Lecture 7-8[1] - ACTL3004: Weeks 7-8 Financial Economics...

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Unformatted text preview: ACTL3004: Weeks 7-8 Financial Economics for Insurance and Superannuation: Weeks 7-8 Discrete Time Derivative Valuation 1 / 33 ACTL3004: Weeks 7-8 Binomial Lattice Model: European Option Valuation Introduction Binomial Lattice Model: European Option Valuation: Introduction Consider an investment world where we can only invest in two financial instruments: 1. risky stock/share which pays no dividends We shall denote by S ( t ) the price of this stock at time t where t = , 1 , 2 ,... S ( t ) is random. 2. risk-free zero-coupon bond/cash account We shall denote by Bt the value of this cash account at time t per unit invested at time 0. Assume r is the risk-free rate compounded continuously so that Bt = e rt . In addition, we shall assume: 1. We can hold arbitrarily large amounts (positive or negative) of stocks or cash. 2. The securities market is arbitrage-free. 3. There are no trading costs, i.e. market is also considered frictionless. 4. There are no minimum/maximum units of trading. 2 / 33 ACTL3004: Weeks 7-8 Binomial Lattice Model: European Option Valuation Principle of No Arbitrage Principle of No Arbitrage I Arbitrage means a risk-free trading profit. I We say that an arbitrage opportunity exists in the capital/securities market if either: 1. an investor is able to make a deal that would give him or her an immediate profit, with no risk of future loss; or 2. an investor is able to make a deal that has zero initial outlay (or cost), no risk of future loss, and a non-zero probability of a future profit. I In the capital market, the principle of no arbitrage is generally assumed. No arbitrage opportunities must exist in the market. I The Law of One Price states that in a no arbitrage situation, any two securities or combination of securities that give exactly the same payments must have the same price. 3 / 33 ACTL3004: Weeks 7-8 Binomial Lattice Model: European Option Valuation Binomial Branch Model Binomial Branch Model Consider one time tick - start at time 0, and one tick (of length t ) later we arrive at time 1 The stock is assumed to either go up to s 3 or down to s 2 . The probability of the stock rising is p. The bond starts off as 1 and rises to e r t , where r is the risk free rate Suppose we want to price a derivative that pays f 3 at node 3 or f 2 at node 2. Eg for a forward f 3 = s 3- K f 2 = s 2- K 4 / 33 ACTL3004: Weeks 7-8 Binomial Lattice Model: European Option Valuation Binomial Branch Model Stock and Bond Strategy We know from forward pricing that pure expected value pricing, ie E P e- r t f ( S ( 1 )) is not good enough. Lets see what we can do by holding a combination of stocks and bonds, namely of stock s 1 of bond B The value of this portfolio today is V ( ) = s 1 + B and the value of this portfolio at time 1 is s 3 + B e r t if the stock goes up s 2 + B e r t if the stock goes down Now remember that we want to price a derivative that pays f ( ) ....
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This note was uploaded on 08/07/2011 for the course ACTL 3004 at University of New South Wales.

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Lecture 7-8[1] - ACTL3004: Weeks 7-8 Financial Economics...

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