Solution of Derivative

For the knock out call the likelihood of getting

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Unformatted text preview: 217 2.2667 2 7.4398 2.4505 3.0360 3 10.2365 2.8529 3.5881 4 12.6969 3.1559 4.0232 5 14.9010 3.4003 4.3823 100 39.9861 5.3112 7.5286 Question 14.8. See Table Four for the values. This highlights the trade-off increasing the time to maturity has on the knock out call option. When time to maturity increases, the standard call has the interest on the strike as well as the higher dispersion of ST making it more valuable. For the knock out call, the likelihood of getting knocked out can offset this effect. Table Four (Problem 14.9) Months Black-Scholes U & O Call 1 0.1727 0.1727 2 0.5641 0.5479 3 0.9744 0.8546 4 1.3741 1.0384 5 1.7593 1.1243 6 2.1304 1.1468 7 2.4886 1.1316 8 2.8353 1.0954 9 3.1718 1.0482 10 3.4991 0.9962 11 3.8180 0.9430 12 4.1293 0.8906 Ratio 1.0003 1.0296 1.1401 1.3233 1.5649 1.8577 2.1991 2.5885 3.0260 3.5124 4.0488 4.6365 Question 14.10. Using σ = 30%, r = 8%, and δ = 0. See Figure Two on the next page. When we are close to maturity (e.g. T = 1/52) we see large variations in delta. The discontinuity at K2 can require deltas greater than one. The value of the option can go to close to zero to close $10 with little movement in the price (if ST is close to K1 . If T 0, delta will be close zero for S < 100, enormous for S = 100, and close to one if S > 100. This problem does not occur as T becomes larger. 107 Part 3 Options Figure 2 (Problem 14.14) 1.6 1.4 1.2 Delta of Gap Call 1 0.8 0.6 6=1/52 6=1/4 6=1 0.4 0.2 0 90 92 94 96 98 100 5 102 104 106 108 110 Question 14.12. Under Black Scholes the standard 40-strike call on S will be BSCall (40, 40, .3, .08, 1, 0) . (1) For the exchange option on S using 2/3 of a share of Q as the strike, we use a strike of (2/3) 60 = 40, a volatility of .32 + .52 − 2 (.5) (.3) (.5) = .43589, and an “interest rate” of .04: BSCall (40, 40, .43589, .04, T , 0) . (2) For, all but very long time to maturities, the higher volatility will offset the lower “interest” and the exchange option will be worth more. With T = 1, we have the standard option is worth 6.28 and the exchange option is worth 7.58. Question 14.14. XYZ will have a “natural hedge” when x ($ price of Euro) and S ($ price of oil) move in together. For example, if x rises (the Euro appreciates implies “good news” for XYZ) and S rises (“bad news” for XYZ) the two risks offset. Similarly if x falls and S falls. When the two move opposite, the company is either “win-win” (x ↑ and S ↓) or “lose-lose” (S ↑ and x ↓). An exchange option paying S − x is therefore natural for XYZ. They will give up upside to hedge against downside. This is likely to be cheaper than treating the two risks separately due to ρ > 0 implying the exchange option will have a lower (implied) volatility. 108 Chapter 14 Exotic Options: I a) Since the options will be expiring at t1 , we have the payoff of a put if ST < K and the payoff of a call if ST > K . This is equivalent to a K strike straddle. b) Using put-call parity at t1 , the value of the as-you-like-it option at t1 will be: max C (S1 , K, T − t1 ) , C (S1 , K, T − t1 ) + Ke−r(T −t1 ) − Se−δ(T −t1 ) (3) = C (S1 , K, T − t1 ) + max 0, Ke−r(T −t1 ) − Se−δ(T −t1 ) (4) = C (S1 , K, T − t1 ) + e−δ(T −t1 ) max 0, Ke(δ−r)(T −t1 ) − S . (5) The first term is the value of a call with strike K and maturity T ; the second term is the payoff from holding e−δ(T −t1 ) put options that expire at t1 with strike Ke(δ−r)(T −t1 ) . a) In 6 months, a 3 month at-the-money call option will be worth 6.9618 if S = 100, 3.4809 if S = 50, and 13.9237 if S = 200. Note it is always 6.9618% of the stock price. b) In six months (t1 = 1/2), we will need .069618ST ; this can be done by buying .069618 shares of stock (since there are no dividends). c) We should pay $6.9618 for the forward start (the cost of the shares); this is the same as the current value of a 3m at-the-money option. d) Using similar arguments, a 3m 105% strike is always worth 4.7166% of the stock price. We should then pay $4.7166 for a forward start 105%-strike option. a) $6.0831 b) The current price of a 1m 95-strike put is 1.2652. In fact, a 1m put with a strike equal to 95% of the stock price will always be equal to 1.2652% of the stock price. Therefore, the present value of twelve of these 1m 95% strike puts is 12 (1.2652) = 15.182. c) Technically, and perhaps non-intuitively, the rolling insurance strategy costs more because it is more expensive to replicate. Note that one strategy doesn’t dominate another. If the price never falls less than 5% in month, all of the 12 one month options will be worthless; yet the price in 1 year could have fall by more than 5%. Interest aside, the rollover options will give the holder 11 i =0 max (Si +1 − .95Si , 0); whereas, the simple insurance gives the holder max(S12 − .95S0 , 0). The rollover strategy has the advantage of being able to provide payoffs (insurance) for each month regardless of the...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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