Simple Exotics

11 Simple Exotics The purpose of this part of the book is to introduce the reader to the most important types of equity derivatives and to illustrate the pricing techniques which have been introduced in the last two parts. Exotic options can mostly be priced using classical statistical techniques, although we will see in Part 4 of this book that some of the analysis can be simplified (or at least rendered more elegant) using stochastic calculus. There is no firm definition of an exotic option and we usually take it to mean anything that is not a simple European or American put or call option. We start with a chapter on simple extensions of the Black Scholes methodology, which should really be understood by anyone involved with options, whether or not they have a specific interest in exotics. 11.1 FORWARD START OPTIONS (i) Suppose we buy an option with maturity T which only starts running at time τ . If the strike price is set now, pricing becomes fairly trivial: the price of a European option depends only on the final stock price and the strike price so there is no difference whatsoever between “starting now” and “starting at time τ ”; with an American option we must take into account the fact that we cannot exercise the option between now and time τ , but this is easily accommodated within a tree or a finite difference scheme. But the type of options considered here are those that are at-the-money or 20% out-of-the-money at some future starting date. (ii) Homogeneous Functions: This is an important mathematical property of option prices which we use freely in the following chapters. The concept is so intuitive that most people use it instinctively without placing a name to it. If the reader is already working within a derivatives environment, it is quite likely that his option model has the initial stock price preset to 100; this yields option prices directly as a percentage of the stock price. We know that the initial stock price will not be 100 but we also know that things move proportionately: if an option is priced at 5.5 on our preset model with S 0 = 100 and strike price X = 120, we know immediately that the price of a similar option with S 0 = 40 and X = 48 would be 2.2. It is immediately apparent to most people that the strike price has to move in line with the stock price for this reasoning to work, just as it is apparent to most that we should not change the time to maturity or the interest rate. An equally obvious conclusion is reached concerning the number of shares on which an option is written. Suppose we own a call option on one share and the company suddenly declares a 2 for 1 stock split. We know that the share price would fall in half, but we would be kept whole if our call option were replaced by two options of the same matu- rity, each on one of the new shares, and with the strike price equal to half the original strike price. 11 Simple Exotics Let f (nS 0 , nX) be the value of an option on n shares, where S 0 is the stock price and X is the strike. The homogeneity condition just described may be written f (nS 0 , nX) = S 0 f  n, n X S 0  = Xf  n S 0 X , n  = nf(S 0 , X) = nS 0 f  1, X S 0  (11.1) It should be pointed out that this property holds true for most options we encounter, although sometimes with modification: for example, barrier options are homogeneous in spot price, strike price and barrier value. However there are exceptions such as power options which are described later in this chapter. (iii) Forward Start with Fixed Number of Shares: Consider an option starting in some time τ in the future, maturing in time T and with a strike price equal to a predetermined percentage α of the starting stock price. Using the homogeneity property, the value of this option in time τ is f (S τ ,αS τ , T − τ ) = S τ f (1,α,T − τ ) The term f (1,α,T − τ ) is non-stochastic and may be calculated immediately. If we buy f (1,α,T − τ ) units of stock today at a cost of S 0 f (1,α,T − τ ), the value of this stock in time τ will be S τ f (1,α,T − τ ); but this is the same as the future value of the forward starting option. Today’s value of the forward starting option must therefore be S 0 f (1,α,T − τ ) = f (S 0 ,αS 0 , T − τ ) i.e. the valueof the forward starting option is the same as if the option started running today, with the time to maturity set equal to the length of time between the start and maturity (Rubinstein, 1991c). A further refinement is needed if the stock pays a dividend. Remember that if we hold a packet of stock from now to time τ , we will receive a dividend but if we hold an option, we will not. The adjustment to the formula can be made by the usual substitution S τ → S τ e −qτ for continuous dividends to give f forward start = e −qτ f (S 0 ,αS 0 , T − τ ) (iv) Forward Start with Fixed Value of Shares: The last subparagraph dealt with a forward starting option on a fixed number of shares. But suppose we were asked to price a forward starting option on $1000 of shares. The value of this option in time τ will be f (n τ S τ ,αn τ S τ , T − τ ) = n τ S τ f (1,α,T − τ ) where n τ S τ = $1000 in our example. Therefore, the value of this option in time τ is completely determinate (non-stochastic); today’s value is simply obtained by present valuing this sum: f forward start = e −rτ f (S 0 ,αS 0 , T − τ) (v) The contrasting results of the last two subsections are well illustrated in the foreign exchange market: r For an option to buy £1 for dollars, forward start means discounting back by the sterling interest rate. r For an option to buy sterling for $1, forward start means discounting back by the dollar interest rate. 146 11.2 CHOOSERS (vi) Cliquets (Ratchets): As the name implies, this type of option was first used widely in France. It is designed for an investor who likes the basic idea of a call option but is concerned that the stock price might spend most of its time above the strike price, only to plunge just before maturity. In such a case, the cliquet would capture the effect of the early price rise. It is really a series of forward starting options strung together. The option has a final maturity T (typically 1 year) and a number of re-set dates τ 1 ,τ 2 , ··· (typically quarterly). The payoff and re-set sequence is as follows: r At τ 1 , the option pays max[0, S τ 1 − S 0 ]. r At τ 2 , the option pays max[0, S τ 2 − S τ 1 ], etc. Clearly, each of these is the payoff of an at-the-money forward starting call option. The fair value is therefore given by f cliquet = C(S 0 , S 0 ,τ 1 ) + e −qτ 1 C(S 0 , S 0 ,τ 2 − τ 1 ) +···+e −qτ n−1 C(S 0 , S 0 ,τ n − τ n−1 ) Common variations on the structure have the effective strikes slightly out-of-the-money, or have the payouts rolled into a single payment at final maturity. 11.2 CHOOSERS (i) The 1990 Kuwait invasion led to a jump in the price of crude oil. Speculators were then faced with a dilemma: if a withdrawal were negotiated, the oil price would fall back; but a declaration of war by the US would lead to a further jump upwards. A ready-made strategy for this situation is the straddle, consisting of both an at-the-money put and an at-the-money call. This has a positive payoff whichever way the oil price moves; but it has the great drawback of being very expensive. (ii) The Simple Chooser: This option has a strike X and a final maturity T. The owner of the option has until time τ to declare whether he wants the option to be either a call option or a put option. The chooser is sold as an option which has the benefits of a straddle, but at a much lower cost. Clearly, at the limit τ = T the option becomes a straddle while at the limit τ = 0 it becomes a put or call option. The pricing of this option is surprisingly easy (Rubinstein, 1991b): at time τ , the holder of the option will choose put or call depending on which is more valuable. The payoff at time τ can therefore be written Payoff τ = max[P(S τ , X, T − τ ), C(S τ , X, T − τ )] Using the put–call parity relationship of Section 2.2(i) gives Payoff τ = max  C(S τ , X, T − τ ) + X e −r(T −τ ) − S τ e −q(T −τ) , C(S τ , X, T − τ )  = C(S τ , X, T − τ ) + e −q(T −τ) max  X e −(r−q)(T −τ ) − S τ , 0  Taking these two terms separately, the instrument which has a value C(S τ , X, T − τ ) in time τ when the stock price is S τ is obviously a call option maturing in time T; its value today is C(S 0 , X, T ). 147 11 Simple Exotics The form of the second term in the payoff is that of a put option maturing in time τ . Its value today may be written e −q(T −τ) P(S 0 , X e −(r−q)(T −τ ) ,τ). Putting these together gives f simple chooser = C(S 0 , X, T ) + e −q(T −τ) P  S 0 , X e −(r−q)(T −τ ) ,τ  (iii) Complex Chooser: The concept of the chooser can be very simply extended so that the put and call options have different strike prices and maturities. Unfortunately, the mathematics of the pricing does not extend so simply and we therefore defer this until Section 14.2. 11.3 SHOUT OPTIONS (i) Like cliquets, these options are for investors who think that the underlying stock price might peak at some time before maturity. The shout option is usually a call option, but with a difference: at any time τ before maturity, the holder may “shout”. The effect of this is that he is guaranteed a minimum payoff of S τ − X , although he will get the payoff of the call option if this is greater than the minimum. (ii) Payoffs: By definition, the final payoff of the option is max[0, S τ − X, S T − X ]. In practice, S τ − X is always greater than zero; if not, we would have S τ < X which means that the holder of the option had shouted at a time when the effect was to turn the shout option into a simple European call option, for no economic benefit in exchange. The payoff at time T can therefore be written max[S τ − X, S T − X ] = S τ − X + max[0, S T − S τ ] At time τ if a shout is made, the value of this payoff is e −r(T −τ ) (S τ − X ) + C(S τ , S τ , T − τ) (iii) Shout Pricing: This option is easily priced using a binomial model as we would for any American option (Thomas, 1993). The final nodes in the tree are max[0, S T − X ] as they would be for a call option, i.e. if we get as far as the final nodes, it means that no shout took place. At each node before the final column, the holder has the choice of shouting or not shouting. To decide which, compare the value obtained by discounting back the values of the two subsequent nodes with the time τ payoff produced by shouting, i.e. e −r(T −τ ) (S τ − X ) + C(S τ , S τ , T − τ); we enter whichever value is greater at that node. In spirit this is the same as the binomial method for pricing American options which was explained in Chapter 7; in that case we rolled back through the tree and at each node we selected the greater of the payoff value or the calculated discounted average. The present procedure calls for the Black Scholes value of a call option to be calculated at each node. However, with the assumption of constant volatility, the Black Scholes formula only needs to be calculated once for each time step. The homogeneity property described in Section 11.1(ii) says that the price of an at-the-money option is proportional to the stock price, so that for an entire column of nodes we only need to calculate the constant of proportionality C(1, 1, T − τ ) once. 148 11.4 BINARY (DIGITAL) OPTIONS (iv) Put Shout: A precisely analogous put option with shout feature can be constructed. A gener- alized payoff at time T can be written as max[φ(S τ − X ),φ(S T − X )] = φ(S τ − X ) + max[0,φ(S T − S τ )] where  φ =+1 for a call φ =−1 for a put At each node in the tree, a shouted value would be e −r(T −τ) φ(S τ − X ) + Option(S τ , S τ , T − τ ) where the option is either a put or a call option. (v) Strike Shout: The shout options described above locked in a minimum payout. Another version of this type of option locks in a new strike price when the shout is made, and is even simpler to price than the previous ones. The payoff at time T for a calll option with strike shout is max[0, S T − X, S T − S τ ] = max[0, S T − S τ ] This is the same as in subsection (ii) above, but without the minimum payout. The rest of the analysis is as before. 11.4 BINARY (DIGITAL) OPTIONS (i) Recall the simple derivation of the Black Scholes formula which was given in Section 5.2. In its simplest form, this may be written C(S 0 , X, T ) =  ∞ 0 F(S T ) max[0, S T − X ]dS T =  ∞ X F(S T )(S T − X )dS T = S 0 e −qT N[d 1 ] − X e −rT N[d 2 ] where F(S T ) is the (lognormal) probability distribution of S T . The two terms in this equa- tion will now be interpreted separately, rather than together as they were before (Reiner and Rubinstein, 1991b). Cash f X P T S Figure 11.1 Cash or nothing option (ii) Cash or Nothing Option (Bet): In the term  ∞ X F(S T )X dS T , the factor X appears in two un- related roles: as a constant multiplicative factor in the integrand and again as the lower limit of inte- gration. The first role is trivial and we will drop X from the integrand. Then e −rT  ∞ X F(S T )dS T = e −rT N[d 2 ] is the present value of the risk-neutral probability that X < S T , and can be interpreted as the arbitrage-free value of an option which pays out $1 if S T is above X, and $0 otherwise (Figure 11.1). This option is essentially a bet: “I will give you $1 if the stock price is over $100 in 6 months”. Its value is given by the second term in the Black Scholes formula. 149 11 Simple Exotics Asset f X T S Figure 11.2 Asset or nothing option (iii) Asset or Nothing Option: By the same reasoning as in the last section, the first term in the Black Scholes formula is the price of an option which delivers one unit of stock if the maturity price is above X and nothing otherwise (Figure 11.2). (iv) Gap Options: The last two options can be combined to give the so-called gap options whose price is given by f gap = S 0 e −qT N[d 1 ] − P e −rT N[d 2 ] = f asset − f cash Pay Later f X T S Figure 11.3 Pay later option There are two special cases of this option which are of interest: first, if P = X , we obviously have a European call option; second, if the asset-or-nothing and the cash-or-nothing components have the same value, then the fair value of the gap option is zero, i.e. f gap = f asset − f cash = 0. This composite option is known as a pay-later option for obvious reasons: the initial pre- mium for the option is zero and any payments either way are made at maturity. The payoff is shown in Figure 11.3. (v) Greeks: The binary option formulas are basically the Black Scholes formula pulled in half; it might therefore seem that there is little new to say about these options. However, in some respects they display pathological behavior which teaches us some important lessons. Imagine a trader trying to dynamically hedge a short cash-or-nothing option (bet). Shortly before maturity he would be trying to replicate the option shown in Figure 11.4. The trader is sitting with S T = X − , i.e. just below the strike price. Gamma will be highish positive and delta moderate. S T starts to rise slightly and gamma shoots up; but more alarmingly, delta goes to astronomical levels. For a simple option,  never rises above 100%; but in this case  can become 1000%. In fact, at the moment of expiry,  →∞as S T → X . It takes quite a brave trader to buy 10 times the underlying stock as a hedge; if the price just zig-zags about S T = X , the trader could lose the entire option premium in transaction costs. cash f X T S + X - X Figure 11.4 Hedging a bet If S T moves to S T = X + , things become calmer again. Delta returns to a low level, but the sign of gamma has reversed. At the strike and at the moment of expiry, an infinitesimal move in S T across X would cause gamma to change from +∞ to −∞; again, not comfortable. A trader’s reaction might be just to sit tight and do nothing as S T moves from X − to X + . But then where would he get the payoff from, when the bet is exercised against him? This intense trading activity over a tiny range is what gener- ates the income to make the payoff: hence the occasional need to trade very large quantities of 150 11.5 POWER OPTIONS stock in the vicinity of the strike price. In practice, market practitioners tend to avoid bet options in anything but small amounts. 11.5 POWER OPTIONS (i) These may take an infinite variety of forms, but the two most common ones encountered in the marketplace have payoffs given by {max[0, S T − X ]} 2 and max  0, S 2 T − X 2  In fact, we can write {max[0, S T − X ]} 2 =  0 S T < X (S T − X ) 2 = S 2 T − X 2 − 2X (S T − X ) X < S T = max  0, S 2 T − X 2  − 2X max[0, S T − X ] so the two options are simply connected by a call option. These options are mostly the domain of leverage junkies and are mathematically rather untidy. Unlike most options we deal with, they are not homogeneous in S T and X. A reflection of this is that the size of the payment depends on the unit of currency used. As an exercise, the reader should try to imagine how the payoff would have been handled in those countries that adopted the Euro during the life of a power option. (ii) We now look for a formula to price an option whose payoff is max[0, S λ T − X ]. Recall from equation (3.7) that S T = S 0 e (r−q)T − 1 2 σ 2 T +σ W T where W T is a Browian motion. Then S λ T = S λ 0 e λ{(r−q)T − 1 2 σ 2 T +σ W T } = S λ 0 e (r−Q)T − 1 2 ν 2 T +νW T where Q = λq − (λ − 1)  r + 1 2 λσ 2  ; ν = λσ Bundle f h …… 1 X h 2 0 n T S Figure 11.5 Equally spaced calls The form of S λ T is the same as the form of S T so we can simply use our Black Scholes model, substituting Q for q and v for volatility, while using S λ 0 where we would have used S 0 . (iii) Bundles of Call Options: Suppose we buy a package of call options as shown in Figure 11.5. The package has the following properties: • Each option on 2 × h shares. • The strike price of the first option is X, and the strike prices of each successive option is h higher than the last. 151 11 Simple Exotics (The first point implies that h is a pure number while the second implies that it is a dollar amount; this is just another reflection of the dimensional awkwardness of these options.) Suppose that at the maturity of the bundle, the stock price is S T = X + nh which is above X.Ifh is small, the payoff of this package is (2h)(h + 2h + 3h +···+nh) = (2h) n(n + 1) 2 h = (nh)(nh + h) ≈ (S T − X ) 2 This is the payoff of the first square power option mentioned above. The power option can be simulated by a bundle of options, with the approximation becoming exact as the spacing between the strikes of the options shrinks to zero. Similarly, the second power option is approximated by the same bundle plus a single call option with strike X,on2X units of stock. (iv) Soft Strike Options: There are high risks associated with hedging an option which is close to maturity, when the stock price is close to the strike price. At this point, gamma blows up. In fact, it is concern over the gamma near maturity that is often the decisive factor when deciding how large an option position can be hedged. Soft Strike f X T S wX+ w X- Figure 11.6 Soft strike call The gamma is the second differential of the derivative price with respect to the stock price; it is therefore constant for a square power option near its maturity. Suppose that a large call option (dotted line in Figure 11.6) is requested by an investor, but the bank is uncomfortable with the potential gamma exposure close to maturity. The bank can instead propose the payoff shown in the graph. For most values of S T this has the same payoff as a straight- forward call option; but for a distance ω on either side of the strike price, the call payoff function is replaced by f soft strike = 1 4ω (S T − X + ω) 2 This option is said to have a soft strike over the range X − ω to X + ω, where it has a constant terminal gamma of 1 2 ω. Using the analysis of the last subparagraph, this soft strike option is equivalent to a bundle of call options, with strike prices infinitesimally spaced between X − ω and X + ω. 152 . 11 Simple Exotics The purpose of this part of the book is to introduce the reader. take it to mean anything that is not a simple European or American put or call option. We start with a chapter on simple extensions of the Black Scholes