DERI L2.pdf - DERIVATIVES AND RISK MANAGEMENT Op#on Valua#on I(Lecture 2 Ma Suominen January 2018 To value [email protected](C0 we rst need a model of stock prices

DERI L2.pdf - DERIVATIVES AND RISK MANAGEMENT Op#on...

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DERIVATIVES AND RISK MANAGEMENT Op#on Valua#on I (Lecture 2) Ma/ Suominen January 2018
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To value [email protected] (C 0 ) we first need a model of stock prices: STOCK PRICES AS A RANDOM WALK In efficient markets prices reflect all past public [email protected] ð The expected returns depend only on beta ð Prices move when unexpected news come to the market Normal [email protected] (with a mean that depends on beta) is a good [email protected] for the [email protected] of stock returns.
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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 60 70 80 90 100 110 0 50 100 150 200 250 50 60 70 80 90 100 110 120 130 140 150 0 50 100 150 200 250 50 60 70 80 90 100 110 120 130 140 150 0 50 100 150 200 250
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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 . S 0 u S d S ud S u 2 S d 2 S T Δ t If we select u and d “correctly”, then as Δ t 0 this discrete [email protected] approaches a [email protected] 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 [email protected] expires. In the next lecture we will extend the analysis to [email protected] periods.
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EXACT OPTION PRICING Basic idea: [email protected] (and other [email protected]) are priced by no arbitrage. Find a por\olio [email protected] the payoff of the [email protected] A simple case: Δ t = 1 S 0 = 50, u = 1.1, d = 0.9 S 0 = 50 uS 0 = 55 dS 0 = 45 What is the price of a call [email protected], expiring at t = 1, with EX = 50 when r f = 5% ? C 0 = ? C u = max(S-­‐EX;0) = 5 C d = max(S-­‐EX;0) = 0
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[email protected] Pricing by No-­‐Arbitrage Given our model of stock prices, we price [email protected] by [email protected] a replica#ng porMolio . [email protected] por\olio: buy h 0 stocks borrow B 0 So that: 55 h 0 – 1.05 B 0 = 5 45 h 0 – 1.05 B 0 = 0
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[email protected] Pricing by No-­‐Arbitrage This leads to two [email protected] in two unknowns, h 0 and B 0 55 h 0 – 1.05 B 0 = 5 45 h 0 – 1.05 B 0 = 0 => h 0 = 0.5 [Hedge [email protected]] B 0 = 21.43 C 0 = This is the basic idea in [email protected] pricing: The price of the [email protected] must equal the value of the [email protected] por\olio. h 0 S – B 0 = 0.5 x 50 – 21.43 = 3.57 By no-­‐arbitrage, the current value of the [email protected] must equal the current value of the [email protected] por\olio.
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How to take advantage of mispricing? What if the call [email protected] is trading at 3.80 in the market? Buy low: Sell high: t = 0 t = 1 ‘up’ state ‘down’ state Buy ½ share Borrow 21.43 Write call Net +21.43 -­‐25 +3.80 +0.23 -­‐ 22.5 +27.5 -­‐5 0 -­‐22.5 = (1+r)21.43 +22.5 0 0 = 3.57 = 5 = 0 [email protected] por\olio
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More generally: Number of shares bought: h 0 = Δ = C u C d S u S d =
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