stock and a put option on the stock.
Example 4.47
Spreads are portfolios using options of the same type, some in long and some in short positions. A popular example is the butterfly spread, which is designed for investors who believe that large fluctuations in the price of the underlying stock are unlikely in [0, N]. One buys calls with strikes K1
and K3 > K1and at the same time sells two calls at strike K2 = 12(K1+ K3). The initial outlay is thus I(0)= C1(0)+C3(0)−2C2(0), which is positive, as the call price is a convex function of the strike. The payoff is 0 outside the strike interval (K1, K3], S (N) − K1on (K1, K2] and K3− S (N) on (K2, K3]. An investor who believes that the stock price will not vary much from X2 is thus likely to profit from this position and is well protected against large unforeseen price changes.
Example 4.48
As we saw in Exercise 3.8, the bottom straddle consists of long positions in both a call and a put with common strike K and expiry date N. This is a simple example of a combination, using options of different types on the same underlying. We buy a call and a put, both with strike K and expiry N. The payoff is |S (N) − K|, against an initial outlay of C(0) + P(0). Thus the holder will lose money if the price of the stock stays close to K but can gain substantially if the final price differs significantly from K. This investment thus suits an investor who expects large price changes, by taking options with strike close to S (0).
Path-dependent options
Recall, that a European derivative security H is path dependent if H(N)= h(S (0), S (1), . . . , S (N)), where h: RN+1→ R is Borel-measurable. We shall consider some common path-dependent options below.
Example 4.49
Barrier options are instruments that are designed to hedge against move-ments in the stock price crossing a given level, seen as a ‘barrier’ Y say. If the stock price reaches (or crosses) Y, a knock-out barrier option becomes
worthless: if S (n) reaches Y from below we call it up-and-out, otherwise down-and-out. On the other hand, a knock-in option only comes into ex-istence once the barrier is reached, although the option premium has been paid at the outset. For a single barrier Y we can devise eight European-style barrier options by using the terms call and put, up and down, in and out in combination. In addition, one can develop two-sided barrier options, using both an upper and a lower barrier, with different outcomes depending on whether activation (or deactivation) occurs when either or both barriers are breached.
The somewhat lengthy computation of closed-form formulae giving ra-tional prices for the different option premia (within the standard Black–
Scholes model setting) is now part of the standard literature. Fortunately there are again some shortcuts: in each case the sum of the payoffs of an in-option and an out-option with the same strike, expiry and barrier is ob-viously the same as that of the ordinary option of that type, since only one of the barrier options survives and then has the same payoff as the standard option.
Further complications for computation can arise in practice, where the option holder may be given a rebate when the option is knocked out, or, as for Parisian options, the option is knocked out only once the barrier has been breached for a specified length of time. Barrier options are attractive to a buyer seeking to minimise the cost of option premia by excluding sce-narios she considers unlikely, while accepting the risk that they may occur.
For example, a down-and-out call should be cheaper than an ordinary one, since the option is knocked out if S (n) reaches Y from above, irrespective of whether the stock price recovers to above Y by time N. The writer of an up-and out call, on the other hand, limits her potential liabilities in the event that the stock rises sharply.
Example 4.50
Lookback options come in several guises, using the maximum Mlbor min-imum mlb of the underlying stock price over a preset ’lookback period’
within T. We have the floating strike variety, where the strike becomes a random variable. Thus the lookback call allows one to buy the stock at the minimum value achieved in the lookback period, hence the payoff is
(S (N)−mlb)+. For the put the holder uses the maximum instead, with payoff (Mlb−S (N))+. There are also fixed strike versions, with payoffs (Mlb−K)+ and (K−mlb)+, and even partial lookbacks, using a preset fraction λMlbor λmlbinstead. Clearly all these are designed to ‘eliminate regret’ over lost opportunities, and their premia will reflect this.
Example 4.51
The Asian call option, in a model with time setT = {0, 1, 2, . . . , N}, has payoff H(N) = (A(N) − K)+, where A(N) is either the arithmetic aver-age N1 N
i=1S (i) or the geometric average ["N
i=1S (i)]1/nof the stock prices, together with a fixed strike K. Clearly these are path-dependent options.
They are usually traded as of European type, to avoid early exercise occur-ring before the averaging period is completed, thus losing the protection it can provide.
To see why one may wish to use such options in certain situations, con-sider a fixed strike K = 1.5, N = 6 and A(N) as the arithmetic average.
Suppose that the price evolution of the underlying S is given by
n 1 2 3 4 5 6
S (n) 1.5 1.6 1.7 1.8 1.6 1.4
Then H(6)= 0.1, whereas for a European call C(6) = 0. The Asian option may be preferred by a company which regularly buys some asset and wants to hedge against high prices, for example, to guard against price spikes near the trading horizon. It can buy a series of vanilla call options with different maturities or a single Asian call on the average price. The latter is usually satisfactory from the point of view of securing the company’s business.
Pricing formulae have been extended to cover many other exotic op-tions, including choosers, where the option holder can decide at an inter-mediate time 0 < N1 < N whether he wishes to treat his option as a put or a call, and compound options, where the underlying is not a stock but an option with earlier expiry. Methods for determining rational prices and market applications for such derivatives can be developed in various model settings.
4.9 Proofs
Theorem 4.31
If Mj(n) are martingales and the processes Hj(n) are predictable, j = 1, . . . , d, then the process X(n) defined by
is a martingale, where X(0)= x0is an arbitrary real number.
Proof First note
The theorem is formulated for any probability space hence we use the gen-eral symbol for expectation. We take the conditional expectation of the above equality to get
where we used the property of conditional expectations called ‘taking out what is known’ – the process H is predictable so the value at time n+ 1 is measurable with respect toFn. The final result:E(X(n + 1) − X(n)|Fn)= 0
proves that X is a martingale
Theorem 4.32
For M(n) to be a martingale it is sufficient that for each predictable process H(n)
E N
n=1
H(n)ΔM(n))
= 0.
Proof We will show
E(M(k + 1)|Fk)= M(k).
Take A∈ Fkand write H(n)=
1A for n= k + 1, 0 otherwise, and then the sum reduces to just one term
N n=1
H(n)ΔM(n) = 1AΔM(k + 1).
By the assumption E N
n=1
H(n)ΔM(n)
= E(1AΔM(k + 1))
= E(1A[M(k+ 1) − M(k)]) = 0,
so the definition of conditional expectation completes the proof.