lecture 6.pdf

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Derivative Products and Markets:FINA2204

BUSINESS SCHOOL

Lecture 6 Sundaram and Das: Chapters 9 & 10

Prepared and delivered by Dr. Mahmoud Agha, CFA

The University of Western Australia

Chapter 9. No-Arbitrage Restrictions on Option Prices

The Objective:

Like other derivatives, an option has a value called option price or option premium.

To price the option we should first understand the boundaries within which the option should trade in the absence of arbitrage.

In this chapter, we will discuss the followings:

Bounds on option prices.

Decomposition of option price

The importance of early exercise.

Put-Call Parity.

This chapter examines the first two items on the list; Chapter 10

examines the other two.

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Preliminary Considerations

Option price bounds depend on several factors:

Type: Call or put?

Style: American or European? Dividends?

Important because options are not "payout protected." Dividends reduce stock prices, so benefit puts and hurt calls.

We distinguish between two cases:

Non-Dividend-Paying or NDP underlying: this is the case when the underlying stock pays no dividends during the life of the option.

Dividend-Paying or DP underlying: this is the case when the underlying stock pays dividends at some point during the option's life.

Note: any dividend payable after the expiration date of the option is irrelevant to option pricing and can be totally ignored.

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Notation

Option features: K : strike price of option. T : expiration date of option.

Underlying:

S : current price of underlying. ST : price of underlying on date T.

Option Prices:

CA , PA : American call and put, respectively. CE , PE : European call and put, respectively. If a property holds for both American and European options, we

simply write C or P.

PV (K ) : is the present value of the strike price. Continuous interest rate is usually used in option pricing, hence:

eKKPVrT

.)(

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Regarding Dividends ...

When there are dividends, we assume that timing and size of dividend

payments are known.

PV (D ) will denote the present value (viewed from today) of the dividends receivable over the life of the option.

For notational simplicity, we assume a single dividend payment.

Size: D. Timing: TD < T. hence:

It is an easy matter to extend the derivations to the case of multiple

dividend payments; just calculate the present value of each, then sum

them all as PV(Dt).

eDDPV DrT

.)(

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A Useful Observation

An American option can always be held to maturity.

Therefore, an American option must be worth at least as much as its

European counterpart:

CA > CE and PA > PE.

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Upper Bound on Call Prices:

We identify bounds on call prices first.

Upper-bound: Price of call must be less than current price of underlying:

C < S.

Why pay more than S for the right to buy it when it is cheaper to buy it directly?

To see why the option price cannot be worth more than the underlying, assume C >S, in this case we have a money-machine arbitrage as follows:

Sell the call and buy the stock.

The profit from this arbitrage is certain positive at date zero. And, if the stock price sky-rocketed and the buyer of the call decided to exercise the call all

you need to do is just to deliver the stock you already hold.

If every one does this, the huge sale of the call will drive its price below S and the arbitrage opportunity is eliminated.

Bounds on Option Prices

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Lower Bounds on Call Prices

We derive three separate lower bounds.

The first lower bound. A call confers a right without an obligation, so its price cannot be negative:

C > 0.

The second bound applies to American calls.

Such a call can be exercised at any time.

The value of immediate exercise is (S K). This value is called the

intrinsic value or the exercise value.

Therefore, the call must be worth at least (S K) i.e., CA (S K ), otherwise an instant arbitrage opportunity would exist.

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Example: an American call option with a strike price of $120 is written on a

stock currently worth $125. If this call option trades for $3, is there any

arbitrage opportunity?

Since $3 < S0 K = $5, there is an instant arbitrage opportunity as follows:

Buy the option at the observed price for $3 Exercise the option and get the stock at the strike price of $120 Sell the stock immediately in the stock market for $125. Your gain = 125-120-3 = $2 (Bon appetite!)

The third lower bound is a little trickier. We proceed in several steps:

First: European call on a NDP asset.

Then: European call on a DP asset. Finally: American calls.

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A Portfolio Comparison

Consider a European call on an NDP underlying.

Consider the following two portfolios:

Portfolio A: Long one call option. Portfolio B: Long 1 unit of underlying; borrow PV (K ) for repayment at T.

Initial value:

Portfolio A: CE. Portfolio B: S PV (K ).

Values of the portfolios at time 0 and T :

t=0 Payoff at expiration T

ST K ST > K

Portfolio A CE 0 ST K

Portfolio B S PV (K ). ST - K ST - K

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The Third Lower Bound

Note that portfolios A and B have the same performance if ST > K.

Portfolio A does strictly better if ST K because its payoff is zero, whereas

portfolio B payoff is negative. Portfolio B is like a synthetic forward.

Neither portfolio involves any interim cash flows.

Therefore, Portfolio A must be worth at least as much as Portfolio B:

CE > S PV (K ).

This is the third lower bound.

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Extending the Third Lower Bound

Extension to DP underlying?

If the stock pays dividends, there is an intermediate cash inflow in Portfolio

B at the time of the dividend, but there is no corresponding cash flow in

Portfolio A.

So we create an interim cash outflow in B that eliminates this cash in flow

and restores comparability. Consider:

Portfolio A: Long one call option. Portfolio B: Long one unit of underlying, borrowing of PV (K ) for

repayment at T, borrowing of PV (D ) for repayment on the dividend

date TD.

Initial values:

Portfolio A: CE Portfolio B: S PV (K ) PV (D )

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The Third Lower Bound with Dividends

By construction, neither portfolio involves interim cash flows.

The payoffs at T are exactly those derived earlier:

The portfolios do identically if ST > K.

Portfolio A does strictly better if ST K.

So Portfolio A must be worth at least as much as Portfolio B:

CE > S PV (K ) PV (D ).

This is the third lower bound extended to dividends.

Note that for portfolio B, the dividend received from holding the underlying will be

used to payoff the future value of PV(D) initially borrowed, hence, the cash flow

from portfolio B at TD = 0.

t=0 t = TD Payoff at expiration T

ST K ST > K

Portfolio A CE 0 ST K

Portfolio B:

S PV (K ) PV (D ) 0 ST K ST K

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The Third Lower Bound for American call options

Remember that we must always have CA > CE.

We have just shown that

CE > S PV (K ) PV (D ).

Therefore, we must also have

CA > S PV (K ) PV (D ).

Summing up, the third bound holds for both American and European calls

and we simply write

C S PV (K ) PV (D ).

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Bounds on Call Prices: Summary

Upper-bound: C < S.

Lower-bounds for European calls:

CE > 0.

CE > S PV (K ) PV (D ).

Combined : CE > Max ( 0, S PV (K ) PV (D ))

Lower-bounds for American calls:

CA > 0

CA > S K

CA > S PV (K ) PV (D )

Combined: CA > Max ( 0, S - K, S PV (K ) PV (D ))

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Call Pricing Bounds: Summary for D = 0

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The University of Western Australia 17

What happens if the lower bound is breached?

In this case, an arbitrage opportunity exists.

Example: The current XYZ stock price is $125. A European call option on this stock has a strike price of $120. The risk-free interest rate is 5% p.a and the option will expire in 45 days. Assume the stock pays no dividends. What is the lower bound for this option?

The lower bound is = Max[0,S0 K.e-rT ]

= Max[0,125 120e-0.05(45/365) ] = $5.74

If the market price of this option is $3, is there any arbitrage opportunity?


Since $3 < $5.74, yes there is an arbitrage opportunity because the option price has breached its lower bound.

To exploit this arbitrage opportunity, we buy the undervalued option on the LHS and sell the overvalued portfolio on the RHS.

Lecturers note: when we sell a portfolio, we reverse the signs of its items. After the reversal, items with positive signs should be bought

and items with negative signs should be sold.

In our example, we have to short sell the stock, and to have a riskless profit we should invest (buy) an amount equals to the present value of the

strike price in a riskless bond that pays K at maturity and have the same

expiration date as that of the option.

The next slide shows how this arbitrage opportunity can be exploited.

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Actions and cash flows at time 0 Actions and Cash flows at expiration

Action Cash flow Action Cash flow

Scenario 1

ST K

Cash flow

Scenario 2

ST > K

Buy the call -3 Decide 0 ST - K

Short sell the

stock

+125 Buy back the

stock

-ST

-ST

Buy a riskless

bond that pays K

at maturity for

K.e-rT

--119.26

=120.e-0.05(45/365)

Get the face

value of your

investment in

the riskless

bond

+120 +120

Net cash flow +$2.74 120 ST 0 0

Since this opportunity provides a certain positive cash flow at time 0 and a non-

negative cash flow at expiration, every one observes this opportunity will dive in

to exploit it until the increased demand for the option drives its value to at least

its lower bound and the arbitrage opportunity is eliminated.

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Bounds on Put Prices

Upper bound on put prices?

PA K . This is the maximum payoff you receive from exercising an

American put and occurs if S falls to zero.

PE PV(K). Because a European put options cannot be exercised,

the upper bound at any time prior to the expiration date is the PV(K).

Two simple lower bounds:

Lower Bound 1: P 0.

Lower Bound 2: PA K S, otherwise an instant arbitrage would exists.

A third lower bound that takes into consideration the effect of dividends:

PE PV (K ) + PV (D ) S.

And Since PA PE, PA PV (K ) + PV (D ) S.

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Deriving the Third Lower Bound

Compare:

Portfolio C: Long one put with strike K and maturity T.

Portfolio D: Short one unit of underlying, Invest PV (K ) for maturity at T,

Invest PV (D ) for maturity at TD.

Values at time 0 and expiration (T):

It follows that P > PV (K ) + PV (D ) S.

The interpretations:

Portfolio C: Right to sell.

Portfolio D: Obligation to sell (short forward)

t=0 t =TD Payoff at expiration T

ST < K ST K

Portfolio A P K- ST 0

Portfolio B:

PV (K ) + PV (D ) S 0 K - ST K - ST

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Bounds on Put Prices: Summary

Upper-bound: P A < K, and PE PV(K)

Lower-bounds for European puts:

PE > 0.

PE > PV (K ) + PV (D ) S.

Combined: PE > Max(0, PV (K ) + PV (D ) S)

Lower-bounds for American puts:

PA > 0

PA > K S

PA > PV (K ) + PV (D ) S.

Combined: PA > Max(0, K- S, PV (K ) + PV (D ) S)

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Put Pricing Bounds: Summary for D = 0

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What happen if the put option price breached its lower bound?

In this case, an arbitrage opportunity would exist.

Example: The current XYZ stock price is $125. A European put option on this stock has a strike price of $130. The risk-free interest rate is 5% p.a and

the option will expire in 45 days. Assume no dividends, what is the lower

bound for this option?

The lower bound = Max[0, K.e-rT S0 ]

= Max[0, 130.e-0.05(45/365) -125 ] = $4.20

If the market price of this option is $3, is there any arbitrage opportunity?

Since $3 < $4.20, there is an arbitrage opportunity as shown in the next slide.

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Actions and cash flows at time 0 Actions and Cash flows at expiration

Action Cash flow Action Cash flow

Scenario 1

ST < K

Cash flows

Scenario 2

ST K

Buy the put option -3 Decide K - ST 0

Buy the stock -125 Sell the stock +ST

+ST

Sell (borrow) a

riskless bond that

pays K at maturity

for K.e-rT

+129.20

=130.e -0.05(45/365)

Buy back

(repay or close

out) your

borrowed

riskless bond

-130 -130

Net cash flow +1.20 0 ST - 130 0

Since this opportunity provides a certain positive cash flow at time 0, and a non-

negative cash flow at expiration, everyone observes this opportunity will dive in to

exploit this opportunity until the increased demand for the option drives its value to

at least its lower bound and the arbitrage opportunity is eliminated.

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The Insurance Value of an Option

An option provides protection against unfavorable price movements.

The option's insurance value measures the value of this protection.

Question: How do we identify the portion of option's value attributable to insurance value?

Consider Portfolios A and B again. The only difference between

the portfolios is at T.

If ST > K, the two portfolios have identical payoffs.

But if ST K :

Portfolio A has a value of 0 (the option is not exercised).

Portfolio B has a negative value of (ST K ).

That is, Portfolio A is protected against a fall in the asset price

below K, while Portfolio B is not.

This is precisely the insurance provided by the call.

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Measuring the Insurance Value

The value of this protectionthe "insurance value" of the call, denoted

IV(C)is therefore the difference in the value of the two portfolios.

IV (C ) = C [S PV (K ) PV (D)].

Analogously, the insurance value of a put is defined by

IV (P ) = P [PV (K ) + PV (D ) S ].

For American options, the insurance value includes not only the insurance

value of the corresponding European option, but also the early exercise

premium.

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Chapter 10. Early Exercise and Put-Call Parity The Objective:

This Chapter examines three questions:

1. Composition of option value.

2. American versus European options.

3. Put versus call options.

Regarding the second question:

We identify when the right to early exercise may be important and the

conditions that make it "more" important.

Regarding the third question: we show that

For European options, there is a precise relation, called put-call parity, between the prices of otherwise identical puts and calls.

For American options, there is no parity relationship, but we can derive

inequalities that relate the prices of calls to puts.

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A decomposition of Option Prices


A decomposition of Option Prices

The Insurance Value of a Call

In Chapter 9, we derived the inequality

C S PV (K ) PV (D )

Recall that the left-hand side of this inequality is the value of an option to buy the underlying for K.

The right-hand side is the value of a synthetic long forward that represents

an obligation to buy the underlying for K.

This gives us a natural definition of the insurance value of a call:

IV (C) = C [S PV (K ) PV (D )]

Of course the insurance value is always non-negative.

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Decomposing Call Values

Rewrite the insurance value expression:

C = S PV (K ) + IV (C ) PV (D ).

Add and subtract K to obtain

C = (S K ) + (K PV (K )) + IV (C ) PV (D ).

Term Label and interpretation

S - K Intrinsic value. Measures current moneyness

K - PV(K) Time value. Interest savings from deferred purchase

IV(C) Insurance value. Value of downside protection

PV(D) Payout during calls life


Insurance Value of a Put

In Chapter 9, we derived the inequality

P PV (K ) + PV (D ) S

Recall that the left-hand side of this inequality is the value of an option to sell the underlying for K.

The right-hand side is the value of a synthetic short forward that

represents an obligation to sell the underlying for K.

This gives us a natural definition of the insurance value of a put:

IV (P ) = P [PV (K ) + PV (D ) S ]

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Decomposing Put Values

Rewriting the last expression,

P = PV (K ) S + IV (P ) + PV (D ).

Adding and subtracting K:

P = (K S ) (K PV (K )) + IV (P ) + PV (D ).

Term Label and interpretation

K S Intrinsic value. Measures current moneyness

K - PV(K) Time value. Interest losses from deferred sale at K.

IV(P) Insurance value. Value of downside protection

PV(D) Payout during puts life


Note that

Time value: Positive for calls (save interest on purchase), negative for puts (lost interest from deferred sale).

Impact of payouts: Negative for calls, positive for puts.

Call Price

=

Intrinsic

+

Time

+

Insurance

+

Impact of

Value

value Value

Payouts

Put Price

=

Intrinsic

+

Time

+

Insurance

+

Impact of

Value

value Value

Payouts

The Decomposition: In Words

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Comments on the Decomposition - I: Intuition

Four sources of value for a call.

1. Ceteris paribus, it is better to start "more" in-the-money.

This is intrinsic value.

2. The call gives the right to buy the asset for K on date T.

The higher are interest rates (or the longer is maturity), the lower is the present value of the amount K we must pay.

This is time value.

3. If the price of the underlying falls, we can also let the call lapse and buy

the underlying at a cheaper price.

This is insurance value, the value of downside protection.

4. Higher dividends on the underlying lowers the distribution of stock prices

at maturity, hurting the call holder.

This is the impact of payouts.


Comments on the Decomposition - I: Intuition

The put has exactly the same four sources of value except that

Time value is negative: Exercise of the put results in a cash inflow of

K. The higher are interest rates or the longer is maturity, the lower is

the present value of the amount received.

The impact of payouts is positive: Higher dividend payouts depress

the growth rate of the stock price benefitting the holder of a put.

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Comments on the Decomposition - II: How Factors Matter

The decomposition suggests the routes by which different factors could

affect option values.

Intrinsic value is affected by moneyness (current depth-in-the-money).

Time value (interest rate savings or losses) is affected primarily by

interest rates and maturity.

Insurance value (downside protection) is affected primarily by volatility

and maturity.

These observations can be used to gauge the qualitative impact on option

values of a change in different factors:

Interest rates. Time to maturity.

Volatility.


Comments on the Decomposition - II: The Effect of Time-to-

Maturity

Time-to-maturity affects option values in two ways:

1. Time value: Lower time-to-maturity

lowers time value for calls, but

increases time value for puts (makes it less negative).

2. Insurance value: Lower time-to-maturity lowers value of downside

protection for both calls and puts.

So lower time-to-maturity

reduces call values, but has an ambiguous effect on put values (could be positive or

negative)

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Comments on the Decomposition - II: The Role of

Moneyness

Deep-in-the-money options derive of their value from intrinsic value. There

is some time value but little insurance value since there is not much chance

of the option falling out of the money.

Thus, for deep-in-the-money options, we have

For deep-out-of-the-money options, most of the value comes from

insurance value, the hope that volatility will push the option into the money.

For such options:

For options that are at- or near-the-money, both time value and insurance

value matter.


The Optimality of Early Exercise

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The Decompositions and Early Exercise

We now turn to the question of optimality of early exercise of American

options.

In this context, note that to monetize an American option, we can either

sell the option or exercise it.

Early exercise of an option means its intrinsic value is realized.

Selling the option means the option value is realized.

Of course, we could also just retain the option but this has the same value

as selling it.

Thus, early exercise is suboptimal if

Option value > Intrinsic value.

So: Compare the option value to the intrinsic value.


The Procedure

We proceed in four steps:

1. Early exercise of calls when there are no dividends.

2. Early exercise of calls when dividends exist.

3. Early exercise of puts when there are no dividends.

4. Early exercise of puts when dividends exist.

Note that the statements "no dividends" or "dividends exist" refers only to

dividends paid on the underlying during the option's life.

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Early Exercise with No Dividends: Calls

If D = 0, the difference between call value and its intrinsic value is

CA (S K) = (K PV(K)) + IV (C).

Time value and insurance value are both positive, so the difference is

always positive.

This means that the market price of the call option is worth more than the

intrinsic value realized from exercising it. And, so you will receive more

money from selling the call than exercising it.

Subsequently, a call on an NDP asset should never be exercised early!

Based on above argument, it follows that in the absence of dividends, American call value = European call value.


Early Exercise with Dividends: Calls

If dividends are positive, possible countervailing effect.

Difference between call value and intrinsic value is now

CA (S K ) = (K PV (K )) + IV (C ) PV (D ).

Right-hand side may not be strictly positive Early exercise may be optimal for calls on dividend-paying assets if PV (D ) > (K PV (K )) + IV (C )

Factors that make early exercise "more" likely:

1. Low interest rates.

Lowers time value lost on account of early exercise. 2. Low volatility.

Lowers insurance value lost on account of early exercise. 3. High dividends.

Increases gain from early exercise.

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Early Exercise with No Dividends: Puts

With D = 0, difference between put value and intrinsic value is

PA (K S ) = (K PV (K )) + IV (P ).

Time value is negative but insurance value is positive.

So difference need not be strictly positive Early exercise may be optimal for a put even if the underlying pays no dividends.


1. High interest rates increase the time value gained from early exercise 2. Low volatility reduces the insurance value lost on early exercise


Early Exercise with Dividends: Puts

With dividends, the difference between put value and intrinsic value is

PA (K S ) = (K PV (K )) + IV (P ) + PV (D )

Time value is negative but the other terms on the RHS are positive.

So, LHS need not be strictly positive Early exercise may be optimal for a put in the presence of dividends also.


1. High interest rates.

2. Low volatility. 3. Low dividends (cost of early exercise is reduced).

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Put-Call Parity

One of the most important results in all of option pricing theory.

Relates the prices of call options to otherwise identical put options.

"Otherwise identical" Same underlying, T, K.

We proceed in four steps:

1. European options when the underlying pays no dividends.

2. European options when the underlying pays dividends.

3. American options, no dividends. 4. American options, dividends.


European Options, No Dividends

Consider two portfolios:

Portfolio A: Long call with strike K and maturity T, investment of PV (K ).

Portfolio B: Long stock, long put with strike K, maturity T.

Values of these portfolios today:

Portfolio A : CE + PV (K )

Portfolio B: PE + S

Value at t = 0 Payoff at expiration, t = T

ST < K ST >K

Portfolio A CE + PV(K) 0 + K = K ST K + K=ST

Portfolio B PE + S K ST +ST =K 0 + ST =ST

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Put-Call Parity: European Options, No Dividends

These payoffs are identical!

So the portfolios must have the same value today (else arbitrage results):

This expression is putcall parity for European options when there are no

dividends.

Following similar arguments as we have done before, we can derive the put

call parity for European options with dividends as follows:

PE +S = CE +PV (K)

PE + S = CE + PV (K ) + PV (D )


Put-Call Parity: Uses

The put call parity can be re-arranged to create synthetic security. For example a synthetic call is given by the following:

i.e., a synthetic long call is equivalent to a portfolio composed of:

Long put + Long underlying + Borrowing of PV (K ) + borrowing of PV(D)

Since we are dealing with a parity, if the parity or any synthetic security derived from it are breached, there is an arbitrage opportunity.

CE = PE + S - PV (K ) - PV (D )

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Example: A European call option has the following information: Ce = $5, K =$50, S0 = $52, rT = 6% p.a, T=33/365 years.

What is the price of a European put option written on the same stock and has a

time to expiration and a strike price as those of the call option?

Solution: The put-call parity says that;

S0 + Pe = Ce + K.e-rT , Solving for Pe

Pe = Ce + K.e-rT - S0 = 5 + 50.e

-0.06(33/365) - 52 = $2.73

This is the theoretical price of the put option.

Now, assume that the observed market price of the put option is $4? Is there

any arbitrage opportunity? Show how to exploit it if any.

Because $4 > Ce + K.e-rT - S0 = $2.73, the European put-call parity is breached.

Therefore, an arbitrage opportunity exists as shown in the next slide.

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Actions and Cash flows at time 0 Actions and Cash flows at expiration = T

Action Cash flow Action Scenario 1

ST < K

Scenario 2

ST > K

Sell the put +4 The buyer

will decide

-( K- ST) 0

Buy the call -5 You Decide 0

(ST - K)

Buy a riskless bond

that pays K at

maturity for K.e-rT

--49.73

=50.e-0.06(33/365)

Get the face

value of your

riskless bond

+50 +50

Short sell the stock +52 Buy back the

stock

-ST -ST

Net cash flow +1.27 0 0

Arbitrageurs will sell the put until its price converges to its theoretical value

and the arbitrage opportunity is eliminated.

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Put-Call "Parity" and American Options

For American options with no dividends, an approximate version of parity

obtains:

CA + PV (K ) < PA + S < CA + K

When there are dividends, the inequality must be modified as follows:

CA +PV (K ) < PA + S < CA + K + PV (D )


Tutorial Questions

Sundaram & Das 2011 Derivatives, 1st edn:

Chapter 9: Q6,Q13,Q21

Chapter 10:Q1,Q2,Q3,Q13

lecture 6.pdf

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