6 Options II

Fig 9 1 let z n0 1 be a standard normal random

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Unformatted text preview: OPTIONS II p. 10 of 18 D. Binomial (Two- State) Option Pricing Trees {Hull Ch. 12 (omit §12.9); BKM §21.3} 1. Call Option Pricing (1- Period): [Fig. 3] uS0 Fig. 3 ST−K $200 a) A no- dividend stock sells for S0 = $100 $75 today. At the end of one year, its price will S0 c either: dS0 $100 • rise by factor u = 2 to uS0 = $200, or $0 $50 • decline by factor d = ½ to dS0 = $50. b) Consider a call option on this stock with K = $125 and T = 1 year. In one year, it will be worth its payoff ST − K = 200 − 125 = $75 (if ST = 200 > K) or $0 (if ST = $50 < K). c) Now construct the following perfect hedge portfolio H: • Buy ½ share of stock (pay $50) • Write/sell one call option (receive $c) 1) Cost today H0 = $50 − c 2) Value in 1 year {HT}: • If ST = $200, buy ½ share for 100 and sell 1 share to option holder for K = $125. You are left with HT = $125 − $100 = $25. • If ST = $50, option not exercised. You own stock worth HT = 50(1/2) = $25. 3) By replication principle #3, H0 = HTe−rT. Let rf = 10%/year. • 50 – c = 25e−0.10(1) = $22.62 ⇒ c = 50 − 22.62 = $27.38 d) Summarizing the above result. 1) We know the current stock price S0 and option strike price K and time to maturity T. 2) If we also know the only two possible values of the stock price ST, we can use replica- tion to deduce the value of the call option and its premium c. 2. The Way it Works – Arbitrage: {Hull §12.1 - 12.2} a) By constructing the perfect hedge portfolio, we invest H0 = ½S0 – c = $50 – c at the beginning, and get $25 back for certain (i.e., with no risk) at the end of one year. 1) This means that our investment grew at a risk- free rate of return rf, and there can’t be more than one risk- free rate at a given time. 2) So if rf = 10%, then (50 – c)e0.10(1) = $25, so 50 – c = 25e- 0.10(1) = $22.62, and c = $27.38. 3) That is, ½S0 – c = $22.62 invested now grows at rf = 10% to $25 in one year. b) Suppose that c = $30 > $27.38 (and hold S0 = $100 and rf = 10% constant)....
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