28C00400_lecture_2 - Aalto University Derivatives LECTURE 2 Matti Suominen STOCK PRICES AS A RANDOM WALK In efficient markets prices reflect all past

# 28C00400_lecture_2 - Aalto University Derivatives LECTURE 2...

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Aalto University Derivatives LECTURE 2 Matti Suominen
2 STOCK PRICES AS A RANDOM WALK In efficient markets prices reflect all past public information. ð༏ The expected returns depend only on beta ð༏ Prices move when unexpected news come to the market Normal distribution (with a mean that depends on beta) is a useful approximation for the distribution of stock returns.
3 Examples of stock price paths: real and simulated 50 60 70 80 90 100 110 0 50 100 150 200 250 50 60 70 80 90 100 110 0 50 100 150 200 250 50 70 90 110 130 150 170 0 50 100 150 200 250 50 70 90 110 130 150 0 50 100 150 200 250 50 70 90 110 130 150 0 50 100 150 200 250 50 60 70 80 90 100 110 0 50 100 150 200 250
4 BINOMIAL MODEL OF STOCK PRICES We model stock prices using a binomial tree: Each period the stock price can go up by a factor u or down by a factor of d . If we select u and d “correctly”, then as Δ t 0 this discrete approximation approaches a continuous model of stock prices where the stock returns are normally distributed. To begin with, we will assume that there is only one period before the option expires. In the next lecture we will extend the analysis to multiple periods. u ² S uS S udS dS d ² S Δ t T
5 EXACT OPTION PRICING Basic idea: Options (and other derivatives) are priced by no arbitrage. Find a portfolio replicating the payoff of the option. A simple case: Δ t = 1 S 0 = 50, u = 1.1, d = 0.9 50 What is the price of a call option, expiring at t = 1, with EX = 50 when r f = 5% ? 1.1 x 50 = 55 0.9 x 50 = 45
6 Assume traders can borrow and lend at the risk free rate. Find a portfolio composed of the underlying stock (buy h 0 stocks) combined with riskless lending/borrowing (borrow B 0 ) that replicates the payoff of the call. t = 0 t = 1 Value of call at t = 1 Value of replicating portfolio at t = 1 55 C u = 50 45 C d = h 0 and B 0 must satisfy the following two conditions: * * Value of call: This is the basic idea in derivatives pricing: The price of the derivative must equal the value of the replicating portfolio.
7 More generally: Number of shares bought: Ratio Hedge or Delta Option 45 55 0 5 0 = = = d u d u S S C C h Amount borrowed: f d d r C S h B + = 1 0 0 Value of the call: C 0 = h 0 S 0 - B 0 Note that call price does not depend on the probability that the stock price goes up; nor on the traders’ risk preferences !!!
8 VALUE OF PUT OPTION With similar procedure, consider a put option with EX = 50 with expiration date at t = 1: t = 0 t = 1 Value of put at t = 1 Value of replicating portfolio at t = 1 55 P u = max{EX-S u ,0} = 0 55h 0 + 1.05B 0 50 45 P d = max{EX-S d ,0} = 5 45h 0 + 1.05B 0 Hedge ratio: 5 . 0

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