2. Monte Carlo Simulation
• A method that numerically ‘imagines’ many possible
scenarios to solve deterministic and probabilistic
problems
• A numerical method which enables the modeling of the
future value of a variable by simulating its behavior
over time.
• It calculates statistical properties such as
expectations, variances or probabilities of certain
outcomes.
• The method is usually quite simple to implement in
basic form and so is extremely popular in practice.

3. Finance Application
• represents the future behavior of equities, exchange
rates, interest rates etc., for two reasons
– study the possible future performance of a portfolio
– price derivatives
Exploring portfolio or
Pricing Derivatives
cash flow statistics
• to determine quantities
• to calculate the present
such as expected
value of the expected
returns, risk, possible
payoff of an option under
downsides, probabilities
a risk-neutral random
of making certain profits
walk
or losses, etc.

4. Monte Carlo Simulation and Option Pricing
• The Monte Carlo technique on option pricing, first
proposed by Boyle (1977), simulates the process
generating the returns on the underlying asset and
invokes the risk-neutrality assumption.
• The method is “simple and flexible … it can easily be
modified to accommodate different processes
governing the underlying asset returns.”
• The accuracy of the results depends on the number of
simulations

5. MCS to Option Pricing: Pros and Cons
Advantages Disadvantages
• May be computationally
intensive
• Flexible
• Calculations can take much
• Can generally be easily
longer than analytical
extended and developed as
models
required
• Solutions are not exact, but
• Easily understood
depend on the number of
repeated runs

6. Stages of a Monte Carlo Simulation
• Identify the probability distribution of the input variables.
• Imitate the movement of the input variables by repeatedly
drawing random numbers, which are adjusted to have the
same probability distribution as the underlying variables.
• Simulate the underlying variable by combining the input
variables together according to the logic of the system.
• Repeat this process many times to get the simulated future
value.
• Increase accuracy by applying variance reduction techniques

7. Application: Option Pricing
Suppose we want to value a 1-year European
call option on the FTSE Index
Given:
= 1,000 ( )
= 1,000 ( )
= 6% . ., ( − )
=1 250 ( ℎ )
= 15.9% ( )

8. Application: Option Pricing
Example  Stage1  Stage2  Stage3  Stage4  Stage5
Stage 1: Identify the probability distribution
Transform uniformly distributed random variables into a random variable with
a probability distribution that matches the empirical distribution of the
option’s underlying asset (FTSE index)
Relative frequency distribution : FTSE Index returns (1984-1992) Cumulative frequency distribution : FTSE Index returns (1984-1992)
250 1.00
200 0.80
150 0.60
100 0.40
50 0.20
0 0.00

9. Application: Option Pricing
Example  Stage1  Stage2  Stage3  Stage4  Stage5
Stage 2: Imitate the input variables
Cumulative frequency distribution : FTSE Index returns (1984-1992)
• Generate a large number of
1.00
uniformly distributed
0.80
random numbers ranging
0.60 from 0 to 1
0.40
0.20
• Each random variable is
located on the y-axis of the
0.00
cumulative density function
and the corresponding
For example, random number 0.472
return on the x-axis is taken

10. Application: Option Pricing
Example  Stage1  Stage2  Stage3  Stage4  Stage5
Stage 3: Simulate the underlying variable
Adjust the observed daily return given the assumption of a risk neutral
framework, i.e., we assume that the return on the FTSE index is
equivalent to the risk-free rate
We need to solve for r, such that
σ2 2
er e 0.06 250
r 0.01006942 2 0.06 250
r 0.000189
where σ2 is the variance of FTSE index daily returns

11. Application: Option Pricing
Example  Stage1  Stage2  Stage3  Stage4  Stage5
Stage 3: Simulate the underlying variable
Compound the current asset price by the random daily returns for
each of the trading days during the life of the option. This is equal
to 1 simulation run.
 r250
ST 1000 e r1 e r2  e r250 1000 e r1 r2
where rn = random observation of the 1-day continuously
compounded return drawn from the same empirical
probability distribution as the underlying data
n = number of trading days

12. Application: Option Pricing
Example  Stage1  Stage2  Stage3  Stage4  Stage5
Stage 4: Repeat the process
Repeat Stages 1 to 3 many times (say, 125,000 times) and compute
the value of the call option at time T, (CT).
• Simulate future values (ST)
1
• Calculate CT = max [ST - X, 0]
2
• The average of all 125,000 CT , when discounted is the option
3 value

13. Application: Option Pricing
Example  Stage1  Stage2  Stage3  Stage4  Stage5
Stage 5: Increase accuracy
Reduce the variability of the mean to be compatible with your
accuracy requirements:
1 n
1 n 2 varS j
ST Sj varS j Sj ST SEST
n j 1 n j 1 n
Rule of thumb for determining the number of runs:
2
St
n
SE
It follows that to reduce the standard error by a factor of 10, the number of
simulations must be increased by a one-hundred fold

14. Application: Option Pricing
Example  Stage1  Stage2  Stage3  Stage4  Stage5
Stage 5: Increase Accuracy
Variance Reduction Techniques:
Antithetic variate technique Control variate technique
Each time random variable
r is drawn, its complement 1-r
calculated and used to drive a Find an option that is both
parallel run of the simulation. highly correlated to the one
you’re trying to estimate
This tends to lead to negatively and has a definite value.
correlated output values;
hence lower variance

15. Examples: Path Dependent Options
 Asset: S = 100  Drift: 5%
 Step: 0.01
 Strike price : X = 105  Volatility: σ = 20%
Notes: Simulation:
δS = rS δt + σS δt φ
Taking relatively small then transformed into lognormal
number of paths S(t + δt) = S(t) + δS
S(t + δt) = S(t) exp((r − 1/2σ2) δt + σsqrt(δt) φ)
250 simulations for a one
year pay off XLS formula: Sim = So*EXP((rate-0.5*σ*σ)*step
+σ*SQRT(step)*NORMSINV(RAND()))
Pay off: Option Value:
Call Payoff = MAX(ST - X,0) option value = e−r(T−t) E [payoff(S)]
Put Payoff = MAX(X -ST ,0) XLS formula: PV = Mean*EXP(rate*ST)

16. Examples: Path Dependent Options
Time Sim1 Sim2 Sim3 Sim4 … Sim249 Sim250
0 100.00 100.00 100.00 100.00 … 100.00 100.00
0.01 100.71 100.36 99.84 99.26 … 97.41 98.32
0.02 101.31 100.66 98.61 101.81 … 97.17 94.99
0.03 105.43 99.23 97.72 105.42 … 95.68 96.04
0.04 108.38 97.93 96.81 108.78 … 95.18 96.79
0.05 107.00 93.64 97.27 109.51 … 97.23 96.57
0.06 106.35 94.31 98.53 107.92 … 95.21 96.16
0.07 108.95 95.88 99.76 108.67 … 95.22 96.12
. . . . . . . .
0.08 109.61 96.85 98.34 109.81 … 92.74 101.18
0.98 111.24 122.08 124.02 139.08 … 127.40 89.89
0.99 110.41 118.50 122.21 141.00 … 128.76 86.23
1.00 109.82 120.11 119.42 139.42 … 129.31 87.60
CALL Payoff 4.82 15.11 14.42 34.42 24.31 0.00
Mean 8.08
PV 8.50
PUT Payoff 0.00 0.00 0.00 0.00 0.00 17.40
Mean 8.90
PV 9.35

17. Examples: Exotic Options
Simulation at run one:
Asian Lookback Barrier
A knock in an
Optimum value for a call option could
ST become active
only when the
stock crosses
this line
average over
entire run
Asian option payoffs are
calculated using the Fixed Lookback option Barrier options either
average over the entire payoffs are calculated knock in or knock out
run rather than the using the optimal value above or below a certain
value of the underlying (maximum for a barrier.
at time t. call, minimum for a put)

18. More on Options Pricing
• Any option that has a payoff formula can be priced with
a Monte Carlo method.
• Very versatile - Multiple underlying, changing
volatility, a different distribution assumption for the
random walk, etc.
• For some options, like the Asian option, variance
reduction strategies can make the Monte Carlo method
very accurate at low trial sizes – meaning that it is both
accurate and quick.
• Besides being used for these particular options, this
method is also often used to double check other
implementations.

19. Conclusion
• The Monte Carlo method is primarily used for pricing of
several kinds of exotic options for which there is no
formula or the formula is difficult.
• It is also good as a way to double check any
implementation of option pricing.
• Unfortunately, there aren’t really any programs that
simply do Monte Carlo approximations of options 
you basically have to write an implementation
yourself, or copy code from someone.

20. References:
• Boyle, P (1977) Options: a Monte Carlo approach. Journal of
Financial Economics 4 323–338
• Glasserman, P (2003) Monte Carlo Methods in Financial
Engineering. Springer Verlag
• Jackel, P (2002) Monte Carlo Methods in Finance. John Wiley
& Sons
• Watsham and Parramore (1997) Quantitative Methods in
Finance
• Wilmott, P. Frequently asked Questions in Quantitative
Finance
• Wilmott,P. Paul Willmot Introduces Quantitative Finance.
p581-604
• Monte-carlo Simulation. http://www.sars.org.uk/old-site-
archive/BOK/Applied%20R&M%20Manual%20for%20Defenc
e%20Systems%20(GR-77)/p4c04.pdf