Ewa Lukasik - JakubLawik - Juan Mojica - Xiaodong Xu Monte Carlo Simulation for Asian Basket Option
The outline of our Project • 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)
Parameter and Algorithm 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
Theoretical background • 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
Theoretical background • 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 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:
Theoretical background • 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
Valuation process • 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
Theoretical background • 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.
Price process • Advantages of geometric Brownian motion as a model for price process
Dividends • No arbitrage argument • Dividend model for stock indices • Price process
Currency interdependence • What is quanto? • How we incorporate currency interdependence into price process • drift rate • Price process
Variance • Measure of statistical dispersion, averaging the squared distance of its possible values from the expected value (mean) • Parameternot observable in themarket.
VarianceAproaches Period with high (low) volatility is usually followed by a period with high (low) volatility • VarianceisConstant in time • Thismethodismore efficent in longer time periods • Increase of computational time and complexity • VarianceisStochastic • Volatilityclustering (autoregresive property) • ARCH & GARCH methods • Autoregressiveconditionalheteroskedastic • Succesful in short termcontracts
VarianceConstant Variance: Historical vs Implied Volatilities • Volatilitycalculatedfromhistorical data • Simplestmethod • Future = Past • Sample SD frompreviousperiod • Sample data shouldbefrom a similar previousrecentperiod***
VarianceImplied vs Historical Volatilities • Volatilitycalculatedfromimplied data • Impliedfromotherderivativescontractstradedonthemarket • 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.
Multi-asset price processBasket Options • Need to model more than one price process! • In the financial world there are thousands of reciprocal relations between different markets • Correlationmethod: Cholesky decomposition • - correlated normally distributed variables [N(0,1)]
Greeks • Estimating error • Approximation error • Unstable correlations and volatilities • Enhancing accuracy • Geometric Brownian motion • Set of random variables