Ewa Lukasik - Jakub Lawik - Juan Mojica - Xiaodong Xu


Valuation of asian basket option
Sampling:
Time period: 13.10.2008 to 13.11.2008 for 24 trading
days
Currency: SEK
Notional amount: 1 SEK
Stock indices involved:
Nikkei 225
FTSE 100
DJIA
(i=1)
(i=2)
(i=3)
Si(tj)-Value of the i-th index being taken as the closing price at the end
of the t-th trading day (excluding Si0)
Si(0)-Value of the i-th index at the opening of the market on 13.10.2008
Ri-Ratio of the mean index to beginning value
S*- indicator of the relative change of the values of indices during the
contract period
i- Number of the index
tj- Number of the trading day

Monte Carlo simulation
 Helps to simplify financial model with uncertainty involved
in estimating future outcomes.
 Be applied to complex, non-linear models or used to
evaluate the accuracy and performance of other models.
 One of the most accepted methods for financial analysis.

Application
 Generating sample paths
 Evaluating the payoff along each path
 Calculating an average to obtain estimation

Mathematically
 If we want to find the numerical integration:
1
S   f ( x)dx
0
 We can simply divide the region [0,1] evenly into M
slices and the integral can be approximated by:
S
1
M

M
n 1
f ( xn )  O(1 / M 2 )
 On the other hand, we can select xn for n=1,...,M from
a random number generator. If M is large enough, xn is
set of numbers uniformly distributed in the region
[0,1], the integration can be approximated by:
S  fn 
1
M

M
n 1
f ( xn )

For example:
 The value of the derivative security:
Pr ice  e  rT E Q  f  S0 , , ST  
 For Monte Carlo method, approximating the
expectation of the derivative’s future cash flows:
Pr ice  e
 rT
1
N

N

n 1
f  S0 ,

, ST  

 The mean of the sample will be quite close to
accurate price of derivate in a large sample
 The rate of convergence is 1/√N

Data selection for the underlying asset
 Our Underlying assets are assumed to follow
geometrical Brownian motion, which begin with:
d(logSi)- change in the natural logarithm of i-th asset’s value
- drift rate for i-th asset
-volatility of i-th asset
dt - time increment
dW - Wiener process

Then to obtain process which is martingale after
discounting, we set drift rate μi to
, as a result:
r- risk free rate
 Therefore,the index value process we obtain the following
form of geometrical Brownian motion:

It leads to 24 simulated time steps in our case for obtaining
required level of accuracy.
Advantages of geometric Brownian motion as a
model for price process
 No arbitrage argument
 Dividend model for stock indices
 Price process
What is quanto?
How we incorporate currency interdependence
into price process
 drift rate
Price process


Measure of statistical dispersion, averaging
the squared distance of its possible values
from the expected value (mean)
Parameter not observable in the market.

Variance is Constant in timePeriod with high (low)

volatility is usually followed
This method is more efficent inbylonger
a period time
with high
(low) volatility
periods
 Increase of computational time and complexity

Variance is Stochastic
 Volatility clustering (autoregresive property)
 ARCH & GARCH methods
▪ Autoregressive conditional heteroskedastic
▪ Succesful in short term contracts

Volatility calculated from historical data
 Simplest method
 Future = Past
 Sample SD from previous period
 Sample data should be from a similar previous
recent period ***

Volatility calculated from implied data
 Implied from other derivatives contracts traded
Pricing
of the option was performed with
on the market
estimates
based
onbehistorical
data.
 Price
of volatility
should
the same for
all traded
assets.
 Remark: There is no exact analytical formula for
implied volatility (or covariance). Values are
obtained by means of numerical algorithms.




Need to model more than one price process!
In the financial world there are thousands of
reciprocal relations between different
markets
Correlation method: Cholesky decomposition
- correlated normally distributed variables
[N(0,1)]

Estimating error
 Approximation error
 Unstable correlations and volatilities

Enhancing accuracy
 Geometric Brownian motion
 Set of random variables
Option value
0,4834
Number of simulations
10000
Variance of results
0,2480
Standard error of simulation
0,00498
Probability of expiring in the money (P)
0,1456
Probability of expiring in the money (Q)
0,4851
Confidence interval
Confidence level
0,4717-0,49494
99%
Width of confidence interval
0,02317
Width of confidence interval (% of price)
0,04911
Greek
Value
0,000149
0,000399
0,000149
0,005879
-0,00139
-0,00847
1,245513
0,109605
0,388600