Monte-Carlo SimulationsSeminar 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. Constant drift 2. 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.

This geometric Brownian motion satisfies the stochastic differential equation given by the formula:

Where:   St = Stock price at time t

µ = drift σ = volatility dwt = Wiener process

Brownian Motion

Brownian Motion in discrete time

The formula is as follows:

Where: ΔSt 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. 

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 will be approximately normally distributed with

mean= (µ - σ2/2 ) and variance= σ2t.

Where:

αt is the drift

ztσt is the stochastic component

Denpendence on t

The initial expected daily drift (αt )to be positive because we assume that historically the expected return of the stock grows over time. Then, the following αt 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 zt is the random shock which is a function of the stock price.

Where: preset drifts

is the standard normal random number at time t.

Euler Scheme in the Heston model type for stochastic volatility

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

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

0

5

10

15

20

25

30

35

40

45

50

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3

29

45

37 37

24

30

7

13

6

2 1 2

57

Stock price at time n

Frequency

Probability Distribution of Stock Price

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

0.1600

0.1800

0.2000

Stock price at time t

Probablity

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.

…. And finally we present the

Implementation in Excel showing our

results

Excel Implementation