Merton s Jump-Diffusion Model

Transcription

1 Merton s Jump-Diffusion Model Empirically, stock returns tend to have fat tails, inconsistent with the Black-Scholes model s assumptions. Stochastic volatility and jump processes have been proposed to address this problem. Merton s jump-diffusion model is our focus. a a Merton (1976). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 698

2 Merton s Jump-Diffusion Model (continued) This model superimposes a jump component on a diffusion component. The diffusion component is the familiar geometric Brownian motion. The jump component is composed of lognormal jumps driven by a Poisson process. It models the sudden changes in the stock price because of the arrival of important new information. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 699

3 Merton s Jump-Diffusion Model (continued) Let S t be the stock price at time t. The risk-neutral jump-diffusion process for the stock price follows ds t S t =(r λ k) dt + σdw t + kdq t. (81) Above, σ denotes the volatility of the diffusion component. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 700

4 Merton s Jump-Diffusion Model (continued) The jump event is governed by a compound Poisson process q t with intensity λ, where k denotes the magnitude of the random jump. The distribution of k obeys ln(1 + k) N ( γ,δ 2) with mean k E (k) =e γ+δ2 /2 1. The model with λ = 0 reduces to the Black-Scholes model. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 701

5 Merton s Jump-Diffusion Model (continued) The solution to Eq. (81) on p. 700 is where S t = S 0 e (r λ k σ 2 /2) t+σw t U(n(t)), (82) U(n(t)) = n(t) i=0 (1 + k i ). k i is the magnitude of the ith jump with ln(1 + k i ) N(γ,δ 2 ). k 0 =0. n(t) is a Poisson process with intensity λ. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 702

6 Merton s Jump-Diffusion Model (concluded) Recall that n(t) denotes the number of jumps that occur up to time t. As k> 1, stock prices will stay positive. The geometric Brownian motion, the lognormal jumps, and the Poisson process are assumed to be independent. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 703

7 Tree for Merton s Jump-Diffusion Model a Define the S-logarithmic return of the stock price S as ln(s /S). Define the logarithmic distance between stock prices S and S as ln(s ) ln(s) = ln(s /S). a Dai (R , D ), Wang (F ), Lyuu, and Liu (2010). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 704

8 Tree for Merton s Jump-Diffusion Model (continued) Take the logarithm of Eq. (82) on p. 702: ( ) St M t ln = X t + Y t, (83) S 0 where X t ( r λ k σ 2 /2 ) t + σw t, (84) Y t n(t) i=0 ln (1 + k i ). (85) It decomposes the S 0 -logarithmic return of S t into the diffusion component X t and the jump component Y t. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 705

9 Tree for Merton s Jump-Diffusion Model (continued) Motivated by decomposition (83) on p. 705, the tree construction divides each period into a diffusion phase followed by a jump phase. In the diffusion phase, X t BOPM. is approximated by the Hence X t canmakeanupmoveto X t + σ Δt with probability p u or a down move to X t σ Δt with probability p d. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 706

10 Tree for Merton s Jump-Diffusion Model (continued) According to BOPM, p u = eμδt d u d, p d = 1 p u, except that μ = r λ k here. The diffusion component gives rise to diffusion nodes. They are spaced at 2σ Δt apart such as the white nodes A, B, C, D, E, F, and G on p c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 707

11 ( l 1) Δt l Δt ( l + 1) Δt p u p d q 1 q 0 q h = γ + δ 2σ Δt White nodes are diffusion nodes. Gray nodes are jump nodes. In the diffusion phase, the solid black lines denote the binomial structure of BOPM, whereas the dashed lines denote the trinomial structure. Here m is set to one here for simplicity. Only the doublecircled nodes will remain after the construction. Note that a and b are diffusion nodes because no jump occurs in the jump phase. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 708

12 Tree for Merton s Jump-Diffusion Model (concluded) In the jump phase, Y t+δt is approximated by moves from each diffusion node to 2m jump nodes that match the first 2m moments of the lognormal jump. The m jump nodes above the diffusion node are spaced at h apart. Thesameholdsforthe m jump nodes below the diffusion node. The gray nodes at time lδt on p. 708 are jump nodes. After some work, the size of the tree is O(n 2.5 ). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 709

13 Multivariate Contingent Claims They depend on two or more underlying assets. The basket call on m assets has the terminal payoff ( m ) max α i S i (τ) X, 0, i=1 where α i is the percentage of asset i. Basket options are essentially options on a portfolio of stocks or index options. Option on the best of two risky assets and cash has a terminal payoff of max(s 1 (τ),s 2 (τ),x). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 710

14 Multivariate Contingent Claims (concluded) a Name Payoff Exchange option max(s 1 (τ) S 2 (τ), 0) Better-off option max(s 1 (τ),...,s k (τ), 0) Worst-off option min(s 1 (τ),...,s k (τ), 0) Binary maximum option I{ max(s 1 (τ),...,s k (τ)) >X} Maximum option max(max(s 1 (τ),...,s k (τ)) X, 0) Minimum option max(min(s 1 (τ),...,s k (τ)) X, 0) Spread option max(s 1 (τ) S 2 (τ) X, 0) Basket average option max((s 1 (τ),...,s k (τ))/k X, 0) Multi-strike option max(s 1 (τ) X 1,...,S k (τ) X k, 0) Pyramid rainbow option max( S 1 (τ) X S k (τ) X k X, 0) Madonna option max( (S 1 (τ) X 1 ) 2 + +(S k (τ) X k ) 2 X, 0) a Lyuu and Teng (R ) (2011). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 711

15 Correlated Trinomial Model a Two risky assets S 1 and S 2 follow ds i S i = rdt+ σ i dw i in a risk-neutral economy, i =1, 2. Let M i e rδt, V i M 2 i (e σ2 i Δt 1). S i M i is the mean of S i at time Δt. S 2 i V i the variance of S i at time Δt. a Boyle, Evnine, and Gibbs (1989). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 712

16 Correlated Trinomial Model (continued) The value of S 1 S 2 at time Δt has a joint lognormal distribution with mean S 1 S 2 M 1 M 2 e ρσ 1σ 2 Δt,where ρ is the correlation between dw 1 and dw 2. Next match the 1st and 2nd moments of the approximating discrete distribution to those of the continuous counterpart. At time Δt from now, there are five distinct outcomes. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 713

17 Correlated Trinomial Model (continued) The five-point probability distribution of the asset prices is (as usual, we impose u i d i =1) Probability Asset 1 Asset 2 p 1 S 1 u 1 S 2 u 2 p 2 S 1 u 1 S 2 d 2 p 3 S 1 d 1 S 2 d 2 p 4 S 1 d 1 S 2 u 2 p 5 S 1 S 2 c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 714

18 Correlated Trinomial Model (continued) The probabilities must sum to one, and the means must be matched: 1 = p 1 + p 2 + p 3 + p 4 + p 5, S 1 M 1 = (p 1 + p 2 ) S 1 u 1 + p 5 S 1 +(p 3 + p 4 ) S 1 d 1, S 2 M 2 = (p 1 + p 4 ) S 2 u 2 + p 5 S 2 +(p 2 + p 3 ) S 2 d 2. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 715

19 Correlated Trinomial Model (concluded) Let R M 1 M 2 e ρσ 1σ 2 Δt. Match the variances and covariance: S 2 1 V 1 = (p 1 + p 2 )((S 1 u 1 ) 2 (S 1 M 1 ) 2 )+p 5 (S 2 1 (S 1M 1 ) 2 ) +(p 3 + p 4 )((S 1 d 1 ) 2 (S 1 M 1 ) 2 ), S 2 2 V 2 = (p 1 + p 4 )((S 2 u 2 ) 2 (S 2 M 2 ) 2 )+p 5 (S 2 2 (S 2M 2 ) 2 ) +(p 2 + p 3 )((S 2 d 2 ) 2 (S 2 M 2 ) 2 ), S 1 S 2 R = (p 1 u 1 u 2 + p 2 u 1 d 2 + p 3 d 1 d 2 + p 4 d 1 u 2 + p 5 ) S 1 S 2. The solutions are complex (see text). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 716

20 Correlated Trinomial Model Simplified a Let μ i r σ2 i /2andu i e λσ i Δt for i =1, 2. The following simpler scheme is good enough: p 1 = p 2 = p 3 = p 4 = p 5 = 1 1 λ 2. a Madan, Milne, and Shefrin (1989). [ ( 1 1 Δt μ 4 λ μ ) 2 + ρ ] λ σ 1 σ 2 λ 2, [ ( 1 1 Δt μ 4 λ μ ) 2 ρ ] λ σ 1 σ 2 λ 2, [ ( 1 1 Δt 4 λ 2 + μ 1 μ ) 2 + ρ ] λ σ 1 σ 2 λ 2, [ ( 1 1 Δt 4 λ 2 + μ 1 + μ ) 2 ρ ] λ σ 1 σ 2 λ 2, c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 717

21 Correlated Trinomial Model Simplified (continued) All of the probabilities lie between 0 and 1 if and only if 1+λ Δt μ 1 σ 1 + μ 2 σ 2 ρ 1 λ Δt μ 1 μ 2 σ 1 σ, (86) 2 1 λ (87) We call a multivariate tree (correlation-) optimal if it guarantees valid probabilities as long as 1+O( Δt) <ρ<1 O( Δt), such as the above one. a a Kao (R ) (2011) and Kao (R ), Lyuu, and Wen (D ) (2014). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 718

22 Correlated Trinomial Model Simplified (concluded) But this model cannot price 2-asset 2-barrier options accurately. a Few multivariate trees are both optimal and able to handle multiple barriers. b An alternative is to use orthogonalization. c a See Chang (B , R ), Hsu (R , D ), and Lyuu (2006) and Kao (R ), Lyuu and Wen (D ) (2014) for solutions. b See Kao (R ), Lyuu, and Wen (D ) (2014) for one. c Hull and White (1990) and Dai (R , D ), Lyuu, and Wang (F ) (2012). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 719

23 Extrapolation It is a method to speed up numerical convergence. Say f(n) converges to an unknown limit f at rate of 1/n: f(n) =f + c ( ) 1 n + o. (88) n Assume c is an unknown constant independent of n. Convergence is basically monotonic and smooth. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 720

24 Extrapolation (concluded) From two approximations f(n 1 ) and f(n 2 ) and ignoring the smaller terms, f(n 1 ) = f + c, n 1 f(n 2 ) = f + c. n 2 A better approximation to the desired f is f = n 1f(n 1 ) n 2 f(n 2 ) n 1 n 2. (89) This estimate should converge faster than 1/n. The Richardson extrapolation uses n 2 =2n 1. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 721

25 Improving BOPM with Extrapolation Consider standard European options. Denote the option value under BOPM using n time periods by f(n). It is known that BOPM convergences at the rate of 1/n, consistent with Eq. (88) on p But the plots on p. 282 (redrawn on next page) demonstrate that convergence to the true option value oscillates with n. Extrapolation is inapplicable at this stage. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 722

26 Call value n Call value n c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 723

27 Improving BOPM with Extrapolation (concluded) Take the at-the-money option in the left plot on p The sequence with odd n turns out to be monotonic and smooth (see the left plot on p. 725). a Apply extrapolation (89) on p. 721 with n 2 = n 1 +2, where n 1 is odd. Result is shown in the right plot on p The convergence rate is amazing. See Exercise of the text (p. 111) for ideas in the general case. a This can be proved; see Chang and Palmer (2007). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 724

28 Call value n Call value n c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 725

29 Numerical Methods c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 726

30 All science is dominated by the idea of approximation. Bertrand Russell c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 727

31 Finite-Difference Methods Place a grid of points on the space over which the desired function takes value. Then approximate the function value at each of these points (p. 729). Solve the equation numerically by introducing difference equations in place of derivatives. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 728

32 c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 729

33 Example: Poisson s Equation It is 2 θ/ x θ/ y 2 = ρ(x, y). Replace second derivatives with finite differences through central difference. Introduce evenly spaced grid points with distance of Δx along the x axis and Δy along the y axis. The finite difference form is ρ(x i,y j )= θ(x i+1,y j ) 2θ(x i,y j )+θ(x i 1,y j ) (Δx) 2 + θ(x i,y j+1 ) 2θ(x i,y j )+θ(x i,y j 1 ) (Δy) 2. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 730

34 Example: Poisson s Equation (concluded) In the above, Δx x i x i 1 and Δy y j y j 1 for i, j =1, 2,... When the grid points are evenly spaced in both axes so that Δx =Δy = h, the difference equation becomes h 2 ρ(x i,y j )=θ(x i+1,y j )+θ(x i 1,y j ) +θ(x i,y j+1 )+θ(x i,y j 1 ) 4θ(x i,y j ). Given boundary values, we can solve for the x i sandthe y j s within the square [ ±L, ±L ]. From now on, θ i,j will denote the finite-difference approximation to the exact θ(x i,y j ). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 731

35 Explicit Methods Consider the diffusion equation D( 2 θ/ x 2 ) ( θ/ t)=0,d>0. Use evenly spaced grid points (x i,t j ) with distances Δx and Δt, where Δx x i+1 x i and Δt t j+1 t j. Employ central difference for the second derivative and forward difference for the time derivative to obtain θ(x, t) t = θ(x, t j+1) θ(x, t j ) +, (90) t=tj Δt 2 θ(x, t) x 2 = θ(x i+1,t) 2θ(x i,t)+θ(x i 1,t) x=xi (Δx) 2 +. (91) c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 732

36 Explicit Methods (continued) Next, assemble Eqs. (90) and (91) into a single equation at (x i,t j ). But we need to decide how to evaluate x in the first equation and t in the second. Since central difference around x i is used in Eq. (91), we might as well use x i for x in Eq. (90). Two choices are possible for t in Eq. (91). The first choice uses t = t j finite-difference equation, to yield the following θ i,j+1 θ i,j Δt = D θ i+1,j 2θ i,j + θ i 1,j (Δx) 2. (92) c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 733

37 Explicit Methods (continued) The stencil of grid points involves four values, θ i,j+1, θ i,j, θ i+1,j,andθ i 1,j. Rearrange Eq. (92) on p. 733 as θ i,j+1 = DΔt (Δx) 2 θ i+1,j + ( 1 2DΔt (Δx) 2 ) θ i,j + DΔt (Δx) 2 θ i 1,j. We can calculate θ i,j+1 from θ i,j,θ i+1,j,θ i 1,j,atthe previous time t j (see exhibit (a) on next page). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 734

38 Stencils c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 735

39 Explicit Methods (concluded) Starting from the initial conditions at t 0,thatis, θ i,0 = θ(x i,t 0 ), i =1, 2,...,wecalculate θ i,1, i =1, 2,.... And then θ i,2, i =1, 2,.... And so on. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 736

40 Stability The explicit method is numerically unstable unless Δt (Δx) 2 /(2D). A numerical method is unstable if the solution is highly sensitive to changes in initial conditions. The stability condition may lead to high running times and memory requirements. For instance, halving Δx would imply quadrupling (Δt) 1, resulting in a running time 8 times as much. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 737

41 Recall that Explicit Method and Trinomial Tree θ i,j+1 = DΔt (Δx) 2 θ i+1,j + ( 1 2DΔt ) θ (Δx) 2 i,j + DΔt (Δx) θ i 1,j. 2 When the stability condition is satisfied, the three coefficients for θ i+1,j, θ i,j,and θ i 1,j all lie between zero and one and sum to one. They can be interpreted as probabilities. So the finite-difference equation becomes identical to backward induction on trinomial trees! The freedom in choosing Δx corresponds to similar freedom in the construction of trinomial trees. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 738

42 Implicit Methods Suppose we use t = t j+1 in Eq. (91) on p. 732 instead. The finite-difference equation becomes θ i,j+1 θ i,j Δt = D θ i+1,j+1 2θ i,j+1 + θ i 1,j+1 (Δx) 2. (93) The stencil involves θ i,j, θ i,j+1, θ i+1,j+1,andθ i 1,j+1. This method is implicit: The value of any one of the three quantities at t j+1 cannot be calculated unless the other two are known. See exhibit (b) on p c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 739

43 Implicit Methods (continued) Equation (93) can be rearranged as θ i 1,j+1 (2 + γ) θ i,j+1 + θ i+1,j+1 = γθ i,j, where γ (Δx) 2 /(DΔt). This equation is unconditionally stable. Suppose the boundary conditions are given at x = x 0 and x = x N+1. After θ i,j has been calculated for i =1, 2,...,N,the values of θ i,j+1 at time t j+1 can be computed as the solution to the following tridiagonal linear system, c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 740

44 Implicit Methods (continued) a a a a a θ 1,j+1 θ 2,j+1 θ 3,j θ N,j+1 = γθ 1,j θ 0,j+1 γθ 2,j γθ 3,j γθ N 1,j γθ N,j θ N+1,j+1, where a 2 γ. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 741

45 Implicit Methods (concluded) Tridiagonal systems can be solved in O(N) time and O(N) space. Never invert a matrix to solve a tridiagonal system. The matrix above is nonsingular when γ 0. A square matrix is nonsingular if its inverse exists. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 742

46 Crank-Nicolson Method Take the average of explicit method (92) on p. 733 and implicit method (93) on p. 739: θ i,j+1 θ i,j = 1 2 Δt ( D θ i+1,j 2θ i,j + θ i 1,j (Δx) 2 + D θ i+1,j+1 2θ i,j+1 + θ i 1,j+1 (Δx) 2 ). After rearrangement, γθ i,j+1 θ i+1,j+1 2θ i,j+1 + θ i 1,j+1 2 = γθ i,j + θ i+1,j 2θ i,j + θ i 1,j 2. This is an unconditionally stable implicit method with excellent rates of convergence. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 743

47 Stencil x i+1 x i x i+1 t j t j+1 c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 744

48 Numerically Solving the Black-Scholes PDE (65) on p. 583 See text. Brennan and Schwartz (1978) analyze the stability of the implicit method. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 745

49 Monte Carlo Simulation a Monte Carlo simulation is a sampling scheme. In many important applications within finance and without, Monte Carlo is one of the few feasible tools. When the time evolution of a stochastic process is not easy to describe analytically, Monte Carlo may very well be the only strategy that succeeds consistently. a A top 10 algorithm according to Dongarra and Sullivan (2000). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 746

50 The Big Idea Assume X 1,X 2,...,X n have a joint distribution. θ E[ g(x 1,X 2,...,X n ) ] for some function g is desired. We generate ( x (i) 1,x(i) 2,...,x(i) n ), 1 i N independently with the same joint distribution as (X 1,X 2,...,X n ). Set Y i g ( x (i) 1,x(i) 2,...,x(i) n ). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 747

51 The Big Idea (concluded) Y 1,Y 2,...,Y N are independent and identically distributed random variables. Each Y i has the same distribution as Y g(x 1,X 2,...,X n ). Since the average of these N random variables, Y, satisfies E[ Y ]=θ, it can be used to estimate θ. The strong law of large numbers says that this procedure converges almost surely. The number of replications (or independent trials), N, is called the sample size. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 748

52 Accuracy The Monte Carlo estimate and true value may differ owing to two reasons: 1. Sampling variation. 2. The discreteness of the sample paths. a The first can be controlled by the number of replications. The second can be controlled by the number of observations along the sample path. a This may not be an issue if the financial derivative only requires discrete sampling along the time dimension, such as the discrete barrier option. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 749

53 Accuracy and Number of Replications The statistical error of the sample mean Y of the random variable Y grows as 1/ N. Because Var[ Y ]=Var[Y ]/N. In fact, this convergence rate is asymptotically optimal. a So the variance of the estimator Y can be reduced by a factor of 1/N by doing N times as much work. This is amazing because the same order of convergence holds independently of the dimension n. a The Berry-Esseen theorem. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 750

54 Accuracy and Number of Replications (concluded) In contrast, classic numerical integration schemes have an error bound of O(N c/n ) for some constant c>0. n is the dimension. The required number of evaluations thus grows exponentially in n to achieve a given level of accuracy. The curse of dimensionality. The Monte Carlo method is more efficient than alternative procedures for multivariate derivatives when n is large. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 751

55 Monte Carlo Option Pricing For the pricing of European options on a dividend-paying stock, we may proceed as follows. Assume ds = μdt+ σdw. S Stock prices S 1,S 2,S 3,... at times Δt, 2Δt, 3Δt,... can be generated via S i+1 = S i e (μ σ2 /2) Δt+σ Δtξ, ξ N(0, 1). (94) c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 752

56 Monte Carlo Option Pricing (continued) If we discretize ds/s = μdt+ σdw directly, we will obtain S i+1 = S i + S i μ Δt + S i σ Δtξ. But this is locally normally distributed, not lognormally, hence biased. a In practice, this is not expected to be a major problem as long as Δt is sufficiently small. a Contributed by Mr. Tai, Hui-Chin (R ) on April 22, c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 753

57 Monte Carlo Option Pricing (continued) Non-dividend-paying stock prices in a risk-neutral economy can be generated by setting μ = r and Δt = T. 1: C := 0; {Accumulated terminal option value.} 2: for i =1, 2, 3,...,N do 3: P := S e (r σ2 /2) T +σ T ξ, ξ N(0, 1); 4: C := C +max(p X, 0); 5: end for 6: return Ce rt /N ; c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 754

58 Monte Carlo Option Pricing (concluded) Pricing Asian options is also easy. 1: C := 0; 2: for i =1, 2, 3,...,N do 3: P := S; M := S; 4: for j =1, 2, 3,...,n do 5: P := P e (r σ2 /2)(T/n)+σ T/n ξ ; 6: M := M + P ; 7: end for 8: C := C +max(m/(n +1) X, 0); 9: end for 10: return Ce rt /N ; c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 755

59 How about American Options? Standard Monte Carlo simulation is inappropriate for American options because of early exercise (why?). It is difficult to determine the early-exercise point based on one single path. But Monte Carlo simulation can be modified to price American options with small biases (pp. 807ff). a a Longstaff and Schwartz (2001). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 756

60 Delta and Common Random Numbers In estimating delta, it is natural to start with the finite-difference estimate rτ E[ P (S + ɛ)] E[ P (S ɛ)] e. 2ɛ P (x) is the terminal payoff of the derivative security when the underlying asset s initial price equals x. Use simulation to estimate E[ P (S + ɛ)] first. Use another simulation to estimate E[ P (S ɛ)]. Finally, apply the formula to approximate the delta. This is also called the bump-and-revalue method. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 757

61 Delta and Common Random Numbers (concluded) This method is not recommended because of its high variance. A much better approach is to use common random numbers to lower the variance: [ ] P (S + ɛ) P (S ɛ) e rτ E. 2ɛ Here, the same random numbers are used for P (S + ɛ) and P (S ɛ). This holds for gamma and cross gammas (for multivariate derivatives). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 758

62 Problems with the Bump-and-Revalue Method Consider the binary option with payoff 1, if S(T ) >X, 0, otherwise. Then P (S + ɛ) P (S ɛ) = 1, if P (S + ɛ) >X and P (S ɛ)]<x, 0, otherwise. So the finite-difference estimate per run for the (undiscounted) delta is 0 or O(1/ɛ). This means high variance. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 759

63 Gamma The finite-difference formula for gamma is [ P (S + ɛ) 2 P (S)+P(S ɛ) e rτ E ɛ 2 ]. For a correlation option with multiple underlying assets, the finite-difference formula for the cross gamma 2 P (S 1,S 2,...)/( S 1 S 2 ) is: [ P e rτ (S1 + ɛ 1,S 2 + ɛ 2 ) P (S 1 ɛ 1,S 2 + ɛ 2 ) E 4ɛ 1 ɛ 2 P (S 1 + ɛ 1,S 2 ɛ 2 )+P (S 1 ɛ 1,S 2 ɛ 2 ) ]. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 760

64 Gamma (continued) Choosing an ɛ of the right magnitude can be challenging. If ɛ is too large, inaccurate Greeks result. If ɛ is too small, unstable Greeks result. This phenomenon is sometimes called the curse of differentiation. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 761

65 Gamma (continued) In general, suppose i θ i e rτ E[ P (S)]=e rτ E [ i P (S) θ i ] holds for all i>0, where θ is a parameter of interest. Then formulas for the Greeks become integrals. As a result, we avoid ɛ, finite differences, and resimulation. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 762

66 Gamma (continued) This is indeed possible for a broad class of payoff functions. a Roughly speaking, any payoff function that is equal to a sum of products of differentiable functions and indicator functions with the right kind of support. For example, the payoff of a call is max(s(t ) X, 0) = (S(T ) X)I { S(T ) X 0 }. The results are too technical to cover here (see next page). a Teng (R ) (2004) and Lyuu and Teng (R ) (2011). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 763

67 Gamma (continued) Suppose h(θ, x) Hwith pdf f(x) forx and g j (θ, x) G for j B, a finite set of natural numbers. Then = h(θ, x) θ R j B R h θ (θ, x) j B + l B h(θ, x)j l (θ, x) 1 {gj (θ,x)>0} (x) f(x) dx 1 {gj (θ,x)>0} (x) f(x) dx 1 {gj (θ, x)>0} (x) f(x), j B\l x=χ l (θ) where ( ) gl (θ, x) gl (θ, x)/ θ J l (θ, x) =sign x k g l (θ, x)/ x for l B. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 764

68 Gamma (concluded) Similar results have been derived for Levy processes. a Formulas are also recently obtained for credit derivatives. b In queueing networks, this is called infinitesimal perturbation analysis (IPA). c a Lyuu, Teng (R ), and Wang (2013). b Lyuu, Teng (R ), and Tzeng (2014). c Cao (1985); Ho and Cao (1985). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 765

69 Biases in Pricing Continuously Monitored Options with Monte Carlo We are asked to price a continuously monitored up-and-out call with barrier H. The Monte Carlo method samples the stock price at n discrete time points t 1,t 2,...,t n. A sample path is produced. S(t 0 ),S(t 1 ),...,S(t n ) Here, t 0 = 0 is the current time, and t n = T is the expiration time of the option. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 766

70 Biases in Pricing Continuously Monitored Options with Monte Carlo (continued) If all of the sampled prices are below the barrier, this sample path pays max(s(t n ) X, 0). Repeating these steps and averaging the payoffs yield a Monte Carlo estimate. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 767

71 1: C := 0; 2: for i =1, 2, 3,...,N do 3: P := S; hit := 0; 4: for j =1, 2, 3,...,n do 5: P := P e (r σ2 /2) (T/n)+σ (T/n) ξ ; 6: if P H then 7: hit := 1; 8: break; 9: end if 10: end for 11: if hit = 0 then 12: C := C +max(p X, 0); 13: end if 14: end for 15: return Ce rt /N ; c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 768

72 Biases in Pricing Continuously Monitored Options with Monte Carlo (continued) This estimate is biased. a Suppose none of the sampled prices on a sample path equals or exceeds the barrier H. It remains possible for the continuous sample path that passes through them to hit the barrier between sampled time points (see plot on next page). a Shevchenko (2003). c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 769

73 H c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 770

74 Biases in Pricing Continuously Monitored Options with Monte Carlo (concluded) The bias can certainly be lowered by increasing the number of observations along the sample path. However, even daily sampling may not suffice. The computational cost also rises as a result. c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 771