2 Modeling Continuous Yields: An Introduction to Non-Stochastic Differential Equations
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which says that the instantaneous percentage rate of return on the bond–which
is defined as the change in the bond price over a very small interval of time,
dB(t,T), divided by what you currently have to pay for the bond, B(t,T)—
equals the instantaneous risk-free rate of r *dt. This shows that if the bond is
priced at B(t,T)=e–r, then its instantaneous rate of return is r*dt.
We can also work backwards by solving (Bond equation) to get B(t,T)=e–r.
This is rather important for understanding the current application as well
as for later ones, so we will do it step by step. Working backwards means
that we start from the assumption that dB(t,T)/B(t,T)=r*dt expressed in (Bond
equation).
We will work with the natural log function of the bond price: ln(·) applied
to B(t,T) and use the ‘chain rule’ of calculus again,
d ln (B(t ,T ))
dt
d ln (B(t ,T )) dB(t,T )
*
dB(t ,T )
dt
1
dB(t,T )
=
*
B(t,T )
dt
dB(t,T )
B(t,T )
=
dt
= r from (Bond equation)
=
From this last equation, after multiplying through by dt we obtain
(Log Bond equation):
d ln (B(t,T )) = r * dt
(Log Bond equation)
This new (Log Bond equation) can be integrated using basic calculus. The
deﬁnite integral of the left-hand side of (Log Bond equation) from t to T is,
∫
T
t
d ln (B(v,T ))
dv
dv = ln (B(T ,T )) − ln (B(t,T ))
⎛ B(T ,T ) ⎞
= ln ⎜
⎟
⎝ B(t,T ) ⎠
(Integral of the
LHS of (Log
Bond equation))
By changing the dt to dv and integrating with respect to v, the deﬁnite
integral of the right-hand side of (Log Bond equation) is,
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∫
FORWARD CONTRACTS AND FUTURES CONTRACTS
T
t
r * dv = r * T − r * t
= r * (T − t )
= r *
(Integral of the
RHS of (Log
Bond equation))
Note that the price of the zero-coupon bond at maturity B(T,T)=$1.0, by
deﬁnition.
Equating the deﬁnite integral of the left-hand side of (Log Bond equation)
to the deﬁnite integral of the right-hand side of (Log Bond equation) we ﬁnd
that,
⎛ 1.0 ⎞
ln ⎜
⎟ = r * .
⎝ B(t ,T ) ⎠
Applying the exponential function, exp(·), to both sides of this equation
we get
⎛ ⎛ 1.0 ⎞⎞
1.0
exp ⎜ ln ⎜
⎟⎟ =
⎝ ⎝ B(t ,T ) ⎠⎠ B(t ,T )
= e r
because the exp function and the ln function are inverse to each other.
Therefore,
1.0
e r
= e −r ,
B(t,T ) =
based on a key property of the exp function.
This is what we wanted to prove. If the instantaneous rate of return is r *dt
then the bond is priced at B(t,T)=e–r. Combining the two results, we have
proved what we set out to prove,
The instantaneous rate of return on a zero-coupon bond is r *dt if and only if the
bond price is given by B(t,T)=e–r where is time to maturity T–t.
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CONCEPT CHECK 2
Consider a 3-month zero-coupon bond with current price per dollar of face
value of B(t,T)=$.0975.
a. Use the relationship ln (1.0/B(t,T))=r * to ﬁnd r.
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4.2.2 Modeling Continuous Dividend Yields for Stocks
Now that we have a good feel for what instantaneous returns mean, we will
next consider an underlying stock with current price St that continuously pays
dividends at a constant, annualized rate . What this means is that the dividend
payable at time t, DIVt=*St*dt where dt is a very small interval of time called
an infinitesimal.
We have to understand how to use this formula, because the idea of paying
dividends at each instant of time is somewhat of an abstraction. A useful
abstraction but nonetheless an abstraction since, in the real world, we usually
deal with time discretely.
We will take a very short interval of time, from time t to time t+⌬t. Keep
in mind that this is a very small interval of time that happens almost
instantaneously. By the time you have thought of it, it has already passed.
Here is a time, cash ﬂow line of the stock and its dividend stream,
St
DIVt
St+∆t
DIVt+∆t
t
t+∆t
Our interval of time is ⌬t and total time intervals are usually measured in
years in continuous-time ﬁnance. So, for example, 1 month would be recorded
as 1/12 of a year or as .0833. The dividend yield is given on an annual basis
as . In order to apply the formula instantaneously to get the instantaneous
dividend, we need the instantaneous dividend yield. We can approximate this
as follows by de-annualizing the annualized dividend yield .
Over an interval of length ⌬t as shown in the time, cash ﬂow line, the
appropriate de-annualized dividend yield is *⌬t. Now, if we make the time
interval ⌬t small enough, then the stock price won’t change much (unless it
has jumps) over the interval [t,t+⌬t], and the dollar dividend payable over this
interval will be roughly equal to *St*⌬t. As ⌬t gets smaller and smaller, it
approaches what we call an infinitesimal amount of time, denoted as dt. The
instantaneous dollar dividend is then equal to *St*dt, or simply as Stdt, where
we have dropped all the multiplication signs.
The bottom line on this discussion is that you can’t simply multiply the
annualized dividend rate by the stock price at time t, St , to get the instantaneous dollar dividend. If dividends were always 3% of the stock price, then
that would be a huge amount of dividends paid out over a year. In reality,
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FORWARD CONTRACTS AND FUTURES CONTRACTS
the instantaneous dollar dividend is a very small amount. Nor can you multiply
the annualized dividend rate by the stock price at time t, St, to get the
annualized dollar dividend. The procedure (formula) applies only to very small
intervals.
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CONCEPT CHECK 3
Suppose that =3% and that the current stock price St=$100.
a. What is the dollar amount of the dividend payable at time t, DIVt , if
⌬t=1 day?
4.3 HOW DIVIDEND PAYMENTS AFFECT STOCK PRICES
Next, we want to more carefully consider the question of how dividend
payments affect stock prices. You have undoubtedly learned that stock prices
are the present values of the payouts generated by owning the stock, discounted
at an appropriate risk-adjusted discount rate (RADR). The only direct cash
payouts that common stocks pay are common dividends. Capital gains appear
in the form of increased stock prices, and are not direct payouts like dividends.
Stock dividends are payable in stock, not cash.
What happens to the stock price after the dividend is paid at time t? In a
perfect capital market, the stock price should drop by the amount of the
dividend because you just received it. The stock price represents the PV of
all subsequent dividends–no dividends already paid in the past. Let’s see if we
can formalize this a bit using the analysis and notation we already have available.
We will again take a very short interval of time, from time t to time t+⌬t.
Keep in mind that this is a very small interval of time that happens almost
instantaneously, as before.
St
DIVt
St+∆t
DIVt+∆t
t
t+∆t
Our strategy is to buy the stock before time t to ensure that we have the
right to receive all subsequent dividends. The ex-dividend date of the stock is
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time t. Let’s say we pay roughly St for the stock with the dividend. When
time t arrives, we receive the dividend DIVt payable at time t, and owed to
us. Assuming that nothing happens in the market to change the stock price
other than the dividend payment, we look at what happened to our stock
price.
It must have dropped by DIVt=*St*⌬t. Looking at our current stock price
after it has paid this dividend, but before the stock pays another dividend, we
ﬁnd that:
St +⌬t = St − DIVt
= St − * St * ⌬t
In other words, St+⌬t–St=–DIVt=–*St*⌬t
Or we can say that the change in stock price, which we will write as ⌬St:
⌬St = St+⌬t − St
= − DIVt
= − * St * ⌬t
We can get this into an integrable, calculus form by dividing both sides of
this equation by St,
⌬St
= − * ⌬t
St
This is a nice notation because then we can think of the instantaneous
% change in the stock price dSt due to the payment of the dividend (this is
just –DIVt ) as,
dSt
= − * dt
St
(–DIVt )
(–DIVt ) is an equation we can easily solve using basic calculus.
We will solve the equation (–DIVt ) using the same procedure we used to
solve
dB(t ,T )
= r * dt,
B(t ,T )
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FORWARD CONTRACTS AND FUTURES CONTRACTS
d ln(St ) ⎛ d ln(St ) ⎞ ⎛ dSt ⎞
=⎜
⎟*⎜ ⎟
dt
⎝ dSt ⎠ ⎝ dt ⎠
⎛ 1 ⎞ ⎛ dS ⎞
= ⎜ ⎟*⎜ t ⎟
⎝ St ⎠ ⎝ dt ⎠
dSt
S
= t
dt
= − from equatiion ( − DIVt )
Or, taking a few mathematical liberties by cross-multiplying again,
d ln(St ) = − * dt
(–Log DIVt )
This new equation (–Log DIVt ) can be integrated using basic calculus and a
few little tricks. The deﬁnite integral of the left-hand side of equation (–Log
DIVt ) is,
∫
t + ⌬t
t
d ln (Sv )
* dv = ln (St +⌬t − ln (St )
dv
⎛S ⎞
= ln ⎜ t +⌬t ⎟
⎝ St ⎠
(Integral of the
(–Log DIVt ))
The last equality follows from the way the ln(·) function works.
The deﬁnite integral of the right-hand side of (–Log DIVt ) is,
∫
t + ⌬t
t
( − ) * dv = ( − ) * (t + ⌬t ) − ( − ) * t
= ( − ) * (t + ⌬t − t )
= − * ⌬t
(Integral of the RHS
of (–Log DIVt ))
Therefore, equating the left hand side to the right hand side, we obtain,
⎛ S + ⌬t ⎞
ln ⎜ t
⎟ = − * dt
⎝ S ⎠
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To get rid of the ln(·) function standing in our way of St+⌬t/St (which is
what we really want), we have to exponentiate both sides of this last equation
and let the ln(·) and exp(·) functions cancel each other out. Since they are
inverse to each other, we know this will happen. Taking the exponential of
both sides we get the solution,
S t + ⌬T
= e − *⌬t or
St
S t + ⌬T = S t e
− * ⌬t
(Dividend Payout Process)
The payment of dividends reduces the stock price process according to this
last (dividend payout process) equation. Basically, the (dividend payout process)
equation says that the continuously paid dividend results in a negative growth
rate of –. This makes sense, because as the stock pays dividends at the rate
, you receive those dividends and can do with them as you please.
For example, you could re invest them in the stock or you could strip them
from the stock and place them in a risk-free account. Since they are no longer
part of the stock (having been paid), the stock price has to decline by the
amount of the dividend.
However, it is important to remember that the reason the stock can pay
dividends at all is because it is earning an overall yield from which those
dividends can be paid. We will call that overall yield . In general, represents the total reward for time preference (the risk-free rate) and for bearing
risk (the capital gains yield).
This solves our problem of modeling the reduction in the stock price process
due to it paying out its dividends, or what is the same thing, modeling the
dividend payout process. The continuous dividend yield of results in a
negative growth rate in the stock of – which is what equation (–DIVt ) says,
dSt
= − * dt
St
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CONCEPT CHECK 4
The current stock price St=$100 and =2%.
a. How much would the stock price be reduced by the payment of dividends
over a period of 3 months?
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FORWARD CONTRACTS AND FUTURES CONTRACTS
4.4 HOW CAPITAL GAINS AFFECT STOCK PRICES
We now model the total return process and the capital gains process associated
with the original stock price process. Let’s assume that every dollar invested
in the stock, before it has paid its dividends, grows continuously at a risk
adjusted rate of . We know from our basic ﬁnance, that is the total yield
of the stock which consists of the dividend yield and the capital gains
component.
In basic ﬁnance we learn that this expected return, , of a common stock
comes from two sources. One is the dividend yield, the other is the capital
gains (growth) component. This is usually written as =DIV1/P0+g in basic
ﬁnance terms. Note that corresponds to DIV1/P0 in our continuous-time
model and g is the capital gains component, (P1–P0)/P0. Together, they add
up to the total return on the stock, =+g.
The stock price, before dividends are paid, grows at the continuously compounded rate and would look like,
dSt
= dt
St
(Total Stock Return Process)
Once the dividend rate of is paid we are left with a stock process, St′,
that earns its capital gains yield g=–, continuously compounded:
dSt′
= ( − ) dt
St′
(Capital Gains Process)
The initial value of this capital gains process is e–**St , which will be
demonstrated below. That is, St′ =e–**St is the capital gains component of
the total stock price, which is St′ . This capital gains component is the total
stock price minus the present value of the dividends paid over the life of the
derivative security.
For example, if the dividend yield =3% and the total yield =10% then
the capital gains yield must be –=10%–3%=7%. In terms of the ending stock
price, since the original stock price is growing at a rate of , we would end
up with a terminal stock price of ST =Ste* were it not for the payment of
the dividends. This is the solution to (Total Stock Return Process).
To remove the dividends, we have to adjust the ending stock price using
the factor e–*. At time T, we look at our ending stock price and we ﬁnd
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that ST′ =Ste*t*e–*=Ste(-)*. This is the solution to (Capital Gains Process)
which corresponds to the total return process minus the dividend yield.
For further details, see the Appendix, section 4.8, to this chapter which
discusses how to model stock returns with and without dividends.
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CONCEPT CHECK 5
For this example St =$150, =6 months, =10%, and =3%.
a. Calculate ST′ and interpret ST′ .
4.5 PRICING FORWARD CONTRACTS ON STOCKS WITH A
DIVIDEND YIELD USING THE NET INTEREST MODEL
We are now in the fortunate position of being able to price forward contracts
on stocks with a continuous dividend yield of .
When you own the underlying stock you get the dividends. If you have a
long position in a forward contract, you don’t get the dividends paid on the
underlying over the life of the forward contract. This is why we spent so much
time looking at the stock price dynamics after the dividend had been paid. A
forward contract is written on the stock without the dividends. This is the key
issue in pricing forward contracts on dividend-paying stocks.
But now we fully understand that the stock price dynamics embeds the
capital gains process, and that this capital gains process is the relevant stock
process for pricing a forward contract.
Suppose that, once we receive the dividends from the stock, we use them
to offset our borrowing cost represented by the riskless rate r. What this means
is that as we receive the dividends on the underlying stock, we strip them
(rather than re-invest them in the stock), and we place the stripped dividends
in a continuously compounded riskless account.
At time T, when we owe the forward price (since we have a long forward
position) we use the amount in the stripped dividends account to reduce our
net borrowing cost to r–. Recall what a synthetic long forward position is.
The fastest, intuitive way to price a forward contract in this scenario is to
just substitute r– for r in our old forward pricing formula Ft,T=er *St to obtain,
F t,T
′ =e(r–)*St .
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Here the ′ indicates the forward price in the presence of dividends.That’s
the only difference between the forward pricing formula with and without
dividends. Note that in both pricing formulas, St is just the normal stock price
at time t.
All we have to remember is that our net borrowing cost is r– since we
simply use the dividends paid by the stock to offset our borrowing cost of r.
Stripping the stock of the dividends between times t and T and investing those
dividends in a riskless savings account generates enough funds to reduce the
continuously compounded borrowing rate to r– .
4.6 PRICING A FORWARD CONTRACT ON A DIVIDEND-PAYING
STOCK USING NO-ARBITRAGE
4.6.1 Arbitrage Definitions
We can also prove the forward pricing formula with a dividend yield using
the usual arbitrage arguments. But ﬁrst we must give the deﬁnitions for arbitrage. One of the most important deﬁnitions in derivatives is that of an arbitrage
opportunity. We give three deﬁnitions. The ﬁrst two are of a risk-free arbitrage
opportunity and the third is of a risky arbitrage.
DEFINITION 1 (RISKLESS ARBITRAGE)
A risk-free arbitrage opportunity is one with the following properties:
1. It generates a positive profit (inﬂow) at time T, subsequent to today, represented by time t.
2. The profit generated at time T is riskless. That is, it is certain.
3. The cost today of generating that risk-free, positive proﬁt at time T is zero.
DEFINITION 2 (RISKLESS ARBITRAGE)
A risk-free arbitrage opportunity is one with the following properties:
1. It generates a positive profit (inﬂow) today, time t.
2. The proﬁt generated today is riskless.
3. There are no subsequent outflows (costs).
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Under Deﬁnition 1, you get something later for certain for nothing today. Under
Deﬁnition 2, you get something today for certain for nothing later. Sometimes,
riskless arbitrage opportunities are called ‘money machines’. If there were such
machines, they wouldn’t last long. Like the mythical perpetual motion
machine which violates the principles of basic physics, arbitrage opportunities
violate the principles of basic ﬁnancial economics.
There is a third very important deﬁnition of arbitrage that does not require
it to be risk-free.
DEFINITION 3 (RISKY ARBITRAGE)
A risky arbitrage opportunity is one with the following properties:
1. It does not cost anything today. This means that its cash ﬂows today are
all positive or zero (no negative cash ﬂows).
2. In all states of the world after today, there are no subsequent outflows (costs).
3. In at least one state of the world subsequent to today, it generates a strictly
positive cash flow.
Under Deﬁnition 3, as in Deﬁnition 1 and in Deﬁnition 2, there are no costs
associated with a risky arbitrage at any time. However, there is only the chance
of a positive cash ﬂow, and you don’t know under which scenario that will
occur. So, you have nothing to lose in a risky arbitrage and the potential to
gain. That potential is certain in the sense that it exists for sure, but the state
of the world, , in which it does occur is uncertain. This is what makes it a
risky arbitrage. An example of a risky arbitrage is an unexpired lottery ticket
that someone lost and that you found at no cost to you.
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CONCEPT CHECK 6
One good way to learn what an arbitrage opportunity is involves looking at
some examples of opportunities that do not permit arbitrage.
a. Consider a long or a short position in an FX forward contract. It involves
no up-front cost. Is it an arbitrage opportunity?
b. Explain why an unexpired lottery ticket that someone lost and that you
found is a risky arbitrage. Check that an unexpired lottery ticket obtained
at no cost, satisﬁes the three conditions for it to be a risky arbitrage.