MSC FINANCIAL ENGINEERING
PRICING I, AUTUMN 2010-2011
LECTURE 2: THE BINOMIAL MODEL, I
RAYMOND BRUMMELHUIS
DEPARTMENT EMS
BIRKBECK
1. One-period Binomial Model
Future values of stock prices (as of any financial or economic variable)
are unknown, and best modelled as a random variable. The simplest
possible type of random variable are the so-called Bernouilli random
variables (the coin-tossing type), which can only take one of two different values. Accordingly, the simplest type of stock-price model is one
where price changes take place in discrete time, and in which, at each
time step, stock prices can go either up or down only by some fixed
percentage. The analogue to the coin-flipping experiments so beloved
by introductory probability texts is evident. In a financial context,
binomial models can moreover be used to provide simple illustrations
of two important financial concepts: hedging and replicating portfolios.
They can also serve as an introduction to both the PDE and the martingale approaches to derivative pricing. We start by explaining things
in a one-period framework.
1.1. One-period binomial model. We begin by supposing that there
is only one trading period or, equivalently, two trading dates, t = 0 and
t = 1, that S0 , the stock price at time 0, is known, and that its price
at time 1 can only take one of two values:
uS0 , probability p
(1)
S1 =
dS0 probability 1 − p ,
where u > d (u stands for ’up’ and d for ’down’). Equivalently, S1 =
X1 S0 , with X1 the Bernouilli- (or coin-tossing-) type random variable
defined
u, probability p
(2)
X1 =
d probability 1 − p .
Note that X1 can be interpreted as the gross-return on S1 as computed
ex-post (i.e. at time 1).
We will suppose that besides investing in the stock, we can also put
money in a bank (or, equivalently, sell a bond) earning interest at a
known rate of r; putting a negative amount in the account (equivalently,
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buying the bond) is interpreted as borrowing. We assume an idealized
market in which the borrowing rate is the same as the deposit rate.
We now want to price a European call with time of exercise T = 1
and strike K. Let Vt be the value of the call at time t; we already know
its value at time 1 is equal to its pay-off:
(3)
V1 := max(S1 − K, 0),
which is a function V1 (S1 ) of S1 . The call will therefore pay off V1 (uS0 )
with probability p, and V1 (dS0 ) with probability 1 − p. What should
the call’s price V0 at time 0 be?
If you consider investing in the call being a straight bet, you might
be tempted to simply take the discounted expectation:
V0 = (1 + r)−1 Ep (C1 )
= (1 + r)−1 (pC(uS0 ) + (1 − p)C(dS0 )) .
This is however false, in general. A slightly more sophisticated approach would be to adapt the probability p to take into account the
investor’s risk-aversion: if p represents the actual, ’physical’, probability, investors might be pessimistic about the probability of a positive
outcome, and replace p by some lower probability p∗ < p when computing the price, giving more weight to the unfavorable outcome1.
However, it turns out that the correct (’rational’) price of the option
does not depend on p at all. This is due to the fact that we can replicate
the option using the two ’basic’ assets at our disposal: the underlying
stock itself, and the savings account.
1.2. Pricing by Replication. We have two possibilities for the option’s pay-off at time 1. We also have two different financial assets at
our disposal: the underlying stock, and the savings account which turns
£1 at time 0 into £ 1 + r at time 1. We can combine these two assets
into portfolios, which then will also take two different values at time 1.
Perhaps we can find some clever combination of stock + savings such
that at time 1, that combination will have a pay-off which is identical
to the option’s pay-off. If so then, by absence of arbitrage, the value of
the option at time 0 will have to be the same as that of the portfolio,
and the latter is known, using the market price of the underlying asset
at time 0. This, in a nutshell, is the idea behind pricing by replication.
To flesh out this idea, suppose that at time 0 we form a portfolio Π
consisting of ϕ0 shares and an amount of ψ0 in the savings account (or,
equivalently, an amount of ψ0 bonds with a face-value2 of 1 + r, sold
1In micro-economics, this line of thought is formalized through the introduction and use of
utility functions.
2that is, any such bond is guaranteed to pay off £ 1 + r at time 1.
PRICING I, LECTURE 2
3
for £1 each at time 0). The value of the portfolio at time 0 is therefore
(4)
Π0 = ϕ0 S0 + ψ0 ,
while its value at time 1 is
(5)
Π1 = ϕ0 S1 + (1 + r)ψ0 =
ϕ0 Su + (1 + r)
ϕSd + (1 + r),
with probability p and 1 − p, respectively. Here we have written
(6)
Su := uS0 , Sd := dS0 .
We will similarly use the notation
(7)
Vu := V1 (uS0 ), Vd := V1 (dS0 ),
for the option price in the two states at time 1.
We would like to find ϕ0 , ψ0 such that Π1 = V1 . This is equivalent
to asking that
ϕ0 Su + (1 + r)ψ0 = Vu
(8)
ϕ0 Sd + (1 + r)ψ0 = Vd ,
which is a system of 2 equations in 2 unknowns. Solving this system
(using your favorite method) we find that
Vu − Vd
1 Su Vd − Sd Vu
, ψ0 =
.
Su − Sd
1 + r Su − Sd
By absence of arbitrage, since both the portfolio and the option have
identical pay-offs whatever the state of the market (”up” or ”down”)
at time 1, we have to have V0 = Π0 = ϕS0 + ψ0 . Substituting ϕ0 and
ψ0 , this then gives the following expression for the time-0 price of our
call option:
Vu − Vd
1 Su Vd − Sd Vu
(10)
V0 =
S0 +
.
Su − Sd
1 + r Su − S d
(9)
ϕ0 =
A few remarks are in order here:
• On deriving this formula, we never used the particular form of
the call’s pay-off. It follows that the same formula remains valid
in much greater generality, for any derivative assets whose payoff at time 1 is given by some function V1 (S1 ) of the price S1 of
the underlying at time 1. For example, for puts we would take
V (S1 ) = max(K −S1 , 0), and similarly for derivatives with more
complicated pay-offs, like binary options, straddles, butterflies,
etc. - cf. exercises.
• The probability p of having un up-move does not enter into the
pricing formula (10). Whether the underlying stock has a bigger
or smaller probability of going up at time 1 has no effect whatsoever on the fair price of the call at 0, despite the fact that
this would increase the probability of having a larger pay-off
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from holding the option. This surprising and somewhat counterintuitive conclusion can perhaps best be understood from the
hedging point of view (cf. below): the combination of 1 option and - ϕ0 stock3 will have a value of ψ0 (1 + r) at time 1,
irrespective of whether the stock has gone up or down.
• In the one-step binomial market model (as well as in the multistep model to be introduced below), the call option, or, for that
matter, any derivative, is a redundant asset, since its behavior
can be completely mimicked by a suitable portfolio of underlying + bank account. Whenever this is the case, we speak
of a complete market. Why would options exist if they are redundant? Possible answers are: (i) market demand plus the fact
that putting an options trade together requires specialized skills,
and (ii) actual markets are not complete.
1.3. Pricing by hedging. Here is another idea to price options. The
stock is risky because at time 1 it can take two different values, and
it is not know ex-ante which one. The option written on the stock is
risky for the same reason. Perhaps one can find a clever combination
of option + stock which will take on only one single value at t = 1,
thereby eliminating the risk. Such a portfolio would then be a risk-free
investment. Since all risk-free investments have to earn the same rate,
namely the risk-free rate r of the savings account (why?), the time 0
value of this portfolio would just be its time 1 value divided by 1 + r.
From this, the option’s price at time 0 could then be determined.
To carry out this idea, we try to find an amount ∆0 of stock, to be
sold short at time 0, such that the combined portfolio of buying the
option and selling ∆0 stock at time 0 will have the same value in both
states (”u” and ”d”) of the market at time 1. That is, we require that
(11)
V (Su ) − ∆0 Su = V (Sd ) − ∆0 Sd .
Solving this immediately gives the following expression for the hedging
ratio ∆0 :
V (Su ) − V (Sd )
Vu − Vd
(12)
∆0 =
=
.
Su − S d
Su − Sd
Note that by (9), ∆0 exactly equals ϕ0 , the amount of stock which we
had to buy for our replicating portfolio!
The portfolio consisting of one option and - ∆0 is now risk-free, and
its value at time 1 is therefore (1 + r)× its value at time 0; in formulas:

V (uS0 ) − ∆uS0

 |
{z
}
k
(13)
(1 + r)(V0 − ∆0 ) = V (S1 ) − ∆0 S1 =

}|
{
 z
V (dS0 ) − ∆dS0 ,
3that is, ϕ stock sold short, if ϕ > 0
0
0
PRICING I, LECTURE 2
5
the two expressions on the right being equal by construction. Solving
for V0 , we find that (using the notations (6), (7))
1
(Vu − ∆0 Su ) + ∆0 S0
1+r 1
V (Su ) − V (Sd )
=
Vu −
Su + ∆0 S0
1+r
Su − S d
1 Su Vd − Sd Vu V (Su ) − V (Sd )
+
S0 ,
=
1 + r Su − Sd
Su − S d
V0 =
which is the same as the expression we found before in formula (10).
(It would of course have been embarrassing, to say the least, had we
found a different price: this would have indicated either an error in our
reasoning, or arbitrage opportunities in our market model.)
2. Risk-neutral pricing
2.1. Interpreting the pricing formula. Rearranging expression (10)
as a linear combination of Vu and Vd :
S0
1
Sd
1
Su
S0
V0 =
−
−
Vu +
Vd ,
Su − S d 1 + r S u − Sd
1 + r Su − Sd S u − Sd
and remembering that Su = uS0 and Sd = dS0 , we find the following
formula for the call’s price at 0:
1
u − (1 + r)
1+r−d
(14)
V0 =
Vu +
Vd .
1+r
u−d
u−d
If we put
(15)
q :=
1+r−d
,
u−d
then
u − (1 + r)
,
u−d
(check this!), and the pricing formula (14) simplifies to:
1−q =
1
(qVu + (1 − q)Vd )
1+r
1
(16)
=
(qV (Su ) + (1 − q)V (Sd )) .
1+r
If we would have that 0 ≤ q ≤ 1, then we could interpret q as a
kind of new probability for an up-move in the price. With such an
interpretation, (16) is nothing but the discounted expected value of
the option’s pay-off at time 1 but computed using this new probability.
It turns out that the assumption that 0 ≤ q ≤ 1 is economically
and financially innocuous, in the sense that the market model presents
V0 =
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arbitrage opportunities if either q < 0 or q > 1 (cf. exercise 2.4). We
can therefore consider q as a bona-fide probability, and write (16) as
1
(17)
V0 =
Eq (V (S1 )) ,
1+r
where Eq stands for ”expected value, as computed using q as probability for an up-move”. Remains the question of the economic/financial
interpretation of q.
The key to the latter is provided by computing the expected return
of the option under q:
Eq (V (S1 )) − V0
V0
(1 + r)V0 − V0
=
V0
= r.
Expected return under q =
So we see that the expected return, under this new but artificial probability q, of any derivative asset is equal to r, the return on the risk-free
asset, whatever derivative’s pay-off V (S1 )! In particular, if we take
V (S1 ) = S1 , the risky asset itself, its expected return under q is r, also,
and this property in fact determines q:
Eq (S1 ) − S0
= r⇔
S0
quS0 + (1 − q)dS0 − S0
= r⇔
S0
1+r−d
q=
.
u−d
Under q, risky assets would, on average, not earn more than the
risk-free asset. Since investors in general like to be rewarded for taking
risk by earning a higher expected return, the only investors who would
invest in a risky asset whose mean return is r are investors who would
be neutral with respect to risk. The probability q is therefore called the
risk-neutral probability, and formula (17) is summarized in the following
statement:
2.2. Risk-Neutral Pricing Principle: The price of a derivative asset is the discounted expected value of its pay-off using the risk-neutral
probability, the latter being defined as the probability under which the
underlying asset has a mean expected return equal to the risk-free return, r.
Although for the moment we only established this principle for the oneperiod binomial model, it actually holds in much greater generality,
including for certain classes of continuous time models, as we will see
during these lectures. Two conditions have to be met, though:
PRICING I, LECTURE 2
7
• We have to assume absence of arbitrage. In fact, absence of
arbitrage (with a suitably sharpened formulation in the case of
continuous-time models) is equivalent to the risk-neutral pricing
principle.
• As stated, the risk-neutral pricing principle only holds for complete markets. The point is that in the case of incomplete markets, the risk-neutral probability is not unique, but there can be
several of them - cf. exercise 2.5.
In the latter case, investors’ attitude towards risk re-enters the picture
through a back-door, when the market decides which of the possible
risk-neutral probabilities to use for computing prices.
Real-life investors are of course not risk-neutral, but want to be rewarded for the risk they are taking: consider e.g. the choice between
paying £10 today for receiving a guaranteed pay-off of £11 in one year’s
time, or paying £10 today for a bet which would earn £1000, 022.−
with probability 1/2, or loose £1000, 000.− with probability 1/2. The
latter also has an expected pay-off of £11, but only a madman (or
possibly someone not trading with his own money) would enter such a
deal. The risk-neutral probability q which we introduced above is an
artificial probability. That it occurs in option pricing ultimately derives
from the fact that suitable combinations of option and underlying can
be made risk-free.
3. The multi-period binomial model
Suppose that we now have more than one period, let us say N periods, with trading times t = 0, 1, , . . . , N . During of these periods, the
stock-price can jump up or down by a percentage-amount of u respectively d (the same for each period), with probabilities p and 1 − p. Its
possible values at time t = n (n ≤ N ) are therefore
un S0 , un−1 dS0 , . . . , un−j dj S0 , . . . dn S0 .
n
To determine the probabilities, note that there are
possible ways
j
of choosing j down-moves from a total of n up or down moves. Hence:
n
n−j j
(18)
P(Sn = u d S0 ) =
pn−j (1 − p)j .
j
Within this model, we now consider a European derivative with payoff VN (SN ) at the final time, t = n (to fix ideas, you might think of
this option as being a European call, but everything below holds for
pay-offs). What should its price at time 0 be? There are several ways
we can approach this.
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R. BRUMMELHUIS
3.1. Multi-period binomial option prices, I. First, we can blandly
and boldly generalize the Risk-Neutral Pricing Principle established
for the one-period case, and take the discounted expected pay-off of
the option but replacing the ”objective” probability p for an up-move
by the risk-neutral one, q given by (15). If we denote risk-neutral
probabilities by Q instead of P, the risk-neutral probability of having
SN equal to uN −j dj S0 would then be:
N
N −j j
(19)
Q(SN = u
d )=
q N −j (1 − q)j ,
j
and the price of the option at time 0 would compute as:
1
EQ (VN (SN ))
1(1 + r)N
N X
1
N
=
q N −j (1 − q)j V (uN −j dj S0 ).
j
(1 + r)N
V0 =
(20)
j=0
This is intuitively quite compelling, and turns out be true: after all,
we can split up the N -period model in lots of little one-step binomial
models
n−j+1 j
u
d S0
n−j j
u d S0 →
n−j j+1
u d S0 ,
and for each of these the probability p should be replaced by q when
computing prices as expected values. However, a more mathematicallyminded individual might ask for a formal proof.
3.2. Multi-period binomial option prices, II. A formal proof can
be given by back-ward induction. Suppose we are at the next-to-final
time N − 1, at which SN −1 will have some realized value SN −1 = S
with S ∈ {un−1 S0 , un−2 dS0 , . . . , dn−1 S0 }. In the next, final, time step,
S can either go up to uS or down to dS. We are therefore in the
situation of having N one-period models (one for each allowed value of
S), and hence we know that the price of option at time N − 1 given
that SN −1 = S will be equal to:
VN −1 (SN −1 = S) =
1
(qV (uS) + (1 − q)V (dS)) .
1+r
Equialently,
(21)
VN −1 (SN −1 ) =
1
(qVN (uSN −1 ) + (1 − q)VN (dSN −1 )) ,
1+r
where now both sides, as well as SN −1 are now, technically speaking,
random variables (from the perspective of time 0). So we now know
the option price at time N − 1, as a function of the price SN −1 of the
underlying at that time.
PRICING I, LECTURE 2
9
We now place ourselves at time N − 2, and consider the option as
an asset which pays off VN −1 (SN −1 ) at N − 1. By the same argument,
1
(qV (uSN −2 ) + (1 − q)V (dSN −2 )) .
1+r
Using (21) twice with, respectively, SN −1 = uSN −2 and SN −1 = dSN −2 ,
we find:
(22)
VN −2 (SN −2 ) =
VN −2 (SN −2 )
1
(qVN −1 (uSN −2 ) + (1 − q)VN −1 (dSN −2 ))
=
1+r
1
2
=
q
qV
(u
S
)
+
(1
−
q)V
(duS
)
+
N
N
−2
N
N
−1
(1 + r)2
(1 − q) qVN (udSN −2 ) + (1 − q)VN (d2 SN −1 )
1
q 2 VN (u2 SN −2 ) + 2q(1 − q)VN (udSN −2 ) + (1 − q)2 VN (d2 SN −2 ) .
=
2
(1 + r)
2
We recognize the binomial coefficients
, j = 0, 1, 2 on the right.
j
Continuing in this way we would find that
n X
1
n
VN −n (SN −n ) =
q n−j (1 − q)j VN (un−j dj SN −n ).
n
j
(1 + r) j=0
(A formal proof can be given by induction on n; we leave this to the
interested reader). For n = N , this then gives
N X
1
N
(23)
V0 (S0 ) =
q N −j (1 − q)j VN (uN −j dj S0 ),
N
j
(1 + r) j=0
which is (20).
Exercises to lecture 2.
Exercise 2.1 A European digital call with strike K pays off £ 1 if the
price ST of the underlying at exercise time is greater or equal than K,
and 0 otherwise. A digital put has pay-off equal to 1 minus that of a
digital call.
(a) Graph the pay-offs of a digital call and put.
(b) Price these two options in a one-period binomial model.
Exercise 2.2 A butterfly spread is obtained by going long a call option
with strike K1 , going long another call option with strike K2 , and going
short 2 call options with strike 21 (K1 + K2 ) midway between K1 and
K2 .
(a) Determine the pay-off and the profit & loss of a butterfly spread,
and present these in a graph.
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(b) Determine the price of a butterfly spread in a one-period binomial
model.
Exercise 2.3 A straddle, which is obtained by simultaneously buying
a European put and a European call with identical maturity and strike.
What would you be speculating on when you buy such a straddle?
Exercise 2.4 Explain why we should have that
d ≤ 1 + r ≤ u,
for the one-period binomial market to be free of arbitrage.
Exercise 2.5 Consider a one-period trinomial model, in which the
stock-price at time 1 can either jump to uS0 , stay the same at S0 , or
jump down to dS0 . Let qu , qd be a risk-neutral probability for for an
up-jump, and a down-jump, respectively. Show that
(24)
(u − 1)qu + (d − 1)qd = r,
while 0 ≤ qu + qd ≤ 1.
Exercise 2.6 Suppose that, in the multi-period binomial model, the
probability of having an up-move at time n would depend on n: P(Sn =
uS|Sn−1 = S) = pn . How would this affect the option pricing formula?
Exercise 2.7 Compute the mean and variance with respect to the
risk-neutral probability of the return of Sn over each period.