Question

Assume that the (annual) interest rate is 2% (continuous compounding), the stock has a volatility of 60%, there is 1 year until expiration of the contract, and the underlying stock is currently traded at $50 in the market. For a call struck at $55, use the Black-Scholes formula to calculate

(a) the value of the call,

(b) the delta and vega of the call. Based on the delta and vega calculated above,

(c) approximately how much does the value of the call go or down if the underlying goes up $1?

(d) approximately how much does the value of the call go or down if the volatility goes down by 2%?

Answer 1

a)

Risk-free rate r = 2%

Strike price K = $55

Current stock price S = $50

T=1 year

Volatility s = 60%

N is the cumulative standard normal distribution function

Substituting in BSM for call price c, we get

d1 = (ln(50/55)+(0.02+(0.6*0.6/2))*1) / ( 0.6*1) = 0.1745

N(d1) = 0.5693

d2 = 0.1745 - 0.6*1 = -0.4255

N(d2) = 0.3352

c= 50*0.5693 - 0.3352*55*e^(-0.02*1) = 10.39

Hence the value of the call c = $10.39

b) Delta of a call option is given by N(d1) = 0.5693 ( as calculated in part a)

Vega of a call option is given by

where is the probability density function

d1 = 0.1745

= 0.39287

= 50*1*0.39287

Hence Vega = 19.643

c)

Delta of a call option is the amount an option price is expected to go up based on a $1 change in the underlying stock.

Hence, a $1 increase in stock price will increase the call option price by 1 delta ie. $0.5693

d)

Vega measures how much the option price will increase or decrease in case of an increase or decrease of volatility and is quoted as price change of the option for every 1 percentage point change in volatility.

The call price goes down with a decrease in volatility. Hence, a 2% decrease in volatility will result in 0.02*19.64 = $0.3928 decrease in the value of the call option