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2 Dividends, foreign interest and cost of carry

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140

Part One mathematical and ﬁnancial foundations

then the ‘dividend yield’ is just the cost of carry (with a minus sign, we beneﬁt from dividends

but must pay out the cost of carry).

To recap, if the underlying receives a dividend of DS dt in a time step dt when the asset

price is S then

∂V

∂V

∂ 2V

+ 12 σ 2 S 2 2 + (r − D)S

− rV = 0.

∂t

∂S

∂S

However, if the underlying is a stock, then the assumption of constant and continuously-paid

dividend yield is not a good one.

8.3 DIVIDEND STRUCTURES

Typically, dividends are paid out quarterly in the US and semi-annually or quarterly in the UK.

The dividend is set by the board of directors of the company some time before it is paid out

and the amount of the payment is made public. The amount is often chosen to be similar to

previous payments, but will obviously reﬂect the success or otherwise of the company. The

amount speciﬁed is a dollar amount, it is not a percentage of the stock price on the day that

the payment is made. So reality differs from the above simple model in three respects:

• the amount of the dividend is not known until shortly before it is paid

• the payment is a given dollar amount, independent of the stock price

• the dividend is paid discretely, and not continuously throughout the year.

In what follows I am going to make some assumptions about the dividend. I will assume that

• the amount of the dividend is a known amount, possibly with some functional dependence

on the asset value at the payment date

• the dividend is paid discretely on a known date.

Other assumptions that I could, but won’t, make because of the subsequent complexity of

the modeling are that the dividend amount and/or date are random, that the dividend amount

is a function of the stock price on the day that the dividend is set, that the dividend depends

on how well the stock has done in the previous quarter . . .

8.4 DIVIDEND PAYMENTS AND NO ARBITRAGE

How does the stock react to the payment of a dividend? To put the question another way, if

you have a choice whether to buy a stock just before or just after it goes ex-dividend, which

should you choose?

Let me introduce some notation. The dates of dividends will be ti and the amount of the

dividend paid on that day will be Di . This may be a function of the underlying asset, but it

then must be a deterministic function. The moment just before the stock goes ex-dividend will

be denoted by ti− and the moment just after will be ti+ .

The person who buys the stock on or before ti− will also get the rights to the dividend. The

person who buys it at ti+ or later will not receive the dividend. It looks like there is an advantage

simple generalizations of the Black–Scholes world Chapter 8

140

120

Stock price

100

80

60

40

20

0

0

0.2

0.4

0.6

0.8

1

Time

Figure 8.1 A stock price path across a dividend date.

in buying the stock just before the dividend date. Of course, this advantage is balanced by a fall

in the stock price as it goes ex-dividend. Across a dividend date the stock falls by the amount

of the dividend. If it did not, then there would be arbitrage opportunities. We can write

S(ti+ ) = S(ti− ) − Di .

(8.1)

In Figure 8.1 is shown an asset price path showing the fall in the asset price as it goes exdividend; the drop has been exaggerated.

This jump in the stock price will presumably have some effect on the value of an option.

We will discuss this next.

8.5

THE BEHAVIOR OF AN OPTION VALUE ACROSS

A DIVIDEND DATE

We have just seen how the underlying asset jumps in value, in a completely predictable way,

across a dividend date.

Jump conditions tell us about the value of a dependent variable, an option price, when there

is a discontinuous change in one of the independent variables. In the present case, there is a

discontinuous change in the asset price due to the payment of a dividend but how does this

affect the option price? Does the option price also jump? The jump condition relates the values

of the option across the jump, from times ti− to ti+ . The jump condition will be derived by a

simple no-arbitrage argument.

To see what the jump condition should be, ask the question: ‘By how much do I proﬁt or lose

when the stock price jumps?’ If you hold the option then you do not see any of the dividend,

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Part One mathematical and ﬁnancial foundations

120

B

Stock price

100

80

A

60

40

20

0

0

0.1

0.2

0.3

0.4

0.5

Time

0.6

0.7

0.8

0.9

1

30

Option value

25

C

20

15

10

5

0

0

0.2

0.4

0.6

0.8

1

Time

Option Value After

70

Option Value Before

60

50

40

V

142

30

C

20

10

A

0

0

20

40

60

80

B

100

120

140

S

Figure 8.2 Top picture, a realization of the stock price showing a fall across the dividend date.

Middle picture, the corresponding realization of the option price (in this example a call). Bottom

picture, the option value as a function of the stock price just before and just after the dividend date.

simple generalizations of the Black–Scholes world Chapter 8

that goes to the holder of the stock not you, the holder of the option. If the dividend amount

and date are known in advance then there is no surprise in the fall in the stock price. The

conclusion must be that the option does not change in value across the dividend date, its path

is continuous. Continuity of the option value across a dividend date can be written as

V S(ti− ), ti− = V S(ti+ ), ti+

(8.2)

or, in terms of the amount of the dividend,

V S, ti− = V S − Di , ti+ .

(8.3)

The jump condition and its effect on the option value can

be explained by reference to Figure 8.2. In this ﬁgure, the top

picture shows a realization of the stock price with a fall across

the dividend date. The middle picture shows the corresponding realization of a call option price.

The bottom picture shows the option value as a function of the stock price just before and just

after the dividend date. Observe the points ‘A’ and ‘B’ on these pictures. ‘A’ is the stock price

after the dividend has been paid and ‘B’ is the price before. On the bottom picture we see the

values of the option associated with these before and after asset prices. These option values

are the same and are denoted by ‘C.’ Even though there is a fall in the asset value, the option

value is unchanged because the whole V versus S plot changes. The relationship between the

before and after values of the option are related by (8.3). I will give two examples.

Suppose that the dividend paid out is proportional to the asset value, Di = DS. In this case

S(ti+ ) = (1 − D)S(ti− ).

Equation (8.3) is then just

V S, ti− = V (1 − D)S, ti+ .

The two option price curves are identical if one stretches the after curve by a factor of (1 − D)−1

in the horizontal direction. Thus, even though the option value is continuous across a dividend

date, the delta changes discontinuously.

If the dividend is independent of the stock price then

S(ti+ ) = S(ti− ) − Di ,

where Di is independent of the asset value. The before curve is then identical to the after curve,

but shifted by an amount Di .

8.6

COMMODITIES

Convenience yield is the beneﬁt or premium associated with holding an underlying product or

physical good, rather than a future position in that product. For example, there is an obvious

beneﬁt to the actual holding of barrels of oil. This naturally leads us to thinking of commodities

as being of two types.

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Part One mathematical and ﬁnancial foundations

• Investment commodities: Commodities held for investment, such as gold.

• Consumption commodities: Commodities held for consumption, such as oil or wheat.

Since they are held for consumption, and have a value associated with this, they may not

be sold when the price rises. Crucially, this can make arbitrage arguments one sided.

8.6.1 Futures Prices and Arbitrage

The standard argument for the relationship between spot and futures prices does not hold for

consumption commodities. Only an upper bound can be found. This is because there may be

constraints on the selling of consumption commodities; the oil is needed for heating, transport

etc. and cannot be reasonably sold.

The no-arbitrage argument can still be applied to investment commodity futures.

8.6.2 Storage Costs

No storage costs

If there are no storage costs then the relationship between spot and forward prices is

F = S er(T −t) .

Storage costs proportional to spot price, for investment commodities

If there are storage costs, and they are proportional to the price of the commodity then

F = S e(r+u)(T −t) .

This is equivalent to there being a negative dividend yield.

Storage costs proportional to spot price, for consumption commodities

For consumption commodities all we can say is

F ≤ S e(r+u)(T −t) .

This is because holders of commodities will be reluctant to sell the commodity (at the spot

price) which they are keeping for consumption. They may buy more oil if the spot price is low,

but they won’t sell it if the spot price is high since then they would have no fuel.

8.6.3 Convenience Yield

Since

F ≤ S e(r+u)(T −t)

we introduce the convenience yield y such that

F = S e(r+u−y)(T −t) ,

with y ≥ 0. For investment commodities y = 0, of course.

simple generalizations of the Black–Scholes world Chapter 8

8.6.4

Cost of Carry

Cost of carry is the sum of storage cost, interest paid to ﬁnance the asset, less any income

from the asset.

Examples: Usually the drift coefﬁcient in the Black–Scholes equation is just r, but this will

change if there is a cost of carry as follows.

No dividends etc.: r

Dividend yield: r − D

Foreign exchange: r − rf

Commodity: r + u

1.

2.

3.

4.

The convenience yield is not included in this.

8.6.5

Effect on Options

If S is the spot price then options on the spot satisfy

∂V

∂V

∂ 2V

+ 12 σ 2 S 2 2 + (r + u)S

− rV = 0.

∂t

∂S

∂S

Change variables to

F = S e(r+u−y)(T −t) ,

H (F, t) = V (S, t)

to get

∂H

∂H

∂ 2H

+ 12 σ 2 F 2 2 + yF

− rH = 0.

∂t

∂F

∂F

This is the relevant equation incorporating cost of carry and the convenience yield.

8.7

STOCK BORROWING AND REPO

I have many times referred to selling stock for hedging purposes, going short the stock. But in

practice how can one sell a quantity that one does not own and which is naturally something

you would buy, in ‘positive’ quantities as opposed to negative quantities? To keep it real, let’s

imagine you want to go short a lawnmower. You don’t own a lawnmower to sell, so what can

you do? Easy, just borrow one from your neighbor and sell that! When your neighbor wants

his lawnmower back you have to go out and buy another one to give him. If lawnmower prices

have meanwhile fallen you will make a proﬁt.

In the world of stocks and shares the same idea applies. If you want to go short a stock you

must ﬁrst borrow it. But this is not going to be costless, usually there is some payment to be

made, like an interest charge on the amount you borrow. This should really be factored into

any option pricing model. To quantify this, let’s suppose that you have to pay interest at a rate

of R on the value of the stock that you have borrowed. Now go through the Black–Scholes

argument and include this cost in the analysis.

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Part One mathematical and ﬁnancial foundations

We begin with the lognormal random walk for the stock,

dS = µS dt + σ S dX.

And we set up a portfolio, long the option and short the stock,

= V (S, t) −

S.

This then changes by

d

= dV −

dS,

which can be written, using Itˆo’s lemma, as

d

=

∂V

∂V

∂ 2V

dt +

dS + 12 σ 2 S 2 2 dt −

∂t

∂S

∂S

dS.

Except that this is not quite right. The value of our portfolio changes because of the interest

rate, the repo rate; we must pay for the borrowed stock. So scratch that equation, it should be

d

=

∂V

∂V

∂ 2V

dt +

dS + 12 σ 2 S 2 2 dt −

∂t

∂S

∂S

dS − R max( , 0)S dt.

Let me explain the new term. First, the interest payment is on the value of the short position,

that is S. Second, because it is an interest rate the actual payment is proportional to the time

step, dt. Finally, and this is the interesting part, the payment is only when

is positive, so

that we are short the stock (remember the minus sign in front of the in the portfolio). Hence

the maximum function above.

To eliminate the random terms we still choose

=

∂V

,

∂S

leaving us with

d

=

∂ 2V

∂V

+ 12 σ 2 S 2 2

∂t

∂S

dt − R max( , 0)S dt.

Because this is deterministic we can set the return on the portfolio equal to the risk-free rate

d

=r

dt.

The end result is the partial differential equation

∂V

∂V

∂ 2V

∂V

+ 12 σ 2 S 2 2 + rS

− rV − RS max

, 0 = 0.

∂t

∂S

∂S

∂S

This is the equation for pricing derivatives in the presence

of interest payments for shorting stocks. Note that it is actually

a non-linear equation. The consequences of non-linearity will

discussed in depth later on in the book.

simple generalizations of the Black–Scholes world Chapter 8

8.8

TIME-DEPENDENT PARAMETERS

The next generalization concerns the term structure of parameters. In this section I show how to

derive formulae for options when the interest rate, volatility and dividend yield/foreign interest

rate are time dependent.

The Black–Scholes partial differential equation is valid as long as the parameters r, D and

σ are known functions of time; in practice one often has a view on the future behavior of these

parameters. For instance, you may want to incorporate the market’s view on the direction of

interest rates. Assume that you want to price options knowing r(t), D(t) and σ (t). Note that

when I write ‘D(t)’ I am speciﬁcally assuming a time-dependent dividend yield, that is, the

amount of the dividend is D(t)S dt in a time step dt.

The equation that we must solve is now

∂V

∂V

∂ 2V

+ 12 σ 2 (t)S 2 2 + (r(t) − D(t))S

− r(t)V = 0,

∂t

∂S

∂S

(8.4)

where the dependence on t is shown explicitly.

Introduce new variables as follows:

S = Seα(t) , V = V eβ(t) , t = γ (t).

We are free to choose the functions α, β and γ and so we will choose them so as to eliminate

all time-dependent coefﬁcients from (8.4). After changing variables (8.4) becomes

γ˙ (t)

2

∂V

∂V

2∂ V

˙

+ 12 σ (t)2 S

+ r(t) − D(t) + α(t)

˙

S

V = 0,

− r(t) + β(t)

2

∂t

∂S

∂S

(8.5)

where ˙ = d/dt. By choosing

T

β(t) =

r(τ ) dτ

t

we make the coefﬁcient of V zero and then by choosing

T

α(t) =

(r(τ ) − D(τ )) dτ ,

t

we make the coefﬁcient of ∂V /∂S also zero. Finally, the remaining time dependence, in the

volatility term, can be eliminated by choosing

T

γ (t) =

σ 2 (τ ) dτ .

t

Now (8.5) becomes the much simpler equation

2

∂V

2∂ V

.

= 12 S

2

∂t

∂S

(8.6)

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Part One mathematical and ﬁnancial foundations

and there is no mention of r, D or σ . If we use V (S, t) to denote any solution of (8.6), then

the corresponding solution of (8.5), in the original variables, is

V = e−β(t) V Seα(t) , γ (t) .

(8.7)

Now use VBS to mean any solution of the Black–Scholes equation for constant interest rate

rc , dividend yield Dc and volatility σ c . This solution can be written in the form

VBS = e−rc (T −t) V BS Se−(rc −Dc )(T −t) , σc2 (T − t)

(8.8)

for some function V BS . By comparing (8.7) and (8.8) it follows that the solution of the timedependent parameter problem is the same as the solution of the constant parameter problem if

we use the following substitutions:

T

rc =

1

T −t

Dc =

1

T −t

σ c2 =

1

T −t

r(τ ) dτ

t

T

D(τ ) dτ

t

T

σ 2 (τ ) dτ

t

These formulae give the average, over the remaining lifetime of the option, of the interest

rate, the dividend yield and the squared volatility.

Just to make things absolutely clear, here is the formula for a European call option with

time-dependent parameters:

Se−

T

t

D(τ ) dτ

N (d1 ) − Ee−

T

t

r(τ ) dτ

N (d2 )

where

d1 =

log(S/E) +

T

t (r(τ )

− D(τ )) dτ +

T

t

1

2

T

t

σ 2 (τ ) dτ

1

2

T

t

σ 2 (τ ) dτ

σ 2 (τ ) dτ

and

d2 =

log(S/E) +

T

t (r(τ )

− D(τ )) dτ −

T

t

.

σ 2 (τ ) dτ

There are some conditions that I must attach to the use of these formulae. They are generally

not correct if there is early exercise, or for certain types of exotic option. The question to ask

to decide whether they are correct is: ‘Are all the conditions, ﬁnal and boundary, preserved by

the transformations?’

simple generalizations of the Black–Scholes world Chapter 8

8.9

FORMULAE FOR POWER OPTIONS

An option with a payoff that depends on the asset price at expiry raised to some power is called

a power option. Suppose that it has a payoff

Payoff(S α )

we can ﬁnd a simple formula for the value of the option if we have a simple formula for an

option with payoff given by

Payoff(S).

(8.9)

This is because of the lognormality of the underlying asset.

Writing

S = Sα

the Black–Scholes equation becomes, in the new variable S ,

∂ 2V

∂V

+ 12 α 2 σ 2 S 2

∂t

∂S 2

1 2

2 σ (α

− 1) + r S

∂V

− rV = 0.

∂S

Thus whatever the formula for the option value with simple payoff (8.9), the formula for the

power version has S α instead of S and adjustment made to σ , r and D.

8.10 THE log CONTRACT

The log contract has the payoff

log(S/E).

The theoretical fair value for this contract is of the form

a(t) + b(t) log(S/E).

Substituting this expression into the Black–Scholes equation results in

a˙ + b˙ log(S/E) − 12 σ 2 b + (r − D)b − ra − rb log(S/E) = 0,

where ˙ denotes d/dt. Equating terms in log(S/E) and those independent of S results in

b(t) = e−r(T −t) and a(t) = r − D − 12 σ 2 (T − t)e−r(T −t) .

The two arbitrary constants of integration have been chosen to match the solution with the

payoff at expiry.

This value is rather special in that the dependence of the option price on the underlying asset,

S, and the volatility, σ , uncouples. One term contains S and no σ and the other contains σ and

no S. We brieﬂy saw in Chapter 7 the concept of vega hedging to eliminate volatility risk. It

is conceivable, even though not entirely justiﬁably, that the simplicity of the log contract value

makes it a useful weapon for hedging other contracts against ﬂuctuations in volatility. Having

said that, it’s not exactly a highly liquid contract.

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Part One mathematical and ﬁnancial foundations

The log contract payoff can be positive or negative depending on whether S > E or S < E.

If we modify the payoff to be

max(log(S/E), 0)

then we have a genuine ‘option’ which may or may not be exercised. The value of this option

is

e−r(T −t) σ T − tN (d2 ) + e−r(T −t) log(S/E) + r − D − 12 σ 2 (T − t) N (d2 ).

8.11 SUMMARY

In this chapter I made some very simple generalizations to the Black–Scholes world. I showed

the effect of discretely paid dividends on the value of an option, deriving a jump condition

by a no-arbitrage argument. Generally, this condition would be applied numerically and its

implementation is discussed in Chapter 78. I also showed how time-dependent parameters can

be incorporated into the pricing of simple vanilla options.

• See Merton (1973) for the original derivation of the Black–Scholes formulae with timedependent parameters.

• For a model with stochastic dividends, see Geske (1978).

• The practical implications of discrete dividend payments are discussed by Gemmill (1992).

• See Neuberger (1994) for further info on the log contract.

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