老师 ，这个到期时欧式期权的payoff 为什么是14.88？

问题如下：

Question An analyst gathers the following information about a put option on a non-dividend-paying stock:

The early exercise premium of the American-style put option at Time 0 is closest to:

选项：

A.A.0.54. B.B.0.59. C.C.0.67.

解释：

A Correct because the American-style put option value at Time 0 is calculated as p = PV [πp⁺ + (1 – π)p^–] = (1/1.0205) × [(0.4498 × 0) + ((1 – 0.4498) × 15.88)] ≈ 8.5617.

The European-style put option at Time 0 is calculated as p = PV [πp⁺ + (1 – π)p^–] = (1/1.0205) × [(0.4498 × 0) + ((1 – 0.4498) × 14.88)] ≈ 8.0225

Therefore the early exercise premium of the American-style put option at Time 0 is 8.5617 – 8.0225 ≈ 0.54.

PV = present value factor = 1 / (1 + r) = 1 / 1.0205

π = probability of up move = [ FV(1) – d ] / (u – d) = (1.0205 – 0.6562) / (1.4662 – 0.6562) ≈ 0.4498

For the American-style calculation:

p⁺ = put exercise value at Time 1 with up move = Max [0, Strike – Underlying stock price at Time 1 with up move] = Max [0, 50.00 – 76.24] = 0

p^– = put exercise value at Time 1 with down move = Max [0, Strike – Underlying stock price at Time 1 with down move] = Max [0, 50.00 – 34.12] = 15.88

For the European-style calculation:

p⁺ = put value at Time 1 with up move (given) = 0

p^– = put value at Time 1 with down move (given) = 14.88

where:

u = up factor (total return) = S⁺ / S = 76.24 / 52.00 ≈ 1.4662

d = down factor (total return) = S^- / S = 34.12 / 52.00 ≈ 0.6562

where:

FV(1) = future value at Time 1 = 1 / PV(1) = 1 + r = 1.0205

S⁺ = value of underlying at Time 1 when an up move occurs = 76.24

S^- = value of underlying at Time 1 when a down move occurs = 34.12

S = value of underlying at Time 0 = 52.00