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```Monte Carlo: Option Pricing
Reference: Option Pricing by
Simulation, Bernt Arne Ødegaard
(http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node12.html)
Introduction
Use Monte Carlo to estimate the price of
a Vanilla European option priced by
Black Scholes equation.
 Already a closed form solution, therefore,
no need to simulate,
 But, for an illustrative process.

… Introduction

At maturity, a call option is worth:


CT = max (0, ST – X)
At an earlier date t, the option will be the
expected present value of this:

Ct = E[PV(max (0, ST – X))]
Risk Neutral Result: Simplify
Decision made by a risk neutral investor
 Also modify the expected return of the
underlying asset such that it earns the
risk free rate.



ct = e-r(T – t)E*[(max (0, ST – X))]
Where E*[.] is a transformation of the
original expectation.
Monte Carlo

One way to estimate the value of the call is to
simulate a large number of sample values of
ST according to the assumed price process,
and find the estimated call price as the
average of the simulated values.
 According the “law of large numbers”, this
average will converge to the actual call value,
depending on the number of simulations that
are performed.
Lognormally distributed randoms
Let x be normally distributed with mean
zero and variance one.
 If St follows a lognormal distribution, the
one-period-later price St+1 is simulated
as
2)+x
(r-½
 St+1=Ste

…..Lognormally distributed randoms

Or more generally,
ST  St e
1 2
( r   )(T t )  T t x
2
Pricing of European Call Options
 ct = e-r(T – t)E*[(max (0, ST – X))]
 Note that here one merely needs
to
simulate the terminal price of the
underlying, ST,
 the price of ST at time between t and T is
not relevant for pricing.
…Pricing of European Call Options
Proceed by simulating lognormally
distributed random variables.
 Let ST,1, ST,2, …. ST,n denote the n
simulated ST values

…Pricing of European Call Options

We estimate E*[max (0, ST – X)] as the
average of option payoffs at maturity,
discounted at the risk free rate.
n

 r (T  t )
ct  e
((  max( 0, ST ,i  X ) / n)
i 1
Price of the Call ( and r constant)
Ct  SN (d1 )  Xe
S
2
log( )  (r  )(T  t )
X
2
d1 
 T t




d 2  d1   T  t
1
N (d ) 
2

d


e

x2

2
dx


 r (T  t )
N (d 2 )
C=Price of the Call
S=Current Stock Price
T=Time of Expiration
X=Strike Price
r=Risk-free Interest Rate
N()=Cumulative normal
distribution function
e=Exponential term (2.7183)
=Volatility
Results:

S = 100; X = 110; r = 0.1;
 sigma = 0.4; t = 6
 Exact ct = 53.4636
 Monte Carlo:

# sims: 10
# sims: 1,000
# sims: 1,000,000
ct = 15.4533
ct = 54.9804
ct = 53.5126

# sims: 100,000,000
ct = 53.4593

# sims: 1,000,000,000
ct = 53.4722


```