Multi-step general models
Theorem 4.42 The law of one price
If H, H are two European path-independent derivative securities with the same payoff
H(N)= H (N) then for each n≤ N their values agree:
H(n)= H (n).
Proof This is routine: if for oneω0, at some time n0we have H(n0, ω0)<
H (n0, ω0), say, then we do nothing up to time n0, take zH(n0, ω0) = 1, zH (n0, ω0)= −1, y(n0, ω0)= H (n0, ω0)− H(n0, ω0) to obtain an arbitrage
opportunity.
Forwards and futures
Recall that a forward contract (long position) with delivery price K is a derivative security with payoff H(N) = S (N) − K. Then, in a complete model free of arbitrage
H(n)= (1 + R)−(N−n)EQ(S (N)− K) = S (n) − K(1 + R)−(N−n). As in Chapter 2, we can give a model-independent proof of this formula.
Proposition 4.43
The time n value of a forward contract initiated at time 0 with delivery price K is H(n)= S (n) − K(1 + R)−(N−n).
Proof We build a portfolio such that at time N, V(x,y,z)(N) = S (N) − K taking x(n) = 1, y(n) = −A(0)1 K(1+ R)−N. Hence by no arbitrage at each time n we must have V(x,y)(n) = H(n) by a straightforward argument. At time n, V(x,y)(n)= S (n)+y(n)A(n) = S (n)−K(1+R)−(N−n), which completes
the proof.
Generally, we define the forward price F(n, N) as the delivery price that gives zero value to a contract entered into at time n, with delivery at time N.
Corollary 4.44
No-arbitrage implies that the forward price at time n< N is F(n, N) = S (n)(1 + R)N−n.
Proof From H(n)= 0 we get the result at once. In particular F(0, N) = S (0)(1 + R)N (as we observed in Chapter 2 for N= 1).
Exercise 4.9 Suppose that you have entered a long forward contract with forward price F(0, 2) = 124.75. At time 1 you would like to close this position. How much money would you have to pay (or receive) if S (1)= 120 and R = 9%?
In addition to stock prices markets also quote so-called futures prices.
The principal difference between them and forwards is that futures are quoted on an exchange, whereas forwards are traded ‘over-the-counter’
(OTC), by mutual agreement between the two parties concerned. The ex-change seeks to minimise the risk of default, so investors are required to hold a margin account with the exchange, whose funds can be adjusted daily to reflect the fluctuations in futures prices.
We restrict our attention to discrete-time situations. Use f (n, N) to de-note the futures price of a commodity, which is established at time n when the contract is initiated. This provides a series of derivative securities with underlying asset S (n) defined by means of the following axioms:
• f (N, N) = S (N).
• f (n, N) is Fn-measurable, whereFn is the field generated by the stock prices S (1), . . . , S (n).
• At time n a long futures position generates the cash flow f (n, N) − f (n − 1, N), n = 1, . . . , N.
• A time n a short futures position generates the cash flow f (n− 1, N) − f (n, N), n = 1, . . . , N.
• It costs nothing to enter any futures position at any time.
The cash flows generated in the margin accounts by changes in the fu-tures position are called marking-to-market. If we neglect the time value of money then the holder of the long futures position has the total cash flow
N n=0
[ f (n, N) − f (n − 1, N)] = f (N, N) − f (0, N) = S (N) − f (0, N), (4.6)
which looks very similar to the long forward payoff. The only difference, f (0, N) in place of F(0, N), is irrelevant if the risk-free returns are known for the future periods, as is shown in the next result. Again we require extended strategies, this time involving either forwards or futures as the derivative security. Our no-arbitrage assumption remains in place for both cases.
Theorem 4.45
If the interest rate is constant, then f (0, N) = F(0, N).
Proof We restrict to the special case N = 2. One forward contract gives the payoff S (2) − F(0, 2). We will design a futures-based strategy with zero cost, which will generate the sum f (0, 2) − S (2). Since the combined balance is f (0, 2) − F(0, 2), which is a deterministic quantity produced at zero cost, it must be zero.
To build the required strategy we write (as in (4.6)) f (0, 2) − S (2) = f (0, 2) − f (2, 2)
= ( f (0, 2) − f (1, 2)) + ( f (1, 2) − f (2, 2)) and then replicate each bracketed term on the right.
The latter is the futures short position cash flow at time 2, so to generate it we have to open this contract at time 1.
The former after discounting becomes 1+R1 ( f (0, 2) − f (1, 2)), which is the payoff at time 1 of the fraction1+R1 of the short futures position opened at time 0. If we open such a fraction (for free), and invest the proceeds obtained at time 1 (or borrow rather than invest if the amount is negative) risk free, this will give us f (0, 2) − f (1, 2) at time 2 as required.
More formally, at time 0 we take x(1)= 0, y(1) = 0,
zforward(1)= 1, zfutures(1)= −1+R1 ,
which is free, and at time 1, at zero cost, we change this to x(2)= 0, y(2) = 1+R1 ( f (0, 2) − f (1, 2)),
zforward(2)= 1, zfutures(2)= −1.
(The key point is the knowledge of the future interest rate for the period starting at time 1 which allows us to figure out the size of the initial futures
position.)
Exercise 4.10 Suppose that the stock prices on consecutive days are 100, 100.33, 100.33, 100.15, 99, 101. Find the cash flow of the long futures position if the annual risk-free rate is 9%.
Complex derivatives
Call and put options are often called (plain) vanilla options, as they have no special features and are regarded by market traders as very familiar. More complex options are often called exotic, although this terminology is not used with much precision.
Vanillas can be combined in various ways to form trading strategies whose payoffs are suited to investors’ various perceived needs. We offer some simple examples, reflecting different investment needs:
Example 4.46
To construct a covered call we write a call, strike K, (receiving the call premium C(0)) and hold one share in the stock. The initial value of this position is S (0)−C(0) > 0 (else the buyer of the call might just as well buy the share to begin with!) and we are ‘covered’ against having to relinquish our share for K at time N: our final position is C(0)− S (0) + min(S (N), K), which is positive: by parity, and as P(0)> 0 and (1 + R)−N≤ 1 we obtain
C(0)> S (0) − (1 + R)−NK ≥ S (0) − K. (4.7) Note, however, that our profit is capped at C(0)− S (0) + K, whereas for the call holder (who has invested only the call premium) the potential profit S (N)− K − C(0) is unbounded above.
Exercise 4.11 Conduct a similar analysis of the protective put,