FBE559.slides.07

# FBE559.slides.07 - Financial Risk Management Prof Scott...

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Financial Risk Management Prof. Scott Joslin USC Marshall FBE 559 Spring 2013

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Outline of the Lecture Outline: Binomial Option Pricing Model Readings: Hull Chapter 12 [8th edition] Chapter 11 [6th, 7th edition] Chapter 10 [5ht edition]
Our goal is to see how we can replicate the risks in options by holding a position in the underlying stock and a bond. In the binomial model there will only be two choices for the stock: the price will go up or down: S 0 = 100 150 50 How do we price the option, such as a call struck at 125? C 0 =??? 25 0 How to replicate risk across random future states?

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An Old Example Suppose we want to replicate bond risk ”across time”: Bond A, price = P A : - 0 1 2 time 6 6 50 150 Bond B, price = P B : - 0 1 2 time 6 6 100 100 Bond C, price = ???: - 0 1 2 time 6 6 0 25
An Old Example (continued) We can see as before that if we buy x units of Bond A and y units of Bond B, then the cash flows are: CF at t = 1 is 50 x + 100 y CF at t = 2 is 150 x + 100 y So if we choose ( x , y ) so that: 0 = 50 x + 100 y (match t = 1 ) 25 = 150 x + 100 y (match t = 2 ) We will replicate Bond C. The solution has (subtracting the first equation from the second): 25 = 100 x x = 1 4 And also we have y = - 1 8 . So we must have P C = 1 4 P A - 1 8 P B .

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Summary of old example We have two future times ( t = 1 and t = 2) and two assets with different cash flows at those times (Bond A and Bond B), so we can replicate any future cash flows. We do this by solving two equations and two unknowns.
Binomial Stock Example I Stock: 100 150 50 Bond: 95 100 100 There are two random states of nature bond and stock have different payoffs in each states C 0 =??? 25 0

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Binomial Stock Example I (continued) The previous solution carries over by changing t = 1 and t = 2 to ”up” state and ”down” state: Stock: 100 150 50 Bond: 95 100 100 Call: C 0 25 0
Binomial Stock Example I (continued) If we buy x units of Bond A the stock and y units of Bond B the zero coupon bond, then the cash flows are: CF in the down state is 50 x + 100 y CF in the up state is 150 x + 100 y So if we choose ( x , y ) so that: 0 = 50 x + 100 y (match down state) 25 = 150 x + 100 y (match up state) We will replicate the option. The solution has (subtracting the first equation from the second): 25 = 100 x x = 1 4 Substituting in, we also get y = - 1 8

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Binomial Stock Example I (continued) So we see we have the replicating portfolio: 1 4 × 150 50 - 1 8 × 100 100 = 25 0 So we must have C 0 = 1 4 × 100 - 1 8 × 95 = \$13 . 125 Jump to Method #2 Jump to Method #3
Expressing the Time-value of Money We could state the discount rate in a few ways: 1 90 100 100 2 The annualized interest rate (with no compounding) is 11.11% 100 111.10 111.10 The price of a 1-year zero coupon bond with face value \$1 is 1 / 1 . 111 = 0 . 90 3 The continuously compounded rate is - log( . 9) = 10 . 54%, so that the price of a 1-year zero coupon bond is e - . 1054 = 0 . 90. Hull uses the last method. For now, all we need is the price of the zero coupon bond, so I use the first method.

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One-period Binomial Model Assumptions: Binomial price movement Frictionless market (no cost for buying or selling) Bd < 1 < Bu (Why do we need this?) Notation: S 0 : initial stock price S 0 u : stock price if it goes up S 0 d : stock price if it goes down B : present value of \$1 in the future If you don’t like B , use 1 1+ r instead Basic Idea of Pricing: find a replicating portfolio, and use the no arbitrage argument
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