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Ch. 4: *

Chapter 4: Option Pricing Models:
The Binomial Model

Options traders can get by with less math than you think. Tour de France cyclists don’t need to know how to solve Newton’s laws in order to bank around a curve. Indeed, thinking too much about physics while riding or playing tennis may prove a hindrance. But good traders do have to have the patience to understand the essential mechanism of replicating the factors they’re trading.

Emanuel Derman

The Journal of Derivatives, Winter 2000, p. 62

An Introduction to Derivatives and Risk Management, 10th ed.

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An Introduction to Derivatives and Risk Management, 10th ed.

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Important Concepts in Chapter 4

The concept of an option pricing model
The one- and two-period binomial option pricing models
Explanation of the establishment and maintenance of a risk-free hedge
Illustration of how early exercise can be captured
The extension of the binomial model to any number of time periods
Alternative specifications of the binomial model

An Introduction to Derivatives and Risk Management, 10th ed.

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An Introduction to Derivatives and Risk Management, 10th ed.

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Definition of a model
A simplified representation of reality that uses certain inputs to produce an output or result
Definition of an option pricing model
A mathematical formula that uses the factors that determine an option’s price as inputs to produce the theoretical fair value of an option.

An Introduction to Derivatives and Risk Management, 10th ed.

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An Introduction to Derivatives and Risk Management, 10th ed.

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One-Period Binomial Model

Conditions and assumptions
One period, two outcomes (states)
S = current stock price
u = 1 + return if stock goes up
d = 1 + return if stock goes down
r = risk-free rate
Value of European call at expiration one period later
Cu = Max(0, Su – X) or
Cd = Max(0, Sd – X)
See Figure 4.1

An Introduction to Derivatives and Risk Management, 10th ed.

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An Introduction to Derivatives and Risk Management, 10th ed.

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One-Period Binomial Model (continued)

Important point: d < 1 + r < u to prevent arbitrage
We construct a hedge portfolio of h shares of stock and one short call. Current value of portfolio:
V = hS – C
At expiration the hedge portfolio will be worth
Vu = hSu – Cu
Vd = hSd – Cd
If we are hedged, these must be equal. Setting Vu = Vd and solving for h gives

An Introduction to Derivatives and Risk Management, 10th ed.

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An Introduction to Derivatives and Risk Management, 10th ed.

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One-Period Binomial Model (continued)

These values are all known so h is easily computed
Since the portfolio is riskless, it should earn the risk-free rate. Thus
V(1+r) = Vu (or Vd)
Substituting for V and Vu
(hS – C)(1 + r) = hSu – Cu
And the theoretical value of the option is

An Introduction to Derivatives and Risk Management, 10th ed.

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An Introduction to Derivatives and Risk Management, 10th ed.

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One-Period Binomial Model (continued)

This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors.
The probabilities of the up and down moves were never specified. They are irrelevant to the option price.

An Introduction to Derivatives and Risk Management, 10th ed.

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One-Period Binomial Model (continued)

An Illustrative Example
S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07
First find the values of Cu, Cd, h, and p:
Cu = Max(0, 100(1.25) – 100)
= Max(0, 125 – 100) = 25
Cd = Max(0, 100(0.80) – 100)
= Max(0, 80 – 100) = 0
h = (25 – 0)/(125 – 80) = 0.556
p = (1.07 – 0.80)/(1.25 – 0.80) = 0.6
Then insert into the formula for C:

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An Introduction to Derivatives and Risk Management, 10th ed.

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One-Period Binomial Model (continued)

A Hedged Portfolio
Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure 4.2.
Value of investment: V = 556($100) – 1,000($14.02)
$41,580. (This is how much money you must put up.)
Stock goes to $125
Value of investment = 556($125) – 1,000($25)
= $44,500
Stock goes to $80
Value of investment = 556($80) – 1,000($0)
= $44,480

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One-Period Binomial Model (continued)

An Overpriced Call
Let the call be selling for $15.00
Your amount invested is 556($100) – 1,000($15.00)
= $40,600
You will still end up with $44,500, which is a 9.6% return.
Everyone will take advantage of this, forcing the call price to fall to $14.02

You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate.

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An Underpriced Call
Let the call be priced at $13
Sell short 556 shares at $100 and buy 1,000 calls at $13. This will generate a cash inflow of $42,600.
At expiration, you will end up paying out $44,500.
This is like a loan in which you borrowed $42,600 and paid back $44,500, a rate of 4.46%, which beats the risk-free borrowing rate.

One-Period Binomial Model (continued)