1. Ewa Lukasik - Jakub Lawik - Juan Mojica - Xiaodong Xu

2.  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 (i=1) FTSE 100 (i=2) DJIA (i=3)

3. S i (t j )-Value of the i-th index being taken as the closing price at the end of the t-th trading day (excluding S i0 ) S i (0)-Value of the i-th index at the opening of the market on 13.10.2008 R i -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 t j - Number of the trading day

4.  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

5.  Mathematically  If we want to find the numerical integration:  We can simply divide the region [0,1] evenly into M slices and the integral can be approximated by:  On the other hand, we can select x n for n=1,...,M from a random number generator. If M is large enough, x n is set of numbers uniformly distributed in the region [0,1], the integration can be approximated by:

6.  For example:  The value of the derivative security:  For Monte Carlo method, approximating the expectation of the derivative’s future cash flows:  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

7.  Data selection for the underlying asset  Our Underlying assets are assumed to follow geometrical Brownian motion, which begin with: d(logS i )- 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

8.  Then to obtain process which is martingale after discounting, we set drift rate μ i 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.

9.  Advantages of geometric Brownian motion as a model for price process

10.  No arbitrage argument  Dividend model for stock indices  Price process

11.  What is quanto?  How we incorporate currency interdependence into price process  drift rate  Price process

12.  Measure of statistical dispersion, averaging the squared distance of its possible values from the expected value (mean)  Parameter not observable in the market.

13.  Variance is Constant in time  This method is more efficent in longer time 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 Period with high (low) volatility is usually followed by a period with high (low) volatility

14.  Volatility calculated from historical data  Simplest method  Future = Past  Sample SD from previous period  Sample data should be from a similar previous recent period ***

15.  Volatility calculated from implied data  Implied from other derivatives contracts traded on the market  Price of volatility should be 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. Pricing of the option was performed with estimates based on historical data.

16.  Need to model more than one price process!  In the financial world there are thousands of reciprocal relations between different markets  Correlation method: Cholesky decompositionCholesky decomposition  - correlated normally distributed variables [N(0,1)]

17.  Estimating error  Approximation error  Unstable correlations and volatilities  Enhancing accuracy  Geometric Brownian motion  Set of random variables

18. Option value0,4834 Number of simulations10000 Variance of results0,2480 Standard error of simulation0,00498 Probability of expiring in the money (P)0,1456 Probability of expiring in the money (Q)0,4851 Confidence interval0,4717-0,49494 Confidence level99% Width of confidence interval0,02317 Width of confidence interval (% of price)0,04911

19. Greek Value 0,000149 0,000399 0,000149 0,005879 -0,00139 -0,00847 1,245513 0,109605 0,388600

20. Thank You!!!