We generalise the above analysis by extending the scheme to N steps. We assume that trading occurs at finitely many specified dates. Denote the time interval between trading dates by h, so that trading occurs at times t= nh, n= 1, 2, . . . , N.
We introduce the sample space for the N-step model exactly as was done for n= 2, 3. Namely, we set Ω = {u, d}N, and, in keeping with the above cases, forω in Ω the probability will take the form
P(ω) = pk(1− p)N−k.
Eachω in Ω (called a path) is a sequence of length N consisting of u’s and d’s. All pathsω with k occurrences of u have the above probability, and the probability of this set of paths is N
k
pk(1− p)N−k.
For n≤ N the binomial splittings induce a partition of Ω into 2ndisjoint sets:Pn = {Buu...u, ..., Bdd...d}, where each set in the partition has an n-tuple subscript consisting of u’s and d’s, so that in each set all pathsω have their first n entries in common. We see (as is obvious in the earlier examples) that for n≤ N, each set in the partition Pnis a (disjoint) union of two sets inPn+1– just as for N= 3, Bu = Buu∪ Bud. More generally, we say that a sequence of partitions (Pn)n≤N is refining if, for each n≤ N, Pnis a finite union of sets inPn+1.
Fix two numbers−1 < D < U to represent the random returns in each step defined by
Kn =
U with probability p, D with probability 1− p.
Again, Knacts only on the nth entry of the sequence defining the pathω.
We note that these returns are independent, in the sense described in the next remark.
This is again easy to see from the definition of the Kn (see also Exercise 3.2).
Assume that S (0) is given, while
S (n)= S (n − 1)(1 + Kn).
which means that the stock prices follow a recombining binomial tree.
In addition to holding stock we can invest in a money market account, which we take as a security manufactured by a series of single risk-free deposits, with return R per period, assumed constant throughout. So we have a sequence of numbers, known in advance, with A(0) given (we will frequently take A(0)= 1 for simplicity), and
A(n)= A(n − 1)(1 + R)
for n= 1, . . . , N. To avoid arbitrage we assume that D < R < U, as we did in the single-step case.
The key property observed for a three-step model can be easily extended.
Theorem 3.10
The sequence of discounted stock prices in the binomial model is a martin-gale with respect to the filtrationPnand probability Q.
Proof The key argument, repeated in each case above, is concerned with examining the two next step prices at each node and computing their expec-tation exactly as for a single step. This can be summarised in the following way: for B∈ Pnat n= 0, 1, 2, . . . , N − 1,
Q(S (n+ 1) = S (n)(1 + U)|B) = q, Q(S (n+ 1) = S (n)(1 + D)|B) = 1 − q, and since q= UR−D−D, and hence 1 − q = UU−D−R we obtain
EQ(S (n+ 1)|Pn)= S (n)(1 + U)R− D
U− D+ S (n)(1 + D)U− R U− D
= S (n)(1 + R)
which gives the result upon discounting.
The above observations lead us to the following theorem Theorem 3.11
In the binomial model the discounted prices of a derivative security with given payoff H(N) are given, under the martingale probability, by the re-cursive relations
H(n˜ − 1) = EQ( ˜H(n)|Pn−1)
for n= 1, 2, . . . , N. In particular, the initial price is given by H(0)= (1 + R)−NEQ(H(N)).
Proof In the binomial model we can replicate the final payoff by a series of steps, moving backwards in time. The values of such a strategy clearly give no-arbitrage prices at any time and any position of the tree. For, if the price H(n) were different from V(n) at some ω, we would construct an ar-bitrage by doing nothing till n (taking x(k) = 0, y(k) = 0 for k < n), and at time n by shorting the expensive and buying the cheap security. Holding this till maturity we would maintain the difference since, at maturity, repli-cation ensures that we would break even. (Strictly speaking, the surplus should be invested in the money market account.)
A replicating strategy is self-financing by its very construction. To see that it is ‘predictable’, note that the number x(1) computed at time 0 is the fraction
x(1)= Hu− Hd Su− Sd.
So x(1) is a deterministic number. Then, by induction, x(n+ 1) will involve all the values of H and S at time n+ 1 and so it will only depend on the position on the tree at time n when the computation is performed. This means that x(n+ 1) will depend on n first elements of the N-tuple ω and so it will be known at time n.
Finally, we know that V(n) and so H(n) must be a martingale under Q and the claim follows from the martingale condition
EQ( V(n)|Pn−1)= V(n− 1)
after multiplying both sides by (1+ R)n−1. If the random variable H(N) has the following form
H(N)= h(S (N)) then we may write
H(N)= h(S (0)(1 + U)Y(1+ D)N−Y)
where Y is a random variable giving the number of upward movements in N steps. Hence
Example 3.12The CRR formula
In particular, for the European call option with strike K this summation begins at the first integer at which the payoff is non-zero
H(0)= (1 + R)−N
This formula can be given a concise form. First recognise two terms on the The second term can be written as
−(1 + R)−NK(1− Fq(l− 1))
where Fq is the cumulative distribution function of the binomial distribution, The first term can be rearranged as
S (0) The first term above then takes the form
S (0) using the distribution function again. Finally
C(0)= S (0)(1 − Fq1(l− 1)) − (1 + R)−NK(1− Fq(l− 1)).
This is the Cox–Ross–Rubinstein (CRR) formula for the initial price of a European call option. (Many texts prefer to state this formula in terms of
the complementary binomial distributionΨ, where Ψ(l, N, r) = 1 − Fr(l− 1) =
N k=l
Nk
k
rk(1− r)N−k for 0< r < 1.)
The coefficient at S (0) has a clear meaning: it is the probability that the option will be exercised, computed by means of the modified martingale probability q1.
In the second term the exercise price is discounted, which is logical since the exercise price K is valid at time N, but the formula is concerned with the current prices. The coefficient at the discounted K gives the martingale probability that the option will be exercised.
The next three exercises illustrate further applications of the above analysis.
Asian options come in many varieties. The guiding principle is that the payoff should represent an averaging of the prices of the underlying up to the trading horizon N. Such options are popular on commodities markets, for example, in guarding against price manipulation near the trading hori-zon. We consider an example of a fixed strike option where the underlying is given as the arithmetic average, A(N)= N1 N
k=0S (k), of the asset values, so that the payoff takes the form H(N) = max(A(N) − K, 0).
Exercise 3.7 Find the process of prices of the Asian option with pay-off H(5) = max{165
k=0S (k)− K, 0} where S (0) = 60, U = 12%, U = −6%, R = 4%, K = 62.
Exercise 3.8 A popular combination of a call and a put is a bottom