Question

(Black-Scholes)

Assume S = $22.00, σ = 0.30, r = 0.05 p.a. continuously compounded, the stock pays a 1.0% p.a. continuous dividend and the option expires in a month.

(a) What is the price of a $20 strike call? (Leave 2 d.p. for the answer)

(b) What is the delta? (Leave 3 sig. fig. for the answer)

(c) If the stock price changes to $22.10 immediately, what is the approximate change of the call price based on delta? (Leave 3 sig. fig. for the answer)

(d) What is the percentage change in the option for a 1% change in the stock? (Leave 3 sig. fig. for the answer)

(e) What is the volatility of the call option? (Leave 2 d.p. for the answer)

(f) If the Sharpe ratio of the stock is 0.3. What is the Sharpe ratio of the call?

Answer 1

Answer: To find the price we have to use the following formula of Black Scholes:

C= S N(d1) - K N(d2)

where, C = price of the call option = Find?

S = Spot Price = $20

K = Strike Price = $22

ln = ln is the natural logarithm

r = risk-free interest rate = 0.05

= given = 0.30

t = time of expire = 1 month

d = dividend =0.01

=1.182

= 1.096

C= 22(1.182) - 20(1.096)

=26.004 -21.96

= 4.004

a) Price pf Call option is = $4.004

b) delta = 0.30 which is given

c) Assuming that the current price of the stock is $20, and it changes to $22.1. It mean $ 2.1 has been increase

change of price x N(d1) =+2.1 x 1.182 = 1.154. The call option will be increase by $1.154.

d) Assuming that the current price of the stock is $22 and assuming a change of 1% as increasing

1% of $22 =.$22 i.e, $22.22 after increase

change of price x N(d1) =+.22 x 1.182 =0.26004. The call option will be increase by $ 0.26004.

as decreasing i.e, price change from $22 to$21.78, change of price x N(d1) = -0.22 x 1.182 =(-0.26004). The call option will be decreased by $ 0.26004

e) delta  =0.30 which is given in the question indicates volatility.