Barriers - Advanced Options

16 Barriers: Advanced Options The last chapter laid out the principles of European barrier option pricing. This chapter con- tinues the same analysis, applied to more complicated problems. The integrals get a bit larger, but the underlying concepts remain the same. Lack of space prevents each solution being given explicitly; but the reader should by now be able to specify the integrals corresponding to each problem, and then solve them using the results of Appendix A.1. 16.1 TWO BARRIER OPTIONS These are options which knock out or in when either a barrier above or a barrier below the starting stock price is crossed. The analysis is completely parallel to what we have seen for a single barrier option (Ikeda and Kunitomo, 1992). (i) In the notation of this chapter, F 0 (x T , T ) is the normal distribution function for a particle starting at x 0 = 0. The function is given explicitly in Section 15.1(i). F non-abs is the probabil- ity distribution function for particles starting at x 0 = 0 which have not crossed either barrier before time T. Two expressions have been derived for this function, which are given by equa- tions (A8.9) and (A8.10) of the Appendix. They are both infinite series although there is no correspondence between individual terms of the two series: 1. F non-abs (x T , T ) = 1 σ √ 2π T +∞  n=−∞  exp  + mu n σ 2  exp  − 1 2σ 2 T (x T − mT − u n ) 2  − exp  + mv n σ 2  exp  − 1 2σ 2 T (x T − mT − v n ) 2  2. F non-abs (x T , T ) = exp  mx T σ 2  ∞  n=1  a n e −b n T sin nπ L (x T + b)  L = a + b; u n = 2Ln; v n = 2(Ln − b) a n = 2 L sin nπ b L ; b n = 1 2  µ σ  2 +  nπσ L  2  (ii) The reasoning of Appendix A.8(iv) and (v) demonstrates that the distribution functions of particles which start at x 0 = 0, then cross either the barrier at −bor+a, and then return to the region −b to +a can be written F return = F 0 − F non-abs 16 Barriers: Advanced Options -b X a return F 0 F 0 F 0 S Figure 16.1 Double barrier-and-in call The total probability distribution function for all those particles that cross one of the barriers can now be written F crossers =      F 0 (x T , T ) x T < −b F return (x T , T ) −b < x T < a F 0 (x T , T ) a < x T (iii) As an example, we will look at the knock-in call op- tion shown in Figure 16.1. We do not really need to worry about whether we move up or down to the barrier: C ki = e −rT  +∞ 0 (S T − X ) + F crossers dS T = e −rT  +∞ X (S T − X )F crossers dS T = e −rT  a X (S T − X )F return dS T + e −rT  ∞ a (S T − X )F 0 dS T (16.1) The second integral is completely standard. The first depends on which form of series is used for F non-abs . The sine series is completely straightforward to integrate while the other alternative is handled using the procedures of Section 15.1. (iv) The question of which of the two series to use and how many terms to retain is best handled pragmatically. Set up both series and see how fast convergence takes place in each case. Both series dampen off regularly, so it is for us to choose how accurate the answer needs to be. We should expect to perform the calculation with one series or the other within four to six terms, and often less. 16.2 OUTSIDE BARRIER OPTIONS The barrier options described so far have been European options which are knocked in or knocked out when the price of the underlying variable crosses a barrier. An extension of this is a European option which knocks in when the price of a commodity other than the underlying stock crosses a barrier. For example, an up-and-in call on a stock which knocks in when a foreign exchange rate crosses a barrier. These options are called outside barrier options, as distinct from inside barrier options, where the barrier commodity and the commodity underlying the European option are the same. The reason for the terminology is anybody’s guess (Heynen and Kat, 1994a). We could repeat most of the material presented so far in this chapter, adapted for outside barriers rather than inside barriers. However, these options are relatively rare so we will simply describe a single-outside-barrier up-and-in call option; the reader should be able to generalize this quite easily to any of the other options in this category. (i) Outside Barrier, Up and In: The general principle remains as before; it is merely the form of some of the distributions that is different. The price of the option is the present value of the risk-neutral expectation of the payoff (Figure 16.2): C outside u−i = e −rT  +∞ S T =0  +∞ Q T =0 (S T − X ) + F jo int dS T dQ T (16.2) 190 16.2 OUTSIDE BARRIER OPTIONS 0 S X Q K return F 0 F 0 Figure 16.2 Outside barrier, up-and-in call where S T is the maturity value of the stock un- derlying the call option and Q T is the maturity value of the barrier commodity. The form of this is the same as for inside barriers; but we need to find an expression for F jo int which is the joint probability distribution for the two price vari- ables. The large topic of derivatives which de- pend on the prices of two underlying assets is at- tacked in Chapter 12. The material of that chapter and of Appendix A.1 is used to solve equation (16.2). (ii) Separate Distributions of the Two Variables: As before, we transform to the logs of prices: x T = ln(S T S 0 ); y T = ln(Q T /Q 0 ). The distribution of x T is normal and the variable z T = [ln(S T /S 0 )− mT]/ σ √ T is a standard normal variate (mean 0, variance 1); σ is the volatility of the stock and m − r − q − 1 2 σ 2 . The variate y T has a more complex distribution. As explained in Section 13.1(i), y T is distributed as F crossers (y T , T ) which has different forms above and below the barrier at Q T = K or y T = ln(K /Q 0 ) = b. b < y T : F crossers = F 0 (y T , T ) which is the distribution at time T of a particle which started at y 0 = 0 and has drift m Q = r − q Q − 1 2 σ 2 Q and variance σ 2 Q . The variable w T = ln(Q T /Q 0 ) − m Q T σ Q √ T is a standard normal variate. y T < b: F crossers = F return = AF 0 (y T − 2b, T ) where A = exp(2m Q b/σ 2 Q ) = (K /Q 0 ) 2m Q /σ 2 Q and F 0 (y T − 2b, T ) is the distribution function for a particle which started at y 0 = 2b and has drift m Q . The variable w  T = ln(Q T /Q 0 ) − m Q T − 2b σ Q √ T is therefore a standard normal variate. (iii) Equation (16.2) may be rewritten C outside u−i = e −rT   +∞ S T =X  K Q T =0 A(S T − X )F 1joint dQ T dS T +  +∞ S T =X  +∞ Q T =K (S T − X )F 2joint dQ T dS T  and transforming to the variables Z T , w T and w  T , this last equation can be written more 191 16 Barriers: Advanced Options precisely as C outside u−i = e −rT  A  +∞ Z X  W  K −∞ (S 0 e mT+σ √ Tz T − X )n 2 (z T ,w  T ; ρ)dz T dw  T +  +∞ Z X  +∞ W K (S 0 e mT+σ √ Tz T − X )n 2 (z T ,w T ; ρ)dz T dw T  Z X = ln(X/S 0 ) − mT σ √ T ; W  K = ln(K /Q 0 ) − m Q T − 2b σ Q √ T ; W K = ln(K /Q 0 ) − m Q T σ Q √ T n 2 (z T ,w  T ; ρ) is the standard bivariate normal distribution describing the joint distribution of the two standard normal variates z t and w  t , which have correlation ρ. n 2 (z T ,w T ; ρ)isthe standard bivariate normal distribution describing the joint distribution of the two standard normal variates z t and w t , which have correlation ρ. Note that the correlations between z t and w  t are the same as between z t and w t ; w  T and w T essentially refer to the same random variable Q T , and differ only in their means, which does not affect the correlations. Using the results of equations (A1.20) and (A1.21), this last integral is evaluated as follows: C outside u−i = A  S 0 e −qT N[(σ √ T − Z X )] − X e −rT N[−Z X ] −  S 0 e −qT N 2 [−(σ √ T − Z X ),−(ρσ √ T − W  X ); ρ]−X e −rT N 2 [−Z X ,−W  K ; ρ]  +  S 0 e −qT N 2 [−(σ √ T − Z X ),−(ρσ √ T − W  X ); ρ] − X e −rT N 2 [−Z X ,−W K ; ρ]  (16.3) 16.3 PARTIAL BARRIER OPTIONS In the foregoing it was always assumed that a barrier is permanent. However, the barrier could be switched on and off throughout the life of the option. Such a pricing problem is usually handled numerically, but the simplest case can be solved analytically using the techniques of the last section (Heynen and Kat, 1994b). This is an option on a single underlying stock at two different times, as described in Chapter 14. The specific case we consider is an up-and-in call of maturity T, which knocks in if the barrier is crossed before time τ , i.e. the barrier is switched off at time τ . Its value can be written analytically as C partial u−i = e −rT   +∞ S T =0  +∞ S τ =0 (S T − X ) + F jo int dS τ dS T F jo int is the joint probability distribution of two random variables S τ and S T , where S τ is subject to an absorbing barrier. This problem is almost precisely the same as the outside barrier option problem solved in the last section. The formula given in equation (16.3) can therefore be applied directly, with the following modifications: r Q 0 → S 0 , σ Q → σ and m Q → m. r T → τ in the formulas for w  K and w K . r The correlation between S τ and S T is shown in Appendix A.1(vi) to be ρ = √ τ/T . 192 16.4 LOOKBACK OPTIONS 16.4 LOOKBACK OPTIONS These are probably the most discussed and least used of the standard exotic options. The problem is that on the one hand they have immense intuitive appeal and pricing presents some interesting intellectual challenges; but on the other hand they are so expensive that no-one wants to buy them. However, this book would not be complete without an explanation of how to price them (Goldman et al., 1979). 0 S max S min S T X T S Figure 16.3 Notation for lookbacks (i) Floating Strike Lookbacks: Lookback options are quoted in two ways. The most common way is with a floating strike, where the payoffs are defined as follows: Payoff of C fl str = (S T − S min ) Payoff of P fl str = (S max − S T ) The lookback call gives the holder the right to buy stock at maturity at the lowest price achieved by the stock over the life of the option. Similarly, the lookback put allows the holder to sell stock at the highest price achieved. The form of the payoff is unusual in that it does not involve an expression of the form max[0, .], since (S T − S min ) can never be negative; it has therefore been suggested that this is not really an option at all, although this is largely a matter of semantics. However, it does make the pricing formula straightforward to write out: risk neutrality gives C fl str = e −rT {ES T −ES min } = e −rT {F 0T − v min } (16.4) P fl str = e −rT {v max − F 0T } where F 0T is the forward price. (ii) Fixed Strike Lookbacks: As the name implies, these options have a fixed strike X. Referring to Figure 16.3, the payoffs of the fixed strike call and put are given by Payoff of C fix str = max[0, S max − X ] Payoff of P fix str = max[0, X − S min ] These are sometimes referred to as lookforward options. They give the option holder the right to exercise not at the final stock price, but at the most advantageous price over the life of the option. The payoffs look more like normal option payoffs, containg the familiar “max” function. However, in practice, the payoff can be further simplified, since the options are usually 193 16 Barriers: Advanced Options quoted at-the-money, i.e. with X = S 0 . This implies that X ≤ S max or S min ≤ X, so that C fix str = e −rT {ES max −X} = e −rT {v max − X} (16.5) P fix str = e −rT {X − v min } (iii) Distributions of Maximum and Minimum: The prices of both floating and fixed strike lookback options depend on the quantities v min and v max , which are defined in the last two subsections. It is shown in Appendix A.8(viii) that the distribution functions for x max = ln(S max /S 0 ) and x min = ln(S min /S 0 ) are F max (x max , T ) = 2 σ √ 2π T exp  − 1 2σ 2 T (x max − mT) 2  − 2m σ 2 exp  + 2mx max σ 2  N  − 1 σ √ T (x max + mT)  (16.6) F min (x min , T ) = 2 σ √ 2π T exp  − 1 2σ 2 T (x min − mT) 2  + 2m σ 2 exp  + 2mx max σ 2  N  + 1 σ √ T (x min + mT)  t 0 S max S H ; previous max path A path B max path A path B t=0 S Figure 16.4 Previous maximum (iv) When we derive the formula for the price of an option, we do not usually have to concern ourselves with what happened in the past: if a call option was issued for an original maturity of 3 months, its price after 2 months is exactly the same as the price of a newly issued 1-month option. However, the pricing of a lookback is a little more difficult: after 2 months, the maximum or minimum value of S t for the whole period may already have been achieved. Let us assume that a previous maximum H has been established and we wish to find the value of v max at time t = 0. Consider the two paths shown in Figure 16.4: path A establishes a new maximum at S max while path B does not make it so that the established maximum remains at H. This generalized definition, accommodating a previous maximum, is expressed in the general definition v max = Emax[ H, S max ]=H PS max < H+ES max   H < S max  194 16.5 BARRIER OPTIONS AND TREES or v max = H  H 0 F max dS max +  ∞ H S max F max dS max There is an analogous expression for v min in terms of a previously established minimum L. (v) Expressions for v max and v min can be obtained by using equations (16.6) for F max and F min , making the substitution S max = S 0 e x max . The resulting integrals are performed using the results of Appendix A.1(v), item (E); the algebra is straightforward but very tedious. The expressions can be combined to give the generalized formula v max / min = H  N[ψ Z K ] − σ 2 2(r − q) exp  2mb σ 2  N[ψ Z  K ]  +F 0T  1 + σ 2 2(r − q) N[−ψ(Z K − σ √ T )]  (16.7) b = ln K /S 0 ; Z K = (ln K /S 0 − mT) σ √ T ; Z  K = Z K − 2b for max: K = H,ψ=+1; for min: K = L,ψ=−1 (vi) Strike Bonus: Using equation (16.7) to obtain an expression for v min , substituting this into equation (16.4) and rationalizing gives (Garman, 1989) C fl str = e −rT {F 0T N[−(Z L − σ 2 T )] − L N[−Z L ]} + σ 2 S 0 2(r − q)  e −rT exp  2(r − q)b σ 2  N[−Z  L ] + e −qT N[Z L − σ 2 T ]  (16.8) The first term is simply the Black Scholes formula for a call option with strike X = L, the previously achieved minimum. At t = 0 we consider two possibilities: r No new minimums are formed below L before the maturity of the option. The payoff of C fl str is then equal to the payoff of C(S 0 , L , T ), i.e. max[0, S T − L]. r A new minimum is established at S min which is below L. The payoff of the option is then max[0, S T − S min ]. Comparing these two possible outcomes, it is clear that the second term in equation (16.8) prices an option to reset the strike price of a call option from L down to the lowest value achieved by S t before maturity. This option is called the strike bonus. 16.5 BARRIER OPTIONS AND TREES (i) Binomial Model (Jarrow–Rudd): Binomial and trinomial trees are a standard way of solving barrier option pricing problems which are not soluble analytically, such as American barrier options. However, these methods do display some special features which will be illustrated with the example of a European knock-out call C u−o (X < K ). The example uses S 0 = 100, X = 110, K = 150, r = 10%, q = 4%, σ = 20%, T = 1 year. This is similar to the example 195 16 Barriers: Advanced Options examined in some detail in Section 7.3(v) and again in Section 8.6, except for the existence of a knock-out barrier. In the previous investigations we examined the variation of the values obtained from the binomial model, as a function of the number of time steps. In order to accommodate the knock-out feature in a binomial tree, we simply set the option value equal to zero at each node for which the stock price is outside the barrier. Consider the above knock-out option, priced on a three-step binomial tree. Using the Jarrow–Rudd discretization, there is no node with a stock price higher than 150; therefore there is no node at which we would set the stock price equal to zero. We are therefore unable to price this option – or alternatively put, this model gives the same value for a barrier at K = 150 and for K =∞. K=150 2.21 112.24 100.00 100.00 125.98 141.40 89.09 79.38 112.24 89.09 70.72 31.40 2.21 0 0 17.92 1.21 0 10.19 2.41 5.79 100.00 112.24 141.40 89.09 100.00 79.38 112.24 89.09 70.72 0 0.96 1.21 0 1.03 .65 .84 K=130 K=140 125.98 0 0 Figure 16.5 Knock-out barriers in binomial trees If we now look at the same model with the barrier at K = 140 as in the second diagram of Figure 16.5, we see how the tree is modified, giving a very different value for the option. But the tree would give exactly the same answer for K = 130; the barrier would have to be below K = 125.98 for the tree to be modified any further. In general, the value of a knock-out option is a step function of the barrier level, with a jump each time the barrier crosses a line of nodes. (ii) Price vs. Number of Steps: Figure 16.6 shows the value of our knock-out call option plotted against the number of steps in the binomial tree; the analytical value of this option is 3.77. It is instructive to compare this graph with Figure 7.11 for a similar option but without a knock-out barrier. The European option shows the characteristic oscillations which are gradually damped away; by the time we reach about 300/400 steps (not shown), the answer obtained is stable enough for commercial purposes. The knock-out option on the other hand shows three different features of interest (Boyle and Lau, 1994): 196 16.5 BARRIER OPTIONS AND TREES 0 200 400 600 800 1000 1200 3.50 3.70 3.90 4.10 4.30 4.50 4.70 4.90 1400 1600 Number of Steps Option Price (Stock = 100) 3.77 ± 1% Figure 16.6 Up-and-out knock-out call price vs. number of binomial steps (Jarrow–Rudd discretization) r The graph continues its relentless sawtooth pattern long after the 1500 steps which we have shown. Convergence to the analytical answer is difficult to achieve. r However, within about 150 steps, we find the bottom of the zig-zags within 1% of the analytical answer; certainly, within 300/400 steps, the envelope of low points gives an almost perfect answer. r The answers converge to the theoretical answer from above, i.e. apart from a few outliers, the tree always gives answers greater than the analytical value. We start by turning our attention to the first two features. Row N x x x x x x o o o o o o o o o o o o x x x x x x o o o o o o x x x x x x K K Row N 1 extra binomial step Figure 16.7 Effect of increasing number of binomial steps In subsection (i) we saw that a knock-out op- tion value calculated with a binomial tree is a step function of the barrier level. This same ef- fect causes oscillations in the calculated value of a knock-out option plotted against the number of steps. If we use the Jarrow–Rudd discretiza- tion with N time steps, each proportional up-jump is given by u = e σ √ δ t = e σ √ T / N . Therefore as N increases, the spacing between adjacent rows of nodes decreases; the rows of nodes become pro- gressively compressed together and at a certain point an entire row of nodes crosses the barrier. At this point there is a jump in the value of the option calculated by the tree. (iii) As the number of steps N is increased, we would expect the most accurate binomial calculation to occur when a row is just above the knock-out barrier; the option value at these nodes is put at 197 16 Barriers: Advanced Options zero. Increasing the number of steps by one would then push the row down through the barrier and change this row of zeros into positive numbers (see Figure 16.7). Let N c be a critical number of steps such that row n of nodes in the binomial tree lies just above K, i.e. S 0 e nσ √ T / N c is greater than K;butS 0 e nσ √ T /( N c +1) is less than K.WefindN c from S 0 e nσ √ T / N c < K < S 0 e nσ √ T /( N c +1) or N c = round down  n 2 σ 2 T (ln K /S 0 ) 2  where “round down” means round down to the nearest integer. Note: It is important to be accurate at this point since N c will give a best answer while N c + 1 gives a worst answer. Figure 16.8 is just a blown up detail taken from Figure 16.6. Use of the formula just given gives the following results: nN c 28 190 29 204 30 218 which correspond precisely to the jumps in the diagram. The rippling effect of the option values between jumps is the residual effect of the oscillations always observed in binomial calculations. Number of Steps Calculated Price 3.7 3.8 3.9 4.0 4.1 184 186 188 190 192 194 196 198 200 202 204 206 208 210 212 214 216 218 220 222 224 Figure 16.8 Knock-out option price vs. steps (iv) Alternative Discretizations: The above sawtooth effect is particularly pronounced when we use the Jarrow–Rudd discretization, since all nodes lie along horizontal levels: entire rows of nodes then cross the barrier at once as the number of time steps is increased. If we use a discretization which does not have horizontal rows of nodes, then only a few grid points cross the barrier each time the number of steps is increased. Figure 16.9 is analogous to Figure 16.6, but using the Cox–Ross–Rubinstein discretization. It continues to display the sawtooth effect, but with much reduced amplitude; the envelope of the low points no longer coincides with the 198 [...]... performance is discretely monitored It is of course possible (indeed very probable) that in our example of an up-and-in 199 16 Barriers: Advanced Options put, the stock price rises above the barrier only to fall back again before the next discrete monitoring point A discretely monitored up-and-out call should always be more valuable than one which is continuously monitored Moreover, the less frequently...16.5 BARRIER OPTIONS AND TREES analytical value for the option The smooth curve which seems to run through the middle of the zig-zags in the diagram is explained later 4.90 Option Price (Stock = 100) 4.70 4.50 4.30 4.10 3.90 3.77 ? 1% 3.77 ± 3.70 Number of Steps 3.50 0 200 400 600 800 1000 1200 1400 1600 Figure 16.9 A: Up-and-out knock-out call price vs number of binomial steps... the context: it would be + in our up-and-out call example This approximation holds fairly well over a wide range and is the heavy curve in Figure 16.9 Finally, the general form of this correction explains one point which may have puzzled the reader: the binomial results of Figure 16.6 refer to a series of discretely monitored barrier options, notwithstanding the zig-zag errors which have already been... that was priced in the last section, using analytical methods? It is now rare to find barrier options which knock in or out if ever the barrier is crossed before maturity: there have been too many disputes about whether the barrier really was crossed and whether a bit of market manipulation helped it over Barrier options contracts now specify a precise time each day (or week) when the market is observed... us from a discretely monitored to a continuously monitored barrier option Therefore, if the center of the sawtoothed pattern in Figure 16.6 is a best estimate of the value of a discretely monitored up-and-out call (Cheuk and Vorst line), then the envelope of the bottom of the sawtooth will approximately give the continuously monitored value 200 ... is generally necessary to use trinomial trees There is nothing to say that these must have horizontal nodes, but they are certainly easier to program if they do An alternative way of overcoming the zig-zag problem, while still retaining nodes in horizontal rows, is to use interpolation This method starts with a fixed number of steps and regards the barrier K as a variable Referring to the diagram of . written F return = F 0 − F non-abs 16 Barriers: Advanced Options -b X a return F 0 F 0 F 0 S Figure 16.1 Double barrier-and-in call The total probability. outside barriers rather than inside barriers. However, these options are relatively rare so we will simply describe a single-outside-barrier up-and-in call