Simulating the value of Asian Options Vladimir Kozak

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Simulating the value of Asian
OptionsVladimir Kozak
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Variance Reduction Techniques for Asian Options

• Some of the possible methods
• Control Variates
• Stratified Sampling
• Importance Sampling

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Definition

• Discretely sampled Asian Call Option has payoff
  at maturity
• Where
• Our objective is to find the value of the Asian
  option where
• And the stock price dynamics under this measure
  is
• It follows that the distribution of is
• where

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• The value of the the Asian Option is given by
  
• where
• We want to reduce the variance of the value of
  the Asian option calculated using Monte Carlo
  simulation
• From our discussion on control variates we know
  that we can estimate unbiasedly by
  estimator
• where
• The idea is to find a random variable such
  that it is close to so that the variance
  of
• is small
• Moreover, we must be able to evaluate
  analytically

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• Observe that if random variables Xi i1,2n are
  lognormal, then geometric mean
• has also lognormal distribution
• Assuming that geometric mean is close to
  arithmetic mean a good control variate is the
  random variable
• where
  

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• The Closed form solution for E(V2) is easily
  obtained since it can be written in the form
• An this is the same integral over lognormal
  density that leads to the Black-Scholes formula

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Asian Options and Stratified Sampling

• For many options, the terminal value of the stock
  has a lot of influence on option price
• By stratifying the terminal value of a stock
  price, much of variability in the options payoff
  can be eliminated
• In the case of the asset price dynamics described
  by the geometric Brownian Motion, we can stratify
  the terminal value of the stock

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• The idea is to stratify the terminal value of a
  Brownian Motion W(T) and then randomly generate
  the rest of the path W(1), W(2) W(T-1) using
  the Brownian Bridge interpolation
• We know that under the risk-neutral measure,
• Where ,
• To stratify into K strata of equal probability
  for ST, we generate ZT

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• We can interpolate the rest of each stock price
  path using the fact that the distribution of
  Brownian Motion increments
• W(jk)-W((j-1)k), j1,2N, NkT conditionally
  on the values W(t(j-1)k) and W(T) is Normally
  distributed with
• Mean (((N-j)/N )W((j-1)k)jW(T)/N)
• and
• Variance(N-j)/(N-j1)

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Importance Sampling and Pricing of Asian Options

• Suppose that an Asian call option is well out of
  money, so that most of the randomly generated
  values for are below the strike K,
  contributing 0 to option price.
• To decrease variance we can generate stock prices
  from a geometric Brownian Motion with a drift
  larger than the the original drift so that it is
  more likely that
• and multiply the result by the Radon Nikodym
  derivative of one process with respect to the
  other

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• To price an Asian option, we need to evaluate
• Under this measure we assume
• (1)
• But since
• 
  (2)
• We may simulate the stock prices assuming a
  different stock price dynamics
• 
  (3)

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• We simulate the path recursively
  through
• Where
• If we change the Stock price dynamics from (1)
  to (3), then
• Where is the pdf of under measure Q and
  is the pdf of under the new measure P