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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