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m48-ch22

# m48-ch22 - Chapter 22 Exotic Options II Question 22.1 With...

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Chapter 22 Exotic Options: II Question 22.1. With a premium of P paid at maturity if S T > K , the COD will have the same value (which will initially be set to zero) as a regular call minus P cash or nothing call options. That is, 0 = BSCall(S 0 , K, σ, r, T, δ) P × CashCall(S 0 , K, σ, r, T, δ). (1) a) Given the inputs and pricing the above options, P , must satisfy 0 = 10 . 45 P ( 0 . 5323 ) (2) which implies P = 10 . 45 /. 5323 = 19 . 632. b) The delta of the COD is 0 . 637 19 . 632 × . 01875 = . 2689 (3) and the gamma of the COD is . 019 19 . 632 ( . 00033 ) = 2 . 55%. (4) c) As the option approaches maturity, the gamma will explode close to the money making delta hedging difficult. Question 22.2. In the same way as the COD, the paylater is priced initially using 0 = BSP ut(S 0 , K, σ, r, T, δ) P × DR(S 0 , K, σ, r, T, δ, H). Thus, the amount to be paid if the barrier is hit is P = BSP ut(S 0 , K, σ, r, T, δ) DR(S 0 , K, σ, r, T, δ, H) = 2 . 3101 0 . 7590 = 3 . 0436 . At subsequent times prior to hitting the barrier, the value of the paylater put is BSP ut(S 0 , K, σ, r, T t, δ) P × DR(S 0 , K, σ, r, T t, δ, H) 271

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Part 5 Advanced Pricing Theory 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 S t gets close to the barrier. Question 22.3. If S = H , d 6 = d 8 implying N (d 6 ) = N (d 8 ) . This makes the probability given in equation (22.7) N (d 2 ) = P (S T > K) which satisfies condition 1. If at time T , S T H and S T K the probability will be N (d 2 ) = 1 since the barrier has been hit and the option is in the money; hence condition 2 is satisfied. For the last condition we have to examine the probability at time T when S T > H or S T < K and verify it will be zero. If S T < K the barrier has been hit since K < H and the probability will be N (d 2 ) which will equal zero. When S T > H , S T must be greater than both H and K . This implies N (d 2 ) = 1, N (d 6 ) = 1 and N (d 8 ) = 0. This implies equation (22.7) will equal zero (i.e. condition 3 is satisfied). Question 22.4. We must show the formula is a solution to e r(T t) P ( S T H and S T < K ) where P stands for risk neutral probability.
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m48-ch22 - Chapter 22 Exotic Options II Question 22.1 With...

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