Question

2- a. What is the price of a European put option on a non-dividend-paying stock when the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the historical volatility is 35% per annum, and the time to maturity is six months?

b. WITHOUT using the B-S-M formula, please calculate the call option price on the same stock with $70 strike price and a time to maturity of 6 months? Explain how and show the process. (Simply showing a number without explanation and process is not a valid answer.)

c. For the same put option, if the actual market trading price is $5, what is the implied volatility?

Answer 1

B.) Even though the option value can be easily calculated using the Black-Scholes Option pricing formula, we can make use of the Monte Carlo Simulation technique to achieve the same results.

Step 1

The role of Monte Carlo simulation is to generate several future value of the stock based on which we can calculate the future value of the call option. The changes in the stock prices can be calculated using the following formula:

In this equation, ε represents the random number generated from a standard normal probability distribution. In our example, we will calculate this number using the Rand () function in excel. A standard normal variable can be approximated using the Excel formula “= Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() + Rand() – 6.0”.

For the purpose of this example, we will generate 1000 random paths. In reality, the higher the number of trials, the more accurate will be the result. For Instance, if the random number is 0.576548, ΔS will be 19.29978.

Step 2

Once we have ΔS, we can calculate the future value of the stock price (S + ΔS). We need to do this for each path.

The stock value at expiry will be 69 + 19.29978 = 88.29978

Step 3

The option value at expiry will be given by the formula =MAX (0, S-X)

= MAX (88.29978 – 70) = 18.29978.

The above three steps will be repeated 1000 times to get 1000 option values.

Step 4

We will take the average of these 1000 option values. In our example, this value comes to approx. 7.13895. Please note that this value will change every time the spreadsheet is recalculated, so you may never get the same answer.

Step 5

The option value will be discounted to the present value by multiplying it with exp(-r*t)

In our example, this value comes to 7.13895.

European
Call or Put (C/P)	C
Spot Price	69
Strike Price	70
Time to Maturity	0.5
Volatility	35.00%
Risk-Free Rate	5.00%
Dividend Yield	0.00%
No of Steps	10
No of Simulation	10000
Option Price	7.13895