This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ACTL3004: Weeks 78 Financial Economics for Insurance and Superannuation: Weeks 78 Discrete Time Derivative Valuation 1 / 33 ACTL3004: Weeks 78 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. riskfree zerocoupon 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 riskfree 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 arbitragefree. 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 78 Binomial Lattice Model: European Option Valuation Principle of No Arbitrage Principle of No Arbitrage I Arbitrage means a riskfree 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 nonzero 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 78 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 78 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 ( ) ....
View
Full
Document
This note was uploaded on 08/07/2011 for the course ACTL 3004 at University of New South Wales.
 '10
 BRIAN

Click to edit the document details