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LECTURE 4 PART 1 INTRODUCTION TO OPTIONS An option is the right, but not the obligation, to do something Usually the something is the act of either buying or selling some asset or financial instrument. Options are derivative securities. This term means that the value of the contract is derived from (i.e. depends on) the value of some other underlying asset. Financial Options are options over financial instruments such as shares, bonds, share price indexes, bank bills etc. These are options to either buy or to sell an asset / financial instrument. There are 2 basic types: CALL OPTIONS AND PUT OPTIONS CALL OPTIONS A Call Option is a contract which gives the owner of the contract (the holder ) the right to buy the asset. Example: Suppose you have a call option over 100 ounces of gold. The contract gives you as the investor the right, but not the obligation, to buy 100 ounces of gold, in exactly 1 years time, for a price of $US 300.00 per ounce. Then x The underlying asset is 100 oz of gold. x term of the contract is 1 year x the exercise price or strike price of the option is $300 per

ounce, which for 100 ounces is $30,000.00 You as the investor have the right to buy gold from the writer of the option. The writer of the option is the person or organisation

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which originally sold the option and which takes on the obligation to sell 100 ounces of gold in 1 years time in return for receiving $30,000.00. At the time when you buy the option, you do not know what the price of gold will be in 12 months time. If, in 1 years time the gold price is $250.00 per ounce then you could buy the 100 oz of gold in the open market for $25,000 instead of buying the gold under the option contract for $30,000.00. The word option means that you do not have to buy the gold for $30,000 in 12 months unless you choose to. In this case you would be paying too much if you did exercise your rights under the contract and could do better by buying gold in the open market for $5,000 less. If you do not exercise your right to buy then the option is said to lapse. If on the other hand, the price of gold were to go up to $500 per ounce in 12 months time then you would (and should) exercise your right to buy gold at a price of $300 per ounce under the terms of the option contract. You could then sell the gold in the open market for $500 per ounce and make a profit of $200 per ounce on 100 ounces which is a profit of $20,000. The date by which you have to choose whether to exercise the option is called the exercise date or the expiry date or the maturity date these terms all mean the same thing. PUT OPTION Another kind of option is an option to sell an asset / instrument on some future date for an agreed price. This is called a put option. For instance you may have bought the right (but not the obligation) to sell 100 ounces of gold for a price of $350 per ounce in 12 months time.

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You would exercise the right to do this if and only if there is a profit (and not a loss) to be had from doing so. This would be true if and only if the price of gold on the maturity date is less than the exercise price of $350 per ounce. European and American Options Another way to classify options is according to the exercise rights of the contract. Some contracts will only allow you to exercise the right to buy or sell on the maturity date itself. These are called European options. Other types of contract will allow you to exercise the right to buy or sell the asset on any date up to and including the maturity date. These sorts of options are called American Options. Note that the terms European and American refer only to the type of exercise rights the option contracts have and not the geographic location where the options are traded. For instance we have European Options traded in American Financial Markets and American options traded on European Financial Markets. Bermudan Option can be exercised early on a specific date or set of dates

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USES OF OPTIONS Option contracts are used by banks, investors, governments and corporations for a range of different purposes. The main reasons for using them are: x for hedging (the motivation being to reduce financial risk) This

is like buying insurance against financial risk x for speculation (the motivation being to make trading profits

from taking a bet on which way market prices may move) this is like gambling. There are many different strategies that can be implemented with options more on this later.

x for arbitrage by exploiting differences in prices between different markets or different instruments we may be able to make a risk free profit.

x Fee income: to make money from selling options for banks and other option writers, it is like selling an insurance policy and they want to sell it for more than it is going to cost them (at least on the average)

INTRINSIC VALUE The intrinsic value of an option is defined as the payoff from exercise of the option assuming it could be exercised immediately. Market Value = Time Value + Intrinsic Value Option contracts have a market value. We can think of the market value as being made up of 2 components:

1. the intrinsic value and 2. the time value.

The time value is defined to be the difference between the market value of the option contract and the intrinsic value of the option, provided this difference is positive, and zero otherwise.

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For American options, the time value represents the value of the opportunity presented by the possibility that the asset will increase above its current level. American options can be exercised immediately but European options cannot be exercised until their maturity date. Notation x S = market price of the underlying asset x TS = market price of the underlying asset at time T x X = exercise price of the option contract x T = Term to maturity of the option x c = market value of a European call option contract over

S with exercise price X and term to maturity T x p = market value of a European put option contract over

S with exercise price X and term to maturity T x C = market value of American call option contract over

S with exercise price X and term to maturity T x P = market value of American put option contract over S

with exercise price X and term to maturity T The intrinsic value of a call option is max( ,0)S X because: If S X! then we could

x buy the asset for price X by exercising our rights under the contract

x then sell the asset in the market for S x make a profit of S X as a result

If S Xd then we would x not exercise our rights under the contract and the profit would

be 0.00

This is written more compactly as max( ,0)payoff S X S X Note: max( ,0)Y Y

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The intrinsic value of a put option is max( ,0)X S X S because: If S X then we could

x buy the asset for price S in the open market x sell the asset for a price of X by exercising our rights under

the put option contract x The resulting profit is X S

If S Xt then we would

x not exercise our rights under the contract and the profit would be 0.00

This is written more compactly by writing max( ,0)payoff X S Example: Consider an American Put Option with

x a term to maturity of 1 year, x the underlying asset is 1000 shares in company XYZ. x exercise price of the option is $1000

Suppose the interest rate is 10% p.a. and the company XYZ has just gone into liquidation with its liabilities exceeding its assets. Answer the following questions: (a) what is the intrinsic value of the put option? (b) when should you exercise the option should you do it

immediately or should you wait until some later time? Why? (c) If the option were a European Option then what would the

value of the option be and why? (d) Why is an American option always worth at least as much as

and possibly more than the equivalent European Option?

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Answers (a) the shares of XYZ will have a value of $0.00 they will have

no value. Thus the intrinsic value of the put option is max( ,0) max(1000 0,0) 1000payoff X S .

(b) this put option is an american option ? you can exercise it

immediately and you should exercise immediately. Doing so gives you an immediate payoff of $1000 now. Delaying the decision means you receive $1000 at some later time. It is better to have money now rather than later so exercise immediately.

(c) If the option were European then you could (and should)

exercise your right to sell the shares in 1 years time for a price of $1000 and the profit to be had from doing this is $1000 received at the end of a year. The option is european and cannot be exercised before then. In this situation the value of the option is the present value (@ 10%) of $1000 received in 1 years time. This is $1000.00 $909.91

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(d) An American option is equivalent to a European one plus the

additional feature of being able to exercise the option early. This additional feature can only add to its value and not reduce its value.

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Payoff and Profit Diagrams A payoff diagram for an option shows the payoff from exercising the option as a function of the price of the underlying asset. This is distinguished from the profit diagram which shows the profit to the investor from exercising the option profit is the payoff received minus the price paid for the option(s). example: call option with: x spot price S= $3.00, strike price X=$2.50 x term T = 1 x Cost = $0.72 We shall show the calculation of the payoff at maturity from exercising this option, and the profit at maturity as well. The price we agree to pay in return for the stock is called the "strike price" or "exercise price" and for this option it is X=$2.50. We use X to denote it. The term to maturity is denoted by T and for this option it is T= 1 year The variability of the return on the stock is called the "volatility" and it is an annualised measure. This option gives us the right, but not the obligation to buy the stock for a price of $2.50 in 1 year's time. We will do this if it is financially worthwhile at the time, but not if we will make a loss from doing so. The price of the option is called the "premium" and for this option it is $0.72.

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We have to pay $0.72 to buy the option contract (which gives us the right to buy the stock in 1 year for a price of $2.50). The current price of the stock is called the "spot price" (S) and it is $3.00 for this option. The following table shows the payoff at maturity and the profit at maturity for a range of values for the underlying asset. If S = 3.00 then

x the payoff is max(3.00-2.50,0)=0.50 but x the profit is payoff - cost = 0.50-0.72 = -0.22. x We would exercise the option because not doing so makes

the profit lower at -0.72 If S = 2.00 then

x the payoff is max(2.00-2.50,0)=0.00 x the profit is payoff - cost = 0.00-0.72 = -0.72. x We would not exercise the option because doing so makes

the profit even lower - the act of buying at 2.50 a stock worth 2.00 is a loss making transaction.

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Table Showing Payoff from exercise and Profit from Exercise

S Payoff Profit $2.00 $0.00 -$0.72$2.10 $0.00 -$0.72$2.20 $0.00 -$0.72$2.30 $0.00 -$0.72$2.40 $0.00 -$0.72$2.50 $0.00 -$0.72$2.60 $0.10 -$0.62$2.70 $0.20 -$0.52$2.80 $0.30 -$0.42$2.90 $0.40 -$0.32$3.00 $0.50 -$0.22$3.10 $0.60 -$0.12$3.20 $0.70 -$0.02$3.30 $0.80 $0.08$3.40 $0.90 $0.18$3.50 $1.00 $0.28$3.60 $1.10 $0.38$3.70 $1.20 $0.48$3.80 $1.30 $0.58$3.90 $1.40 $0.68$4.00 $1.50 $0.78

If S = 3.40 then Payoff = max(3.40-2.50,0)=0.90 but profit = 0.90-0.72 = 0.18 = revenue cost If S = 2.90 then Payoff = max(2.90-2.50,0)=0.40 but profit = 0.40-0.72 = -0.32 = revenue cost If S = 2.00 then Payoff = max(2.00-2.50,0)=0.00 but profit = 0.00-0.72 = -0.72 = revenue cost

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call option payoff and profit at maturity

-$1.00

-$0.50

$-

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$1.

00

$1.

20

$1.

40

$1.

60

$1.

80

$2.

00

$2.

20

$2.

40

$2.

60

$2.

80

$3.

00

$3.

20

$3.

40

$3.

60

$3.

80

$4.

00

$4.

20

$4.

40

$4.

60

$4.

80

$5.

00

spot price

$val

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Option Terminology: In the money The option is said to be "in the money" if it has a positive intrinsic value, i.e. if it would be worth exercising (if you could exercise it). x a call is in the money if S > X, while x a put is in the money if S

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Out of the money The option is said to be "out of the money" if it is not worth exercising x a call is out of the money if S < X x a put is out of the money if S >X ETO's Options can be bought and sold like any other financial instrument. Some options are traded on exchanges. These are known as "exchange traded options", or ETO's. OTC's Some options are not traded on exchanges but are individually negotiated between buyer and seller. The seller in these circumstances is the party who accepts the obligations to provide the payoff to the buyer of the option should the buyer choose to exercise the option. These sorts of options are called "over the counter options" or OTC options.

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Buying / selling / writing options If you have bought an option then you are (while you own it) the "holder" of the option and you can sell it to some other investor. The party that has the obligation to provide the payoff on the option is called the option writer. If you sell an option that you bought, you dont have the obligation to make the option payoff on the maturity date, the option writer does. If you do not own the option you can "short sell" the option or "write" the option. This (writing or short selling) means you accept the liability involved for the other side of the deal. The payoff to the option writer is the reverse of the payoff to the option holder. For a call option x the payoff to the option holder at maturity is max( ,0)TS X x the payoff to the option writer at maturity is max( ,0)TS X Writing an option is equivalent financially to short selling an option. In some books they don't distinguish between selling an option and writing (short selling) an option. Usually by selling they mean writing. This can lead to confusion.

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Variables that impact on option values The variables which affect the option prices and the effect of changes (increases) in them are as follows: variable effect on call effect on put spot price S increases decreases Strike price X decreases increases dividends paid decreases increases Term to maturity increases increases risk free interest rate

increases decreases

Volatility increases increases The effect of increases in either term to maturity or volatility is to increase the values of both call and put options. The basic reason is that an increase in either volatility or term will increase the likelihood that the option will be worth exercising at maturity. What is the volatility? The term "volatility" is a measure of the variability of the stock price. It is actually defined as the annualised standard deviation of the continuously compounded return on the stock. We measure / define it as V , where

2

0

1 var log TeS

T SV

or equivalently, 0

1 var log TeSST

V

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Note that x the ratio

0

TSS

is called the price relative. If you invest $1.00 at

time 0, it will grow in value to an amount of 0

TSS

at time T

x 0

log TeSRS

is the continuously compounded return on the

stock over the period from time 0 to time T

x 0

1log TeSrS T

u

is the continuously compounded rate of return

(per year) on the stock x This return is random, since we do not know in advance what

the future value TS of the stock will be x The volatility is the square root of the variance of the

annualised rate of return on the stock European vs American Options American option is always worth at least as much as the equivalent European option. (equivalent meaning same strike price, underlying asset, term to maturity etc) The reasons are 1. American option is equal to a European option plus the right to

exercise early 2. the right to exercise early must have a value t 0

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CALL OPTIONS It can be shown that for x an american call option x over a non dividend paying stock, x it is never optimal to exercise the option early. Therefore it follows that American call value = European call value in this case. This does not necessarily apply in the case of dividend paying stocks. American calls on dividend paying stocks may be worth exercising early. By dividend paying we mean the stock pays a dividend during the term of the option. PUT OPTIONS For put options over non dividend paying stocks it is sometimes optimal to exercise the option early. Therefore american put options have a value that is european put options.

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Part 2: the Black Scholes Formula There is an analytic formula for European call and put options over an asset that does not pay a dividend during the life of the option. It is called the Black Scholes formula. This analytic formula does not apply to American options in general. This is one of the deficiencies of the BS formula. Black Scholes Formula for a European Call Option over a non dividend paying stock

1 2( ) ( )rTc S N d Xe N d u u

where x S = "spot price" (i.e. the current price) of the stock x X = the exercise price of the option x r = the risk free interest rate per annum x T = term to maturity of the option x V = volatility of the stock x c = value a european call option

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1 1log2e

Sd r TXT

VV

2

2 11 1log

2eSd r T d TXT

V VV

21

212

xz

N x e dzS

f

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N(x) is the cumulative density function of the standard normal distribution. You do not have to calculate it. We use tables of the normal distribution to obtain the value of N(x). Excel has a built in function that computes it, called the "normsdist" function. What is N(x) The following graph shows both x the probability density function n(x) (the bell shaped curve) and x the cumulative density function N(x) (the S shaped curve) for

the standard normal distribution. x N(x) is the area under the curve n(x) to the left of the value x. x N(x) varies between 0 and 1. x It is the probability that Z

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Black Scholes Formula for a European Put Option over a non dividend paying stock

2 1( ) ( )rTp Xe N d S N d u u

where the definitions of the various variables are as above for a call option. Under the Black Scholes Model, the logarithm of the price relative has a normal distribution. x

0

TSS

is the price relative, the ratio of the value of the stock at the

maturity date to its initial value

x 0

log TeSS

is the natural logarithm of the price relative

x the BS model is based on the assumption that 0

log TeSS

has a

normal distribution with

1. 212

mean r TV

2. 2variance TV u 3. standard deviation TV u

Note we usually write just S instead of 0S for the initial value of the stock.

This distribution of the log of the price relative implies that the stock price has what is called a "log normal distribution". The expected rate of return on the stock is the risk free interest rate under this distribution. The distribution is called the "risk neutral distribution" of the stock price.

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This is because a "risk neutral investor" would be indifferent between a risk free bond returning rate r and a risky stock with the same expected return of r. Worked example of call option valuation We shall calculate the value of a 3 month European call option with a strike price of $9.50 over a non dividend paying stock worth $10.00. The details are x S = $10.00 = spot price of asset x X = $9.50 = exercise price (strike price) of option x r = 10% = risk free rate of interest x V = 20% = volatility of asset x T = 0.25 years = 3 months We shall also calculate the value of a put option and the intrinsic value and time value of the call and of the put For the call option, the calculation procedure is as follows: Step 1: compute d1 and d2

10.00 1.0526329.50

SX

log log 1.052632 0.051293e eSX

2 21 10.10 0.20 0.25 0.032 2

r TV u

0.20 0.25 0.10TV u

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21

1 1log2

0.081293 0.812930.10

eSd r TXT

VV

2 1 0.81293 0.10 0.71293d d TV Step 2: Calculate (or lookup in tables) the values of 1 2( ) & ( )N d N d To 4 decimal places these values are:

1( ) 0.7919N d 2( ) 0.7621N d

These can be computed in excel or looked up in tables of the standard normal distribution. Step 3: Now we have the values of 1( )N d and 2( )N d we plug these values into the BS formula

> @> @ > @

1 2( ) ( )

10.00 0.7919 9.50 0.975310 0.76210.8579

rTc S N d Xe N d

cc

u u u u u

The value of the call option is 85.79 cents. Note: xe is exp(x) in excel, the function loge x is ln x in excel. The intrinsic value of the call option here is $0.50. The value of the option using BS model was $0.8579 The difference between the option value and the intrinsic value is called the "time value" of the option ($0.3579 for this example).

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Put Option valuation We shall now calculate the value of a 3 month European put option with a strike price of $9.50 over a non dividend paying stock worth $10.00. The details are as above. We have already done the hard work of computing the d's and N(d)'s when we valued the call option. The formula for the put option is

2 1( ) ( )rTp Xe N d S N d u u

Now that we have the value of 1 2( ) & ( )N d N d we can compute the values of 1 2( ) & ( )N d N d using the relationship

( ) ( ) 1.0N x N x from which it follows that

( ) 1.0 ( )N x N x This allows us to compute the values 1 2( ) & ( )N d N d without having to redo the calculations above.

> @

2 1

2 1

( ) ( )

1 ( ) 1 ( )

9.50 0.975310 1 0.7621 10.00 1 0.7919

0.1234

rT

rT

p Xe N d S N d

Xe N d S N d

p

u u u u

u u u

The intrinsic value of this option is 0.00 as it would not be worth exercising the right to sell the stock for $9.50 when its market value is $10.00. The time value of the option is thus $0.1234, which is 100% of the option's value.

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ADJUSTING BLACK SCHOLES FORMULA FOR STOCKS THAT PAY DIVIDENDS We can adapt the BS formula to cope with options over shares that pay a dividend during the life of the option. We consider 2 different adjustments: Case 1: known dollar dividend amount Let D be the amount of dividends paid at time W where 0

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We use this share price instead of $50.00 as the spot price input to the BS formula. The calculations are as follows: spot price S $49.44strike price X $45.00term T 0.3333volatility v 24.49%int rate r 3.00%S/X 1.098729ln(S/X) 0.094154(r+0.5vol^2)T 0.0200vol*root(T) 0.1414d1 0.8072d2 0.6658Nd1 0.7902Nd2 0.7472C $5.7804 The option price is found to be $5.78. Exercise: check this numerically using a spreadsheet or otherwise. Case 2: known dividend yield Let the dividend yield on the underlying asset (stock) be q. We take this to mean a continuous dividend yield.

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BS formulae for options over an asset with a continuous dividend yield at rate q: The formulae for European Calls and Puts are

1 2( ) ( )qT rTc Se N d Xe N d u u

2 1( ) ( )rT qTp Xe N d Se N d u u

Where

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1 1log2e

Sd r q TXT

VV

22 1

1 1log2e

Sd r q T d TXT

V VV

Note that the dividend yield q appears as part of a discount factor applying to the stock price and also in the definitions of the d's here. Exercise: Show algebraically that we get exactly the same value for both calls and puts if we use the standard black scholes model with an adjusted stock price of ' qTS Se and use this adjusted stock price as the spot price input to the BS model

> @1 2, , , , , ( ) ( )rTc c S X r q T SN d Xe N dV where 21 1 1 1, , , , , log 2e

Sd d S X r q T r q TXT

V VV

2 1d d TV Write an expression for ' , , , , ,c S X r q TV

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Numerical Worked example: Consider the above option on IBM but now value it assuming a 2% p.a. dividend yield.

spot price S $50.00 strike price X $45.00

term T 0.3333 volatility v 24.49% int rate r 3.00%

div yield q 2.00% S/X 1.1111

ln(S/X) 0.1054 (r-q+0.5vol^2)T 0.0133

vol*root(T) 0.1414 d1 0.8393 d2 0.6979

Nd1 0.7993 Nd2 0.7574

C $5.9592

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PART 3: Put Call Parity and Bounds on option values Bounds means upper and lower limits for option values Option values are influenced by a range of factors as reflected in the parameters of the Black-Scholes option valuation formula. With European options the payment of a dividend can make a longer maturity European call option less valuable than a shorter maturity contract since the dividend results in a fall in the share price ex-dividend usually around 60-70% of the amount of the dividend because of tax and other factors. The bounds we cover here are general in that they only rely on simple arbitrage arguments. If they do not hold then there exists a strategy that can be used to guarantee a profit without taking any risk. It is important to realise that minimal assumptions are made in order to derive these results. notation: Notation European American

Call C C Put P P

note that

0 max ,0X S Xd d 0 max ,0S X Sd d

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For upper bounds when there are no dividends on the asset:

,C S c Sd d Both American and European call option values are less than the stock price. S is an upper bound for these option values. The right to buy the stock cant be worth more than the stock itself. The reason is that If c S! then we can

x sell the call for c, buy the stock for S, x time t = 0 cashflow = c-S>0 x wait till time T, if the call is exercised,

give the stock to the option holder and receive X for it, otherwise sell stock for S.

time 0 payoff = c-S>0 time T payoff = min(X,S)>0 we make a risk free profit

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,P X p Xd d Both American and European put option values

are less than the exercise price. X is an upper bound for these option values Because: If p X! then we can

x sell the put at time 0 for amount p, x immediately invest amount X in a risk free

bond for term T at rate r, x keep the difference p-X x hold this position until either the put is

exercised or till time T (maturity) x if the put is exercised the value of our

position is , max ,0 0rtX

X

Xe X St d

t

x if the put isnt exercised then our position at maturity is zero

overall we make a profit at time 0 and may make a profit at some future time too but not a loss

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For a European put option the bound can be improved:

rTp Xed European put option values are less than the present value of the exercise price rTXe .

rTXe is an upper bound for these option values Otherwise we would have rTp Xe! If so we could write (sell) the put option and invest the proceeds at the risk free interest rate to make a guaranteed profit. If you did this you get a payoff of p at time 0. By investing it to time T at the risk free rate you have rTpe at time T and this is more than X, rT rT rTpe Xe e X ! If the option holder who you sold the option to did exercise the option, then the payoff to them would be max ,0X S and your payoff as the option writer would be max ,0X S . Our put option liability is max ,0X S X d . So the worst case for you at maturity is a payoff of , max ,0 0rT

X X

pe X S! d

!

This happens only if the put option is exercised. If the put is not exercise then you get a payoff of p at time 0 which has an equivalent value of

rTpe X ! at time T

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For lower bounds when there are no dividends on the underlying: European call option: max( ,0)rTc S Xet max( ,0)rTS Xe is a lower bound for the option value Proof: we compare the pay-offs at maturity of 2 portfolios: 1. a European call option (c) with 2. a portfolio of one share and borrowing equal to the present

value of the exercise price rTXe , borrowed at rate r for term T. The value of this is rTS Xe at time 0.

At maturity the borrowing has a value of X, because rTXe invested at rate r for term T accumulates to rT rTXe e X u At the maturity date we must have either TS X! or TS Xd . We look at what happens to the 2 payoffs under these 2 scenarios. This is done in tabular form below. scenario for stock price at maturity

call option payoff

portfolio payoff which payoff is higher?

TS X! TS X TS X same TS Xd 0 TS X option is higher

It can be seen that the call option pay-off is always greater or equal to that on the portfolio which means that the value of the option must be greater than the value of the portfolio. Hence

rTc S Xet . But we also have 0c t ( the value of the option cant be negative). Hence it follows that max( ,0)rTc S Xet

32

For the European put option max ,0rTp Xe S! max ,0rTXe S is a lower bound for the put

Proof: This result can be derived by comparing 2 portfolios:

1. a European put option with 2. a portfolio made up of of cash equal to the present value of

the exercise price rTXe and a short (sold) position in one share.

At maturity x our risk free investment accumulates to rT rTXe e X u x our liability under the short sale agreement is TS scenario for stock price at maturity

put option payoff

portfolio payoff which payoff is higher?

TS X! 0 0TX S put is higher TS Xd TX S TX S same

The method of taking a short position in a share in practice is to borrow the share from an existing holder, under a share borrowing arrangement or a buy-back agreement, and to sell this share. Alternatively an equivalent short position can be established using a forward contract or a futures contract. For the American put option A lower bound is max ,0p X St This is true because of the possibility of early exercise This lower bound is stronger than the one for European puts which was max ,0rTp Xe St

33

Put Call Parity (when the underlying has no dividends) This is a very important relationship between the values of a European call option, the value of a European put option, and the value of a long forward contract. All 3 contracts:

x are over the same underlying asset, x have the same delivery / exercise price, and x have the same maturity date.

This relationship does not depend on the validity of the Black Scholes formula. The relationship says that a combination of a long call option and a short put option is equivalent to a long forward contract.

rTc Xe p S or equivalently , ,long long forwardshortcall contractput

rTc p S Xe

Proof: Let S and X be 2 numbers. Then

max ,0 max ,0S X X S S X Because: if S Xt then

0

max ,0 max ,0S X

S X X S S X

if S X then

0

max ,0 max ,0 0X S

S X X S X S S X

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Alternatively, Consider 2 portfolios: Portfolio 1 = a European call option plus

a cash holding equal to the present value of the exercise price

Portfolio 2 = a European put option plus a holding of 1 unit of the stock

Now consider the value at maturity of these portfolios. This is the same for each of them, because at maturity the first portfolio has value

,max( ,0) rT rTTaccumulationcall cash factorpayoff

S X Xe e u max( ,0) max( , )T TS X X S X

and the second has value of max( ,0) max( , )T T TS X S X S note the results max( ,0) max( , )S X X S X and max( ,0) max( , )X S S S X this is easily proven (try it for yourself) If the pay-offs are the same at maturity then the values must be equal, (this is the law of one price) otherwise there is the opportunity to make arbitrage profits. Otherwise we could buy the cheaper portfolio and sell the more expensive one to make a risk free profit. Hence it follows that rTc Xe p S This is called put call parity. It is a relationship that holds regardless of whether the BS model holds. It is enforced by arbitrage.

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This relationship is sometimes written instead as rTc p S Xe can rewrite this as rTc p S Xe interpretation: long call = long put plus long forward can rewrite as rTp c S Xe interpretation: short put = short call plus long forward can use put call parity to see how to synthetically create some securities from combinations of others. More on this later American Calls and European Calls on no dividend stock: Assume that there are no dividends payable during the term of the option. In this case it is never optimal to exercise an American call early and therefore C=c. Proof: Assume that it is optimal to exercise the call early at some time when the option has a remaining term of T years to maturity. Then: 1. we already showed that max( ,0)rTc S Xe! 2. it follows that rTc S Xet because max( ,0)rT rTS Xe S Xe t 3. we observed earlier that an American option is equivalent to a

European one plus the right to exercise early. From this it follows that C ct

4. hence we have rTC S Xet 5. the interest rate r is always >0. Hence we have rTXe X and so

C S X! as well 6. if it were optimal to exercise early then it would be true that

C S X at some time before the maturity date.

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Thus we reach a contradiction because C S X! and C S X cannot both be true. So it cant be possible for it to be optimal to exercise early. Accordingly it must be true that C=c (American call option value is the same as European call option value) If there are no dividends paid during the term of the option then it is never optimal to exercise an American call early and therefore C=c. Implication: We can value an American call using the Black Scholes formula for a European call when there are no dividends paid on the stock during the term of the option. American Puts and European Puts on no dividend stock: This result above does not hold for put options. Early exercise for a put might be optimal since

x P pt as American option = European option plus right to exercise early

x rTp Xe S! was proved earlier x rTP Xe S? ! x but P X St on early exercise. x if the stock price gets sufficiently low (e.g. close to zero) the

payoff from exercising early is higher than the possible payoff from waiting until later to exercise. In the extreme case where the stock price falls to zero we get X now by exercising early and with interest this accumulates to rtXe at time t. If we wait till later the stock price may recover and the payoff would be less.

x it follows that there are some circumstances where P>p

37

Put Call Parity for American Options on a no dividend asset The Put Call Parity relationship is rTS X C P S Xe d d This is an inequality that gives a range of values for C P , both an upper and a lower bound for it The put call parity relationship for European Options is an equality instead, which says that rTC P S Xe Proof:

1. from put call parity for European options we have rT rTc p S Xe p c Xe S ?

2. we know that P>p (American put worth more than European put)

3. from 1 and 2 we have rTP c Xe S! 4. with no dividends we have C = c 5. from 3 and 4 we have rTP C Xe S! and from this it

follows that rTP C S Xe and hence that rTC P S Xe . This proves part of the inequality

To prove the other part, namely S X C P d we have to consider 2 portfolios: Portfolio 1 = 1 unit of a European Call + cash of amount X Portfolio 2 = 1 unit of an American Put + 1 unit of the share Where both options have the same term to maturity and the same exercise price X. Assume the cash in portfolio 1 is invested at the risk free rate r for term T. The value of this cash at time is rXe W , where 0 TWd d .

38

The put option is an American put. It can be exercised early. We look at 2 separate cases. x Case 1: the put is not exercised early and x Case 2: the put is exercised early at time , where 0 TWd d Consider the value of portfolio 1 at time T under case 1.

10max ,

max ,0 max ,0rT rT

S X

V S X Xe S X X Xe XD !

1 max , max ,V S X S XD t Consider the value of portfolio 2 under case 1.

2 max ,0 max , max ,V X S S X S S X

1 2max , max ,V S X S X VD? t We see that the value of portfolio 1 is more than the value of portfolio 2. Consider the value of portfolio 2 at time under case 2. The put is only worth exercising at that time if S < X. ,2put payoff

V X S S X

Consider the value of portfolio 1 at time under case 2.

10max ,

max ,0 max ,0r r

S X

V S X Xe S X X Xe XW WD !

1 2max , max ,V S X S X X VD? t !

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We see that the value of portfolio 1 is more than the value of portfolio 2. So this holds for both case 1 and case 2, so it is true under all possible cases. It follows that c X P S

C X P SC P S X

t ? t ? t

Put-Call parity (where underlying asset has dividends) If D denotes the present value of dividends between the current date and the option expiration date then the put-call parity result needs to be modified to allow for the fact that options are not dividend protected (this means that if a dividend is paid no adjustment is made to the terms of the option resulting from the change in the share price after the dividend payment). The relationship for put-call parity then needs to allow for dividends and becomes For European options: ,

value of a long long a call &forward contractshort a put

rT rT

f

c D Xe p S c p S D Xe

For American options:

rTS D X C P S Xe d d

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Proof of put call parity : European options on dividend paying stock

rTc D Xe p S Portfolio A = long a European Call + investment of cash equal to D X invested at rate r

for term T in a risk free asset

Initial value of Portfolio A = 0AV c D X Portfolio B = long a European Put + holding of 1 unit of the asset

Initial value of Portfolio B = 0BV p S The underlying asset pays dividends with present value D during the term of the options. Thus a holding of the stock will produce a flow of dividends to the holder during the term of the option. If these dividends are reinvested at the risk free interest rate and accumulated to time T, the accumulated value at time T of the dividends will be rTDe The value at time T of the two portfolios will be:

max ,0

max ,0 max ,

rT rTA T

rT rTT T

V T S X D Xe e

S X X De S X De

max ,0 max ,rT rTB T T TV T X S S De X S De

We see that the payoff at time T from the 2 portfolios is the same. By the law of one price the values of the 2 portfolios must be the same at time t=0 as well. From this it follows that rTc D Xe p S

41

Put call parity : American options on dividend paying stock The put call parity result in this case is rTS X D C P S Xe d d This is an inequality which defines an upper and a lower bound for the difference C P Proof: exercise for students to do. Lower bound for European Call on a dividend paying asset We can show that 0 rTc S D Xet Proof consider 2 portfolios: x portfolio A = 1 unit of a European Call plus cash of amount rTD Xe

invested at rate r for term T in a risk free asset. x Portfolio B = 1 unit of the asset At time T the value of portfolio B is

, ,accumulatedvalue value of dividendsof 1 unit reinvested at rate rof Asset S

rTB TV T S De

The value at time T of portfolio A is

payoff on 1unit accumulated value of the call option at time T

of cash invested

max ,0 rT rTA TV T S X Xe D e

max ,0 D max , DrT rTA T TV T S X X e S X e

since it is true that max ,T TS X St it follows that A BV T V Tt If this is true at time t=T it must also be true at time t = 0

42

If not you could buy portfolio A and short sell B and hold till maturity for a guaranteed risk free profit. It follows that 0 0A BV Vt and hence 0rTc D Xe S t so that 0 rTc S D Xet Lower bound for European Put on a dividend paying asset Can show that 0rTp D Xe St : Consider 2 portfolios: x portfolio A = 1 unit of a European put plus 1 unit of the asset x Portfolio B = cash of amount rTD Xe invested at rate r for term T in

a risk free asset. At time T the value of portfolio B is

accumulated value at time Tof cash invested

rT rT rTBV T Xe D e X De

The value of portfolio A is , ,accumulated valuepayoff on 1unit reinvested of assetof the put option dividends

max ,0 rTA T TV T X S S De

,accumulated reinvested dividends

max , rTA TV T X S De

since max ,TS X Xt it follows that A BV T V Tt If this is true at time t=T it must also be true at time t = 0 If not you could buy portfolio A and short sell B and hold till maturity for a guaranteed risk free profit. It follows that 0 0A BV Vt and hence 0 rTp S D Xe t so that 0rTp D Xe St

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PART 4: BS Model and the replicating and self financing portfolio The B.S. model was based on the idea that you can create a dynamically adjusted portfolio comprised of a position in the stock and a position in a risk free bond that will x provide the same payoff as the option at maturity x cost a certain amount of money to establish x not cost any extra money as the portfolio is adjusted over time.

If we need to "rebalance" the portfolio by changing the mix of bonds and shares then the cost of the new portfolio is provided by liquidating the old portfolio. The strategy is "self financing"

The model is based on the assumption of perfect markets, along with some other assumptions: x no taxes x no transaction costs x can borrow or lend any amount of money at the risk free rate x not restrictions on short selling shares or bonds x all assets are perfectly divisible (you can hold fractional or

negative amounts) x stock price follows a continuous process called geometric

brownian motion (no jumps in share price) The number of units of the underlying share you should hold in the replicating portfolio is called the "option delta" or the "hedge ratio" For a long (i.e. bought) call option the hedge ratio is the coefficient of S in the BS formula

44

,1 2( ) ( )rT

hedge bondratio value

c S N d Xe N d u

The terms in the BS formula can be interpreted as follows: x 1( )N d is the hedge ratio, which is the number of units of the

share to hold in the hedge portfolio that dynamically replicates the option

x 2( )N d can be thought of as the probability that the call option is worth exercising at maturity

x 2( )rTXe N d u is the value of the bonds held in the hedge portfolio and a minus sign indicates borrowing (a short position in the bond)

For a long position in a put option the formula is

> @2 1. ( ) . ( )rTp Xe N d S N d here the hedge ratio (the coefficient of S) is 1( )N d which means we short sold 1( )N d units of the stock, and invested 2. ( )rTXe N d dollars in risk free bonds.

45

PART 5: WORKED EXAMPLES EX 1: Explain how you could construct a sold put synthetically using the following prices / market information:

x $5 call option at a premium of $0.20, (X = $5.00) x $5 put option at a premium of $0.25, (X = $5.00) x and a forward price of $5.00 for the asset

Is there an arbitrage opportunity in these prices? If so, describe the transactions that would achieve an arbitrage SOLUTION c - p = S Xe-rT = f = value of forward contract. Thus p = c - f, so -p = f c. Hence we can synthesize a sold put by selling calls and buying the forward contract (the minus sign means short selling and a plus sign means a long position). If the forward price is F = $5 then we can enter into a forward contract with delivery price X = $5 for no outlay of cash, as either the holder of a long or the holder of a short position. This means that f = 0 when X = $5. Therefore we can do the following and make a risk free profit (arbitrage profit): create a synthetic long put position by x selling the forward and buying the call, for a net cost of $0.20 x sell the put for $0.25 This creates an up front profit of $0.05 per put option. At maturity the cashflows on all of the contracts cancel out.

46

This analysis ignores transaction costs and bid offer spreads. When these are taken into account, it may eliminate or at least reduce the arbitrage profit.

EX2: A CAPITAL GUARANTEED INVESTMENT PRODUCT An Investment Company has guaranteed (in return for an extra fee) to pay an earnings rate of 5%pa, over the next 2 years, on money invested into a unit trust / mutual fund. The guarantee only applies at the end of the second year. This means that the investor is entitled to cash in the investment for his / her share of the value of the funds assets in 2 years time and is guaranteed to make at least a 5 % return on the investment. If the assets backing the fund make a return above 5% p.a. then the investor gets this higher return, but if the asset value should not grow at that rate, the investor still gets at least a 5% return on the money invested over the 2 year period. The financial institution will charge a price to the investor for this investment guarantee we will look at how they could quantify the price to be charged, using the black scholes formula for a put option. Question 1 Assuming a volatility of 10%pa and a risk free rate of 9%pa continuously compounded, determine the premium to be charged for this investment guarantee using the appropriate Black Scholes Formula. Ignore expenses and tax.

47

Question 2 Redo the above calculations using (a) a revised volatility of 20%pa (& risk free rate =9%) (b) a revised interest rate of 19%pa (& volatility of 10%).

Comment on the results Solution

(1) If the amount paid in by the customer was S0 and the value of the unit fund at time 2 is ST then the guaranteed maturity payment in 2 years time is

max(ST, S0u1.052) = ST + max(ST - ST, S0 u1.052 - ST) = ST + max(0, S0 u1.052 - ST) = ST + max(0, X - ST) where X = S0 u1.052 This is a payoff comprised of

x the value of the unit fund with no guarantee ( the ST part of the payoff) plus

x the payoff from a put option on the unit fund with a strike price of X = S0 u1.052 (the payoff max(0, S0 u1.052 - ST))

We can use the Black Scholes put option formula to derive a formula for the cost of the guarantee G The formula is

2 0.09 20 2 0 1S 1.05G e N d S N d u u or

2 0.09 20 2 11.05 1.00G S e N d N d u u u

48

Where

20 2

01

1ln 0.09 0.1 21.05 2

0.1 2

SS

d

u u u

22 1ln 1.05 0.09 0.1 220.1 2

u u

which is independent of S, and

2 1 0.1 2d d u which is also independent of S Calculations: We shall do the calculation assuming the unit fund has a (spot) value of $1.00 at the start of the 2 year period i.e. the investor started off with a $1.00 investment. The value of the put option with a strike price of $1.1025 is $0.0235, as shown below. Thus we find that G = Pu0.0235 i.e. the guarantee is going to cost the customer an additional 2.35% on top of the amount invested

(2) The calculations for 3 sets of valuation assumptions are set

out below.

Looking at the results we see that increasing the volatility increases the guarantee charge substantially and increasing the interest rate lowers it. The result is more sensitive to a change in volatility than to a change in the interest rate.

49

INPUTS INPUTS INPUTS

risk free int rate 9.00% 9.00% 19.00% volatility 10.00% 20.00% 10.00% term to maturity 2.0000 2.0000 2.0000 spot price $1.00000 $1.00000 $1.00000 strike price $1.10250 $1.10250 $1.10250

Results Results Results d1 0.65351 0.43282 2.06772 d2 0.51208 0.14998 1.92630 N(d1) 0.74328 0.66743 0.98067 N(d2) 0.69570 0.55961 0.97297 N(-d1) 0.25672 0.33257 0.01933 N(-d2) 0.30430 0.44039 0.02703 C 0.10262 0.15209 0.24709 P 0.02351 0.07298 0.00105

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Reverse Engineering Case Study: GOLD REPURCHASE AGREEMENT A large bank is also a gold bullion dealer and it is considering introducing a five year 100 ounce gold repurchase contract to support its bullion sales. Under this contract the buyer buys gold at the current market price and the bank / dealer guarantees to: x repurchase the gold at the original purchase price at the

end of 5 years if the market price of gold at that time is less than the original purchase price, but

x If the market price of gold is above the original purchase price then the dealer has the right under the contract to buy 40% of the gold at the original purchase price and the remaining 60% of the gold at the current market price.

Each repurchase agreement will be for 100 ounces of gold and all prices and payments are made in US dollars. For example an investor buys 100 ounces of gold at US$340 per ounce (i.e. $US34,000 for a 100 ounce contract) x if the price rose to US$500 / oz then the dealer would buy

40 oz @ $US$340 and 60 oz @ US$500. x If the price fell below US$340 /oz then the dealer would

buy the full 100 oz at US$340 per oz.

51

Scenario: You are an investor and you have been approached by a representative (a Mr Devious) of the bank about investing in this derivative security. Mr Devious is extremely keen on selling this security to you so much so that you become suspicious that this is not really such a great deal after all. Assume that x current gold price is US$340 / oz x 5 year AAA rated zero coupon bonds are trading at a yield

of 13.5%pa x 5 year at the money 100 oz gold call options are available

in the market at a price of $US24,500 x the dealer meets any holding and insurance costs

associated with physical holding of the gold QUESTION: Based on the above information determine whether or not you consider the terms of the repurchase agreement to be fair to you as the purchaser. (are you being ripped off if you accept this deal on these terms?)

52

APPROACH TO SOLVING THE PROBLEM:

x we write down an expression for the payoff to a purchaser of the contract at the end of 5 years,

x then we try to re express it as something involving a call option payoff and / or other payoffs we know how to value.

x then see whether the purchaser could reproduce the

payoff more cheaply by synthetically creating the repurchase agreement with other available market instruments

x the issue to consider is whether the purchaser being charged a fair price for the payoff they receive

Solution: From the perspective of a purchaser the contract represents (a) an initial investment of S = X = $34000

(S is the current cost of 100 oz of gold) AND

(b) a payoff in 5 years of x X if ST < X x 0.4 u X + 0.6 u ST if ST >X

Where ST = value of 100 oz of gold at the end of 5 years

53

We can rewrite this payoff as X + 0.6 u max(0, ST -X) Because the payoff is 0.4 u X + 0.6 u X if ST < X 0.4 u X + 0.6 u ST if ST > X = 0.4 u X + 0.6 u max(X, ST) = 0.4 u X + 0.6 u (X + max(X-X, ST -X)) = 0.4 u X + 0.6 u X + 0.6 u max(0, ST - X) = X + 0.6 u max(0, ST -X) This payoff is equivalent to the payoff on x a zero coupon bond with a face value of X = $34000 and x 0.6 units of an at the money call option on 100 oz of gold with a

strike price of X = $34000 As such we can value the payoff of the contract and then compare it with the price to be charged to enter the contract. The value of the payoff is

5 5

340000.60 0.6 24500 32750.91.135 1.135

X C u u

x This is less than the price being charged by the bullion dealer,

which was $34000. x This means the purchaser could achieve the same payoff more

cheaply by investing in a 5 year zero coupon bond and investing in an at the money call option on gold.

x The repurchase agreement upfront price means the purchaser is paying too much for the option contract and could do better by accessing markets directly. Thus the terms of the repo are not fair to the purchaser.