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Levy and Post, Investments © Pearson Education Limited 2005

Slide 20.1

Investments

Chapter 20: Derivatives Valuation


Slide 20.2

Derivatives Valuation

Derivatives valuation relies heavily on the no-arbitrage rule:

As long as an instrument can be replicated using investments with known prices, then arbitrage will ensure that the price of the instrument will be equal to that of the ‘synthetic replica’.


Slide 20.3

Static and Dynamic Hedges

Two ways to conduct an arbitrage trade:

1. Static Hedge (Using a portfolio that remains riskless by construction.)

2. Dynamic Hedge(Using a portfolio that must constantly be rebalanced to remain riskless.)


Slide 20.4

Futures:The Cost-of-Carry Model: I

The futures price differs for different underlying assets because it is equal to the spot price plus the net benefits of owning the spot asset (the ‘costs’ of carrying the asset).

Investors often calculate this ‘cost’ as the difference between the current spot price and the current futures price. This measure is also known as the ‘basis’.


Slide 20.5

Futures:The Cost-of-Carry Model: II

The basis may be positive or negative:

If the basis is positive, the market is said to be normal. This situation is also called backwardation.

If the basis is negative, the market is said to be inverted. This situation is also called contango.


Slide 20.6

Valuing Stock Index Futures

Exhibit 20.1 Determining the equilibrium price of stock index futuresSource: From Introduction to Investments, 2nd edn, by Levy. © 1999. Reprinted with permission of South-Western, a division of Thomson Learning: www.thomsonrights.com. Fax 800 730-2215.


Slide 20.7

Valuing Currency Futures

Exhibit 20.2 Determining the equilibrium price of foreign currency futuresSource: From Introduction to Investments, 2nd edn, by Levy. © 1999. Reprinted with permission of South-Western, a division of Thomson Learning: www.thomsonrights.com. Fax 800 730-2215.


Slide 20.8

Valuing Commodity Futures

Exhibit 20.3 Determining the equilibrium price of commodity futuresSource: From Introduction to Investments, 2nd edn, by Levy. © 1999. Reprinted with permission of South-Western, a division of Thomson Learning: www.thomsonrights.com. Fax 800 730-2215.


Slide 20.9

Options:Option Boundaries

• The value of options at expiration must lie between certain boundaries: the option value boundaries.

• These bounds are NOT influenced by assumptions regarding the distribution of returns.


Slide 20.10

Options:Call Option Boundaries

Exhibit 20.6 Call option boundariesSource: From Introduction to Investments, 2nd edn, by Levy. © 1999. Reprinted with permission of South-Western, a division of Thomson Learning: www.thomsonrights.com. Fax 800 730-2215.


Slide 20.11

Options:Put Option Boundaries

Exhibit 20.8 Put option boundariesSource: From Introduction to Investments, 2nd edn, by Levy. © 1999. Reprinted with permission of South-Western, a division of Thomson Learning: www.thomsonrights.com. Fax 800 730-2215.


Slide 20.12

Put-Call Parity

Put-call parity establishes an exact relationship among the current stock price, the call price and the put price at any given moment. It can be written as:


Slide 20.13

Options Valuation Models:Discrete vs. Continuous Time

1. The Binominal ModelDiscrete time approach: option prices are assumed

to only change at predetermined moments.

2. The Black-Scholes option-valuation model (BSmodel)Continuous time approach: option prices are

assumed to change all the time – i.e. continuously.


Slide 20.14

Option Valuation Models:The Binominal Model: I

Assumptions:

1. The capital market is characterized by perfect competition.

2. Short selling is allowed, with full use of the proceeds.

3. Investors prefer more wealth to less.

4. Borrowing and lending at the risk-free rate is permitted.

5. Future stock prices will have one of two possible values.


Slide 20.15

Option Valuation Models:The Binominal Model: II

The binominal model is developed in five steps:

Step 1: Determine the stock price distribution.Step 2: Determine the option price distribution.Step 3: Create a hedged portfolio.Step 4: Solve for the hedge ratio.Step 5: Solve for the call price using net present

value.


Slide 20.16

Option Valuation Models:The Binominal Model: III

The binominal model can be extended to an infinite number of periods. This results in a so-called binominal tree:

u · d · S0 = Sud

t2 0 t1 time

S0

u · S0 = Su

d · S0 = Sd

u2 · S0 = Suu

d2 · S0 = Sdd


Slide 20.17

Three Period Binomial Option Pricing Example

• There is no reason to stop with just two periods.• Find the value of a three-period at-the-money call

option written on a $25 stock that can go up or down 15 percent each period when the risk-free rate is 5 percent.


Slide 20.18 Three Period Binomial Process: Stock Prices

$25

28.75

21.25

2/3

1/3

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2)15.1(00.25$

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)15.1()15.1(00.25$ 2

2)15.1()15.1(00.25$

3)15.1(00.25$

33.06

24.44

2/3

1/3

18.06

2/3

1/3

15.35

2/3

1/3

38.02

2/3

1/3

20.77

2/3

1/3

28.10


Slide 20.19

$25

28.75

21.25

2/3

1/3

15.35

2/3

1/3

38.02

28.10

2/3

1/3

20.77

2/3

1/3

33.06

24.44

2/3

1/3

18.06

2/3

1/3

]0,25$02.38max[$),,(3 UUUC

13.02

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),,(

33

3

DUUCUDUC

UUDC

3.10

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33

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4.52

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C

Three Period Binomial Process: Call Option Prices


Slide 20.20

Option Valuation Models:The BS Model: I

Assumptions:

1. The capital market is characterized by perfect competition.

2. Short selling is allowed, with full use of the proceeds.

3. Investors prefer more wealth to less.

4. Borrowing and lending occur at the risk-free rate.

5. Price movements are such that past price movements cannot be used to forecast future price changes.


Slide 20.21

Option Valuation Models:The BS Model: II

The BS model formula for a call options is:

where


Slide 20.22

Option Valuation Models:The BS Model: III

The formula for put options can be found using the BS model for call options and put-call parity:


Slide 20.23

Option Valuation Models:The BS Model: IV

Exhibit 20.11 Reaction of option price to increases in input parameters


Slide 20.24

Option Valuation Models:The BS Model: V

Empirical issues:

1. The log-normal return distribution that is assumed in the BS model is often violated.2. The fact that the BS model does NOT allow for jumps in the underlying stock prices.3. The issue of non-constant volatility of an option during its lifetime.

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