Technical Note 6

Technical Note No. 6* Options, Futures, and Other Derivatives, Seventh Edition John Hull Differential Equation for Price of a Derivative on a Stock Providing a Known Dividend Yield Define f as the price of a derivative dependent on a stock that provides a dividend yield at rate q. We suppose that the stock price, S, follows the process dS = µS dt + σS dz where dz is a Wiener process. The variables µ and σ are the expected growth rate in the stock price and the volatility of the stock price, respectively. Because the stock price provides a dividend yield, µ is only part of the expected return on the stock.1 Because f is a function of S and t, it follows from Ito’s lemma that   ∂f 1 ∂2f 2 2 ∂f ∂f µS + + σ S dt + σS dz df = 2 ∂S ∂t 2 ∂S ∂S Similarly to the procedure of Section 13.6, we can set up a portfolio consisting of −1 : ∂f + : ∂S

derivative stock

If Π is the value of the portfolio, Π = −f +

∂f S ∂S

(1)

and the change, ∆Π, in the value of the portfolio in a time period ∆t is as given by equation (13.14):   1 ∂2f 2 2 ∂f ∆Π = − − σ S ∆t ∂t 2 ∂S 2 In time ∆t the holder of the portfolio earns capital gains equal to ∆Π and dividends on the stock position equal to ∂f qS ∆t ∂S Define ∆W as the change in the wealth of the portfolio holder in time ∆t. It follows that   ∂f 1 ∂2f 2 2 ∂f ∆W = − − σ S + qS ∆t (2) ∂t 2 ∂S 2 ∂S c * Copyright John Hull. All Rights Reserved. This note may be reproduced for use in conjunction with Options, Futures, and Other Derivatives, seventh edition. 1 In a risk-neutral world µ = r − q as indicated in equation (15.3) of the book.

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Because this expression is independent of the Wiener process, the portfolio is instantaneously riskless. Hence ∆W = rΠ ∆t (3) Substituting from equations (1) and (2) into equation (3) gives 

so that

1 ∂2f 2 2 ∂f ∂f − σ S + qS − 2 ∂t 2 ∂S ∂S





 ∂f ∆t = r −f + S ∆t ∂S

∂f 1 ∂2f ∂f + (r − q)S + σ 2 S 2 2 = rf ∂t ∂S 2 ∂S

This is equation (15.6) in the book.

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