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
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