Solution of Derivative

032 08455 18 which equals 50e04 901 e 0552 2 note

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Unformatted text preview: y b) If denominated in cash, we could make the bet fair by setting the strike price equal to 2 K = Se(r −δ−.5σ )T , which is the median (50% of the probability is above this value). This will make d2 = 0 and the bets worth e−rT /2 which is not a surprise since the sum of the two bets must 2 be worth e−rT . Using T = 1, r = 6%, σ = 30%, we have K = 100e.06−.3 /2 = 101.51. c) If denominated in shares, we could make the bet fair by setting the strike price equal to 2 2 K = Se(r −δ+.5σ )T = 100e.06+.3 /2 = 111.07, which is above the forward price. This makes d1 = 0 and the bets worth S/2 = 50. Question 21.10. If we purchase one unit of the claim, −VS shares, and invest W in the risk free bond, our investment is worth I = V (S, t) − VS S + W = 0. By purchasing one claim, we will receive a dividend of dt that will be added to dI . The change in the investment value is dI = dt + Vt dt + VS dS + = σ 2 S 2 VSS dt − VS dS − VS δSdt + rW dt 2 1 + Vt + σ 2 S 2 VSS − VS δS + rW d t. 2 (12) (13) Since this is risk free and is (initially) a zero investment, both the drift and I must be zero. This implies W = VS S − V and 1 + Vt + σ 2 S 2 VSS − VS δS + r (VS S − V ) = 0, 2 (14) 1 + Vt + σ 2 S 2 VSS + (r − δ) VS S = rV . 2 (15) hence Note if we assume is a continuous yield of the claim (rather than a $ per unit rate), the first term would be V rather than . Question 21.12. Setting b = −1 and using Proposition 21.1, we change the dividend yield of S to η = .02 − .2 (.3) (.5) = −.01. The prepaid forward price, i.e. V in equation (21.35), is S0 e−ηT . Letting δ ∗ = .06 + (.06 − .01) − .52 = −.14, we have the value of the claim being 1 .2(2) e 50e.01(2) = 0.8455. 90 134 (16) Chapter 21 The Black-Scholes Equation Using Proposition 20.4, the claim should be worth S0 e−δT Qb e 0 2 b(r −δQ )+.5b(b−1)σQ 2 ebρσ σQ T (17) e.03(2) = 0.8455. (18) which equals 50e−.04 90−1 e −.05+.52 2 Note that Proposition 20.4 derives the forward price; upon discounting, the forward price of S becomes S0 e−δT and the forward price of Qb terms does not get discounted. Question 21.14. Using Proposition 21.1, since b = 1, the insurance payoff should be worth Qe(r −δQ )T V (S, K, σS , r, T , δ − ρσ σS ) (19) hence we should use a dividend yield of .02 + .2 (.3) (.5) = .05 making the put relatively more valuable. For K = 50, V = 7.09 hence the insurance is worth 90e(.06−.01)2 (7.09) = 705.21. If we wanted to insure 90e(.06−.01)2 = 99.465 units, it would cost 90e(.06−.01)2 (6.05) = 601.77. This is intuitive since ln (S) and ln (Q) are negatively correlated. When Q is high, S is more likely to be low making the insurance payout larger (the holder has the right to sell more units for K ). 135 Chapter 22 Exotic Options: II Question 22.2. In the same way as the COD, the paylater is priced initially using 0 = BSP ut (S0 , K, σ, r, T , δ) − P × DR(S0 , K, σ, r, T , δ, H ) Thus, the amount to be paid if the barrier is hit is P= 2.3101 BSP ut (S0 , K, σ, r, T , δ) = = 3.0436. DR(S0 , K, σ, r, T , δ, H ) 0.7590 At subsequent times prior to hitting the barrier, the value of the paylater put is BSP ut (S0 , K, σ, r, T − t, δ) − P × DR(S0 , K, σ, r, T − t, δ, H ) The paylater premium has the potential to be much lower than the COD premium. Compare a paylater with H = K to a COD. You’ll find in many cases that the COD premium is approximately twice as great. This is a consequence of the reflection principle—once you have hit the barrier, there is approximately a 50% chance that the option will move out of the money, which means that half the time, you’ll pay the premium without the option paying off. Thus, the premium is half that of the COD, where the premium is always paid when and only when the option is in the money. The initial delta will be −.1903 − 3.0436 (−.0439) = −0.0567. The DR has a gamma very close to zero hence, initially, there is little difference between the paylater’s gamma and a regular put option’s gamma. As time evolves the behavior of delta and gamma becomes similar to the COD, since in each case a small move can trigger a discrete payment; the main difference being that the discrete payment is likely to occur before expiration when St gets close to the barrier. Question 22.4. We must show the formula is a solution to e−r(T −t) P S T ≥ H and ST < K where P stands for risk neutral probability. We begin with the case when H ≥ K (i.e. the top equation). If the barrier is hit, i.e. S = H , −d4 = −d2 implying the probability is N (−d2 ) = P (ST < K) (i.e. the risk neutral probability of receiving one dollar). If at time T , S T ≥ H and ST < K , the barrier has been hit and the probability is equal to N (−d2 ) = 1. Lastly, if at time T , S T < H or ST > K we must check the probability is zero. If ST > K and the barrier has been hit, the probability becomes N (−d2 ) = 0. If the barrier 136 Chapter 22 Exotic Options: II has not been hit (as a reminder ST > K ), then ST < H and H 2 / (ST K) > 0 implying d4 → ∞. k The probability will be H N (−d4 ) = 0. S For the case when H < K , if the barrier is hit d6 = d8 and H drops out of the probability leaving N (−d2 ) = P (ST < K). If at time T , S T ≥ H and ST < K , the barrier has been hit and the probability is equal to N (−d...
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