Diapositiva 1
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Transcript Diapositiva 1
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