Question

Stock-A has current price, SA = $90 and stock-B has current price, SB = $80.

Put options on A and B have 30 (trading) days to maturity and are currently both at-the

money. The risk-free rate is r = 2% pa (continuously compounded).

You do not have closed form solutions (e.g. Black-Scholes) for these option

prices but the portfolio deltas and gammas for long positions in these put options

are change A= -400 Gamma A = 30

change B= -200 Gamma B = 20

and   .

The mean growth rate and volatility for stock-A are 5% pa and 15% pa, respectively

and for stock-B the figures are 10% pa and 20% pa, and the correlation between the

two stock returns is 0.7.

a). Carefully, outline the steps you would take to simulate stock prices A and B over

10 trading days using Monte Carlo Simulation (MCS), and how you would calculate

the possible outcomes for the change in value of your option’s portfolio (over 10 trading

days).

b). Assume you only hold options-A. Using your own formulas in Excel, perform a

MCS (for m=1000 simulations) over 10 trading days. Show the possible changes in the

option’s premium and present the results in a histogram.

Now, calculate the change in value of your option’s position at the “5th percentile”.

Briefly comment and interpret your results.

c). Use MCS (in Excel) to price option-A and measure the error in your estimate.

Set out the steps in your analysis and briefly comment on your results by noting any

changes in your MCS analysis between b). and c).

Provide the Excel file with the above “illustrative simulations”.

Answer 1

The two components that results in any asset price change are Drift ad Volatility. These two are the most imortant building blocks of Monte Carlo Simulation.

Now using Excel we can calculate the periodic daily return using historical price data by the formula as given below -:

Periodic Daily Return=ln(Day’s Price/ Previous Day’s Price​)

Now in order to calculate the Drift use he following formula

Drift= Average Daily Return- Variance/2

Where average Daily return can be calculated from Excel using Average formula to the daily return calculated above and variance can be calculated using Var.P function

Next Step is to obtain a Random input using the following formula -:

Random Input= Standard Deviation (Sigma) * NORMSINV(RAND())

Therefore the next day prices can be calculated by -:

Next Day Price= Day's Price * e^ (Drift+Random value)