Elements of financial risk management chapter 10

• 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 p

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

Option Pricing

Elements of Financial Risk Management

Chapter 10 Peter Christoffersen

• 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.

Basic Definitions

• 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

• S t is the price of the underlying asset today

• is the price of the underlying at maturity

Basic Definitions

• A European put option gives the owner the option the right to sell a

unit of the underlying asset days from now at the price X

• 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 only around 252 trading

days.

T ~

T ~

Basic Definitions

• 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

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

Basic Definitions

• 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

Basic Definitions

• The portfolio values underlying this argument are shown in the

following

Basic Definitions

• 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 S t

• 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

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

Option Pricing Using Binomial Trees

• 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

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

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

from the Current Stock Price

Step 2: Compute the Option Pay-off at

Maturity

• 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,

Step 2: Compute the Option Pay-off

at Maturity

• 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.

Step 3:Work Backwards in the Tree to

Get the Current Option Value

• 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

Step 3:Work Backwards in the Tree to

Get the Current Option Value

• 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

Step 3:Work Backwards in the Tree

to Get the Current Option Value

• 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

Table 10.3: Working Backwards in the Tree

Step 3:Work Backwards in the Tree

to Get the Current Option Value

• 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

Step 3:Work Backwards in the Tree to Get

the Current Option Value

• 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

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

Step 3:Work Backwards in the Tree to Get

the Current Option Value

• 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

Risk Neutral Valuation

• We have constructed a risk-free portfolio that in the absence of

arbitrage must earn exactly risk-free rate

• From this portfolio we can back out European option prices

• For example, for a call option at point B we used the formula

• which we used to find the call option price at point B

using the relationship

Risk Neutral Valuation

• Using the ∆B formula we can rewrite Call B formula as

• where the risk neutral probability of an up move is defined as

price appears as a discounted expected value when using

RNP in the expectation

• RNP can be viewed as the probability of an up move in a

world where investors are risk neutral

Risk Neutral Valuation

Risk Neutral Valuation

• The new formula can be used at any point in the tree

• For example at point A we have

• It can also be used for European puts

• We have for a put at point C

RNP is constant throughout the tree

Pricing an American Option using

the Binomial Tree

• American options can be exercised prior to maturity

• This added flexibility gives them potentially higher fair market

values than European-style options

• Binomial trees can be used to price American options

• At the maturity of the option American- and European-style

options are equivalent

• But at each intermediate point in the tree we must compare

European option value with early exercise value and put the

largest of two into tree at that point

Pricing an American Option using

the Binomial Tree

• Let us price an American option that has a strike price of 1,100 but

otherwise is exactly the same as the European option considered before

• If we exercise American put option at point C we get

• We have the risk-neutral probability of an up-move RNP =

0.4618 from before

• So that the European put value at point C is

• which is lower than the early exercise value $291.14

Pricing an American Option using

the Binomial Tree

• Early exercise of the put is optimal at point C as fair market value

of the American option is $291.14 at C

• This value will now influence the American put option value at

point A, which will also be larger than its corresponding European put option value.

• Table 10.4 shows that the American put is worth $180.25 at point A

• The American call option price is $90.25, which turns out to be the

European call option price as well

• American call stock options should only be exercised early if a

large cash dividend is imminent

American Put is red 0.00

Dividend Flows, Foreign Exchange

and Future Options

• When the underlying asset pays out dividends or other cash flows

we need to adjust the RNP formula

• Consider an underlying stock index that pays out cash at a rate of

q per year In this case we have

• When underlying asset is a foreign exchange rate then q is

set to interest rate of the foreign currency

so that RNP = (1-d) / (u-d) for futures options

Option Pricing under the Normal

Distribution

• We now assume that daily returns on an asset be independently

and identically distributed according to normal distribution

• Then the aggregate return over days will also be

normally distributed with the mean and variance

appropriately scaled as in

• and the future asset price can be written as

Option Pricing under the Normal

Distribution

• The risk-neutral valuation principle calculates option price as the

discounted expected payoff, where discounting is done using free rate and where the expectation is taken using risk-neutral

risk-distribution:

• Where is the payoff function and rf is

the risk-free interest rate per day

• The expectation is taken using the risk-neutral

distribution where all assets earn an expected return equal

to the risk-free rate

Option Pricing under the Normal

Distribution

• In this case the option price can be written as

• where x* is risk-neutral variable corresponding to the

underlying asset return between now and maturity of

option

• f (x*) denotes risk-neutral distribution, which we take to

be normal distribution so that

Option Pricing under the Normal

Distribution

• Thus, we obtain the Black-Scholes-Merton (BSM) call option price

variable, and where

Option Pricing under the Normal

Distribution

• Interpretation of elements in the option pricing formula

• is the risk-neutral probability of exercise

• is the expected risk-neutral payout when exercising

• is the risk-neutral expected value of the stock

acquired through exercise of the option

∀ Φ(d) is the delta of the option, where is the first

derivative of the option with respect to the underlying asset price

Option Pricing under the Normal

Distribution

• Using the put-call parity result and the formula for c BSM , we can get the put price formula as

• Note that the symmetry of the normal distribution implies

that for any value of z

Option Pricing under the Normal

Distribution

• When the underlying asset pays out cash flows such as dividends,

we discount the current asset price to account for cash flows by replacing S t by

• Where q is the expected rate of cash flow per day until maturity of

the option

• This adjustment can be made to both the call and the put price

formula, and the formula for d will then be

Option Pricing under the Normal

Distribution

• The adjustment is made because the option holder at maturity

receives only the underlying asset on that date and not the cash flow that has accrued to the asset during the life of the option

• This cash flow is retained by the owner of the underlying asset

Option Pricing under the Normal

Distribution

• We now use the Black-Scholes pricing model to price a European

call option written on the S&P 500 index

• On January 6, 2010, the value of index was 1137.14

• The European call option has a strike price of 1110 and 43 days to

maturity

• The risk-free interest rate for a 43-day holding period is found

from the T-bill rates to be 0.0006824% per day (that is,

0.000006824)

• The dividend accruing to the index over the next 43 days is

expected to be 0.0056967% per day

• Assume volatility of the index is 0.979940% per day

Option Pricing under the Normal

Distribution

• Thus we have:

• from which we can calculate BSM call option price as

Model Implementation

• BSM model implies that a European option price can be written as

a nonlinear function of six variables

• The stock price is readily available, and a treasury bill rate

with maturity is used as the risk-free rate

• The strike price and time to maturity are known features

of any given option contract

• Volatility can be estimated from a sample of n options on

the same underlying asset, minimizing the mean-squared dollar pricing error (MSE):