Applications of short-time asymptotics to the statistical estimation and option pricing of Lévy-driven models

Transcription

1 Applications of short-time asymptotics to the statistical estimation and option pricing of Lévy-driven models José Enrique Figueroa-López 1 1 Department of Statistics Purdue University CIMAT and Universidad de Guanajuato Guanajuato, Mexico May 13-17, 2013

2 Overview of the Lecture Series 1 Lecture I Introduce the elements of continuous-time stochastic modeling. Basic building blocks: Brownian Motion, Poisson Process, etc. Semiparametric Estimation of finite-jump activity Lévy jump-diffusions (Joint work with Jeff Nisen). 2 Lecture II Options and derivatives. Elements of arbitrage-free option pricing. Quick primer of Lévy processes Short-time asymptotics for option prices in exponential Lévy models (Joint work with Martin Forde, Christian Houdré, and Ruoting Gong). 3 Lecture III An adaptive Monte Carlo valuation method for Barrier options in exponential Lévy models. (Joint work with Peter Tankov).

3 Overview of the Lecture Series 1 Lecture I Introduce the elements of continuous-time stochastic modeling. Basic building blocks: Brownian Motion, Poisson Process, etc. Semiparametric Estimation of finite-jump activity Lévy jump-diffusions (Joint work with Jeff Nisen). 2 Lecture II Options and derivatives. Elements of arbitrage-free option pricing. Quick primer of Lévy processes Short-time asymptotics for option prices in exponential Lévy models (Joint work with Martin Forde, Christian Houdré, and Ruoting Gong). 3 Lecture III An adaptive Monte Carlo valuation method for Barrier options in exponential Lévy models. (Joint work with Peter Tankov).

4 Overview of the Lecture Series 1 Lecture I Introduce the elements of continuous-time stochastic modeling. Basic building blocks: Brownian Motion, Poisson Process, etc. Semiparametric Estimation of finite-jump activity Lévy jump-diffusions (Joint work with Jeff Nisen). 2 Lecture II Options and derivatives. Elements of arbitrage-free option pricing. Quick primer of Lévy processes Short-time asymptotics for option prices in exponential Lévy models (Joint work with Martin Forde, Christian Houdré, and Ruoting Gong). 3 Lecture III An adaptive Monte Carlo valuation method for Barrier options in exponential Lévy models. (Joint work with Peter Tankov).

5 Lecture III: Outline 1 Motivation: Barrier Options 2 Small-time asymptotics for stopped Lévy bridges Formulation of the Problem The Main Result 3 Monte Carlo Methods for stopped Lévy processes Bridge MC Simulation Adaptive simulation with bias control Numerical Illustration 4 Conclusions

6 Barrier options Set-up: Market consisting of a money market account with constant interest rate r and a risky asset with prices process {S t } 0 t T ; European Barrier Options: Option whose payoff at maturity is triggered or cancelled when the stock price hits" a certain domain of price values. Up-and-in call: Given a barrier" value B > S 0, { (ST K ) X = +, if sup 0 t T S t B, = (S T K ) + 1 {supt T S t B}. 0, otherwise, Up-and-out call: Given a barrier" value B > S 0, ) X = (S T K ) + (1 1 {supt T S t B} = (S T K ) + 1 {supt T S t <B}. Down-and-out call: Given a barrier" value A < S 0, ) X = (S T K ) + (1 1 {inft T S t A} = (S T K ) + 1 {inft T S t >A}.

7 Barrier options Set-up: Market consisting of a money market account with constant interest rate r and a risky asset with prices process {S t } 0 t T ; European Barrier Options: Option whose payoff at maturity is triggered or cancelled when the stock price hits" a certain domain of price values. Up-and-in call: Given a barrier" value B > S 0, { (ST K ) X = +, if sup 0 t T S t B, = (S T K ) + 1 {supt T S t B}. 0, otherwise, Up-and-out call: Given a barrier" value B > S 0, ) X = (S T K ) + (1 1 {supt T S t B} = (S T K ) + 1 {supt T S t <B}. Down-and-out call: Given a barrier" value A < S 0, ) X = (S T K ) + (1 1 {inft T S t A} = (S T K ) + 1 {inft T S t >A}.

8 Barrier options Set-up: Market consisting of a money market account with constant interest rate r and a risky asset with prices process {S t } 0 t T ; European Barrier Options: Option whose payoff at maturity is triggered or cancelled when the stock price hits" a certain domain of price values. Up-and-in call: Given a barrier" value B > S 0, { (ST K ) X = +, if sup 0 t T S t B, = (S T K ) + 1 {supt T S t B}. 0, otherwise, Up-and-out call: Given a barrier" value B > S 0, ) X = (S T K ) + (1 1 {supt T S t B} = (S T K ) + 1 {supt T S t <B}. Down-and-out call: Given a barrier" value A < S 0, ) X = (S T K ) + (1 1 {inft T S t A} = (S T K ) + 1 {inft T S t >A}.

9 Barrier options Set-up: Market consisting of a money market account with constant interest rate r and a risky asset with prices process {S t } 0 t T ; European Barrier Options: Option whose payoff at maturity is triggered or cancelled when the stock price hits" a certain domain of price values. Up-and-in call: Given a barrier" value B > S 0, { (ST K ) X = +, if sup 0 t T S t B, = (S T K ) + 1 {supt T S t B}. 0, otherwise, Up-and-out call: Given a barrier" value B > S 0, ) X = (S T K ) + (1 1 {supt T S t B} = (S T K ) + 1 {supt T S t <B}. Down-and-out call: Given a barrier" value A < S 0, ) X = (S T K ) + (1 1 {inft T S t A} = (S T K ) + 1 {inft T S t >A}.

10 Barrier options Set-up: Market consisting of a money market account with constant interest rate r and a risky asset with prices process {S t } 0 t T ; European Barrier Options: Option whose payoff at maturity is triggered or cancelled when the stock price hits" a certain domain of price values. Up-and-in call: Given a barrier" value B > S 0, { (ST K ) X = +, if sup 0 t T S t B, = (S T K ) + 1 {supt T S t B}. 0, otherwise, Up-and-out call: Given a barrier" value B > S 0, ) X = (S T K ) + (1 1 {supt T S t B} = (S T K ) + 1 {supt T S t <B}. Down-and-out call: Given a barrier" value A < S 0, ) X = (S T K ) + (1 1 {inft T S t A} = (S T K ) + 1 {inft T S t >A}.

11 Barrier options. Cont... Call-put parity relationship: up-and-in call + up-and-out call = regular call Payoff of a general double-barrier option of out-type": where X := f (S T )1 {St (A,B), for all t [0,T ]} (0 A < S 0 < B ) = F (X T ) 1 {Xt (a,b), for all t [0,T ]} = F (X T ) 1 {τ>t }, X t = ln S t S 0 S t = S 0 e Xt, (Log-Return Process), F(x) = f (S 0 e x ), a = ln(a/s 0 ), b = ln(b/s 0 ), ( a < 0 < b ), τ := inf{t > 0 : X t / (a, b)}, (Exit or Hitting Time).

12 Arbitrage-Free Pricing 1 Under the absence of arbitrage, Π 0 (X ) = E Q ( e rt X ) = E Q ( e rt F (X T )1 {τ>t } ), where Q is a risk-neutral probability measure; i.e., for any t < T, ( ) E Q e r(t t) S T Su, u t = S t. 2 In the Black-Scholes model, there exists a unique risk-neutral measure Q. Furthermore, under Q, 3 Hence, the barrier premium is S t := S 0 e α Qt+σW t with α Q := r σ2 2. Π(X ; T ) = E Q ( e rt F (α Q T + σw T ) 1 {αq t+σw t (a,b), for all t T }). 4 There is no closed formula. Need to use on numerical methods.

13 Traditional (sequencial) Monte Carlo (MC) Method Algorithm: 1 Selection of the Sampling Scheme: 0 = t 0 < < t n = T (e.g., uniform sampling t i = it /n); 2 Sample simulation: X t1,..., X tn 3 Approximation of the exit time: τ := inf{t [0, T ] : X t / (a, b)} τ n := min {t k : X tk / (a, b)} ; 4 Evaluation of the approximate final discounted payoff: Y := e rt F (X T )1 {τ>t } Ỹ := e rt F(X tn )1 { τn>t }. 5 Repeat (1)-(4) to generate m independent copies of Ỹ: Ỹ 1,..., Ỹm. 6 MC estimate: Π 0 := 1 m m Ỹ i. i=1

14 Traditional (sequencial) Monte Carlo (MC) Method Error analysis: There are two types: Discretization Error and Statistical Error. The former is due to approximation Y Ỹ (equiv., τ n τ), while the latter is due to the approximation 1 ) m m i=1 Ỹi E (Ỹ Q. Drawback of sequential MC for stopped processes: 1 Highly biased due to the possibility of exiting the interval (a, b) between sampling observations. 2 The discretization error is of order 1 n for diffusions (Asmussen, Glyn, and Pitman; 1995) and diffusions with finite jump activity (Dia and Laberton; 2007); 3 Unknown for general Lévy processes, but it is expected to be much higher for infinite jump activity Lévy processes.

15 Improved MC for Markov processes (Baldi, 1995) 1 Suppose one can compute the exit probability of the bridge" process: p(x, y, t) := P (X u / (a, b), for some u [s, s + t] X s = x, X s+t = y). 2 By the Markov Property, for any fixed times 0 = t 0 < < t n = T, E [ ] F (X T )1 {Xu (a,b), u [0,T ]} [ ] n 1 = E F(X T ) = E [ [ = E F(X T )E n 1 F (X T ) 1 {Xu (a,b), u (t i 1,t i ]} i=0 [ n 1 1 {Xu (a,b), u (t i 1,t i ]} i=0 i=0 ]] X t 1,..., X tn E [ ] ] 1 {Xu (a,b), u (t i 1,t i ]} X ti 1, X ti [ n 1 ( = E F(X T ) 1 p(xti, X, t ti+1 i+1 t i ) ) ]. i=0

16 Improved MC for Markov processes (Baldi, 1995) 1 Suppose one can compute the exit probability of the bridge" process: p(x, y, t) := P (X u / (a, b), for some u [s, s + t] X s = x, X s+t = y). 2 By the Markov Property, for any fixed times 0 = t 0 < < t n = T, E [ ] F (X T )1 {Xu (a,b), u [0,T ]} [ ] n 1 = E F(X T ) = E [ [ = E F(X T )E n 1 F (X T ) 1 {Xu (a,b), u (t i 1,t i ]} i=0 [ n 1 1 {Xu (a,b), u (t i 1,t i ]} i=0 i=0 ]] X t 1,..., X tn E [ ] ] 1 {Xu (a,b), u (t i 1,t i ]} X ti 1, X ti [ n 1 ( = E F(X T ) 1 p(xti, X, t ti+1 i+1 t i ) ) ]. i=0

17 Improved MC Method. Cont... Algorithm: 1 Simulation of a discrete skeleton of the process: {(t 1, X t1 ),..., (t n, X tn )} 2 Compute the (discounted) conditional expected payoff given the skeleton: n 1 Ỹ := e rt ( F (X tn ) 1 p(xti, X, t ti+1 i+1 t i ) ). i=0 3 Repeat (1)-(2) to generate m independent copies: Ỹ 1,..., Ỹm. 4 MC estimate: Π 0 := 1 m Advantage: m Ỹ i. There is no discretization error and the only error is statistical (which is of order n 1/2 by the CLT). i=1

18 Formulation of the Problem Important question: How to find the exit probability p(x, y, t)? Closed form available for the Black-Scholes model X t = αt + σw t ; Small-time approximation known for diffusions (Baldi, 1995) dx t := α(t, X t )dt + σ(t, X t )dw t ; Unknown approximation for processes with jumps. The Key Problem: Given a domain (a, b) with a < 0 < b and initial and final points x, y (a, b), we want to characterize the small-time asymptotics of the exit probability for Lévy bridges: p(x, y, t) := P (X u / (a, b), for some u [s, s + t] X s = x, X s+t = y), where X := (X t ) t 0 is a general Lévy Process".

19 Exponential Lévy Model 1 Definition: X 0 = 0; X has memoryless property (equiv., independent increments); X has stationary increments: X t X s D = X t s; The paths of X may exhibit jumps; 2 Hereafter, we assume that E ( e iux ) 1 = e iub σ2 u R\{0}(e iux 1 iux1 x 1)s(x)dx. for some b R, σ [0, ), and s : R\{0} [0, ) with 0< x 1 x 2 s(x)dx < and s(x)dx <. x >1 3 Interpretation: The function s governs the intensity of jumps; σ is the volatility of the continuous component; b is related to a deterministic drift or expected rate of grow of the stock.

20 Short-time asymptotics of stopped Lévy bridges 1 Problem: Characterize the small-time asymptotics of the exit probability: p(x, y, t) := P ( u [s, s + t] : X u / (a, b) X s = x, X s+t = y). 2 Note that P (X u / (a, b), for some u [s, s + t] X s = x, X s+t = y) = P (X u / (a x, b x), for some u [0, t] X 0 = 0, X s+t = y x) So, it suffices to study the small-time asymptotics of the exit probability: P (X u / (a, b), for some u [0, t] X t = y, X 0 = 0) = P (τ t X t = y, X 0 = 0), where τ := inf {u 0 : X u / (a, b)}, y (a, b), a < 0 < b.

21 Short-time asymptotics of stopped Lévy bridges 1 Problem: Characterize the small-time asymptotics of the exit probability: p(x, y, t) := P ( u [s, s + t] : X u / (a, b) X s = x, X s+t = y). 2 Note that P (X u / (a, b), for some u [s, s + t] X s = x, X s+t = y) = P (X u / (a x, b x), for some u [0, t] X 0 = 0, X s+t = y x) So, it suffices to study the small-time asymptotics of the exit probability: P (X u / (a, b), for some u [0, t] X t = y, X 0 = 0) = P (τ t X t = y, X 0 = 0), where τ := inf {u 0 : X u / (a, b)}, y (a, b), a < 0 < b.

22 Short-time asymptotics of stopped Lévy bridges 1 Problem: Characterize the small-time asymptotics of the exit probability: p(x, y, t) := P ( u [s, s + t] : X u / (a, b) X s = x, X s+t = y). 2 Note that P (X u / (a, b), for some u [s, s + t] X s = x, X s+t = y) = P (X u / (a x, b x), for some u [0, t] X 0 = 0, X s+t = y x) So, it suffices to study the small-time asymptotics of the exit probability: P (X u / (a, b), for some u [0, t] X t = y, X 0 = 0) = P (τ t X t = y, X 0 = 0), where τ := inf {u 0 : X u / (a, b)}, y (a, b), a < 0 < b.

23 Some useful known results 1 For any x > 0, P (X t x) t 2 Intuition: Think of X t = σw t + bt + N t j=1 ζ j: x s(w)dw, (t 0); P (X t x) = P (σw t + bt x) e λt + tp (σw t + bt + ζ 1 x) e λt λt +... t P (ζ 1 x) λ = t x s(w)dw. 3 In terms of the probability density f t of X t (when it exits), 1 lim t 0 t P (X t x) = lim t 0 x 1 t f t(w)dw = b s(w)dw. 4 [Léandre(1987)]. If f t (x) is the probability density of X t, then 1 lim t 0 t f t(x) = s(x), (x 0).

24 Some useful known results 1 For any x > 0, P (X t x) t 2 Intuition: Think of X t = σw t + bt + N t j=1 ζ j: x s(w)dw, (t 0); P (X t x) = P (σw t + bt x) e λt + tp (σw t + bt + ζ 1 x) e λt λt +... t P (ζ 1 x) λ = t x s(w)dw. 3 In terms of the probability density f t of X t (when it exits), 1 lim t 0 t P (X t x) = lim t 0 x 1 t f t(w)dw = b s(w)dw. 4 [Léandre(1987)]. If f t (x) is the probability density of X t, then 1 lim t 0 t f t(x) = s(x), (x 0).

25 Some useful known results 1 For any x > 0, P (X t x) t 2 Intuition: Think of X t = σw t + bt + N t j=1 ζ j: x s(w)dw, (t 0); P (X t x) = P (σw t + bt x) e λt + tp (σw t + bt + ζ 1 x) e λt λt +... t P (ζ 1 x) λ = t x s(w)dw. 3 In terms of the probability density f t of X t (when it exits), 1 lim t 0 t P (X t x) = lim t 0 x 1 t f t(w)dw = b s(w)dw. 4 [Léandre(1987)]. If f t (x) is the probability density of X t, then 1 lim t 0 t f t(x) = s(x), (x 0).

26 The Main Result Theorem. [F-L & Tankov (2012)] For a.e. y (a, b)\{0}, we have, as t 0, In particular, P (τ t X t = y) = t 2 2 = t 2 p(x, y, t) = t 2 s(v)s(y v) (a,b) f c t (y) s(v)s(y v) (a,b) s(y) c s(v)s(y x v) (a x,b x) s(y x) c dv + O(t 3/2 ) dv + O(t 3/2 ). dv + O(t 3/2 ).

27 Intuition from the finite jump-activity case Consider a compound Poisson process X t = N t n=0 ξ n with jump intensity λ = 1: Reason: P(τ t X t = y) δ 1 P( u t : X u / (a, b); X t (y δ, y + δ)) P(X t (y δ, y + δ) t 2 2 P(ξ 1 / (a, b), ξ 1 + ξ 2 (y δ, y + δ)) f t (y)δ δ 0 t 2 s(x)s(y x) dx. 2 (a,b) f c t (y) If, during a small time interval, X exits the interval (a, b) and then comes back to a point y (a, b), this essentially happens with two large jumps: the first one takes the process out of (a, b), while a second jumps brings it back. We show that this logic extends to a large class of infinite jump activity Lévy processes.

28 Illustration Left: Cauchy bridge when it cross level b = 2 during [0, 1]; Right: Cauchy bridge when it cross level b = 2 during [0, 0.1].

29 Back to the Baldi s sequential MC Method Algorithm: 1 Generation of the sample: X t1,..., X tn (e.g., from the generation of the increments n i X := X t i X ti 1 ); 2 Compute the expected payoff conditional on the discrete skeleton: n 1 ( Ỹ := F(X tn ) 1 p(xti, X, t ti+1 i+1 t i ) ). i=0 3 Repeat (1)-(2) to generate m copies of approx. payoffs: Ỹ 1,..., Ỹm. 4 MC estimate: Π(0) := 1 m m i=1 Ỹi. Proposed solution: Short-time approximation. p(x, y, t) := P ( u [0, t] : X u / (a x, b x) X 0 = 0, X t = y x) t 2 s(v)s(y x v) dv =: p(x, y, t), (x y). 2 (a x,b x) f c t (y x)

30 Back to the Baldi s sequential MC Method Algorithm: 1 Generation of the sample: X t1,..., X tn (e.g., from the generation of the increments n i X := X t i X ti 1 ); 2 Compute the expected payoff conditional on the discrete skeleton: n 1 ( Ỹ := F(X tn ) 1 p(xti, X, t ti+1 i+1 t i ) ). i=0 3 Repeat (1)-(2) to generate m copies of approx. payoffs: Ỹ 1,..., Ỹm. 4 MC estimate: Π(0) := 1 m m i=1 Ỹi. Proposed solution: Short-time approximation. p(x, y, t) := P ( u [0, t] : X u / (a x, b x) X 0 = 0, X t = y x) t s(v)s(y x v) dv =: p(x, y, t), (x y). 2 (a x,b x) s(y x) c

31 Controlling the bias 1 We propose to generate the approximated expected payoff: n 1 ( F(X tn ) 1 p(xti, X, t ti+1 i+1 t i ) ) n 1 ( F(X tn ) 1 p(xti, X, t ti+1 i+1 t i ) ) i=0 i=0 2 The bias is introduced via the error in the approximation p(x, y, t i+1 t i ) p(x, y, t i+1 t i ), which in principle improves if t i+1 t i is small; 3 How to choose a suitable mesh size between sampling times? 4 If we have at our hand an estimate e p (x, y, t) of the approximation error: p(x, y, t) p(x, y, t) e p (x, y, t), one may control the bias by splitting the subinterval [t i, t i+1 ] into two if e(x ti, X ti+1, t i+1 t i ) γ(t i+1 t i ) for some desired tolerance γ > 0; 5 More suitable with adaptive simulation (i.e. sample more points only when and where is needed) and Bridge Monte Carlo.

32 Bridge Monte Carlo 1 Simulate the final value X T from the marginal law f T ( ) of X T ; 2 Simulate intermediate points using the bridge law: f br t ( s, x, u, y) = Law (X t X s = x, X u = y), (s < t < u). Concretely, e.g., to generate X 0, X T 4 Simulate X T from f T ( );, X T 2 Simulate X T from f br T /2 0, 0, T, X T ); 2 ( ) Simulate X T from f br T /4 0, 0, T, X 4 2 T ; ( 2 Simulate X 3T from f br T /4 T, X 4 2 T, T, X T ); 2 Advantages:, X 3T, X T, we proceed as follows: 4 The trajectory can be adaptively refined only where and when necessary; Variance reduction methods are easy to design by replacing the density of X T with an important sampling distribution;

33 Bridge-based MC method with controlled bias Algorithm Given x and y, the algorithm simulates a discrete skeleton of a Lévy bridge X = {(T i, X Ti )} N i=0 on [0, T ] with T 0 = 0, T N = T, X 0 = x, X T = y s.t. e p (X T(i), X T(i+1), T (i+1) T (i) ) γ(t (i+1) T (i) ), (1) where 0 = T (0) < < T (N) are the order statistics" of {T 0,..., T N } Returns Ñ(X ) := N 1 ( i=0 1 p(xt(i), X T, T (i+1) (i+1) T (i) ) ). FUNCTION N(parameters: x, y, T ) IF x / D OR y / D THEN RETURN 0 IF e p (x, y, T ) γ T THEN RETURN 1 p(x, y, T ) ELSE Sample ˆX from the bridge distribution X T X T = y 2 RETURN N(x, ˆX, T /2) N( ˆX, y, T /2) END IF

34 Illustration Figure: A typical trajectory simulated by the adaptive algorithm (Cauchy Process). The algorithm places more points at the parts of the trajectory which are close to the boundary.

35 Some important issues 1 The ordered sampling times 0 = T (0) < < T (N) = T are random (and non-anticipative") times. Is the decomposition [ E[F (X T )1 τ>t ] = E F (X T ) still true? N 1 i=0 ( 1 p(xt(i), X T (i+1), T (i+1) T (i) ) ) ], 2 Does the algorithm terminate in finite-time? Note that we require for all i = 0,..., N 1. e p (X T(i), X T(i+1), T (i+1) T (i) ) γ(t (i+1) T (i) ), 3 Does the algorithm attains the desired controlled bias?

36 Convergence of the algorithm and bias control Theorem. [F-L & Tankov (2012)] Suppose that X satisfies one of the conditions: 1 X does not hit points; that is, P(τ {x} < ) = 0 for all x, where τ {x} := inf{s > 0 : X s = x} or, equivalently, ( ) 1 R du =, 1 + ψ(u) R 2 X has finite variation (e.g. Variance Gamma Process). Also, assume the approximation error satisfies 1 lim sup e p (x, y, t) = 0, t 0 t x,y (a,b ) a, b (a, b).

37 Convergence of the algorithm and bias control. Cont... Theorem. [F-L & Tankov (2012)] Then, for any T > 0, γ > 0, and F such that E F(X T ) <, we have (i) The previous adaptive algorithm terminates in finite time a.s. (ii) The random skeleton X = {(T i, X Ti )} N i=0 generated by the above algorithm satisfies: E[F (X T )1 {τ>t } ] E[F(X T ) Ñ(X )] γe[ F (X T ) ].

38 Error estimate for self-decomposable Lévy processes Theorem. [F-L & Tankov (2012)] Fix ε > 0 small enough, let (i) λ ε := x ε s(x)dx, b ε := b ε< x 1 xν(dx), σ2 ε := σ 2 + x ε x 2 ν(dx) (ii) a ε := sup x >ε s(x), a ε := sup x >ε s (x), C(η, ε) := (iii) α := b a and y := (b y) (y a) > 0 ( ) eσ 2 η ε ε εη (a,b) c s(v)s(y v) f t (y) For t > 0 small enough, the approx. p(0, y, t) = t 2 2 p(0, y, t) p(0, y, t) 1 ( { e λεt C( y /4, ε)t y 8 4ε + 2a ε t + a ε λ ε t 2 f t (y) y dv is s.t. + 2e λεt a ε C(α/2, ε)t 1+ α 2ε {1 + tλε } + λ2 εa ε 2 t 3 + a ε λ 1 ( ε 1 e λ εt [1 + λ ε t + (λ ε t) 2 /2] ) + e λεt t 2[ 2aε 2 + λ ε a ε ] (σε t 1/2 + b ) ε 2 t). }

39 Numerical Example 1: Cauchy Process Uniform Adaptive True value 5% confidence bound 0.1 Adaptive discretization Uniform discretization Bias Time Time Figure: Computation of P(τ > 1) := P[sup 0 s 1 X s 10 2 ] (a =, b = 0.01, and F( ) 1). Left: Values computed by the uniform discretization algorithm (UDA) and the adaptive algorithm (AA), as function of the computational time (in sec.), for 10 6 paths. Different points on the graph correspond to different numbers of discretization times (n) for the UDA (from 256 to 16384) and different values of the tolerance parameter γ for the AA (from 9 to ). Right: Comparison of the discretization bias for the uniform discretization and the bias for the adaptive algorithm.

40 Conclusions and extensions Main results First-order asymptotics for stopped Lévy bridges with explicitly computable error bounds; Bridge simulation method for general Lévy processes; Application to Monte Carlo simulation of stopped Lévy process with controlled bias via an adaptive bridge Monte Carlo simulation method. Extensions: Simulation of stopping times and overshoots; Multidimensional Lévy processes in confined domains; General Markov jump processes.

41 For Further Reading I Figueroa-López & Tankov. Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias To appear in Bernoulli, Available at Arxiv and at figueroa. Figueroa-Lopez & Houdré. Small-time expansions for the transition distributions of Lévy processes. Stochastic Processes and Their Applications, 119: , Léandre Densité en temps petit d un processus de sauts. Séminaire de probabilités XXI, Lecture Notes in Math. J. Azéma, P.A. Meyer, and M. Yor (eds), 1987.