Math623_Lecture3_Spring2016

Math623_Lecture3_Spring2016 - Computational Finance Lecture...

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Computational Finance - Lecture 3 David Eliezer, lecture notes by Paul M. N. Feehan Lecturer in Mathematics Rutgers, The State University of New Jersey and Numerix, LLC Spring 2016 Rutgers University – Mathematics 623
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Outline Sources of bias in Monte-Carlo estimators
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Sources of bias in Monte-Carlo estimators As well as reducing variance in a Monte Carlo estimator, we shall also need to reduce its bias. Possible sources of bias include I Payoff discretization; I Model discretization; I Payoff non-linearity; I Pseudo-random number generator quality. Almost all payoffs are non-linear functions of S ( t ). Poor choices of pseudo-random number generators lead to poor simulations of the desired probability density functions.
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Bias due to payoff discretization I Payoff discretization is one source of bias in estimators used in Monte Carlo computation of prices of options with path-dependent payoffs. As a first example, discretization of continuous integrals may introduce bias. If we use Monte Carlo simulation to compute the price of an option depending on a continuous average , S Average ( T ) := 1 T Z T 0 S ( u ) du , there will be an error (and so a bias) introduced by any choice numerical integration method. For this reason, practitioners prefer to specify a discrete average when designing an option payoff, ˆ S Average ( T ) := m X i =1 S ( t i ) ,
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Bias due to payoff discretization II where 0 = t 0 < t 1 < · · · < t m = T and the prices, S ( t i ), are unambiguous (for example, daily closing prices). Alternatively, one can partially correct for this discrete monitoring bias using a correction due to Broadie, Glasserman, and Kou [ ? ]. As a second example, discrete monitoring of a continuous barrier may introduce bias. If we use Monte Carlo simulation to compute the price of a barrier option with continuous monitoring of a barrier B > K , E h e - rT ( S ( T ) - K ) + 1 { M ( T ) B } i , there will be a bias due to the fact that numerical schemes can can only simulate the discrete running maximum, ˆ M ( T ) := max 0 i m { S ( t i ) } .
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Bias due to payoff discretization III For this reason, practitioners again prefer to specify a discretely monitored barrier when designing an option payoff for a client. As an exercise, can you tell in which direction the estimator will be biased for each kind of barrier option? Alternatively, one can partially correct for this discrete monitoring bias using a correction due to Broadie, Glasserman, and Kou [ ? ]. See Glasserman [ ? , § 6.4] for an explanation of this idea in the case of payoffs depending on extremes (maxima or minima) or averages. In summary, the Terms and Conditions (T & C) sheet will specify discrete monitoring, but if it specifies continuous monitoring, we can either use a Broadie-Glasserman-Kou correction or offer to revise the T & C sheet.
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Bias due to payoff discretization IV As a third example, for a lookback call option, ( ¯ M ( T ) - K ) + , we would offer to replace the continuous running maximum, ¯ M ( T ) := max u [0 , T ] S ( u ) , by the discrete running maximum, ¯ M ( T ) := max i [0 , 1 ,..., N ] S ( t i ) .
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Model discretization error Model discretization error is introduced when we choose to
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