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Lecture 7-8[1]

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

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ACTL3004: Weeks 7-8 Financial Economics for Insurance and Superannuation: Weeks 7-8 Discrete Time Derivative Valuation 1 / 33

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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 = 0 , 1 , 2 , ... S ( t ) is random. 2. risk-free zero-coupon bond/cash account We shall denote by B t 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 B t = 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

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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 δ 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 δ 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 δ t f ( S ( 1 )) is not good enough. Lets see what we can do by holding a combination of stocks and bonds, namely φ of stock s 1 ψ of bond B 0 The value of this portfolio today is V ( 0 ) = φ s 1 + ψ B 0 and the value of this portfolio at time 1 is φ s 3 + ψ B 0 e r δ t if the stock goes up φ s 2 + ψ B 0 e r δ t if the stock goes down Now remember that we want to price a derivative that pays f ( · ) .

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