Numerical Methods for Option Pricing

1
Numerical Methods for Option Pricing
Prof Olivier Pironneau

• Kimiya Minoukadeh
• Ecole Polytechnique
• M2 Mathématiques Appliquées, OJME

2
Agenda

• Introduction to Monte-Carlo method
• Heston stochastic volatility model using M-C
• Basket option using Monte-Carlo
• Accuracy of Monte-Carlo methods
• Variance Reduction methods
• Conclusion

3
Monte-Carlo Method I

• Based on the expectation of a random variable
  X, given
• N samples X1,X2,,XN

• Price of a European Call option is therefore
  calculated as

where is the ith estimate of the stock
price at time T, the time of maturation, r is the
risk free interest rate and K is the strike
price.
4
Monte-Carlo Method II

• The stock price St follows the stochastic
  differential
• Equation (SDE)
• where
• is the drift term
• is the volatility

5
Agenda

• Introduction to Monte-Carlo method
• Heston stochastic volatility model using M-C
• Basket option using Monte-Carlo
• Accuracy of Monte-Carlo methods
• Variance Reduction methods
• Conclusion

6
Heston Stochastic Volatility I

• Studies have shown that the volatility , if
  held constant, does not reproduce observed market
  data. We therefore consider the model suggested
  by Heston

volatility of stock rate of mean reversion
volatility mean volatility of volatility
The cost of the call at time t 0
7
Heston stochastic volatility II

• Results are consistent with the a priori lower
  bounds known for call options.

8
Heston stochastic volatility III

• Barrier options pose the constraint that a
  certain asset is never allowed to reach outside a
  certain interval a,b.

Expectation of payoff considerably
reduced Price of option reduced
b 130
a 0
9
Agenda

• Introduction to Monte-Carlo method
• Heston stochastic volatility model using M-C
• Basket option using Monte-Carlo
• Accuracy of Monte-Carlo methods
• Variance Reduction methods
• Conclusion

10
Basket options I

• Sometimes a derivative may be based on more
  than one underlying asset. S(1),S(2),,S(p)
• The Black-Scholes equation becomes
  p-dimensional. We consider the case of two
  underlying assets p 2, and once again the
  Brownian motions have a correlation
• Payoff is based on the sum of the two stocks
  at time T

11
Basket options II

• Suppose we use
• L starting prices of each of the two stocks
• N samples of the estimated stock prices
• M intervals for the calculations of the stock
  prices using explicit Eulers method
• Complexity of the program would be O(L2NM).
• To reduce this by a factor M to O(L2N), we use
  Itos Lemma with Yi log(S(i)) to obtain the
  explicit solution to the SDE

12
Basket options III
By using the explicit solution we can observe
that we get desirable results, accuracy similar
to using the Explicit Eulers method, however
time performance improved dramatically.
ERROR ANALYSIS
TIME PERFORMANCE
13
Basket options IV

• Letting K K1 K2 for the respective quasi
  strike prices of stocks S(1) and S(2), we observe
  the following results

By choosing S0(1) K1 100, we observe that
results resemble that of a standard European call
option with one underlying asset
14
Agenda

• Introduction to Monte-Carlo method
• Heston stochastic volatility model using M-C
• Basket option using Monte-Carlo
• Accuracy of Monte-Carlo methods
• Variance Reduction methods
• Conclusion

15
Accuracy of Monte-Carlo method
The central limit theorem shows that the accuracy
of the Monte-Carlo method is controlled by
Thus to halve the error we would need to
quadruple the number of samples N used in the
Monte-Carlo simulation.
16
Agenda

• Introduction to Monte-Carlo method
• Heston stochastic volatility model using M-C
• Basket option using Monte-Carlo
• Accuracy of Monte-Carlo methods
• Variance Reduction methods
• Conclusion

17
Variance Reduction Methods I
IDEA Reduce the variance of the random process
X. For an independent random process Y, we note
that The variance is then given by therefore
we have
18
Variance Reduction Methods II
Need to choose a random variable Y such that it
is closely correlated with X. We adapt a
method suggested by P. Pellizzari 1 for
variance reduction of basket options
1 P. Pellizzari. Efficient Monte-Carlo pricing
of basket options. Finance, EconWPA, 1998
19
Variance Reduction Methods III
We see that we considerably improve the accuracy
of the Monte-Carlo method when using variance
reduction technique.
With variance reduction, we obtain with N 2000
samples, results as accurate as the normal
Monte-Carlo method with N10000 samples.
20
Agenda

• Introduction to Monte-Carlo method
• Heston stochastic volatility model using M-C
• Basket option using Monte-Carlo
• Accuracy of Monte-Carlo methods
• Variance Reduction methods
• Conclusion

21
Conclusion
The Monte-Carlo method is intuitive and extremely
easy to implement It can be used to calculate
call prices when an analytic solution of a PDE
does not exist Data is consistent with observed
data For well estimated expectations we need
many sample simulations. To double accuracy,
number of samples must quadruple.
IMPROVEMENT When analytic solutions do not exist
and we are obliged to use Monte-Carlo methods,
variance reduction can improve the performance of
the calculation.