14 Options on One Asset at Two Points in Time In the last two chapters we have looked at various options involving two or more stochastic assets. The resulting pricing formulas involved the correlation between the asset prices and it was observed that in financial markets, correlation is usually highly unstable. We can de- rive elegant formulas based on the assumption of constant correlation, but in the real world most practitioners handle these products with extreme caution. However, there is one notable exception. Suppose we have an option whose payoff depends on two prices: but instead of these being the prices of two different assets, they are the prices of the same asset at two different times τ and T in the future. It is shown in Appendix A.1(vi) that the correlation between S τ and S T (or to be more precise, between the logarithms of the price movements over the periods τ and T)isρ = √ τ/T . This is just about the only case where we can have confidence in the value for ρ. The most common options in this category are described in this chapter, although other examples will be encountered in later chapters. 14.1 OPTIONS ON OPTIONS (COMPOUND OPTIONS) t T t 0 now maturity of underlying option time t maturity of compound option (i) Definitions: We will consider an option (the com- pound option) on an underlying option. Both the compound and the underlying options can be either put or call options, so that we have four options to consider in all. Half the battle in pricing these options is simply getting the notation straight, and this can be summarized as follows: UNDERLYING STOCK: S t and σ Stock price at time t and volatility UNDERLYING OPTIONS: C u (S t , X, t); P u (S t , X, t) Value at time t of an underlying call/put option. The general case is written U (S t , X, t) T ; X Maturity date and strike of underlying options COMPOUND OPTIONS: C C ; P C ; C P ; P P Value at time t of a call on a call, put on a call, etc. The general case is written  U (S t , K , t) τ ; K Maturity date and strike of the compound options S ∗ τ Critical stock price at time τ , which determines whether or not the compound option is exercised. It is the value of S τ that solves the equation K =  u (S ∗ τ , X,τ) 14 Options on One Asset at Two Points in Time (ii) Payoffs of Compound Options: Before moving on to pricing formulas, it is worth getting an idea of the likely shape of the curve of the compound price. Let us start with a call on a call. At time τ , the price of the underlying call is given by the curve shown in Figure 14.1. The payoff of the compound call option is defined as C C (τ ) = max[C u (τ ) − K , 0] Define S ∗ τ as the value of S τ for which K = C U (S ∗ τ , X,τ). Clearly, the payoff diagram is made up of the x-axis and that part of the curve C U (τ ) which lies above K. This is shown as the solid line in Figure 14.2, together with the compound option price before maturity (dotted curve). t S * t S Xe -r (T-t) K C U (t) Figure 14.1 Underlying option (call) t S * t S C U (t) U C(t) Figure 14.2 Compound option (call on call) Using the same analysis as for the call on call above, a put option on the underlying stock is represented by the curve shown in Figure 14.3. The payoff of the compound put option, shown in Figure 14.4, is P P (τ ) = max[K − P u (τ ), 0] P U (t) t S * t S Xe -r (T-t) K Figure 14.3 Underlying option (put) P U (t) t S * t S C P (t) Figure 14.4 Compound option (put on put) The remaining two compound options have curves shown in Figures 14.5 and 14.6. (iii) Consider the most common case: a call on a call. In order to calculate the value of this compound option, we use our well-established methodology of finding the expected value of the payoff in a risk-neutral world, and discounting to present value at the risk-free rate; but now, the ultimate payoff is a function of two future stock prices (Geske, 1979): 170 14.1 OPTIONS ON OPTIONS (COMPOUND OPTIONS) t S C P (t) Figure 14.5 Compound option (call on put) t S P C (t) Figure 14.6 Compound option (put on call) r S τ – The stock price when the compound option matures. If this is less than some critical value S ∗ τ , it will not be worth exercising the compound option since the underlying option would then be cheaper to buy in the market. This may be written only exercise if S ∗ τ < S τ , where S ∗ τ is the solution to the equation K = S ∗ τ e −q(T −τ ) N[d ∗ 1 ] − X e −r(T −τ ) N[d ∗ 2 ] d ∗ 1 and d ∗ 2 are the usual Black Scholes parameters with the stock price set equal to S ∗ τ . r S T – The stock price when the underlying option matures. The ultimate payoff at time T is the payoff of the underlying call, if (and only if) the condition S ∗ τ < S τ was fulfilled. This ultimate payoff is of course a function of S T . Since the value of a compound option depends on the expected values of both S τ and S T ,we must examine their joint probability distribution. (iv) Following the approach of Section 5.2(i) for the Black Scholes formula, the price of this option may be expressed as C C (0) = PV  E  payoff of underlying option −payment for underlying option    only if compound option was exercised  risk neutral  r “Only if compound option exercised” ≡ S ∗ τ < S τ . r “Payment for underlying” = K at time τ . r “Payoff of underlying option” (at time T) = max[0, S T − X]. Combining this together and simplifying the notation gives C C (0) = e −rT E[max[S T − X, 0]: S ∗ τ < S τ ] − e −rτ E[K : S ∗ τ < S τ ] = e −rT E[S T − X : S ∗ τ < S τ ; X < S T ] − e −rτ K P[S ∗ τ < S τ ] (14.1) These expectations are evaluated explicitly in Appendix A.1(v) and (ix), to give C C (0) = S 0 e −qT N 2 [d 1 , b 1 ; ρ] − X e −rT N 2 [d 2 , b 2 ; ρ] − K e −rτ N[b 2 ] (14.2a) 171 14 Options on One Asset at Two Points in Time where d 1 = 1 σ √ T  ln S 0 e −qT X e −rT + 1 2 σ 2 T  ; b 1 = 1 σ √ τ  ln S 0 e −qτ S ∗ τ e −rτ + 1 2 σ 2 τ  d 2 = d 1 − σ √ T ; b 2 = b 1 − σ √ τ ; ρ =  τ T and S ∗ τ is a solution to the equation K = C U (S ∗ τ , X,τ). The lower limits of integration in the above-mentioned appendices are the values of z τ and z T corresponding to S τ = S ∗ τ and S T = X . (v) General Formula: The put–call parity relationship C C (0) + K e −rτ = P C (0) + C U (0) may be used to calculate the formula for a put on an underlying call. The relationships N[d] + N[−d] = 1 and N 2 [d, b; ρ] + N 2 [d,−b;−ρ] = N[d] [see equation (A1.17)] are used to simplify the algebra, giving P C (0) = X e −rT N 2 [d 2 ,−b 2 ;−ρ] − S 0 e −qT N 2 [d 1 ,−b 1 ;−ρ] + K e −rτ N[−b 2 ] Similar results are obtained for put and call options on an underlying put option. The four possibilities for compound options can be summarized in the general formula  U (0) = φ U φ  {S 0 e −qT N 2 [φ U d 1 ,φ U φ  b 1 ; φ  ρ] − X e −rT N 2 [φ U d 2 ,φ U φ  b 2 ; φ  ρ]} − φ  K e −rτ N[φ U φ  b 2 ] (14.2b) where d 1 = 1 σ √ T  ln S 0 e −qT X e −rT + 1 2 σ 2 T  ; b 1 = 1 σ √ τ  ln S 0 e −qτ S ∗ τ e −rτ + 1 2 σ 2 τ  d 2 = d 1 − σ √ T ; b 2 = b 1 − σ √ τ ; ρ =  τ T ; S ∗ τ solves K =  U (S ∗ τ , X,τ) φ U =  +1 underlying call −1 underlying put φ  =  +1 compound call −1 compound put (vi) Installment Options: When they are first encountered, compound options often look to stu- dents like rather contrived exercises in option theory. However they do have very practical applications, as the following product description indicates: r An investor receives a European call option which expires at time T and has strike X. r Instead of paying the entire premium now, the investor pays a first installment of C C today. r At time τ , the investor has the choice of walking away from the deal or paying a second installment K and continuing to hold the option. This structure clearly has appeal in certain circumstances; it is just the call on a call described in this section, but couched in slightly less dry terms. 172 14.2 COMPLEX CHOOSERS 14.2 COMPLEX CHOOSERS Recall the simple chooser which was described in Section 11.2: we buy the option with strike X at time t = 0 and at t = τ we decide whether the option which matures at t = T is a put or a call. The complex chooser is similar in principle, but the put and call can have different maturities and strikes (Rubinstein, 1991b). The payoff at time τ is therefore written Payoff τ = max[C(S τ , X C , T C − τ ), P(S τ , X P , T P − τ )] For some critical price S ∗ τ at time τ , the value of the put and the call are exactly the same. S ∗ τ is obtained from the equation C(S ∗ τ , X C , T C − τ ) = P(S ∗ τ , X P , T P − τ ) (14.3) using some numerical procedure or the goal seek function of a spread sheet. For S τ < S ∗ τ , the payoff is the put option, while for S ∗ τ < S τ the call option price is larger. Using the reasoning of Section 14.1(iv) gives f complex chooser = e −rT C E  S T C − X C   S ∗ τ < S τ ; X c < S T C  + e −rT P E  X P − S T P   S τ < S ∗ τ ; S T P < X P  The first term here is just the first term of equation (14.1) for a call on a call; similarly, the second term is the first term of the formula for a call on a put. Instead of slogging through a bunch of double integrals again, we just steal the answer from equation (14.2b): f complex chooser =  S 0 e −qT C N 2  d (C) 1 , b 1 ; ρ (C)  − X C e −rT C N 2  d (C) 2 , b 2 ; ρ (C)  −  S 0 e −qT P N 2  −d (P) 1 ,−b 1 ; ρ (P)  − X P e −rT P N 2  −d (P) 2 ,−b 2 ; ρ (P)  (14.4) d (i) 1 = 1 σ √ T i  ln S 0 e −qT i X i e −rT i + 1 2 σ 2 T i  , i = C or P; b 1 = 1 σ √ τ  ln S 0 e −qτ S ∗ τ e −rτ + 1 2 σ 2 τ  S ∗ τ solves equation (14.3): d (i) 2 = d (i) 1 − σ  T i ; b 2 = b 1 − σ √ τ ; ρ (i) =  τ T i 14.3 EXTENDIBLE OPTIONS (i) Consider a European call option with maturity at time τ and strike price K; at maturity, the holder has the choice of exercising or not exercising (Longstaff, 1990). Now suppose that an additional feature is added to this option: the holder is given a third choice at maturity of extending the option to time T at a new strike price X, in exchange for a fee of k. The payoff of this extendible option at time τ is max[0, S τ − K, C(S τ , X, T − τ) − k] The issues are best illustrated graphically. Figure 14.7 shows the value of the extended call option C(S τ , X, T − τ) at time τ . This is just a simple graph of a European call option at time 173 14 Options on One Asset at Two Points in Time T − τ before maturity. The points to note on the graph are: r X e −r(T −τ ) which is the well-known value at which the upper asymptote to the curve for a call option crosses the x-axis. r S ∗ τ is simply defined by C(S ∗ τ , X, T − τ) = k. r k + X e −r (T −τ ) which is obtained simply by construction as shown. (ii) Figure 14.8 is a detail of the previous graph (with a shift of the x-axis upwards of k), to- gether with the dotted line S τ − K which is the payoff of the original (unextended) option at time τ . Payoff E = max[0, S τ − K, C E − k] is represented by the northwest boundary of this composite graph: r S τ < S ∗ τ – The payoff is zero because the other two terms in Payoff E are less than zero. r S ∗ τ < S τ < S ∗∗ τ – The payoff is C E − k, i.e. the holder would logically choose to extend the option. S ∗∗ τ is defined as the point at which the curve and the diagonal straight line in Figure 14.8 intersect: C(S ∗∗ τ , X, T − τ) − k = S ∗∗ τ − K r S ∗∗ τ < S τ – The payoff is S τ − K , i.e. a holder would logically take the payoff of the original (unextended) option. t S C(S , X, T - t) t k+ t * S -r(T-t ) Xe k -r(T-t ) Xe Figure 14.7 Call option at t = τ; maturity at t = T (iii) The position of the dotted line in Figure 14.8 depends on the value of the original strike K relative to the extension fee k and the extended strike X. In addition to the relative positioning shown in the graph and described in the last subsection, two other configurations are possible. These may be determined at time t = 0 when the extendible option is being priced: r K < S ∗ τ – In this case, an extension would never be economically optimal; therefore, the price of this option is just the same as that of a European call option maturing at time τ . r k + X e −r (T −τ ) < K – In this case, extending the option would always be preferred to taking the payoff of the original option; therefore, this is just a compound option (call on a call). Any model we build for an extendible option must therefore test whether K is between S ∗ τ and S ∗∗ τ and if not, just substitute the value of a European call or a compound option. In the remainder of this section we ignore these special cases. 174 14.3 EXTENDIBLE OPTIONS t S , t ** S t * S -r(T-t) k+Xe K C(S t ,X T- )-kt t S -K Figure 14.8 Detail previous (exercise boundary) (iv) Using the same approach as for previous options examined in this chapter, today’s value of an extendible call is obtained by taking the risk-neutral expectation of the payoff: f ext (0) = PV[E[0: S τ < S ∗ τ ] + E[Payoff C E − k : S ∗ τ < S τ < S ∗∗ τ ] + E[S τ − K : S ∗∗ τ < S τ ]] = e −rT E[S T − X : S ∗ τ < S τ < S ∗∗ τ ; X < S T ] − e −rτ k P[S ∗ τ < S τ < S ∗∗ τ ] + e −rτ E[S τ − K : S ∗∗ τ < S τ ] (14.5) The terms in this last equation need some dissection; starting with the second and third terms which depend only on S τ and using equation (A1.7): P[S ∗ τ < S τ < S ∗∗ τ ] = P[S τ < S ∗∗ τ ] − P[S τ < S ∗ τ ] = N[−b ∗∗ 2 ] − N[−b ∗ 2 ] where b ∗∗ 2 = 1 σ √ τ  ln S 0 e −qτ S ∗∗ τ e −rτ + 1 2 σ 2 τ  and similarly for b ∗ 2 . E[S τ − K : S ∗∗ τ < S τ ] ≡ E[:][S τ − S ∗∗ τ : S ∗∗ τ < S τ ] + (S ∗∗ τ − K )P[S ∗∗ τ < S τ ] = e rτ C(S 0 , S ∗∗ τ ,τ) + (S ∗∗ τ − K )N[−Z ∗∗ ] where C(S 0 , S ∗∗ τ ,τ) is the value of a call option with strike S ∗∗ τ and maturity τ. The bivariate conditional expectation in equation (14.5), depending on both S τ and S T , can be decomposed as follows: E[S T − X : S ∗ τ < S τ < S ∗∗ τ ; X < S T ] = E[S T − X : S ∗ τ < S τ ; X < S T ] − E[S T − X : S ∗∗ τ < S τ ; X < S T ] = A ∗ − A ∗∗ where A ∗ appears as the first term in equation (14.1) for the value of a call on a call. Just copying the answer from equation (A1.21) of the Appendix gives A ∗ = S 0 e (r−q)T N 2 [b ∗ 1 , d 1 ; ρ] − X N 2 [b ∗ 2 , d 2 ; ρ] 175 14 Options on One Asset at Two Points in Time where d 1 = 1 σ √ T  ln S 0 e −qT X e −rT + 1 2 σ 2 T  ; d 2 = d 1 − σ √ T and b ∗ 1 and b ∗ 2 are defined as before. There is a similar expression for A ∗∗ . (v) Equation (14.6) has become a bit of a monster, but each term has now been given explicitly in terms of cumulative distributions. It will be apparent to the reader who has studied Section A.1 of the Appendix that there are different ways of expressing the answers, so a term-by-term comparison with the results quoted in other publications may be difficult: for example, there is a term identical to the value of a call option with strike S ∗∗ and maturity T; some other authors show instead a call option with strike K, and with the other terms modified slightly (Longstaff, 1990). 176 . the underlying options can be either put or call options, so that we have four options to consider in all. Half the battle in pricing these options is simply. most common options in this category are described in this chapter, although other examples will be encountered in later chapters. 14.1 OPTIONS ON OPTIONS