Question

Suppose S = 25, sigma = 32%, r = 4%, o = 0%, and T = 182 days. Suppose you buy a 23-strike call. What is delta? If the option is on 100 shares, what investment is required for a delta-hedged portfolio? What is your overnight profit if the stock price tomorrow is S = 24.552 What if the stock price is $26.25?

Answer 1

In the Black Scholes world, the delta of a call option is given by N(d₁)

where ) + + entit

and the price of the call option is given by:

Value of call = S N (d1) - KetN(d2) where () + (6 + d2 = di-o vt

Please look at the snapshot from my model below:

7 Inputs 8 s 25 9t 0.50 182 days = 182 / 365 = 0.50 years 10 0 32.00% 11 K 23 23 strike call, hence strike price = 23 12 r. 4.00% 13 Output Values Formula Used 14 d1 0.570253145 =(LN(C8/C11)+(C12+C10^2/2)*C9)/(C10*SQRT(C9)) 15 d2 0.344289152 =C14-C10*SQRT(09) 16 N(D1)= 0.715746992 =NORMSDIST(C14) 17 N(d2) 0.634685579 =NORMSDIST(C15) 18 Value of the call option, c. 3.58 =C8*C16-C11*EXP(-C12*09)*C17

Hence, delta of the call option = N(d₁) = 0.715747

The option is on 100 shares. Let's say we need to short = Number of shares under call option x Delta of the call option / Delta of the stock = 100 x 0.715747 / 1 = 71.57 number of shares.

Hence, we need to short (sell) 71.57 number of shares to create a delta hedged portfolio.

Hence,  investment required for a delta-hedged portfolio = Cost to buy option on 100 shares - proceed from short sell of 71.57 shares = 100 x C - 71.57 x S = 100 x 3.58 - 71.57 x 25 = - 1,430.95

The negative sign signifies that there will be an inflow of 1,430.95 on creating a delta hedge portfolio.

When S = 24.552:

Your overnight profit = 100 x V_call - 71.57 x S - initial investment = 100 x max (S - K, 0) - 71.57 x 24.552 - (-1,430.95) = 100 x max (24.552 - 23, 0) -  326.35 = 100 x 1.552 - 326.35 = - 171.15 (i.e there is a loss of 171.15)

When S = 26.25:

Your overnight profit = 100 x V_call - 71.57 x S - initial investment = 100 x max (S - K, 0) - 71.57 x 26.25 - (-1,430.95) = 100 x max (26.25 - 23, 0) - 447.89 = 100 x 3.25 - 447.89 = - 122.89 (i.e there is a loss of 122.89)