Let us go back and actually verify that the ratio of the stock to bond process is a martingale. In the case of a two-period market we have
S0
which says that given that the system is in the state S0 at time 1, the ratio of the stock to bond price can be expressed in terms of conditional probabilities
ˆ
p00∗ and ˆp01∗ , and similarly for S1, ˆp10∗ and ˆp11∗ . Writing this in terms of conditional expectation, we then obtain
S1
In fact, using the one-period argument to derive the risk-netural probabilities we have for n≤ N,
or, more generally, by induction we have Sm
Bm
= Em∗
·Sn Bn
¸
(8.19) for m≤ n.
Exercise 8.2 Verify equation (8.19) for the three-period binomial model.
Thus, in the risk-neutral measure the ratio of the share price Sn to the money market account Bn is a martingale. This can be regarded as the definition of the neutral measure. Alternatively, we say that the risk-neutral probability measure P∗ is a martingale measure for the ratio Sn/Bn. The no arbitrage argument for derivative pricing, combined with a back-wards induction, allows us to deduce, in the case of an N-period binomial model, that
fm Bm
= Em∗
"
fn Bn
#
, (8.20)
for 0 ≤ m ≤ n ≤ N. In other words: the ratio of the derivative price to the money market account is a martingale. In particular, if N is the payoff date of the derivative, then
f0
B0
= E∗
"
fN
BN
#
. (8.21)
So, once we show that the ratio of the derivative price to the money market account is a martingale in the risk-neutral measure, that is, equation (8.20) holds, then we can ‘price’ the derivative by use of (8.21). It will be a crucial relation when we come to pricing derivatives in the continuous time case.
9 Binomial Lattice Model
The general kind of tree model that we have been discussing so far is some-times called a ‘bushy tree’, since the number of branches gets large very quickly. At time n there are 2n states, which as n grows larger clearly makes the tree computationally unfeasible. A very useful special model is obtained, however, by letting the branches recombine to form a lattice of prices, and hence is called a ‘lattice model’ or ‘recombining tree’. At time t = n the number of different states is only n + 1 which grows much more slowly than the 2n nodes of the ‘basic’ tree.
At each node, the asset price can move either up in value, with probability p, by a multiplicative factor u, or down, with probability q, by a factor d.
Index the nodes at each time by the number of ‘down’ moves that you need to reach it, for example, S31 is the node at time 3 with one ‘down’ and two
‘up’ movements. We can write the value of the ith node at time n as
Sni = S0un−idi (9.1)
where i = 0, . . . , n and the time index is now necessary. Thus, for example, S20 = S0u2, S21 = S0ud, and so on. A three-period recombining tree is shown below:
S
0©©p©©*S
0u
HHqHHj
S
0d
©©p©©*
S
0u
2HHqHHj
S
0ud
©©p©©* HHqHHj
S
0d
2©©p©©*
S
0u
3HHqHHj
©©p©©*
S
0u
2d
HHqHHj
S
0ud
2©©p©©* HHqHHj
S
0d
3(9.2)
Suppose that at each node the actual probability of an up move is p, and a down move is q. Then, the probability of Sn taking on a specific value Sni is
Prob[Sn = Sni] = Cinpn−iqi, (9.3) where Cin is the standard binomial coefficient, given by
Cin = n!
i!(n− i)!, (9.4)
e.g., C23 = 3!/(2!1!) = 3. The factor Cin is the number of different ways of arriving at the node Sni.
Exercise 9.1 Show that the number of different ways of arriving at the node Sni is Cin.
Recall that Cin is called the binomial coefficient because (x + y)n=
n
X
i=0
Cinxn−iyi. (9.5) Hence if we set x = p and y = q then using the fact that p + q = 1, we see
that n
X
i=0
Cinpn−iqi = 1. (9.6)
This means that if we sum the probabilities (9.3) of each node at the nth time step, then we get one, as we should.
However, in order to price derivatives, we know that we do not need to make any assumptions about the ‘physical’ probabilities, that is, the num-bers p and q, but instead, we must calculate the appropriate risk-neutral probabilities. For the bank account process, we assume for simplicity a con-stant interest rate of r per period, continuously compounded, so Bn= B0ern. Then, for risk-neutrality, we want
S0
B1
B0
= E∗[S1] = p∗S0u + q∗S0d (9.7) at the first node. In fact, the probabilities are governed by essentially the same equation at each node. For example, at the node S21 = S0ud we want
S0udB3
B2
= E∗[S3]
= p∗S0u2d + q∗S0ud2, (9.8) but this reduces to the previous equation. The solution for p∗, q∗ is easily seen to be
p∗ = er− d
u− d and q∗ = u− er
u− d. (9.9)
Exercise 9.2 Verify that equation (9.9) is consistent with the risk-neutral probabilities calculated in the general theory of tree models developed earlier.
In the risk-neutral measure, the lattice is highly structured, which makes calculations easy. In particular, we can price derivatives. The risk-neutral probability of any state Sni is
Prob∗[Sn= Sni] = Cinpn−1∗ q∗i, (9.10) where Prob∗ denotes probability in the risk-neutral measure P∗. Thus, sup-pose f0 is the price of a derivative (at time 0) with specified payoff fni at time n in the ith state. Then, from our general theory of tree models, we know
that f0
B0
= E∗[fn] Bn
, (9.11)
from which it follows that
f0 = e−rn
n
X
i=0
Cinpn−i∗ qi∗fni. (9.12) This is the binomial derivative pricing formula which, together with its vari-ous generalizations, has many useful applications. In the case of a call option with strike K, for example, we would insert
fni = max[Sni − K, 0]
= max[S0diun−i− K, 0] (9.13) for the payoff function fni.
Exercise 9.3 For a three-period model with S0 = $100, p = 0.6, r = 1.01, u = 1.01 and d = 0.99, construct the lattice of stock prices and probabilities, calculate the risk-neutral probabilities for the lattice, and then price a call option with strike $100.
10 Relation to Binomial Model
We shall now show how in a suitable limit the binomial lattice model of chapter 3 can give rise to the Wiener model for asset price movements.
10.1 Limit of a Random Walk
Consider an n-period lattice. Recall that at any node of the lattice we have an ‘up’ and a ‘down’ branch, with probabilities p and q respectively, as shown below From an initial value of S0, the asset price will increase to S0u with probability p, or decrease to S0d with probability q.
S
0©©p©©*S
0u
HH
HHj
S
0d
q (10.1)
.
Let δt be the time-step to the next node, and since there are n steps, the final time will be t = nδt. Suppose moreover that the up and down factors are given explicitly by
u = eµδt+σ˜ √δt and d = eµδt−σ˜ √δt, (10.2) where ˜µ and σ are constants. We shall assume that the actual probabilities p and q (i.e., not the risk-neutral probabilities) are each 12. If we let Xn be the random variable equal to the number of ‘up’ movements after n steps, then the asset price St is
St= S0uXndn−Xn, (10.3) where n− Xnis the number of ‘down’ movements. Substituting in the values of u and d given above then yields
St= S0exp
"
˜
µt + σ√
t 2X√n− n n )
#
. (10.4)
The random variable Xn has a binomial distribution, with mean 12n and variance 14n. We can improve the notation here slightly by defining the new random variable
Zn ≡ 2X√n− n
n , (10.5)
which has a binomial distribution with mean zero and variance one. Hence the asset price process is given by
St= S0exphµt + σ˜ √ tZn
i. (10.6)
Next we use the central limit theorem, which says that if A1, A2, . . . are inde-pendent, identically distributed random variables with mean m and variance V , and Xn is the sum Xn= A1+ A2 +· · · An, then the random variable Zn
defined by Zn= (Xn− nm)/√
nV → N(0, 1) for large n. This tells us that in the limit of large n, Zn converges to a normally distributed random variable with mean zero, and variance 1. Thus, it follows that
n→∞lim St = S0exp[˜µt + σWt] (10.7) where Wt is a normally distributed random variable with mean zero and variance t. Then if we set ˜µ = µ− 12σ2, the asset price process becomes
n→∞lim St= S0exp[µt + σWt− 1
2σ2t], (10.8)
and we are back to the Wiener model. Or so it appears. To show that we have recovered the model entirely, we still need to show that Wtis indeed the Wiener process—we know that it is normally distributed, but now we need to check that it has independent increments. This follows intuitively from the fact that the binomial process is defined as n independent measurements, and hence, by construction, has independent increments. When we take the large n limit, we are tempted to believe that this property is preserved, so (10.8) actually is the Wiener process.