Suppose S = 40, = 0.30, r = 0.06, = 0. Suppose you sell a 45-strike call with 91
ID: 2445200 • Letter: S
Question
Suppose S = 40, = 0.30, r = 0.06, = 0. Suppose you sell a 45-strike call with 91-days (T = 91/365 = 0.24931) to expiration. Suppose the call priced at 0.9207 and = 0.2704 for a single share. If the option is on 100 shares, what investment is required for delta- hedged portfolio? Suppose tomorrow the stock price is 39. The put is now priced at 0.6664 (T = 90/365 = 0.24658) and = 0.2157 for a single share. What is your overnight profit? Suppose you is going to re-balance the portfolio for delta-hedging. What is the value of the new portfolio?
Explanation / Answer
Answer: First, let’s value the call and find delta.
Applying the standard B-S formula, C(S,K, ,r,t, ) = C($40,$45,30%,6%,91/365,0) = $0.971, with d1=-0.5783 and d2=-0.7280. There is no dividend, so the delta is N(d1) = N(-0.5783) = 0.2815 (which answers the first part of the question).
Since we are trying to hedge a sold option, we need to replicate the long option, so will be buying delta shares. Now we can figure out our portfolio cash flows:
Today: Sell call, get premium $0.971 Buy delta shares -0.2815 x $40 = -$11.262
Borrow the net cost $11.262 - $0.971 = $10.291 (this is the net investment required for a delta-hedged portfolio, per the second part of the question)
Net outflow per share (nothing!)
Next day, if price is $39: Liability under option -$0.705 (recalculate, now with C($39,$45,30%,8%,90/365,0))
Value of shares held 0.2815 x $39 = $10.980
Money owed under loan -$10.291 x e 8% x 1/365 = -$10.293
Overnight loss per share -$0.018
Total overnight loss -$1.82
Similarly, next day, if price is $40.50:
Liability under option -$1.105 (recalculate, now with C($40.5,$45,30%,8%,90/365,0))
Value of shares held 0.2815 x $40.5 = $11.403
Money owed under loan -$10.291 x e 8% x 1/365 = -$10.293
Overnight gain per share $0.005
Total overnight gain $0.49
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