# Assume that the (annual) inter

**Assume that the (annual) interest rate is 2% (continuouscompounding), the stock has a volatility of 60%, there is 1 yearuntil expiration of the contract, and the underlying stock iscurrently 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 deltaand vega calculated above,**

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

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

Answer:

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 expectedto go up based on a $1 change in the underlying stock.

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

d)

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

The call price goes down with a decrease in volatility. Hence, a2% decrease in volatility will result in 0.02*19.64 =**$0.3928 decrease** in the value of the calloption