Elements of financial risk management chapter 10

1

Option Pricing
Elements of
Financial Risk Management
Chapter 10
Peter Christoffersen

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Overview

2

• In this chapter we derive a no-arbitrage relationship between put
and call prices on same underlying asset
• Summarize binomial tree approach to option pricing
• Establish an option pricing formula under simplistic assumption
that daily returns on the underlying asset follow a normal
distribution with constant variance
• Extend the normal distribution model by allowing for skewness and
kurtosis in returns
• Extend the model by allowing for time-varying variance relying on
the GARCH models
• Introduce the ad hoc implied volatility function (IVF) approach to
option pricing.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Basic Definitions

3

• An European call option gives the owner the right but not the
obligation to buy a unit of the underlying asset days from now at
the price X
•
is the number of days to maturity
• X is the strike price of the option
• c is the price of the European option today
• St is the price of the underlying asset today
•
is the price of the underlying at maturity

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Basic Definitions

4

• A European put option gives the owner the option the right to sell a
~ now at the price X
unit of the underlying asset
days from
T
• p denotes the price of the European put option today
• The European options restricts the owner from exercising the
option before the maturity date
• American options can be exercised any time before the maturity

date
• Note that the number of days to maturity is counted is calendar
days and not in trading days.
~
• A standard year has 365 calendar days but onlyTaround 252 trading
days.

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Basic Definitions

5

• The payoffs (shown in Figure 10.1) are drawn as a function of the
hypothetical price of the underlying asset at maturity of the option,
• Mathematically, the payoff function for a call option is

• and for a put option it is
• Note the linear payoffs of stocks and bonds and the
nonlinear payoffs of options from Figure 10.1
• We next consider the relationship between European call
and put option prices
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Figure 10.1: Payoff as a Function of the Value of the
Underlying Asset at Maturity Call Option, Put
Option, Underlying Asset, Risk-Free Bond

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

6


Basic Definitions

7

• Put-call parity does not rely on any particular option pricing
model. It states

• It can be derived from considering two portfolios:
• One consists of underlying asset and put option and
another consists of call option, and a cash position equal to
the discounted value of the strike price.
• Whether underlying asset price at maturity,
ends up
below or above strike price X; both portfolios will have
same value, namely
, at maturity
• Therefore they must have same value today, otherwise
arbitrage opportunities would exist
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Basic Definitions
• The portfolio values underlying this argument are shown in the
following

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

8


Basic Definitions

9

• put-call parity suggests how options can be used in risk
management
• Suppose an investor who has an investment horizon of days owns
a stock with current value St
• Value of the stock at maturity of the option is
which in the worst
case could be zero
• An investor who owns the stock along with a put option with a
strike price of X is guaranteed the future portfolio value
, which is at least X
• The protection is not free however as buying the put option requires
paying the current put option price

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


10

Option Pricing Using Binomial Trees
• We begin by assuming that the distribution of the future price of the
underlying risky asset is binomial

• This means that in a short interval of time, the stock price can only
take on one of two values, up and down
• Binomial tree approach is able to compute the fair market value of
American options, which are complicated because early exercise is
possible

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Option Pricing Using Binomial Trees

11

• The binomial tree option pricing method will be illustrated using
the following example:
• We want to find the fair value of a call and a put option with three
months to maturity
• Strike price of $900.
• The current price of the underlying stock is $1,000
• The volatility of the log return on the stock is 0.60 or 60% per year
corresponding to
per calendar day

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


12

Step 1: Build the Tree for the Stock Price
• In our example we will assume that the tree has two steps during

the three-month maturity of the option
• In practice, a hundred or so steps will be used
• The more steps we use, the more accurate the model price will be
• If the option has three months to maturity and we are building a
tree with two steps then each step in the tree corresponds to 1.5
months
• The magnitude of up and down move in each step reflect a
volatility of
• dt denotes the length (in years) of a step in the tree

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


13

Step 1: Build the Tree for the Stock Price
• Because we are using log returns a one standard deviation up
move corresponds to a gross return of

• A one standard deviation down move corresponds to a
gross return of

• Using these up and down factors the tree is built as seen in
Table 10.1, from current price of $1,000 on the left side to
three potential values in three months
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


14

Table 10.1: Building the Binomial Tree Forward
from the Current Stock Price
Market Variables

D

St=

1000

Annual r =

0.05

1528.47

Contract Terms
X=

1100

B

T=

0.25

1236.31

Parameters

Annual Vol=
tree steps

0.6
2

dt=

0.125

u=

1.236311

d=

0.808858

A

E

1000.00

1000.00

C
808.86

F

654.25

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Step 2: Compute the Option Pay-off at
Maturity

15

• From the tree, we have three hypothetical stock price values at
maturity and we can easily compute the hypothetical call option at
each one.
• The value of an option at maturity is just the payoff stated in the
option contract
• The payoff function for a call option is

• For the three terminal points in the tree in Table 10.1,

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Step 2: Compute the Option Pay-off
at Maturity

16

• For the put option we have the payoff function

• and so in this case we get

• Table 10.2 shows the three terminal values of the call and
put option in the right side of the tree.
• The call option values are shown in green font and the put
option values are shown in red font.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


17

Table 10.2: Computing the Hypothetical Option
Payoffs at Maturity
Market Variables

D

St =

1000

1528.47

Annual r =

0.05

628.47
0.00

Contract Terms

X=

900

B

T=

0.25

1236.31

Parameters
Annual Vol=

0.6

tree steps =

2

A

E

1000.00

1000.00

dt=

0.125

u=

1.236311

100.00

d=

0.808858

0.00
C

Stock is black

808.86

Call is green
Put is red
F
654.25
0.00
245.75

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Step 3:Work Backwards in the Tree to
Get the Current Option Value
•
•
•
•
•

18

Stock price at B = $1,236.31 and at C = $808.86
We need to compute a option value at B and C
Going forward from B the stock can only move to either D or E
We know the stock price and option price at D and E
We also need the return on a risk-free bond with 1.5 months to
maturity
• The term structure of government debt can be used to obtain this
information
• Let us assume that the term structure of interest rates is flat at 5%
per year

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Step 3:Work Backwards in the Tree to
Get the Current Option Value

19

• Key insight is that in a binomial tree we are able to construct a

risk-free portfolio using stock and option
• Our portfolio is risk-free and it must earn exactly the risk-free rate,
which is 5% per year in our example
• Consider a portfolio of 1 call option and ∆ B shares of the stock
• We need to find a ∆ B such that the portfolio of the option and the
stock is risk-free

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Step 3:Work Backwards in the Tree
to Get the Current Option Value

20

• Starting from point B we need to find a ∆ B so that

• which in this case gives
• which implies that

• So, we must hold one stock along with the short position
of one option for the portfolio to be risk-free
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


21

Table 10.3: Working Backwards in the Tree
Market Variables
St=

1000

D

Annual r =

0.05

Contract Terms
X=
T=

1528.47
628.47
0.00

900
0.25

Parameters
Annual Vol=
tree steps =
dt=
u=
d=
RNP =
Stock is black
Call is green
Put is red

0.6
2
0.125
1.236311
0.808858
0.461832

B
1236.31
341.92
0.00
A
1000.00
181.47
70.29

E
1000.00
100.00
0.00
C
808.86
45.90
131.43
F
654.25
0.00
245.75

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Step 3:Work Backwards in the Tree
to Get the Current Option Value

22

• The value of this portfolio at D (or E) is $900 and the portfolio
value at B is the discounted value using the risk-free rate for 1.5
months, which is

• The stock is worth $1,236.31 at B and so the option must
be worth
• which corresponds to the value in green at point B in Table
10.3
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Step 3:Work Backwards in the Tree to Get
the Current Option Value

23

• At point C we have instead that

• So that
• This means we have to hold approximately 0.3 shares for
each call option we sell
• This in turn gives a portfolio value at E (or F) of

• The present value of this is
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen


Step 3:Work Backwards in the Tree to
Get the Current Option Value
• At point C we therefore have the call option value

• which is also found in green at point C in Table 10.3
• Now that we have the option prices at points B and C we
can construct a risk-free portfolio again to get the option
price at point A. We get
• which implies that

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

24


Step 3:Work Backwards in the Tree to Get
the Current Option Value
25

• which gives a portfolio value at B (or C) of

• with a present value of
• which in turn gives the binomial call option value of
• which matches the value in Table 10.3
• Once the European call option value has been computed,
the put option values can also simply be computed using

the put-call parity
• The put values are provided in red font in Table 10.3
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen