# Rutgers University MATH 612 - Computing Greeks using Monte Carlo methods (41 pages)

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## Computing Greeks using Monte Carlo methods

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## Computing Greeks using Monte Carlo methods

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Lecture Notes

Pages:
41
School:
Rutgers University- The State University of New Jersey
Course:
Math 612 - Selected Topics in Applied Mathematics
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Computing Greeks using Monte Carlo methods Mike Giles giles comlab ox ac uk Oxford University Computing Laboratory Monte Carlo Greeks p 1 41 Introduction Given a stochastic o d e dS S t dt S t dW subject to initial conditions S 0 S0 the expected value of a payoff V E f S T can be approximated by calculating the average over a number of discrete paths given by some time marching approximation S n 1 S n S n tn t S n tn W n Monte Carlo Greeks p 2 41 Introduction Strengths simple scales well to multiple correlated factors Weaknesses American options are challenging Glasserman Broadie methods to reduce Monte Carlo variance compromise the simplicity stratified sampling importance sampling Latin hypercube sampling quasi MC computing Greeks can be a problem Monte Carlo Greeks p 3 41 Introduction The pathwise approach to computing E f S T S0 relies on and f being Lipschitz continuous and piecewise differentiable so one can interchange the derivative and the expectation to get df e S T E f S T E S0 dS where S T e S t S0 Monte Carlo Greeks p 4 41 Introduction e is approximated by linearising the time discretisation to S t get n n Sen 1 Sen Sen t Sen W S S and the expectation is again approximated by averaging over multiple paths Monte Carlo Greeks p 5 41 Introduction If and f have Lipschitz continuous first derivative and are piecewise twice differentiable then 2 2 E f S T E f S T 2 2 S0 S0 2 d f e2 df ee E S T S T 2 dS dS where 2 S t ee S t S02 can be approximated by again differentiating the discrete stochastic equations Monte Carlo Greeks p 6 41 Introduction What can go wrong Consider geometric Brownian motion dS r S dt S dW and a digital option f S 1 H S 1 K 1 S K 0 S K For this case and are both non zero but we get zero values for 2 df e d f e2 df ee E S 1 E S 1 S 1 2 dS dS dS Monte Carlo Greeks p 7 41 Payoff Regularisation Key idea replace discontinuous payoff H S K by a smooth payoff H S K with width 1 2 1 0 8 h 0 6 0 4 0 2 0 H H 0 2 5 4 3 2 1 0 x 1 2 3 4 5 How should H

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