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 .
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St = Stock price at time t
µ = drift
σ = volatility
dwt = Wiener process
The formula is as follows:
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.
The lognormal random variable will be approximately normally distributed with
mean= (µ - σ2/2 ) and variance= σ2t.
αt is the drift
ztσt is the stochastic component
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
is the standard normal random number at time t.
=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.
And it is disounted by:
is the fair price of the option today.
r is the risk freeinterest rate
n is the total number of nodes
The construction of both Stock Price Distribution and Probability distribution in histogram is very straight forward in Excel.
The Black-Scholes formula is: