By Chern I.-L.

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Ii) On S = 0: p(0, τ ) = Ee−r(T −τ ) . This follows from the put-call parity and c(0, t) = 0. (iii) For call option, at S = ∞: c(S, t) ∼ S − Ee−r(T −t) , as S → ∞. Since S → ∞, the call option must be exercised, and the price of the option must be closed to S − Ee−r(T −t) . (iv) For put option, at S = ∞: p(S, t) → 0, as S → ∞ As S → ∞, the payoff function Λ = max{E − S, 0} is zero. Thus, the put option is unlikely to be exercised. Hence p(S, T ) → 0 as S → ∞. 28 CHAPTER 3. 1 Reduction to parabolic equation with constant coefficients Let us recall the Black-Scholes equation ∂V 1 ∂2V ∂V + σ 2 S 2 2 + rS − rV = 0.

1 ∂V ∂ 2V Vt + σ 2 S 2 2 + (r − D0 )S − rV = 0 2 ∂S ∂S This is the Black-Scholes equation when there is a continuous dividend payment. The boundary conditions are: c(0, t) = 0, c(S, t) ∼ Se−D0 (T −t) The latter is the asset price S discounted by e−D0 (T −t) from the payment of the dividend. The payoff function c(S, T ) = Λ(S) = max{S − E, 0}. To find the solution, let us consider c(S, t) = e−D0 (T −t) c1 (S, t). Then c1 satisfies the original Black-Scholes equation with r replaced by r − D0 and the same final condition.

2. 2 Discrete dividend payments Suppose our asset pays just one dividend during the life time of the option, say at time td . The dividend yield is a constant. At td +, the asset holder receiver a payment dy S(td −). Hence, S(td +) = S(td −) − dy S(td −) = (1 − dy )S(td −). , V (S(td −), td −) = V (S(td +), td +). Reason : Otherwise, there is a net loss or gain from buying V before td then sell it right after td . To find V (S, t), here is a procedure. 1. Solve the Black-Scholes from T to Td + to obtain V (S, td +) (using the payoff function Λ) 2.

### Financial mathematics by Chern I.-L.

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