This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: EC3070 FINANCIAL DERIVATIVES BINOMIAL OPTION PRICING MODEL A One-Step Binomial Model The Binomial Option Pricing Model is a sim- ple device that is used for determining the price c τ | that should be attributed initially to a call option that gives the right to purchase an asset at time τ at a strike price of K τ | . The model supposes a portfolio where the assets are N units of stock, with a spot price of S per unit, and the liability is one call option. The initial value of the portfolio at time t = 0 will be V = NS − c τ | . (1) There will be an direct relationship between the ensuing movements of the stock price and the option price. When the value of the stock, which is the asset, rises, that of the option, which is a liability, will also rise. It is possible to devise a portfolio in which such movements are exactly offsetting. We may begin by envisaging two eventualities affecting the spot price of the stock. Either it increases to become S u τ = S U at time t = τ , where U > 1, or else it decreases to become S d τ = S D , where D < 1. In the hands of its owner, the value of the call option will be S τ − K τ | , if S τ > K τ | , in which case the option will be exercised, or it will be worthless, if S τ ≤ K τ | . Thus c τ | τ = max( S τ − K τ | , 0) . (2) Let the values of the option corresponding to S u τ and S d τ be c u τ and c d τ , respectively. Then, the value of the portfolio at time t = τ will be V τ = Ω NS U − c u τ , if S τ = S u τ = S U ; NS D − c d τ , if S τ = S d τ = S D . (3) The number of units of the asset can be chosen so that the values of the portfolio are the same in these two cases. Then, the portfolio will be riskless; and, according to the argument that there should be no arbitrage opportunities, it should earn the same as the sum V invested for τ periods at the riskless rate of interest. Thus V τ = V e rτ or, equally, V = V τ e − rτ . (4) There are therefore two equations that cover the two eventualities: V τ = S UN − c u τ ⇐⇒ c u τ = S UN − V τ , V τ = S DN − c d τ ⇐⇒ c d τ = S DN − V τ . (5) D.S.G. Pollock: stephen [email protected] BINOMIAL OPTION PRICING MODEL Their solutions for N and V τ are as follows: N = c u τ − c d τ S ( U − D ) and V τ = c u τ D − c d τ U U − D . (6) From (1), it follows that the initial value of the option is c τ | = NS − V = NS − V τ e − rτ , (7) where the second equality follows from (4). Putting the values for N and V τ from (6) into this equation gives c τ | = Ω c u τ − c d τ S ( U − D ) æ S − Ω c u τ D − c d τ U U − D æ e − rτ = e − rτ Ω e rτ ( c u τ − c d τ ) − ( c u τ D − c d τ U ) U − D æ = e − rτ Ωµ e rτ − D U − D ∂ c u τ + µ U − e rτ U − D ∂ c d τ æ ....
View Full Document
This note was uploaded on 03/02/2012 for the course EC 3070 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.
- Spring '12