Efficient Monte Carlo Simulation of Security Prices Explained

Efficient Monte Carlo Simulation of Security Prices By Darrell Duffie and Peter Glynn, Stanford University, 1995 Paper Review by: Niv Nayman Technion- Israel Institute of Technology July 2015

What are we going to speak about ? Background The Paper - Brownian Motion - Wiener Process (SBM) - Introduction - Discretization Schemes - Euler - Milshtein (1978) - Talay (1984, 1986) - Motivation - Abstract - Assumptions - Theorem - Proof - Interpretation - Experiments - More to go - Stochastic Calculus (It s Calculus) - Stochastic Integral (It s Integral) - Stochastic Differential Equation (SDE) - It Process - Diffusion Process - It s Lemma - Example : Geometric Brownian Motion (GBM) - Security Price Models - The Black-Scholes Model (1973) - The Heston Model (1993)

Brownian Motion Definition. We say that a random process, , is a Brownian motion t : 0 t X with parameters ( , ) if 1. For 0 < t1 < t2< < tn-1 < tn ) ( , ) ( ) ( , ... , X X X X X X t t t t t t 2 1 3 2 1 n n are mutually independent. ( ) ( ) 2. For s>0, + 2 , X X N s s s t t 3. Xt is a continuous function of t. We say that Xt is a B( , ) Brownian motion with drift and volatility Remark. Bachelier (1900) Modeling in Finance Einstein (1905) Modeling in Physics Wiener (1920 s) Mathematical formulation

Wiener Process When =0 and =1 we have a Wiener Process or a standard Brownian motion (SBM) . We will use Btto denote a SBM and we always assume that B0=0 Then, ( ) ( ) = 0, B B B N t 0 t t Note that if Xt ~B( , ) and X0=x then we can write = + + X x t B t t Where Bt is an SBM. Therefore see that ( ) + 2 , t X N x t t Since, E B ( ) ( ) = + + = + = = 2 2 E X x t x t Var X Var B t t t t t

What are we going to speak about ? Background The Paper - Brownian Motion - Wiener Process (SBM) - Introduction - Discretization Schemes - Euler - Milshtein (1978) - Talay (1984, 1986) - Motivation - Abstract - Assumptions - Theorem - Proof - Interpretation - Experiments - More to go - Stochastic Calculus (It s Calculus) - Stochastic Integral (It s Integral) - Stochastic Differential Equation (SDE) - It Process - Diffusion Process - It s Lemma - Example : Geometric Brownian Motion (GBM) - Security Price Models - The Black-Scholes Model (1973) - The Heston Model (1993)

ItIntegral Definition. A partition of an interval [0,T] is a finite sequence of numbers T of the form 1 2 1 : , , , , 0 [( ) | T n n t t t t = = = ] t t t t T 1 2 1 n n Definition. The norm or mesh of a partition T is the length of the longest of it s subintervals, that is 1,...,n i = = = : max h t t 1 T i i Recall the Riemann integral of a real function n t ( ) ( )( ) = , lim h , s X ds t X t t 1 1 1 s i i i i 0 0 = 1 i And the Riemann-Stieltjes integral of a real function w.r.t a real function gt=g(t) n )( ) t ( ) ( = , lim h , s X dg t X g g 1 1 s s i i t t 0 1 i i 0 = 1 i Then the It integral of a random process w.r.t a SBM Bt n )( ) t ( ) ( = , lim h , s X dB t X B B 1 1 s s i i t t 0 1 i i 0 = 1 i It can be shown that this limit converges in probability.

Stochastic Differential Equation A stochastic differential equation (SDE) is of the form ( ) ( ) = + , , dX t X dt t X dB t t t t Which is a short-hand of the integral equation t t ( ) ( ) = + + , , X X s X ds s X dB 0 t s s s 0 0 Where Bt is a Wiener Process (SBM). t is a Riemann integral. ( ) 0 , s X ds s is an It integral. ( ) 0 t , s X dB s s

It Process An It process in n-dimensional Euclidean space t X n is a process defined on a probability space and : 0,T X ( ) , ,P n satisfying a stochastic differential equation of the form ( t dX = ) ( ) + , , t X dt t X dB t t t Where T > 0 Bt is a m-dimensional SBM : 0,T n n satisfies the Lipschitz continuity and polynomial growth conditions. : 0,T Existence and Uniqueness Conditions n m n satisfies the Lipschitz continuity and polynomial growth conditions. Such that t y + C D n , : 0,T ; , t x y ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + , , , , ; , , 1 t x t x t y C x y t x t x D x = 2 = 2 2 2 Where and . : i i : ij , i j These conditions ensure the existence of a unique strong solution Xt to the SDE ( ksendal 2003)

Diffusion Process A time-homogeneous It Diffusion in n-dimensional Euclidean space X n t ( ) is a process defined on a probability space : X , ,P n and satisfying a stochastic differential equation of the form ( ) t t dX X dt = ( ) + X dB t t Where Bt is a m-dimensional SBM is Lipschitz continuous. : n n n m n is Lipschitz continuous. : Existence and Uniqueness Conditions Meaning ( ) x ( ) y ( ) x ( ) y + n . . , C st x y C x y = 2 = 2 2 2 Where and . i : : ij i , i j This condition ensures the existence of a unique strong solution Xt to the SDE ( ksendal 2003).

It'sLemma ( ) ( ) Suppose Xt is an It process satisfying the SDE = + , , dX t X dt t X dB t t t t And f(t,x) is a twice- differentiable scalar function. Then 2 2 f t f x f f x = + + + df dt dB t 2 2 x In more detail 2 1 2 f t f x f f x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 = + + + , , , , , , , , df t X t X t X t X t X t X dt t X t X dB t t t t t t t t t 2 x Sketch of proof 2 1 2 f t f x f The Taylor series expansion of f(t,x) is = ) + + + 2 ... df dt dx dx 2 x ( ( ) ( ) Recall that ~ 0, dB B B N dt O dt t dt + t t Substituting and gives t x X Sketch of Proof + dx dt dB t 2 1 2 ( f t f x f ( ) ( ) = + + + + + + 2 2 2 2 2 ... df dt dt dB dt dtdB dB t t t 2 x ) dt As the terms and tend to zero faster. Setting and we are done. t dB dt 1.5 0 dtdB O dt 2 dt t 2

Geometric Brownian Motion dX ( ) ( ) = + , , t X dt t X dB Suppose Xt satisfies following SDE t t t t With and ( , t t X ) ( ) = = , X t X X t t t = X dt + X dB Thus the SDE turns into ( , f t X dX t t t t Since is a twice- differentiable scalar function in (0, ) ) ( ) log t t X = By It Lemma we have 2 2 f t f x f f x = + + + df dt dB t 2 2 x Then 2 2 2 1 X 1 1 X X ( ) = + + + = + log 0 t d X X dt X dB dt dB t t t t t 2 2 2 X t t t 2 2 t t ( ) ( ) ( ) The integral equation = + + = + + log log log X X ds dB X t B 0 0 t s t 2 2 0 0 2 + t B Finally t 2 = t X X e If X0 >0 then Xt > 0 for all t> 0 0

Geometric Brownian Motion t Definition. We say that a random process, , is a geometric : 0 t X Brownian motion(GBM) if for all t 0 2 + t B t 2 = t X X e 0 Where Bt is an SBM. We say that Xt ~GBM( , ) with drift and volatility Note that 2 ( ) t s + + B t s + 2 = X X e t s + 0 2 2 ( ) + + + t B s B B s t + t t 2 2 = X e 0 2 ( ) + s B B s t + t 2 = X e t -a representation which we will later see that is useful for simulating security prices.

Expected Growth Rate of GBM Define , where denotes the information available at time t. : | t t t E E = t ( ) N Recall the moment generating function of 2 , Y 1 2 2 2 s + s ( ) s E e = = sY M e Y Suppose Xt ~GBM( , ) , then 2 ( ) + s B B s t + t 2 = E X E X e t s + t t t ( ) 2 ~ 0, N s s ( ) 2 B B = X e E e s t + t t t 2 2 + s s 2 2 = = X e t s X e t - So the expected growth rate of Xt is

Properties of GBM Note that, 2 ( ) 2 + X X s B B t s + ( ) t 2 = = + + + log t s X t s X e s B B t s + t 2 t t Thus 1. Fix t1 < t2< < tn-1 < tn . Then X X X X X X t t t , ,..., 3 n 2 t t t 1 2 1 n are mutually independent. 2. Paths of Xt are continuous as a function of t, i.e. they do not jump. 3. For s>0, 2 X 2 + log ~ N , t s X s s 2 t

What are we going to speak about ? Background The Paper - Brownian Motion - Wiener Process (SBM) - Introduction - Discretization Schemes - Euler - Milshtein (1978) - Talay (1984, 1986) - Motivation - Abstract - Assumptions - Theorem - Proof - Interpretation - Experiments - More to go - Stochastic Calculus (It s Calculus) - Stochastic Integral (It s Integral) - Stochastic Differential Equation (SDE) - It Process - Diffusion Process - It s Lemma - Example : Geometric Brownian Motion (GBM) - Security Price Models - The Black-Scholes Model (1973) - The Heston Model (1993)

The Black-Scholes Model The Black-Scholes equation is a partial differential equation (PDE) satisfied by a derivative (financial) of a stock price process that follows a deterministic volatility GBM, under the no-arbitrage condition and risk neutrality settings. Starting with a stock price process that follows GBM t : 0 tS 2 + t B t 2 = S dt + S dB = dS S S e 0 t t t t t ( ) ( ) + We represent the price of the derivative of the stock price process as Where a twice- differentiable scalar function at St>K. : f = rt , f t S e S K t t 2 r interest rate K- strick price We apply the It lemma 2 2 2 S f t f s f f s = + + + t df S dt S dB t t t 2 2 s The arbitrage-free condition serves to eliminate the stochastic BM component, leaving only a deterministic PDE. = + + 2 2 2 S f t f s f t df S dt t 2 2 s 2 2 2 S f t f s f Risk neutrality is achieved by setting = r + + = t rS rf t df 2 2 s With we have the celebrated BS PDE: rf dt=

The Heston Model A more elaborated model, dealing with a stochastic volatility stock price process that follows a GBM = S dt + S t dS v S dB t t t t tv Where , the instantaneous variance, is a Cox Ingersoll Ross (CIR) process ( ) t t dv v dt = + v t v S dB t t S v and are SBM with correlation , or equivalently, with covariance dt. , t t B B is the rate of return of the asset. is the long variance, or long run average price variance: as t tends to infinity, the expected value of ttends to . is the rate at which treverts to . is the volatility of the volatility.

What are we going to speak about ? Background The Paper - Brownian Motion - Wiener Process (SBM) - Introduction - Discretization Schemes - Euler - Milshtein (1978) - Talay (1984, 1986) - Motivation - Abstract - Assumptions - Theorem - Proof - Interpretation - Experiments - More to go - Stochastic Calculus (It s Calculus) - Stochastic Integral (It s Integral) - Stochastic Differential Equation (SDE) - It Process - Diffusion Process - It s Lemma - Example : Geometric Brownian Motion (GBM) - Security Price Models - The Black-Scholes Model (1973) - The Heston Model (1993)

Monte Carlo Evaluation t X n Suppose is an It process satisfying the SDE ( ) ( ) = + , , dX t X dt t X dB t t t t and one is interested in computing ( ) E f X l T Sometimes it is hard to analytically solve the SDE or to determine its distribution. Although numerical computations methods are usually available (i.e. the Kolmogorov backward equation (KBE) via some finite-difference algorithm), in some cases it is convenient to obtain a Monte Carlo approximation ( ) 1 N N = ( ) i l f X T 1 i This requires simulation of the system.

The Monte Carlo realizations are done by simulating a discrete-time approximation of Discrete-time Simulation the continuous-time SDE. The time interval [0,T] is sampled at periods of length h>0. h k X Th X Denote as Xt evaluated at t=kh. h Thus is the discrete evaluation of XT. ( ) ( ) E f X E f X Denote the approximation of by . h l hl T Th e l l And the approximation error by . h h e = lim h 0 It can be shown, under some technical conditions, that . h 0 Definition. A sequence has order-k convergence if is bounded in h. he h k he

The Euler Approximation The integral equation form of the SDE t h + t h + ( ) ( ) += + + , , X X s X ds s X dB t h t s s s t t The Euler method approximates the integrals using the left point rule. t h + t h + ( ) ( ( ( ) )( ) t h + t h + ( ) ( ) , , s X dB t X dB , , s X ds t X ds s s t s s t t t t t ) ( ) = , t X B B = , t X h + t t h t t = , t X h With ~N(0,1). t h X Thus, one can evaluate XT with by starting at and proceeding by Th X ( 1 , k k k X X kh X h = x h 0 ) ( ) = + + h h h h k , kh X h + + 1 k For k=0, , T/h-1. Where 1, 2, are N(0,1) i.i.d. h X Note. The processes Xt and are not necessarily defined on the same probability space ( , , ). Th

The Euler Approximation Error Is said to be a first-order discretizaion scheme ( ) ( ) = + + h k h k h k h k , , X X kh X h kh X h + + 1 1 k In which eh has order-1 convergence. There is an error coefficient s.t. has a order-2 convergence. h + e h h h The error coefficient gives a notion of bias in the approximation. Although usually unknown it can be approximated to first order by ( ) 2 l l = 2 h h h h Under the polynomial growth condition of f (Talay and Tubaro, 1990).

The Milshtein Scheme X t h + t h + ( ) ( ) += + + , , X s X ds s X dB The integral equation form of the SDE t h t s s s t t ( ( ) ) ( ( ) ) = = = = , : t X X t t t The Milshtein method apply the It s lemma on . , : t X X t t t 2 2 1 2 = + + + + + 2 ' '' ' t t t t t d dt dB dt dB t t t t t t t t t t t 2 2 t x x x 2 2 1 2 = + + + + + 2 ' '' ' t t t t t d dt dB dt dB t t t t t t t t t t t 2 2 t x x x The integral form 1 2 1 2 s s = + + + 2 ' '' ' du dB s t u u u u u u u t t s s = + + + 2 ' '' ' du dB s t u u u u u u u t t Then the original SDE turns into 1 2 t h + s s = + + + + 2 ' '' ' X X du dB ds t h + t t u u u u u u u t t t 1 2 t h + s s + + + + 2 ' '' ' du dB dB t u u u u u u u s t t t

The Milshtein Scheme As the terms and are ignored. 0 h ( ) dsdu O h ( ) 1.5 2 dsdB O h u Thus t h + t h + t h + s += + + + ' X X ds dB dB dB t h t t t s u u u s Applying the Euler approximation to the last term we obtain t t t t t h + t h + t h + s s ( ) ) = ' ' ' dB dB dB dB B B dB u u u s t t u s t t s t s t t t t t ) ( ( t h + t h + ( ) = = + 2 ' ' B dB B B B B dB B B B t h + t h + t t s s t t t t s s t t t t 1 2 ( ) ( ) ( ) = 2 Y B t = + = , , dY a t Y dt b t Y dB BdB Define and suggest a solution . Indeed, Y dY a t 2 B B B t t t t t t t t t t 2 2 1 2 1 2 Y b Y Y = + + + = + + 1 1 + = 0 1 t t t t t dt b dB dt BdB BdB t t t t t t t t 2 Thus, 1 2 1 2 ( ) t h + t h + = = = = 2 2 B dB dY Y Y Y B B h t h + t h + s s s t t t t t and 1 2 1 2 1 2 1 2 ( ) ( ) t h + s ( ) 2 + = = 2 2 2 2 ' ' ' ' 1 dB dB B B h B B B B B h h t h + t h + t h + u u u s t t t t t t t t t t t t

The Milshtein Scheme Thus 1 2 1 2 1 2 ( ) t h + s ( ) 2 + = 2 2 2 ' ' ' dB dB B B h B B B B B h t h + t h + t h + u u u s t t t t t t t t t t Then t h + t h + t h + s += + + + ' X X ds dB dB dB t h t t t s u u u s t t t t Turns into the Milshtein discretization 1 2 ( ) += + + + 2 ' 1 X X h h h t h t t t t t With ~N(0,1). h X Thus, one can evaluate XT with by starting at and proceeding by Th X = x h 0 ( ) ( ) ( ) ( ) ( h 1 2 ) = + + + 2 h k h k h k h k h k h k ' 1 X X X h X h X X + + + 1 1 1 k k For k=0, , T/h-1. Where 1, 2, are N(0,1) i.i.d.

The Milshtein Scheme ( u dsdB O h ) If the terms and are to be visible. ( ) dsdu O h 32 2 We have Where And Remark. Euler and Milshtein deal with first and second-order discretization schemes respectively for 1-D SDE with . t X as Talay (1984,1986) provides second-order discretization schemes for multidimensional SDE with . : n 1 t X n

What are we going to speak about ? Background The Paper - Brownian Motion - Wiener Process (SBM) - Introduction - Discretization Schemes - Euler - Milshtein (1978) - Talay (1984, 1986) - Motivation - Abstract - Assumptions - Theorem - Proof - Interpretation - Experiments - More to go - Stochastic Calculus (It s Calculus) - Stochastic Integral (It s Integral) - Stochastic Differential Equation (SDE) - It Process - Diffusion Process - It s Lemma - Example : Geometric Brownian Motion (GBM) - Security Price Models - The Black-Scholes Model (1973) - The Heston Model (1993)

Discrete-time Monte Carlo Simulations The Monte Carlo realizations are done by simulating a discrete-time approximation of the continuous-time SDE. Denote, as the i th discrete-time simulation output. Th X ( ) h i iY 1 N = as a discrete-time crude monte carlo approximation of . ( ) ( ) 1 i N ( ) E f X l , l h N f Y T i as the approximation error . ( ) ( ) , , e h N l h N l N h The time required to compute is roughly proportional to . ( ) , l h N 2 n T Where n and T are fixed for a given problem. This paper pursues the optimal tradeoff between N and h given limited computation time.

What are we going to speak about ? Background The Paper - Brownian Motion - Wiener Process (SBM) - Introduction - Discretization Schemes - Euler - Milshtein (1978) - Talay (1984, 1986) - Motivation - Abstract - Assumptions - Theorem - Proof - Interpretation - Experiments - More to go - Stochastic Calculus (It s Calculus) - Stochastic Integral (It s Integral) - Stochastic Differential Equation (SDE) - It Process - Diffusion Process - It s Lemma - Example : Geometric Brownian Motion (GBM) - Security Price Models - The Black-Scholes Model (1973) - The Heston Model (1993)

Abstract This paper provides an asymptotically efficient algorithm for the allocation of computing resources to the problem of Monte Carlo integration of continuous-time security prices. The tradeoff between increasing the number of time intervals per unit of time and increasing the number of simulations, given a limited budget of computer time, is resolved for first-order discretization schemes (such as Euler) as well as second- and higher order schemes (such as those of Milshtein or Talay).

What are we going to speak about ? Background The Paper - Brownian Motion - Wiener Process (SBM) - Introduction - Discretization Schemes - Euler - Milshtein (1978) - Talay (1984, 1986) - Motivation - Abstract - Assumptions - Theorem - Proof - Interpretation - Experiments - More to go - Stochastic Calculus (It s Calculus) - Stochastic Integral (It s Integral) - Stochastic Differential Equation (SDE) - It Process - Diffusion Process - It s Lemma - Example : Geometric Brownian Motion (GBM) - Security Price Models - The Black-Scholes Model (1973) - The Heston Model (1993)

Assumptions C f is and satisfies the polynomial growth condition. ( T f X f X h ) ( ) h 0 as . i. Th ( ) ( ) 2 2 as (i.e. is uniformly integrable ). ( ) T E f X E f X ( l h O h = + + 0 h h h 0 2 f X h 0 h ii. Th Th ) p p hl iii. as , where 0 and p>0. ( ) h f X iv. The (computer) time required to generate is given the deterministic Th ( ) = + q q as , where >0 and q>0. h h O h h 0 Theorem Given t units of computer time. 1 1 If or as then ( ) ( ) 0 t tht tht + + 2 2 q p q p i. p )( e h N ) ( , as . t t + 2 q p t t 1 If , where 0<c< , as then c ( 1 ( ) t tht + 2 q p ( ) 0 tht + 2 q p ii. ) 12 p as , where is N(0,1) and ) , q t t t e h N c )( ( ) t = Var f X + 2 p ( c + 2 q p T

Proof

The Asymptotically Optimal Allocation ( ) Assuming the discritization error alone has order-p convergence with h = + p p hl l O h h t As the computation budget t gets large the period ht should have order-1/(q+2p) convergence with t. 1 ( ) tht c + 2 q p Then the estimation error has order-p/(q+2p) convergence in probability with t. ( ) 12 p )( e h N ) + p ( , t c + 2 q p q c t t 1 ( ) If it the above does not hold, the estimation error does not converge in distribution to zero as quickly (i.e. not at this order ). 0/ tht + 2 q p p )( e h N ) ( , t + 2 q p t t It follows that this rule is Asymptotically Optimal .

Interpertation ( ) ( ) h 1 + Informally, if then , where c t 2 q p = + th c ( ) q q . tht t N N h O h + 2 q p t t t 2p h N Const Thus, t t T Let be the number of time intervals, as ht is the period. t t h n : 2 = = : 4 = For the Euler scheme (p=1) : For the Milshtein or Talay schemes (p=2): : n n h h N N 12 t t t t t t : 2 = = : 16 = : n n h h N N 12 t t t t t t ( ) 1 12 ( ) tht c p )( e h N ) + 2 q p For the asymptotically optimal allocation, if then Thus ) ( , t t t t t e h N h ht e h N = + p ( , t c + 2 q p q c t t ( ) ) ) ( ( 12 p p p )( ) ( ) ( ) 1 1 + p p p p p ( ( ) ( ) , , , h ht e h N h c e h N c + + + 2 2 2 q p q p q p q c t t t t t t t t t Thus, ( ) + p , e h N h + 2 q p t t t c As the root-mean-squared estimation error is and the error bias is . p e h + 2 q p t c For the Euler scheme (p=1) : For the Milshtein or Talay schemes (p=2): : 2 = = = : : n n h h e e 1 1 2 2 t t t t : 2 = = = : : n n h h e e 1 1 2 4 t t t t

Experiments - Settings ( ( ) ) ( ( ) ) = = , t X X rX t t t Consider the SDE with . ( ) ( ) , , t t t t dX t X dt t X dB = + = = , 0 t X X X X t t t t The diffusion process Xt>0 which satisfies the constant-elasticity-of-variance model (Cox,1975) = + X dB dX rX dt t t t t with , can be simulated by the Euler or Milshtein Schemes. 0.5,1 A European call option on the asset with strike price K and expiration date T is , has we would try to estimate its initial price by . ( ) T Y e X K = ( ) ( ) ( ) + + E f X = = rT rT f X e Y X K T T T For =1 we obtain the Black-Scholes model. Technical Details. For which f is and satisfies the polynomial growth conditions everywhere but at x=K. C ( ) 2 + + x K x K ( ) x ( ) Then it can be uniformly well approximated for above by . TechnicalDetails = 0,1 f 2 If we take to be of order-2p convergence we preserve the quality of aproximation. For we have as (Yamada, 1796 and dominated convergence). ( T f X f X ) ( ) ) h 0 h 0.5,1 Th

What are we going to speak about ? Background The Paper - Brownian Motion - Wiener Process (SBM) - Introduction - Discretization Schemes - Euler - Milshtein (1978) - Talay (1984, 1986) - Motivation - Abstract - Assumptions - Theorem - Proof - Interpretation - Experiments - More to go - Stochastic Calculus (It s Calculus) - Stochastic Integral (It s Integral) - Stochastic Differential Equation (SDE) - It Process - Diffusion Process - It s Lemma - Example : Geometric Brownian Motion (GBM) - Security Price Models - The Black-Scholes Model (1973) - The Heston Model (1993)

Experiments - Results For the particular case: We take The following results are obtained, Comments For the Euler scheme (p=1) : For the Milshtein scheme (p=2): : 2 = = : 4 = = = , : n n N N e e 12 = : 2 : 16 , : n n N N e e 14 The assumption hold with other but normal i.i.d increments with zero mean and unit variance uder technical conditions (Talay, 1984,1986).

What are we going to speak about ? Background The Paper - Brownian Motion - Wiener Process (SBM) - Introduction - Discretization Schemes - Euler - Milshtein (1978) - Talay (1984, 1986) - Motivation - Abstract - Assumptions - Theorem - Proof - Interpretation - Experiments - More to go - Stochastic Calculus (It s Calculus) - Stochastic Integral (It s Integral) - Stochastic Differential Equation (SDE) - It Process - Diffusion Process - It s Lemma - Example : Geometric Brownian Motion (GBM) - Security Price Models - The Black-Scholes Model (1973) - The Heston Model (1993)

More to go More monte carlo methods for asset pricing problems: Boyle (1977,1988,1990) Jones and Jacobs (1986) Boyle, Evnine and Gibbs (1989) An Alternative large deviation method for the optimal tradeoff : chapter 10 of Duffie (1992). Extensions ) ( + T X X dt Path dependent security payoffs T t 0 ( ) = Stochastic short-term interest rates (Duffie,1992) r R X t t A path dependent lookback payoff inf X X T t t Extension of the general theory of weak convergence for more general path-dependent functionals More discretization schemes: Slominski (1993,1994), Liu (1993) Adding a memory constraint

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