Equating these last two sets of equations and dropping terms of higher order inδtgives F0δt=S0{pu+(1−p)d}; F02δtσ2δt=S02p(1−p) (u−d)2 (7.3) These are two equations in three unknowns (u,dandp), so there is leeway to choose one of the parameters; is there any constraint in this seemingly arbitrary choice?
From the first relationship, it is clear that ifSu(=u S0) andSd(=d S0) do not straddleF0δt, then eitherpor (1−p) must be negative. Since we wish to interpretpas a probability (albeit in a risk-neutral world), we must impose the conditionSd<F0δt <Su.
The function p(1−p) has a maximum at p= 12. The second of equations (7.3) above therefore yields the following inequality:
F0δtσ√ δt Su−Sd ≤ 1
2 (7.4)
This is really saying that if the spreadSu−Sd is not chosen large enough, the random walk will not be able to approximate a normal distribution with volatilityσ.
(iv)Relationship with Wiener Process:Another way of looking at the analysis of the last para- graph is to say that the Wiener process St+δt−St =δSt =St(r−q)δt+Stσ√
δt z can be represented by one step in a binomial process, wherezis a standard normal variate so that E[Sδt]=S0(1+(r−q)δt) and var[Sδt]=S02σ2δt.
We must now chooseu, dandpto match these, i.e.
E[Sδt]=S0(1+(r−q)δt)=S0(pu+(1−p)d)
var[Sδt]=S02σ2δt =S20p(1−p) (u−d)2 (7.5) The reader may very well object at this point since this seems to be the wrong answer; equations (7.3) and (7.5) are not quite the same. But recall that the entire Ito analysis is based on rejection of terms of order higher thanδt:
F0δt =S0e(r−q)δt=S0{1+(r−q)δt} +O[δt2]; F0δtσ√
δt =S0σ√
δt+O[√ δt3] To within this order, the results of this and the last subparagraph are therefore equivalent.
7.2 THE BINOMIAL NETWORK
(i) The stock price movement over a single step of lengthδtis of little use in itself. We need to construct a network of successive steps covering the entire period from now to the maturity of the option; the beginning of one such network is shown in Figure 7.2.
The procedure for using this model to price an option is as follows:
(A) Select parametersu, dandpwhich conform to equation (7.2). The most popular ways of doing this are described in the following subparagraphs.
(B) Using these values ofuandd, work out the possible values for the stock price at the final nodes att=T. We could work out the stock value for each node in the tree but if the tree is European, we only need the stock values in the last column of nodes.
(C) Corresponding to each of the final nodes at timet=T, there will be a stock price Sm,T
wheremindicates the specific node in the final column of nodes.
77
7 The Binomial Model
(D) Assume the derivative depends only on the final stock price. Corresponding to the stock price at each final node, there will be a derivative payoff fm,T(ST).
(E) Just as each node is associated with a stock price, each node has a derivative price. The nodal derivative prices are related to each other by the repeated use of equations (7.1).
Looking at Figure 7.2 we have
f4=e−rδt{p f7+(1−p)f8} f5=e−rδt{p f8+(1−p)f9}
...
f2=e−rδt{p f4+(1−p)f5} ...
This sequence of calculations allows the present value of the option, f0, to be calculated from the payoff values of the option, fm,T(ST); this is commonly referred to as “rolling back through the tree”.
(ii)Jarrow and Rudd:There remains the question of our choice ofu,d andp. The options are examined for a simple arithmetic random walk in Appendix A.2(v); we now develop the corresponding theory for a geometric random walk.
t = 0 t = T
0 1
3
6
2 4
5
7
8 9 Sm, t
S0
uS0
dS0 2
u S0
S0
S0
2
d S0
3
u S0
2
u dS0
2
u d S0
3
d S0
uS0
dS0 m,T
final stock pricesS
Figure 7.2 Binomial tree (Jarrow–Rudd)
The most popular choice is to putu =d−1, giving the same proportional move up and down.
Writingu =d−1 =e , substituting in equations (7.5) and rejecting terms higher thanδtgives
=σ√
δt. The pseudo-probability of an up-move is then given by p= e(r−q)δt−e−σ√δt
eσ√δt−e−σ√δt ≈ 1+(r−q)δt−
1 − σ√
δt+12σ2δt 1 + σ√
δt+12σ2δt
− 1−σ√
δt+12σ2δt
≈ 1 2+1
2
r−q−12σ2 σ
√δt (7.6)
Apart from its simplicity of form, this choice is popular becauseu=d−1. The effect of this is that in Figure 7.2,S4=ud S0=S0. In other words, the center of the network remains at a constantS0. Compare this formula forpwith the corresponding result for an arithmetic random
7.2 THE BINOMIAL NETWORK
walk given by equation (A2.7). An extra term 12σ2has appeared in the drift, which typically happens when we move from a normal distribution to a lognormal one.
Final stock prices merely take the values
S0e−Nσ√δt, S0e−(N−1)σ√δt, . . . ,0, . . . , S0eNσ√δt whereNis the number of steps in the model.
(iii)Cox, Ross and Rubinstein:An alternative, popular arrangement ofSu andSd is to start the other way round: specify the pseudo-probability asp= 12 and derive a compatible pairuand d. Putting p=12 in equations (7.3) gives
S0
1
2u+12d
=F0δt or S0(u+d)=2F0δt 1
2
1−12
S02(u−d)2=F02δtσ2δt or S0(u−d)=2F0δtσ√ δt The equations on the right immediately yield
u = F0δt
S0 (1+σ√
δt); d = F0δt
S0 (1−σ√
δt) (7.7)
The binomial network for these values is shown in Figure 7.3. The probability of an up-move or a down-move at each node is now 12. The center line of the network is no longer horizontal, but slopes up. At node 4 in the diagram the stock price is
Scenter,2δt =S4 =ud S0=S0e(r−q)2δt(1−σ√
δt)(1+σ√ δt)
=S0e(r−q)2δt
1−12σ22δt
=S0e(r−q)2δt
1−12σ2NT/2 There are N steps altogether so thatδt=T/N, and the center lineScenterhas equation
Scenter,T = S0e(r−q)T
1−12σ2NT/2N/2
→exp
r−q−12σ2
T as N → ∞
t = 0 t = T
Scenter
0
1
2
3 4
5
Final Stock Prices Sm, T Sm, t
Figure 7.3 Binomial tree (Cox–Ross–Rubinstein)
79
7 The Binomial Model
Final stock prices now take valuesS0(1+σ√
δt)Ne(r−q−12σ2)T, S0(1+σ√
δt)N−1(1−σ√
δt) e(r−q−12σ2)T, . . . , S0(1−σ√
δt)Ne(r−q−12σ2)T (iv) For completeness, we list a third discretization occasionally used:
u =exp{(r−q)δt+σ√
δt}; d =exp{(r−q)δt−σ√ δt}
Substituting in equation (7.5) and retaining only terms O[δt] gives p=12(1−12σ√ δt). The center line of the grid now has the equationScenter=S0e(r−q)t, which is the equation for the forward rate (known as theforward curve).