Transcript Monte-Carlo Simulations

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. 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.
Brownian Motion
 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 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
approximately normally distributed with
mean= (µ - σ2/2 ) and variance= σ2t.
Where:
αt is the drift
ztσt is the stochastic component
be
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.
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
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q
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n
c
y
45
45
40
37 37
35
30
29
30
24
25
20
15
13
10
7
5
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
5
2
1
7
2
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165
25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170
Stock price at time n
Probability Distribution of Stock Price
0.2000
P
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a
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0.1800
0.1600
0.1400
0.1200
0.1000
0.0800
0.0600
0.0400
0.0200
0.0000
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165
25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170
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