling J Ψ (x, y) = 2π x. Since u2 + v 2 = 2log(x), according to the change of variables theorem (Φ = Ψ 1 ), one has for F C b (R 2, R), F (Ψ(x, y))dxdy = F (u, v) 1 ]0,1[ 2 R 2 (R + {0}) 2π e u 2 +v 2 2 dudv. hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

41 Gaussian random variables fromtransformation method Figure: Simulation of 5000 pairs of independent N (0, 1) by Box-Muller method hristophe Chorro ([email protected]) (MASTER MMMEF) Monte Carlo Methods and Black Scholes model (some reminder) 22 Janvier / 87

43 Simulation of Brownian motion We consider a probability space (Ω, A, P). Definition Standard Brownian motion (B.M) is a stochastic process (B t ) t [0,T ] fulfilling : a) B 0 = 0 P-a.s. b) B is continuous i.e t B t (w) is continuous for P almost all w. c) B has independent increments: For Si t > s, B t B s is independent of Fs B = σ(b u, u s). d) the increments of B are stationary and gaussian: For t s, B t B s follows a N (0, t s). hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

44 Simulation of Brownian motion We consider a subdivision 0 = t 0 <... < t n = T of [0, T ]. We want to simulate Idea: (B t0,...b tn ). B tk = B tk 1 + B tk B tk 1. } {{ } N (0,t k t k 1 ) B tk 1,...,B 0 Proposition If (G 1,...G n ) are i.i.d N (0, 1), we define X 0 = 0, X i = i tj t j 1 G j i > 0. j=1 Then (X 0,..., X n ) = D (B t0,...b tn ). Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

45 Brownian motion Figure: 4 paths of the Brownian motion on [0, 1] generated using the preceding method with the regular subdivision of step hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

46 Brownian motion If we want to add points to the preceding simulation, the following result is useful. Proposition For t > s, Idea of the proof ( D B t+s 2 ) ( Bt + B s (B t, B s ) = N, t s ). 2 4 where B t+s 2 = B t + B s 2 + Z Z is independent of σ(b u u t) and σ(b u u s). Z N ( ) 0, t s 4. hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

48 Black Scholes model We consider the time interval [0, T ] and r the risk free rate (supposed to be constant) during this period. Non-risky asset: Its dynamic is given by S 0 0 = 0, S0 t = e rt. Risky asset: Under the historical probability P its dynamic is given by the following SDE: ds t = µs t dt + σs t db t (1) with initial condition S 0 = x 0 > 0 and where B is a standard BM under P. Itô formula S t = x 0 e (µ 1 2 σ2 )t+σb t. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

49 Black Scholes model sigma=1, mu=1 sigma=1, mu= sigma=0.5, mu=1 sigma=0.5, mu= Figure: Simulation of a path of the risky asset in the B&S model for different parameters hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

50 Black Scholes model What is, in this model, the price at t of a contingent claim with payoff Φ T at T? Proposition In the B&S model there exists a unique probablity Q P such that the price at t of a contingent claim with payoff Φ T at T is given by P t = E [e r(t t) Φ T F t ]. Moreover the dynamic of the risky asset under Q is given by where W is a standard BM under Q. ds t = rs t dt + σs t dw t (µ r) (2) hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

51 Black Scholes model Examples: The Black Scholes Formulas For Call options (Φ T = (S T K ) + ) one has where E [(S T K) + F t ] = S t N(d 1 (t, S t )) Ke r(t t) N(d 2 (t, S t )) d 1 (t, x) = log( x K ) + (r + σ 2 2 )(T t) σ T t and where N is the distribution function of a N (0, 1). For Put options (Φ T = (K S T ) + ) one has et d 2 (t, x) = log( x K ) + (r σ 2 2 )(T t) σ T t E [(K S T ) + F t ] = S t N( d 1 (t, S t )) + Ke r(t t) N( d 2 (t, S t )). Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

52 Black Scholes model This formulas are fundamental because They are easy to compute in practice They are a Benchmark to test numerical methods In the case where we don t have closed form formulas we may use Monte Carlo Methods price = e rt E [Φ T ] e rt 1 N where the Φ i T are independent realizations of Φ T. Morever we have a control of the error (CLT): with a probability of 95% price [ e rt 1 N N i=1 Φ i T 1.96e 2rT Σ, e rt 1 N N Σ being the (empirical) variance of Φ T. N i=1 N i=1 Φ i T ] Φ i T e 2rT Σ N Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

53 Black Scholes model Examples: We take x = 100, K = 100, σ = 0, 15, r = 0, 05, T = 1, t = 0. Price of a call on mean: e r E[ ( S 1 +S 1 2 ) 2 K Estimated value (N=10000) = 6.05, confidence interval at 95% :[5.89; 6.20] Estimated value (N=100000) = 6.04, confidence interval at 95% : [5.99; 6.09] Price of a call on max: e r E[(Max(S 1, S 1 ) K ) + ] with σ = 0.5 and 2 K = 1. Estimated value (N=10000) = 8.68, confidence interval at 95% : [8.49; 8.87] Estimated value (N=10000) = 8.88, confidence interval at 95% : [8.82; 8.95] ] + For prices, the error only comes from Monte-Carlo approximations Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

55 The greeks We suppose that the payoff of a contingent claim is given by Φ T = f (S T ), thus, where P t = E [e r(t t) Φ T F t ] = F (t, S t ) (Markov process) + F (t, x) = e r(t t) f (xe (r 1 2 σ2 )(T t)+σy T t 1 ) e y2 2 dy. 2π Thus, F (t, x) = E [e r(t t) f (ST x t )]. Proposition Under mild hypotheses on f, F C 1,2 ([0, T [ R, R). Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

56 The greeks The greeks measure the sensitivity of the price with respect to a given parameter. measures the sensitivity of the price with respect to the underlying t (S t ) = F x (t, S t) It is also the quantity of risky asset in the hedging portfolio!!!! Γ measures the sensitivity of the delta with respect to the underlying Γ t (S t ) = 2 F x 2 (t, S t) It is also a measure of the frequence a position must be re-hedged in order to maintain a delta neutral position Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

57 The greeks Θ measures the sensitivity of the price with respect to the underlying Θ t (S t ) = F t (t, S t) ρ measures the sensitivity of the price with respect to interest rate ρ t (S t ) = F r (t, S t) vega (which is not a greel letter!!!) measures the sensitivity of the price with respect to the volatility vega t (S t ) = F σ (t, S t) precautions to take for the estimation of σ! Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

58 The greeks For call and put options ( strike K maturity T ) the greeks at t = 0 are given by: Call Put 0 (x) N(d 1 ) > 0 N( d 1 ) < 0 Γ 0 (x) 1 xσ T N (d 1 ) > 0 1 xσ T N (d 1 ) > 0 Θ 0 (x) xσ 2 T N (d 1 ) Kre rt N(d 2 ) < 0 xσ 2 T N (d 1 ) + Kre rt (N(d 2 ) 1)? ρ 0 (x) TKe rt N(d 2 ) > 0 TKe rt (N(d 2 ) 1) < 0 vega 0 (x) x T N (d 1 ) > 0 x T N (d 1 ) > 0 where d 1 (x) = log( x K ) + (r + σ 2 2 )T σ T and where N is the distribution function of a N (0, 1). et d 2 (x) = log( x K ) + (r σ 2 2 )T σ T (3) Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

59 The greeks In the preceding table, to obtain the values of the greeks at time t we just have to change T into T t They are easy to compute in practice They are a Benchmark to test numerical methods We will restrict ourselves to and Γ. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

61 Finite difference method for Greeks A classical method is to use a finite difference scheme t (x) F (t, x + h) F (t, x h) 2h F (t, x + h) + F (t, x h) 2F (t, x) Γ t (x) h 2 where h is sufficiently small. F (t, x + h), F (t, x h) et F (t, x) are computed by Monte Carlo methods. Reminder: F(t, x) = E [e r(t t) f(s x T t )]. hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

62 Finite difference method for Greeks Contrary to prices, for the greeks there are two factors of approximation two factors of approximation Error } from finite {{ difference } + Error } from {{ Monte Carlo } E 1 E 2 The choice of h may be difficult (cf: Broadie-Glasserman): When h is too big, E 1 may strongly increase. When h is too small, the variance of the Monte Carlo estimator may explode. When we use t (x) F (t,x+h) F(t,x h) 2h since Var(F (t, x + h) F (t, x h)) = Var(F(t, x + h)) + Var(F(t, x + h)) 2Cov(F(t, x + h), F (t, x h)) it is (often) better to use the same random sample for the two Monte Carlo simulations! hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

63 Finite difference method for Greeks A priori (it will be confirmed by numerical results) this methods should be more efficient when the prices are regular in x: When f (y) = 1 y M (Binary options) ( M E [ f(s x+h T ) f(s x T) 2 ] = P x + h < e(r 1 2 σ2 )T+σB T < M ) = O(h). x When f (y) = (y K ) + (Call) E [ f(s x+h T ) f(s x T) 2 ] E [(S x+h T S x T) 2 ] = h 2 E [e 2(r 1 2 σ2 )T+2σB T ] = O(h 2 ). Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

65 Integration by parts method for Greeks The idea is simple: Write the greeks on the following form Greeks = E [PAYOFF weight] with A weight independent of the Payoff. A weight such that the variance of PAYOFF weight is minimal. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

66 Integration by parts method for Greeks Here we suppose that Φ T = f (S T ). The following result is independent of f : Proposition One has [ t (x) = e r(t t) E B T t xσ(t t) f (Sx T t) and [( ) ] Γ t (x) = e r(t t) E B T t x 2 σ(t t) + B2 T t (T t) (σ(t t)x) 2 f (ST x t). ] Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

67 Integration by parts method for Greeks Proof in the case of delta: Let f CK 1 (R, R) (use approximation otherwise). According to the Lebesgue theorem of differentiation under the integral sign, But + t (x) = e r(t t) x f (xe(r 1 2 σ 2 )(T t)+σy T t 1 ) e y2 2 dy. } {{ } 2π g(x,y) g x (x, y) = 1 xσ g (x, y). T t y Thus, using Integration by parts, and t) e r(t t (x) = xσ T t + f (xe (r 1 2 σ2 )(T t)+σy T t y ) e y2 2 dy 2π [ ] t (x) = e r(t t) E B T t xσ(t t) f (Sx T t). Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

68 Integration by parts method for Greeks Advantages: One factor of approximation (Monte Carlo) This method doesn t depend on the Payoff (the weight is independent of f ). Question: Is there a criteria (in terms of variance) to choose among all the possible weights? Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

69 Integration by parts method for Greeks Proposition Let Π being a square integrable weight such that for all Payoffs of the form f (S T ) Greek = E [f (S T ) Π]. Thus, the weight minimzing the variance of f (S T ) Π is given by Π 0 = E [Π F T ]. Rk: The weights in the preceding proposion are optimal. hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

70 Integration by parts method for Greeks Proof: Let Π fulfilling greek = E [f (S T ) Π], we want to minimize Var(f (S T ) Π). One has [ ( ) ] 2 Var(f (S T ) Π) = E f (S T ) Π greek [ ( ) ] 2 = E f (S T )( Π Π 0 ) + f (S T )Π 0 greek [ ( ) ] 2 = E f (S T )( Π Π 0 ) + Var(f (S T )Π 0 ) )] + 2E [(f (S T )( Π Π 0 )(Π 0 f (S T ) greek). The last line being equal to zero, the result follows. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

72 Numerical results Call-Put We take x = 100, K = 100, σ = 0, 15, r = 0, 05, T = 1, t = 0. Delta Call 0,67 0,665 0,66 delta 0,655 Mall DF Valeur Theo:0, ,65 0,645 0, Nbre de Simulations (10E4) Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

73 Numerical results Call-Put Gamma Call 0,0255 0,025 0,0245 Gamma 0,024 Mall DF Valeur theo 0, ,0235 0,023 0, Nbre simulations(10e4) hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

74 Numerical results Call-Put Delta Put -0,334-0,337-0,34 delta Mall DF Valeur Theo -0, ,343-0,346-0, Nbre simulations (10E4) hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

75 Numerical results Call-Put Gamma Put 0,026 0,0255 0,025 gamma 0,0245 0,024 Mall DF Valeur Theo 0, ,0235 0,023 0, Nbre simulations (10E4) hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

76 Numerical results Call-Put Call Ratio of Variances (FD/ IBP) Γ Put Ratio of Variances (FD/ IBP) 0.30 Γ 1.32 The payoff being regular, the finite difference performs quite well. Integration by parts gives better results for Put than for Call (Explosion of weight!!!) hristophe Chorro (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

77 Numerical results Digital A Digital option is characterized by a payoff 1 S x T >K min irregular in K min. We can show easily that F (0, x) = e rt KN(d), and Γ 0 (x) = 0 (x) = e rt xσ T n(d) ( e rt x 2 σ 2 T n(d) d + σ ) T where n is the density of a N (0, 1) and where d = log( x K )+(r σ2 min 2 )(T ) σ T. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

78 Numerical results Digital We take x = 100, K = 100, σ = 0, 15, r = 0, 05, T = 1, K min = 95. Delta Digitale 0,0221 0,0218 0,0215 Delta 0,0212 0,0209 Mall DF Valeur Theo 0, ,0206 0,0203 0, Nbre simulations (10E4) Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

79 Numerical results Digital Gamma Digitale -0,0005-0,0006-0,0007-0,0008-0,0009 Gamma -0,001-0,0011 Mall DF Valeur Theo 0, ,0012-0,0013-0,0014-0, Nbre Simulations (10E4) hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

80 Numerical results Digital Digitale Ratio of Variances (FD/ IBP) 5.31 Γ 2354 hristophe Chorro (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

81 Numerical results Corridor A corridor option is characterized by a payoff 1 Kmax >ST x >K (difference of two min digitals). We take x = 100, K = 100, σ = 0, 15, r = 0, 05, T = 1, K min = 95, K max = 105. Delta Corridor -0,0032-0,0037 Delta -0,0042 Mall DF Valeur Théo: -0, ,0047-0, Nbre de simulation (10E4) hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

82 Numerical results Corridor Gamma Corridor -0,0005 Gamma -0,001 Mall DF Valeur Theo: -0, ,0015-0, Nbre de simulations (10E4) hristophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

83 Numerical results Corridor Corridor Ratio of Variances (FD/ IBP) 134 Γ 5785 Christophe Chorro (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

84 Numerical results The more irregular is the payoff, the more efficient is the integration by parts method. The performance of the integration by parts method increases with the order of derivation. In practice, the weights we found here are polynomials of W T. Thus the integration by parts method will be more efficient for small maturities (small weights). The integration by parts method performs better for a put than for a call. The integration by part method is in fact a variance reduction technique.. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

86 Conclusions For a payoff of the form f (S t0,..., S tn ) one has: Proposition 0 (x) = e r(t t) E [f (S t0,..., S tn ) with λ 1 = 1 xσt 1 et 1 i < n 1 ] n λ i (B ti B ti 1 ) 1 λ i+1 = 1 x i (t j t j 1 )λ j σ j=1 (t i+1 t i )σ. So we may use (with optimality) the integration by parts method in the Black Scholes model for discrete Lookback or asian options. Christophe Chorro ([email protected]) (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87

87 Conclusions What happens for other payoffs in the Black Scholes model??? What happens for other models??? Christophe Chorro (MASTER Monte Carlo MMMEF) Methods and Black Scholes model (some reminder) 22 Janvier / 87