investment analysis and portfolio management lecture 10 gareth myles

Investment Analysis and Portfolio Management

Lecture 10

Gareth Myles

Put-Call Parity

The prices of puts and calls are related Consider the following portfolio

Hold one unit of the underlying asset Hold one put option Sell one call option

The value of the portfolio is

P = S + Vp – Vc

At the expiration date

P = S + max{E – S, 0} – max{S – E, 0}

Put-Call Parity

If S < E at expiration the put is exercised so

P = S + E – S = E If S > E at expiration the call is exercised so

P = S – S + E = E Hence for all S

P = E This makes the portfolio riskfree so

S + Vp – Vc = (1/(1+r)t)E

Valuation of Options

At the expiration date Vc = max{S – E, 0} Vp = max{E – S, 0} The problem is to place a value on the options

before expiration What is not known is the value of the

underlying at the expiration date This makes the value of Vc and Vp uncertain An arbitrage argument can be applied to value

the options


The unknown value of S at expiration is replaced by a probability distribution for S

This is (ultimately) derived from observed data A simple process is assumed here to show

how the method works Assume there is a single time period until

expiration of the option The binomial model assumes the price of the

underlying asset must have one of two values at expiration


Let the initial price of the underlying asset be S The binomial assumption is that the price on

the expiration date is uS with probability p “up state” dS with probability 1- p “down state”

These satisfy u > d Assume there is a riskfree asset with gross

return R = 1+ r It must be that u > R > d


The value of the option in the up state is Vu

= max{uS – E, 0} for a call = max{E – uS, 0} for a put The value of the option in the down state is Vd

= max{dS – E, 0} for a call = max{E – dS, 0} for a put Denote the initial value of the option (to be

determined) by V0 This information is summarized in a binomial

tree diagram


Stock Price SOption Value 0V

Stock Price uSOption Value uV

dVStock Price dSOption Value

Probability p

Probability 1 - p

Risk-free (gross) return R


There are three assets Underlying asset Option Riskfree asset

The returns on these assets have to related to prevent arbitrage

Consider a portfolio of one option and – units of the underlying stock

The cost of the portfolio at time 0 is

P0 = V0 – S


At the expiration date the value of the portfolio is either

Pu = Vu - uS or

Pd = Vd - dS The key step is to choose so that these are

equal (the hedging step) If = (Vu – Vd)/S(u – d) then

Pu = Pd = (uVd – dVu)/(u – d)


Now apply the arbitrage argument The portfolio has the same value whether the

up state or down state is realised It is therefore risk-free so must pay the risk-

free return Hence Pu = Pd = RP0

This gives

R[V0 – S] = (uVd – dVu)/S(u – d)


Solving gives

This formula applies to both calls and puts by choosing Vu and Vd These are the boundary values

The result provides the equilibrium price for the option which ensures no arbitrage

If the price were to deviate from this then risk-free excess returns could be earned

du Vdu

RuV

du

dR

RV

10


Consider a call with E = 50 written on a stock with S = 40

Let u = 1.5, d = 1.125, and R = 1.15

Stock Price SOption Value 0V

Stock Price uS = 60

Vu = max{60 – 50, 0} = 10Probability p

Probability 1 - p

Risk-free (gross) return R = 1.15

Stock Price dS = 45

Vd = max{45 – 50, 0} = 0


This gives the value

For a put option the end point values are

Vu = max{50 – 60, 0} = 0

Vd = max{50 – 45, 0} = 5 So the value of a put is

58.0

0375.0

35.010

375.0

025.0

15.1

10

V

058.45375.0

35.00

375.0

025.0

15.1

10

V


Observe that

40 + 4.058 – 0.58 = 43.478 And that

(1/1.15) 50 = 43.478 So the values satisfy put-call parity

S + Vp – Vc = (1/R)E


The pricing formula is

Notice that

So define

0 ,0

du

Ru

du

dR1

du

Ru

du

dR

du

dRq

du Vdu

RuV

du

dR

RV

10


The pricing formula can then be written

The terms q and 1 – q are known as risk neutral probabilities

They provide probabilities that reflect the risk of the option

Calculating the expected payoff using these probabilities allows discounting at the risk-free rate

du VqqVR

V ]1[1

0


The use of risk neutral probabilities allows the method to be generalized

V0

Vd

Vdd = max{ddS – E, 0} for a call

= max{E – ddS, 0} for a put

Vud= Vdu = max{udS – E, 0} for a call

= max{E – udS, 0} for a put

Vuu = max{uuS – E, 0} for a call

= max{E – uuS, 0} for a putVu

q

q

q

1 – q

1 – q

1 – q


u and d are defined as the changes of a single interval

R is defined as the gross return on the risk-free asset over a single interval

For a binomial tree with two intervals the value of an option is

dduduu VqVqqVqR

V 2220 )1()1(2

1


With three intervals

Increasing the number of intervals raises the number of possible final prices

The parameters p, u, d can be chosen to match observed mean and variance of the asset price

Increasing the number of periods without limit gives the Black-Scholes model

dddudduuduuu VqVqqVqqVqR

V 322330 )1()1(3)1(3

1

investment analysis and portfolio management lecture 10 gareth myles

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