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Notes_7_finance_5108_08 - Notes 7 Finance 5108 Learning...

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Notes 7 Finance 5108 Learning Objectives 1. Modification of the Black-Scholes Formula to accommodate continuously paid dividends 2. Understand how to use the binomial with continuous dividend payments. 3. Valuation of index options using the modified Black-Scholes formula 4. What is portfolio insurance and how options can be used to set up this type of hedge 5. Valuation of currency options 6. Options on the futures contract and their value 7. The Black model for the evaluation of options on futures contracts Bounds of options that pay a continuous dividend 1. The argument that was put forward in previous proofs would hold in the pricing of options on a stock that pays a continuous dividend. It could be shown that the only modification would be to the stock price. Since the dividend will be assumed to have been paid over the life of the option the value of the stock would be 0 0 0 0 = S Where q is the continous dividend this leads us to c max(0, S ) max(0, S ) Which are the lower bounds of the call and put option Put/Call parity is then S - qT adjusted qT rT rT qT qT S e e Ke p Ke e e - - - - - - - - c + p = rT Ke - We can use the same logic to modify the -qT 0 1 2 -qT 2 0 1 2 0 1 2 1 e ( ) ( ) ( ) e ( ) ln + q + 2 = rT rT c S N d Ke N d p Ke N d S N d Where S r T K d T d d T σ σ σ - - = - = - - - - = -
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The same modification can be made to the stochastic differential equation and the risk neutral process by just modifying the interest rate. The logic is that the net interest rate would be the risk free rate minus the continuous dividend yield which is r-q. This follows from the fact that you would receive the dividend rate of q which offset the risk-free rate. In addition this same logic can be used to apply to the binomial and the interest rate would be e (r-q)T . This can be used to value index options using the Black- Scholes or the binomial.
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