LAPLACE TRANSFORMS AND THE AMERICAN STRADDLE

Transcription

1 LAPLACE TRANFORM AND THE AMERICAN TRADDLE G. ALOBAIDI AND R. MALLIER Received 2 October 2001 and in revised orm 12 March 2002 We address the pricing o American straddle options. We use partial Laplace transorm techniques due to Evans et al. (1950) to derive a pair o integral equations giving the locations o the optimal exercise boundaries or an American straddle option with a constant dividend yield. 1. Introduction and analysis Options are derivative inancial instruments which give the holder certain rights. A call option carries the right (but not the obligation) to buy an underlying security at some predetermined price, while a put allows the holder to sell the underlying security. The value V (, t) o many options can be ound using the Black-choles partial dierential equation (PDE) (see, e.g., 6), V t + σ2 2 2 V 2 + ( ) V r D 2 0 rv = 0, (1.1) together with appropriate boundary conditions, where is the price o the underlying security and t<t is the time, with T being the expiry time. The parameters in the above equation are the risk-ree rate, r, the dividend yield, D 0, and the volatility, σ; all o them are assumed constant. In addition, we assume that r>d 0 > 0. I an option is European, it can only be exercised at the expiration date. I an option is American, it can be exercised at or beore expiry, and a rational investor will exercise the option early i it is to his advantage. There are thereore regions where it is optimal to hold the option Copyright c 2002 Hindawi Publishing Corporation Journal o Applied Mathematics 2:3 (2002) Mathematics ubject Classiication: 65R20 URL:

2 122 Laplace transorms and the American straddle and others where exercise is optimal, and the need to ind the boundary between these regions means that American options are more challenging mathematically than their European counterparts. Indeed, apart rom one or two very special cases, closed orm solutions have yet to be ound or most American options, whereas or European options, solutions can usually be ound using error unctions or equivalently the cumulative distribution unction or the normal distribution. Numerical methods and approximations can however be used to value American options. In this study, we consider an American straddle, which in this context gives us the right, but not the obligation, to either buy or sell (but not both) an underlying stock at a predetermined price at or beore expiry. Thus we have both a put and a call with the same expiry and strike price, but we are allowed to use only one o them. For a European straddle, where exercise is only allowed at expiry, this limitation does not constitute a problem, and a European straddle is worth exactly the same as a European put and call combined. It is important to note that a call and a put with the same exercise price cannot be simultaneously in the money, so or a European straddle when exercise is permitted only at expiry, the option which is currently in the money will be exercised. For an American straddle, by contrast, when early exercise is permitted, it is perectly possible that the price o the underlying stock moves in such a way that sometimes the call is in the money while at others the put is in the money; and an investor holding a separate call and put would be able to exercise both at dierent times while an investor holding a straddle can only exercise one o the two, and would thereore have a lower expected return. Because o this limitation, the option value is not simply the sum o the values o a call and a put option. uch an option might be useul i an investor expects a large change in the value o the underlying stock that makes a signiicant move, but is unsure in terms o the direction o the change, which, as an example, might occur i a company were involved in a major lawsuit or when a major bank or corporation is about to ail. This problem involves two ree boundaries: i the option price is suiciently high, + (t), then the holder will exercise the call, while i it is suiciently low, (t), then the holder will exercise the put, and between these two boundaries, (t) + (t), the holder would retain the option or the time being. We will tackle this problem using a modiied Laplace transorm, and the end result o our study is not an exact solution (very ew o which exists or American options), but rather a pair o integral equations or the location o the optimal exercise boundaries. Previously, in 2, we looked at the corresponding problems or the call and put options, and derived in each case an integral equation with a general orm similar to those ound here.

3 G. Alobaidi and R. Mallier 123 The starting point o our analysis is the Black-choles PDE (1.1), together with the pay-o at expiry, V (, T )=max( E,E ). (1.2) For the European straddle, the PDE (1.1) can be solved airly easily. For an American option, we have also the constraint that the price o the option cannot all below the pay-o rom immediate exercise, V (, t) max( E,E ), (1.3) with the PDE (1.1) being valid only where V (, t) > max( E,E ). There is o course a region in which it is optimal to hold the option to expiry rather than to exercise it, and the boundary o this region is known as the optimal exercise boundary. For this particular problem, there are in act both an upper boundary = + (t) and a lower bound- ary = (t). In the present analysis, it is convenient to invert these relations, and write instead a single relation, t = T (). We will use a mod- iied Laplace transorm to arrive at an integral equation giving the location o this ree boundary. Integral equation methods have been used to tackle American options beore, including the early works 3, 5 on calls and the recent paper by Kuske and Keller 1 ontheput,aswellasour own previous work on the put and call 2. We discuss the dierences between those studies and our own in ection 2. everal properties o the ree boundaries are known (e.g., 6). Firstly, we know that the value o the option and its derivative with respect to must be continuous across the ree boundaries, so that V = + (t) E and ( V/ )=1at +, and V = E (t) and ( V/ )= 1at. Continuity o these maximizes the value o an American option. The value o the option must be continuous, as i it were greater than the return rom immediate exercise the holder would not exercise, and i it were less than that, it would result in an arbitrage opportunity, in that an investor could buy an option and immediately exercise it or a risk-ree proit. imilarly, i the delta o the option at the ree boundary were greater than the delta o the pay-o, delaying exercise would lead to a higher expected return, while i the delta o the option was less than the delta o the pay-o, exercising earlier would increase the expected return. econdly, i we evaluate ( V/ t) right at expiry using (1.1), we can deduce that + (T)= 0 = Er/D 0 >Eand (T)=E. In the unusual event that D 0 >r,thetwo locations are reversed. In addition, we know that + moves upwards and downwards as we move away rom expiry. Hence we can deduce that T ()=T or E 0. Thirdly, we know the position o the boundaries

4 124 Laplace transorms and the American straddle as T t. I we consider the perpetual American straddle (which never expires and thereore has no time dependence), the value o this option is V = A α+ + B α 6,where α ± = 1 σ 2 2 ( ) r D 2σ 2 0 ± 4 ( ) r D ( 2 ) 0 + 4σ 2 r + D 0 + σ 4. (1.4) I we denote the upper and lower boundaries or this perpetual option by + and, then we require that V = + E and ( V/ ) =1at = +, while V = E and ( V/ )= 1 at. This yields our nonlinear equations, rom which we ind that the ratio R = + / obeys the equation α +( α 1 )( R α + 1 )( R α+ + R ) = α ( α + 1 )( R α+ + 1 )( R α + R ), (1.5) with + = Eα (R α+ + 1)/(α 1)(R α+ + R). In our terms, we require that T () as + rom below and as rom above. The upper optimal exercise boundary will lie between the limits, 0 + (t) +, while the lower one will lie between the limits (t) E. Having ormulated the problem, we now attempt to solve it using a Laplace transorm in time. This technique is known to work well with European options, but with American options, one perceived diiculty has been that the Black-choles PDE only holds where it is optimal to retain the option. Because o this, we modiy the usual deinition L(G)(p)= 0 g(t)e pt dt (1.6) somewhat, and deine our version as ollows or + : V(, p)= T () V (, t)e pt dt, (1.7) so that the sign o t is reversed rom the usual deinition, and also the upper limit is t = T () rather than t = 0. This is o course equivalent to setting V (, t)=0 in the region where it is not optimal to hold. Because o this deinition, the price o the option V (, t) will obey the Black-choles equation everywhere where we integrate. We require the real part o p to be positive, that is, R(p) > 0, or the integral in (1.7) to converge. We know rom the deinition that V(, p) 0as ±. We also know that as p, we have V(, p) 0 and pv bounded, and in this limit, we

5 G. Alobaidi and R. Mallier 125 can show that lim pv = lim V ( T (), ) e pt (). (1.8) p p We can also deine an inverse transorm V (, t)= 1 2πi γ+i γ i V(, p)e pt dp. (1.9) Given our deinition o the orward transorm, this inverse is only meaningul where it is optimal to hold the option. In the above, we have adopted the convention that T () is the location o the ree boundary or <<Eand 0 << +, while or E<< 0,wesetT = T since there it is optimal to hold the option to expiry. Transorm methods in general can be useul when dealing with linear partial dierential equations such as (1.1), because they can be used to reduce the dimension o the problem. The appropriate transorm to use will obviously depend both on the orm o the equation and the geometry o the domain, and or (1.1) it is well known that taking a Laplace transorm in time o (1.1) will eliminate the temporal derivative, reducing the problem to an ordinary dierential equation; this same technique is regularly used with the heat conduction equation into which the Black-choles equation can be transormed. In addition to our earlier work 2 (and that o Knessl (2001)) in applying Laplace transorms to American options, Laplace transorms have been used or pathdependent options beore, though we believe that our earlier work was the irst to consider an option problem with a ree boundary. Geman and Yor (1996) used Laplace transorms to price barrier options, where there are ixed rather than ree boundaries, and Geman and Yor (1993) used them to price Asian options, where the pay-o motivation or using Laplace transorms was that they reduced the dimension o the problem. Applying this modiied Laplace transorm to the Black-choles PDE (1.1), we arrive at the ollowing (nonhomogeneous Euler) ordinary dierential equation ODE or the transorm o the option price, 1 2 σ ( ) r D 2 0 (p + r) V + F()=0, (1.10) where the nonhomogeneous term F() takes a dierent value in each o the ollowing regions:

6 126 Laplace transorms and the American straddle Region (a) <<E, where we have V ( (t),t)=e, ( V/ )( (t),t)= 1, and T <T, we have F()=(E )e pt () + σ 2 2 ( r D 0 ) (E ) T () σ2 2 (E )T 2 (). (1.11a) Region (b) E<< 0,whereT = T and F()=( E)e pt. (1.11b) Region (c) 0 << +,wherev ( + (t),t)= E, ( V/ )(+ (t),t)=1, T <T, and F()=( E)e pt () + σ 2 2 ( r D 0 ) ( E) T () σ2 2 ( E)T 2 (). (1.11c) The general solution o (1.10) is V = 2 )(2D 0 2r+σ 2 +λ(p)) λ(p) (1/2σ2 C + (p) + 2 λ(p) (1/2σ2 )(2D 0 2r+σ 2 λ(p)) C (p)+ (1/2σ2 )(2D 0 2r+3σ 2 +λ(p)) F()d (1/2σ2 )(2D 0 2r+3σ 2 λ(p)) F()d, (1.12) where λ(p)=4(r D 0 ) 2 + 4σ 2 (r + D 0 + 2p)+σ 4 1/2, and C ± are the constants o integration, which may depend on the transorm variable p. Applying this solution (1.12) to the three separate regions outlined above, we ind that in region (a) in order to get a solution which vanishes as, we have

7 V = 2 1 λ(p) ( ) (1/2σ 2 )(2D 0 2r+3σ 2 ) ( ) λ(p)/2σ 2 G. Alobaidi and R. Mallier 127 ( ) λ(p)/2σ 2 F( )d, (1.13) and similarly in region (c) in order to get a solution which vanishes as +, we have V = λ(p) ( ) (1/2σ 2 )(2D 0 2r+3σ 2 ) ( ) λ(p)/2σ 2 ( ) λ(p)/2σ 2 F( )d, (1.14) while in region (b), we have V = 2 )(2D 0 2r+σ 2 +λ(p)) C (b) λ(p) (1/2σ2 + (p) (1/2σ2 )(2D 0 2r+3σ 2 +λ(p)) F( )d E + 2 )(2D 0 2r+σ 2 λ(p)) λ(p) (1/2σ2 C (b) (p)+ E (1/2σ2 )(2D 0 2r+3σ 2 λ(p)) F( )d. (1.15) We require the transorm V and its derivative with respect to to be continuous at = E as we move rom region (a) to (b), and also at 0,as we move rom (b) to (c), which tells us that E C (b) ± (p)= (1/2σ2 )(2D 0 2r+3σ 2 ±λ(p)) F( )d + = ± (1/2σ2 )(2D 0 2r+3σ2 ±λ(p)) F( )d 0 ± 2σ 2 e pt (1/2σ2 )(2D 0 2r+σ 2 ±λ(p)) 0 E 2D 0 2r + σ 2 ± λ(p) 0 2D 0 2r σ 2 ± λ(p) (1.16) ± 2σ 2 e pt E (1/2σ2 )(2D 0 2r σ 2 ±λ(p)) 1 2D 0 2r σ 2 ± λ(p) 1. 2D 0 2r + σ 2 ± λ(p) Comparing these two pairs o expressions, we require that

8 128 Laplace transorms and the American straddle E (1/2σ2 )(2D 0 2r+3σ 2 ±λ(p)) F ( )d + + (1/2σ2 )(2D 0 2r+3σ2 ±λ(p)) F( )d = 2σ 2 e pt (1/2σ2 )(2D 0 2r+σ 2 ±λ(p)) 0 E 2D 0 2r + σ 2 ± λ(p) 0 2D 0 2r σ 2 ± λ(p) ± 2σ 2 e pt E (1/2σ2 )(2D 0 2r σ 2 ±λ(p)) 1 2D 0 2r σ 2 ± λ(p) 1. 2D 0 2r + σ 2 ± λ(p) 0 (1.17) The reader s attention is drawn to the act that there is a ± inronto λ(p) in the exponent o, so that (1.17) is actually a pair o equations, one or either sign. 2. Discussion This last pair o (1.17) is the main result o this paper. They constitute integral equations or the location o the ree boundary, T (), ormore speciically, Urysohn equations o the irst kind 4. ince these equations involve the variable p, and must be true or each value o p or which R(p) > 0, we can think o them as a orm o integral transorm operating on T (), and inverting this transorm would give T (). However, this inversion would appear to be extremely diicult to do analytically because o the term involving e pt () in F() as given in (1.11a), (1.11b), and (1.11c); i this term were absent, we could regard the equations as aormo(inite) Mellin transorm. In theory, (1.17) could be solved numerically, but that is outside the range o expertise o the present authors. As we mentioned briely in ection 1, other authors have previously used integral equation methods to analyze American options, including the studies 1, 3, 5. However, those studies tackled the problem in very dierent ways to that used here, and ended up with equations o a somewhat dierent orm to (1.17). For example, in their recent study, Kuske and Keller 1 used Green s unctions to solve the Black-choles PDE or the American put, and their result involved an integral equation or (t), whereas we have an integral equation or the inverse o that unction, T (). As is the case here, those authors were unable to obtain exact solutions o their integral equations. tudies similar to the present have been perormed or both the American put and call 2; each o these problems involved a single ree boundary, and in each case the end result was a single integral equation o the same general orm as those ound here.

9 G. Alobaidi and R. Mallier 129 Moving on to the issue o the value o the option, in (1.13), (1.14), and (1.15), we have a series o expressions or V(p,), the transorm o the option price V (, t). In theory, given these expressions, we could apply the inverse transorm (1.10), and then we would arrive at the option price itsel. Unortunately, these expressions involve T (), thelocation o the ree boundary, which we know only abstractly as the solution o the integral equations (1.17); however, i T () were known explicitly, taking the inverse Laplace transorm would give the value o the option. Reerences 1 R. E. Kuske and J. B. Keller, Optimal exercise boundary or an American put option, Appl. Math. Fin. 5 (1998), R. Mallier and G. Alobaidi, Laplace transorms and American options, Appl. Math. Fin. 7 (2000), no. 4, H. P. McKean Jr., Appendix: A ree boundary problem or the heat equation arising rom a problem in mathematical economics, Industrial Management Review 6 (1965), A. D. Polyanin and V. Manzhirov, Handbook o Integral Equations, CRC Press, New York, P. Van Moerbeke, On optimal stopping and ree boundary problems, Arch. Rational Mech. Anal. 60 (1975/76), no. 2, P. Wilmott, Derivatives: The Theory and Practice o Financial Engineering, Wiley University Edition, Chichester, G. Alobaidi: Department o Mathematics and tatistics, University o Regina, Regina, askatchewan, Canada 4 0A2 address: alobaidi@math.uregina.ca R. Mallier: Department o Applied Mathematics, University o Western Ontario, London, Ontario, Canada N6A 5B7 address: mallier@uwo.ca

10 Boundary Value Problems pecial Issue on ingular Boundary Value Problems or Ordinary Dierential Equations Call or Papers The purpose o this special issue is to study singular boundary value problems arising in dierential equations and dynamical systems. urvey articles dealing with interactions between dierent ields, applications, and approaches o boundary value problems and singular problems are welcome. This pecial Issue will ocus on any type o singularities that appear in the study o boundary value problems. It includes: Theory and methods Mathematical Models Engineering applications Biological applications Medical Applications Finance applications Numerical and simulation applications Compostela, antiago de Compostela 15782, pain; juanjose.nieto.roig@usc.es Guest Editor Donal O Regan, Department o Mathematics, National University o Ireland, Galway, Ireland; donal.oregan@nuigalway.ie Beore submission authors should careully read over the journal s Author Guidelines, which are located at Authors should ollow the Boundary Value Problems manuscript ormat described at the journal site Articles published in this pecial Issue shall be subject to a reduced Article Processing Charge o C200 per article. Prospective authors should submit an electronic copy o their complete manuscript through the journal Manuscript Tracking ystem at according to the ollowing timetable: Manuscript Due May 1, 2009 First Round o Reviews August 1, 2009 Publication Date November 1, 2009 Lead Guest Editor Juan J. Nieto, Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de antiago de Hindawi Publishing Corporation