Question

We are in a Black and Scholes world. A stock today has a price of 100. The discretely compounded one-year risk-free interest rate is 0.05. A European put on this stock with a strike price of 100 that expires in one year has a price of 8.893. What is the price of a European call on this stock with a strike price of 110, which expires in one year? Report in two digits behind the comma, i.e. 0.345 +0.35.

Answer 1

Call Premium C= S N(d1) - Ke^-rt N (d2)

Put Premium P = K e ^-rt N(-d2) - S N((-d1)

Where d1= [ln(S/K) + (r+ variance/2)T]/stdev*t^(1/2)

d2 = [ln (S/K) + (r- Variance /2)T]/stdev* t^(1/2)

N(-d2) = 1-N(d2)

N(-d1) = 1 - N(d1)

Hear S= Spot Price

K = Strike Price

r= Risk Free rate

t = Time to Expiry

First we have got all the values of a put option except standard deviation therefore by using the formula for put vauation we get standard deviation from excel:

Stock price	Annual Dividend yield (D/P)	Exercise Price	Risk free Rate	Time to expiration (yrs)	Volatility (Annualized)	Adjusted Stock Price	d1	N'(d1)	d2	Put premium
100	0	100	0.05	1	0.158204	100	0.395149	0.368981	0.236945	8.893598

By using goal seek function we find out that at what level of volatility put will be 8.8935.

Now using volatility we can get the value of a call option by putting all the values in the call formula:

Stock price	Annual Dividend yield (D/P)	Exercise Price	Risk free Rate	Time to expiration (yrs)	Imp. Volatility (Annualized)	Adjusted Stock Price	d1	N'(d1)	d2	Option premium
100	0	110	0.05	1	0.158204	100	-0.2073	0.390462	-0.36551	4.39554

Therefore, the value of a call option is 4.40.