X3Binomial_model_I

X3Binomial_model_I - Binomial option pricing Risk-neutral...

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Slide 10-1 Binomial option pricing Risk-neutral pricing: the basic idea Section 10.1, Appendix 11.A
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Slide 10-2 Binomial Option Pricing Binomial option pricing enables us to determine the price of an option, given the characteristics of some underlying asset. The binomial option pricing model assumes that the price of the underlying asset follows a binomial distribution – that is, the asset price in each period can move only up or down by a constant amount. The binomial model was first proposed by Cox, Ross and Rubinstein in 1979.
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Slide 10-3 A One-Period Binomial Tree Consider a European call option on the stock of XYZ, with a $40 strike and 1 year to expiration. XYZ does not pay dividends, and its current price is $41. The continuously compounded risk-free interest rate is 8%. The following binomial tree depicts possible stock prices over 1 year.
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Slide 10-4 Let’s value a call option Consider two portfolios: Portfolio 1 : Buy one call option. Portfolio 2 : Buy 0.7376 shares of XYZ and borrow $22.405 at the risk-free rate. Costs: Portfolio 1 : The call premium, which is unknown. Portfolio 2 : 0.7376 ! $41 – $22.405 = $7.839.
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Slide 10-5 Payoffs at Maturity Stock Price in 1 Year $32.903 $59.954 Portfolio 1 : Payoff of call 0 $19.954 Portfolio 2 : 0.7376 purchased shares $24.271 $44.225 Repay loan of $22.405 – $24.271 –$24.271 Total payoff 0 $19.954
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Slide 10-6 Computing the option price Portfolios 1 and 2 have the same payoff. Therefore, Portfolios 1 and 2 must have the same cost. Since Portfolio 2 costs $7.839, the price of the call option must be $7.839. In a binomial world the payoff of a call can be replicated by buying shares and borrowing. " S + B is a “ synthetic call.” The call has the same risk as 0.7376 shares. The value 0.7376 is the delta ( " ) of the option: The number of shares required to replicate the option payoff.
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Slide 10-7 The binomial solution How do we find a replicating portfolio consisting of " shares of stock and a dollar investment B , such that the portfolio imitates the payoff of the option? Stock price tree: Option value tree: uS 0 C u S 0 C 0 dS 0 C d uS 0 denotes the stock price when the price goes up, and dS 0 denotes the stock price when the price goes down. Note that u ( d ) is interpreted as one plus the rate of capital gain (loss). We will calculate u and d later.
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Slide 10-8 The binomial solution Form a replicating portfolio of the form " S + B. The value of the replicating portfolio at time h , consisting of " shares of stock and a dollar amount B in lending, is " S h e # h + e r h B.
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X3Binomial_model_I - Binomial option pricing Risk-neutral...

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