Regression estimation in continuous time with a view towards pricing Bermudan options

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1 with a view towards pricing Bermudan options Tagung des SFB 649 Ökonomisches Risiko in Motzen

2 Financial engineering in times of financial crisis Derivate... süßes Gift für die Spekulanten (Die Zeit, ) Derivate & Co.: Die gefährlichen Produkte der Finanzbranche (Die Welt, ) Die Droge Derivat: Verbietet das Kokain des Kapitalismus! (Schweizer WochenZeitung WOZ, ) Pricing derivatives - is this ethical?

3 Financial engineering in times of financial crisis Derivate... süßes Gift für die Spekulanten (Die Zeit, ) Derivate & Co.: Die gefährlichen Produkte der Finanzbranche (Die Welt, ) Die Droge Derivat: Verbietet das Kokain des Kapitalismus! (Schweizer WochenZeitung WOZ, ) Pricing derivatives - is this ethical?

4 Back to basics Definition (Financial Mathematics I) A European put option is a contract giving the owner of the option the right to sell the asset (the underlying) at some fixed time T (the expiry time of the option) in the future at a fixed price K (the strike price). Most of exchange-traded options allow for early exercise. Accurate estimation of the price of an American option when the dimensionality of the underlying process is large remains an important (and ethical) research problem.

5 Back to basics Definition (Financial Mathematics I) A European put option is a contract giving the owner of the option the right to sell the asset (the underlying) at some fixed time T (the expiry time of the option) in the future at a fixed price K (the strike price). Most of exchange-traded options allow for early exercise. Accurate estimation of the price of an American option when the dimensionality of the underlying process is large remains an important (and ethical) research problem.

6 Back to basics Definition (Financial Mathematics I) A European put option is a contract giving the owner of the option the right to sell the asset (the underlying) at some fixed time T (the expiry time of the option) in the future at a fixed price K (the strike price). Most of exchange-traded options allow for early exercise. Accurate estimation of the price of an American option when the dimensionality of the underlying process is large remains an important (and ethical) research problem.

7 Pricing American options by regression Regression-based methods are commonly used for pricing of American-style options. Standard approach: We assume that we have perfect knowledge of the underlying process. 1. Sample discrete data. 2. Use regression estimators based on discrete data. Aim: Analyze regression estimates based on continuous-time data.

8 Pricing American options by regression Regression-based methods are commonly used for pricing of American-style options. Standard approach: We assume that we have perfect knowledge of the underlying process. 1. Sample discrete data. 2. Use regression estimators based on discrete data. Aim: Analyze regression estimates based on continuous-time data.

9 Pricing American options by regression Regression-based methods are commonly used for pricing of American-style options. Standard approach: We assume that we have perfect knowledge of the underlying process. 1. Sample discrete data. 2. Use regression estimators based on discrete data. Aim: Analyze regression estimates based on continuous-time data.

10 Pricing American options by regression Regression-based methods are commonly used for pricing of American-style options. Standard approach: We assume that we have perfect knowledge of the underlying process. 1. Sample discrete data. 2. Use regression estimators based on discrete data. Aim: Analyze regression estimates based on continuous-time data.

11 Pricing American options by regression Regression-based methods are commonly used for pricing of American-style options. Standard approach: We assume that we have perfect knowledge of the underlying process. 1. Sample discrete data. 2. Use regression estimators based on discrete data. Aim: Analyze regression estimates based on continuous-time data.

12 1 Motivation 2 The optimal stopping problem Approximate dynamic programming 3 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling 4

13 The optimal stopping problem Approximate dynamic programming Pricing American options We restrict attention to Bermudan options which represent discrete time approximations of American options, admit only a finite set of exercise dates {t 0, t 1,..., t L }. From a mathematical point of view, the pricing problem can be reduced to one of optimal stopping; optimal stopping problem is solved by approximate dynamic programming.

14 The optimal stopping problem Approximate dynamic programming Pricing American options We restrict attention to Bermudan options which represent discrete time approximations of American options, admit only a finite set of exercise dates {t 0, t 1,..., t L }. From a mathematical point of view, the pricing problem can be reduced to one of optimal stopping; optimal stopping problem is solved by approximate dynamic programming.

15 The optimal stopping problem Approximate dynamic programming The optimal stopping problem R d -valued Markov process (X t ) 0 t T (with X 0 fixed) is assumed to record all relevant financial information. Let X ti X i. If exercised at time t i, i = 0, 1,..., L, the option pays p i (X i ) for some known payoff functions p 0, p 1,..., p L : R d [0, ). T i denotes the set of randomized stopping times taking values in {t i, t i+1,..., t L }. Define V i (x) := sup τ Ti E [p τ (X τ ) X i = x], x R d. Objective: Find V 0 (X 0 ).

16 The optimal stopping problem Approximate dynamic programming The optimal stopping problem R d -valued Markov process (X t ) 0 t T (with X 0 fixed) is assumed to record all relevant financial information. Let X ti X i. If exercised at time t i, i = 0, 1,..., L, the option pays p i (X i ) for some known payoff functions p 0, p 1,..., p L : R d [0, ). T i denotes the set of randomized stopping times taking values in {t i, t i+1,..., t L }. Define V i (x) := sup τ Ti E [p τ (X τ ) X i = x], x R d. Objective: Find V 0 (X 0 ).

17 The optimal stopping problem Approximate dynamic programming The optimal stopping problem R d -valued Markov process (X t ) 0 t T (with X 0 fixed) is assumed to record all relevant financial information. Let X ti X i. If exercised at time t i, i = 0, 1,..., L, the option pays p i (X i ) for some known payoff functions p 0, p 1,..., p L : R d [0, ). T i denotes the set of randomized stopping times taking values in {t i, t i+1,..., t L }. Define V i (x) := sup τ Ti E [p τ (X τ ) X i = x], x R d. Objective: Find V 0 (X 0 ).

18 The optimal stopping problem Approximate dynamic programming The optimal stopping problem R d -valued Markov process (X t ) 0 t T (with X 0 fixed) is assumed to record all relevant financial information. Let X ti X i. If exercised at time t i, i = 0, 1,..., L, the option pays p i (X i ) for some known payoff functions p 0, p 1,..., p L : R d [0, ). T i denotes the set of randomized stopping times taking values in {t i, t i+1,..., t L }. Define V i (x) := sup τ Ti E [p τ (X τ ) X i = x], x R d. Objective: Find V 0 (X 0 ).

19 The optimal stopping problem Approximate dynamic programming Dynamic programming formulation Option values V i satisfy the dynamic programming equations V L (x) = p L (x), V l (x) = max {p l (x), E [V l+1 (X l+1 ) X l = x]} = max {p l (x), C l (x)}, l = L 1,..., 1, 0, where C i (x) := E [V i+1 (X i+1 ) X i = x], x R d, is the so-called continuation value.

20 The optimal stopping problem Approximate dynamic programming Approximate dynamic programming 1 parametric approximation ( Longstaff-Schwartz algorithm) for each l = 1,..., L 1, choose basis functions ψ lk : R d R, k = 0,..., K consider linear combinations as approximation of continuation values critical issue: choice of basis functions 2 nonparametric regression - local polynomial regression - smoothing spline estimation - neural networks

21 The optimal stopping problem Approximate dynamic programming Approximate dynamic programming 1 parametric approximation ( Longstaff-Schwartz algorithm) for each l = 1,..., L 1, choose basis functions ψ lk : R d R, k = 0,..., K consider linear combinations as approximation of continuation values critical issue: choice of basis functions 2 nonparametric regression - local polynomial regression - smoothing spline estimation - neural networks

22 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Nonparametric regression estimators based on iid data Common approach: Fix i {0, 1,..., L 1}, and set X i =: X, V i+1 (X i+1 ) =: V. Given simulated iid data { ( j X, jv) } N j=1 (X, V) Rd R, we wish to estimate nonparametrically. c(x) := E [V X = x], x R d, If c C s (D), D R d, s > 0, minimax optimal rate of convergence for the mean square error of regression estimates is N 2s+d 2s.

23 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Nonparametric regression estimators based on iid data Common approach: Fix i {0, 1,..., L 1}, and set X i =: X, V i+1 (X i+1 ) =: V. Given simulated iid data { ( j X, jv) } N j=1 (X, V) Rd R, we wish to estimate nonparametrically. c(x) := E [V X = x], x R d, If c C s (D), D R d, s > 0, minimax optimal rate of convergence for the mean square error of regression estimates is N 2s+d 2s.

24 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Nonparametric regression estimators based on iid data Common approach: Fix i {0, 1,..., L 1}, and set X i =: X, V i+1 (X i+1 ) =: V. Given simulated iid data { ( j X, jv) } N j=1 (X, V) Rd R, we wish to estimate nonparametrically. c(x) := E [V X = x], x R d, If c C s (D), D R d, s > 0, minimax optimal rate of convergence for the mean square error of regression estimates is N 2s+d 2s.

25 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling I General framework: Suppose Z t = (X t, Y t ), t R, is an R d R-valued stochastic process, defined on a probability space (Ω, A, P). Given the data (Z t ) 0 t T, we wish to estimate the regression function r(x) := E [Y 0 X 0 = x], x R d. Let K : R d R d R be a kernel function, (h T ) T 0 a bandwidth sequence. A Nadaraya-Watson regression estimator is defined by r T (x) := T ( x Xt ) 0 Y t K h dt T ( ), x T R d. 0 K x Xt h dt T

26 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling I General framework: Suppose Z t = (X t, Y t ), t R, is an R d R-valued stochastic process, defined on a probability space (Ω, A, P). Given the data (Z t ) 0 t T, we wish to estimate the regression function r(x) := E [Y 0 X 0 = x], x R d. Let K : R d R d R be a kernel function, (h T ) T 0 a bandwidth sequence. A Nadaraya-Watson regression estimator is defined by r T (x) := T ( x Xt ) 0 Y t K h dt T ( ), x T R d. 0 K x Xt h dt T

27 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling I General framework: Suppose Z t = (X t, Y t ), t R, is an R d R-valued stochastic process, defined on a probability space (Ω, A, P). Given the data (Z t ) 0 t T, we wish to estimate the regression function r(x) := E [Y 0 X 0 = x], x R d. Let K : R d R d R be a kernel function, (h T ) T 0 a bandwidth sequence. A Nadaraya-Watson regression estimator is defined by r T (x) := T ( x Xt ) 0 Y t K h dt T ( ), x T R d. 0 K x Xt h dt T

28 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling II Assumptions on the underlying process: The joint density of (X s, X t ) exists for all s = t, is measurable and satisfies f (Xs,X t ) = f (Xt,X s ) = f (X0,X t s ) =: f [t s], s, t R. We will use the condition: (I) There exists γ 0 > 0 such that f (X0,X u ) (y, z) M(y, z)u γ 0 (y, z, u) R 2d (0, u 0 ), with either M(, ) bounded or M(, ) L 1 (R 2d ) and continuous at (x, x). - Special case: γ 0 < 1 Castellana & Leadbetter (1986). for

29 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling II Assumptions on the underlying process: The joint density of (X s, X t ) exists for all s = t, is measurable and satisfies f (Xs,X t ) = f (Xt,X s ) = f (X0,X t s ) =: f [t s], s, t R. We will use the condition: (I) There exists γ 0 > 0 such that f (X0,X u ) (y, z) M(y, z)u γ 0 (y, z, u) R 2d (0, u 0 ), with either M(, ) bounded or M(, ) L 1 (R 2d ) and continuous at (x, x). - Special case: γ 0 < 1 Castellana & Leadbetter (1986). for

30 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Example (Veretennikov): Diffusion given by an SDE (d = 1) Consider the diffusion X which satisfies the stochastic differential equation (SDE) where dx t = b(x t )dt + σ(x t )dw t, t 0, (1) σ is uniformly non-degenerate, it holds lim sup x x b(x) σ 2 (x) L < 0, where L sup x σ 2 (x) + inf x σ 2 (x) 2 sup x σ 2 (x) > 3 2, σ, b C s (D) for some s > 0, D R d. Then the condition (I) is satisfied for some γ 0 < 1.

31 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Example (Veretennikov): Diffusion given by an SDE (d = 1) Consider the diffusion X which satisfies the stochastic differential equation (SDE) where dx t = b(x t )dt + σ(x t )dw t, t 0, (1) σ is uniformly non-degenerate, it holds lim sup x x b(x) σ 2 (x) L < 0, where L sup x σ 2 (x) + inf x σ 2 (x) 2 sup x σ 2 (x) > 3 2, σ, b C s (D) for some s > 0, D R d. Then the condition (I) is satisfied for some γ 0 < 1.

32 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Example (Veretennikov): Diffusion given by an SDE (d = 1) Consider the diffusion X which satisfies the stochastic differential equation (SDE) where dx t = b(x t )dt + σ(x t )dw t, t 0, (1) σ is uniformly non-degenerate, it holds lim sup x x b(x) σ 2 (x) L < 0, where L sup x σ 2 (x) + inf x σ 2 (x) 2 sup x σ 2 (x) > 3 2, σ, b C s (D) for some s > 0, D R d. Then the condition (I) is satisfied for some γ 0 < 1.

33 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Superoptimal rates of convergence Proposition Assume that X solves (1), and that the kernel K satisfies the usual regularity assumptions. 1 If h T is chosen such that h T T 1/4, then (under some mild further regularity assumptions) the kernel-based estimator r T satisfies T E [ r T (x) r(x)] 2 0, x D. 2 If h T is chosen such that h T (log T) 4 /T, then r T satisfies ( ) T 1/2 r T r log T D 0 almost surely.

34 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Main ingredients of the proof Truncation argument; (uniform) inequalities for strongly mixing sequences (follow from coupling arguments and Bernstein-type inequalities for independent variables); moment bounds.

35 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Admissible sampling In practice: Estimation from discrete-time observations. Construct estimators by replacing Lebesgue integrals by Riemann sums. Definition Consider a process (X t ) t 0 with irregular paths (i.e., (X t ) satisfies (I) for some γ 0 < 1). A sampling is said to be admissible if it corresponds to the minimal number of data which preserve the superoptimal rate (in mean square or uniformly).

36 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Sampling schemes Example Consider a process (X t, Y t ) observed at times δ n, 2δ n,..., nδ n where δ n 0 and T n nδ n, and assume that (X t ) has irregular paths. Let ( r n (x) := n j=1 Y x Xjδn ) jδ n K h n For δ n = T d/4 n, h n T 1/4 n, n j=1 K ( x Xjδn h n ), x R d. E [ r n (x) r(x)] 2 = O(T 1 n ), x D.

37 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Sampling schemes Example Consider a process (X t, Y t ) observed at times δ n, 2δ n,..., nδ n where δ n 0 and T n nδ n, and assume that (X t ) has irregular paths. Let ( r n (x) := n j=1 Y x Xjδn ) jδ n K h n For δ n = T d/4 n, h n T 1/4 n, n j=1 K ( x Xjδn h n ), x R d. E [ r n (x) r(x)] 2 = O(T 1 n ), x D.

38 Estimators based on iid data Superoptimal rates for continuous-time estimators Admissible sampling Sampling schemes Example Consider a process (X t, Y t ) observed at times δ n, 2δ n,..., nδ n where δ n 0 and T n nδ n, and assume that (X t ) has irregular paths. Let ( r n (x) := n j=1 Y x Xjδn ) jδ n K h n For δ n = T d/4 n, h n T 1/4 n, n j=1 K ( x Xjδn h n ), x R d. E [ r n (x) r(x)] 2 = O(T 1 n ), x D.

39 Computing the fair price of an American option - a fairytale Recall that the equilibrium price of the option is given by V 0 (x) := sup τ T 0 E [p τ (X τ ) X i = x], x R d. Estimators are incriminated by approximation error, discretization error, stochastic error.

40 Computing the fair price of an American option - a fairytale Recall that the equilibrium price of the option is given by V 0 (x) := sup τ T 0 E [p τ (X τ ) X i = x], x R d. Estimators are incriminated by approximation error, discretization error, stochastic error.

41 Computing the fair price of an American option - a fairytale Recall that the equilibrium price of the option is given by V 0 (x) := sup τ T 0 E [p τ (X τ ) X i = x], x R d. Estimators are incriminated by approximation error, discretization error, stochastic error.

42 Computing the fair price of an American option - a fairytale Recall that the equilibrium price of the option is given by V 0 (x) := sup τ T 0 E [p τ (X τ ) X i = x], x R d. Estimators are incriminated by approximation error, discretization error, stochastic error.

43 Challenges All-encompassing error analysis of the option pricing problem. Identification of stochastic processes (diffusion processes, Lévy processes) which achieve parametric rates in terms of coefficients of the associated SDEs. Deepened analysis of admissible sampling schemes.

44 Challenges All-encompassing error analysis of the option pricing problem. Identification of stochastic processes (diffusion processes, Lévy processes) which achieve parametric rates in terms of coefficients of the associated SDEs. Deepened analysis of admissible sampling schemes.

45 Challenges All-encompassing error analysis of the option pricing problem. Identification of stochastic processes (diffusion processes, Lévy processes) which achieve parametric rates in terms of coefficients of the associated SDEs. Deepened analysis of admissible sampling schemes.

Lecture 4. Finite difference and finite element methods

Lecture 4. Finite difference and finite element methods Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation