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# LectSlide3 - EC3070 FINANCIAL DERIVATIVES BINOMIAL OPTION...

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EC3070 FINANCIAL DERIVATIVES BINOMIAL OPTION PRICING MODEL A One-Step Binomial Model Suppose that a portfolio consists of N units of stock as assets, with a spot price of S 0 per unit, together with a liability of one call option. The initial value of the portfolio at time t = 0 will be V 0 = NS 0 c τ | 0 . When the value of the stock rises, so will that of the option. We envisage two possibilities. Either the price increases to S u τ = S 0 U at time t = τ , where U > 1, or it decreases to S d τ = S 0 D , where D < 1. The value of the call option will be S τ K τ | 0 , if S τ > K τ | 0 , in which case it will be exercised, or it will be worthless, if S τ K τ | 0 . Thus c τ | τ = max( S τ K τ | 0 , 0) = ( S τ K τ | 0 ) + . 1

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EC3070 FINANCIAL DERIVATIVES Let c u τ and c d τ be the values of the option corresponding to S u τ and S d τ , respectively. Then, the value of the portfolio at time t = τ will be V τ = NS 0 U c u τ , if S τ = S u τ = S 0 U ; NS 0 D c d τ , if S τ = S d τ = S 0 D . The number of units of the asset can be chosen so that the values of the portfolio are the same in these two cases. Therefore, the portfolio is riskless; and there are two equations for V τ : V τ = S 0 UN c u τ ⇐⇒ c u τ = S 0 UN V τ , V τ = S 0 DN c d τ ⇐⇒ c d τ = S 0 DN V τ . Their solutions for N and V τ are N = c u τ c d τ S 0 ( U D ) and V τ = c u τ D c d τ U U D . 2
EC3070 FINANCIAL DERIVATIVES The initial value the riskless portfolio is V 0 = V τ e , and that of the option is c τ | 0 = NS 0 V 0 = NS 0 V τ e , Putting the solutions for N and V τ into this equation gives c τ | 0 = c u τ c d τ S 0 ( U D ) S 0 c u τ D c d τ U U D e = e e ( c u τ c d τ ) ( c u τ D c d τ U ) U D = e e D U D c u τ + U e U D c d τ .

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LectSlide3 - EC3070 FINANCIAL DERIVATIVES BINOMIAL OPTION...

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