1. Monte-Carlo Simulations Seminar Project

2. Task  To Build an application in Excel/VBA to solve option prices.  Use a stochastic volatility in the model.  Plot the histogram of the outcome and calculate the probability to reach the strike.

3. Introduction  Monte Carlo simulation treats randomness by selecting variable values from a certain stochastic model.  We use Brownian motion to model the stock price and the Euler Scheme in the Heston model type to implement stochastic volatility into the model.  The periodic return is expressed in continuous compounding and it is a function of two components: 1. Constant drift 2. Random shock

4. The idea of Monte Carlo It is a well-known method to estimate the Value at Risk (VaR) with regard to the asset class i.e. stocks. It is an application of the geometric Brownian motion also called Weiner process. This process models the random behavior of the stock price in continuous time.

5. This geometric Brownian motion satisfies the stochastic differential equation given by the formula: Where: S t = Stock price at time t µ = drift σ = volatility dw t = Wiener process Brownian Motion

6. Brownian Motion in discrete time The formula is as follows: Where: ΔS t is the Change in the stock price for a unit of time. Δt is the time interval (one trading day). ε is the standard normal random number.

7. Markov Process Implication The price of tomorrow only depends on today´s price and not the past. Provides the sense of market efficiency

8. Lognormal Returns The lognormal random variable will be approximately normally distributed with mean= (µ - σ 2 /2 ) and variance= σ 2 t. Where: α t is the drift z t σ t is the stochastic component

9. Denpendence on t The initial expected daily drift (α t )to be positive because we assume that historically the expected return of the stock grows over time. Then, the following α t will be calculated with the formula: Daily drift – Where the daily drift is the annual drift divided by 252 trading days The reason for this calculation is because the stochastic volatility erodes the returns The stochastic component z t is the random shock which is a function of the stock price.

10. Where: preset drifts is the standard normal random number at time t. Euler Scheme in the Heston model type for stochastic volatility

11. Calculating the European Call Option We run the simulation “m” times for “n” nodes or trading days. Then, we evaluate every stock price that comes out as a final outcome of each simulation. The European call option is: Where: =Last node or time step. = Number of simulations. = The value of the call option at the last node which resulted from each of the simulations or paths. = The final stock price i.e. at the last node for each of the simulations. = The strike price which is a given constant.

12. Calculating the European Call Option The arthimetic mean of the payoff is calculated by And it is disounted by: Where: is the fair price of the option today. r is the risk free interest rate n is the total number of nodes

13. Histogram Representation The construction of both Stock Price Distribution and Probability distribution in histogram is very straight forward in Excel.  The final stock price is grouped in to classes.  In the Excel the histogram assigns the Frequency for each class.  The probability distribution is also calculated depending on the out come of the final Stock price divided by the frequency of each bar.  The Histogram is finally depicted frequency versus the fair price and the probability as well.

14. Distribution of Final Stock Price

15. Probability Distribution of Stock Price

16. Black-Scholes Comparison The Black-Scholes formula is: The B-S model requires that both the risk-free rate and volatility remain constant over the period of analysis. When comparing the call option’s fair price calculated using the B-S formula with our method, it can deviate quite a bit and sometimes get very close to B-S which may mean just a shot of luck.

17. …. And finally we present the Implementation in Excel showing our results Excel Implementation