Monte-Carlo Simulations

Seminar Project

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.

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.

2.

Constant drift Random shock

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.

Brownian Motion

 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 in discrete time

The formula is as follows: Where: ΔS ε t is the Change in the stock price for a unit of time.

Δt is the time interval (one trading day).

is the standard normal random number.

Markov Process Implication

 The price of tomorrow only depends on today´s price and not the past.

 Provides the sense of market efficiency

Lognormal Returns

The lognormal random variable approximately normally distributed with mean= (µ - σ 2 /2 ) and variance= σ 2 t.

will be Where: α t z t σ t is the drift is the stochastic component

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.

Euler Scheme in the Heston model type for stochastic volatility

Where: preset drifts is the standard normal random number at time t.

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.

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

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.

Distribution of Final Stock Price

50

F r e q u e n c y

45 40 35 30 25 20 15 29 45 37 37 24 30 13 10 5 0 7 6 5 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 1 2 20 25 25 30 30 35 35 40 40 45 45 50 50 55 55 60 60 65 65 70 70 75 75 80 80 85 85 90 90 95 95 100 100 105 105 110 110 115 115 120 120 125 125 130 130 135 135 140 140 145 145 150 150 155 155 160 160 165 165 170

Stock price at time n

Probability Distribution of Stock Price

P r o b a b l i t y

0,2000 0,1800 0,1600 0,1400 0,1200 0,1000 0,0800 0,0600 0,0400 0,0200 0,0000 20 25 25 30 30 35 35 40 40 45 45 50 50 55 55 60 60 65 65 70 70 75 75 80 80 85 85 90 90 95 95 100 100 105 105 110 110 115 115 120 120 125 125 130 130 135 135 140 140 145 145 150 150 155 155 160 160 165 165 170

Stock price at time t

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.

Excel Implementation

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