EVALUATION OF AMERICAN STRANGLES

Transcription

1 EVLUTION OF MERICN STRNGLES CRL CHIRELL ND NDREW ZIOGS School of Finance and Economics University of Technology, Sydney PO Box 2 Broadway NSW 200, ustralia Fax: ( 2) 94 BSTRCT. This paper presents a generalisation of McKean s free boundary value problem for merican options by considering an merican strangle position, where the early exercise of one side of the payoff will knockout the outofthemoney side. When attempting to evaluate the price of this merican strangle, it is not correct to simply price the component merican call and put options which make up the strangle, and take the sum of their values. The Fourier transform technique is used to derive the integral equation for the price of our merican strangle. From this expression, a coupled integral equation system for the strangle s call and putside free boundaries is found. While the equation for the price of the strangle is simply the sum of its component merican call and put option equations, the free boundary for each side is shown to have a more complex nature. numerical algorithm for solving the coupled integral equation system for the free boundaries is provided, and the resulting approximations are used to determine the price of the merican strangle position. Numerical comparisons between the strangle price and the price of a portfolio formed from a long position in both an merican call an merican put option are presented. JEL classification C, D Keywords: merican options, Volterra integral equation, freeboundary problem. Date: June, 2002.

2 2 CRL CHIRELL ND NDREW ZIOGS. INTRODUCTION merican options are highly common derivative securities in today s financial markets. merican calls and puts are frequently written on a range of underlying assets, including stocks, futures, and foreign exchange rates. Since the groundbreaking results of Merton (9) and Black and Scholes (9) regarding the analytic pricing of European call and put options, a great deal of research has been conducted into applying the BlackScholes framework to merican options. While McKean (9) and Kim (990) successfully extended the BlackScholes European option pricing methodology to merican calls and puts, the method has never been generalised to allow a broader range of payofffunctions. This paper extends the results of McKean and Kim to a special type of merican strangle position, where the early exercise of one side of the position will knockout the remaining side. Through this example, a general merican option pricing framework is provided. The fundamental difference between merican and European options is that an merican option can be exercised at any up to and excluding the expiry date. lthough this difference is conceptually simple, it adds a large degree of mathematical complexity to the option pricing problem. Using mathematical results from Kolodner (9), McKean (9) first derived the integral equation for both the price and early exercise boundary of an merican call option as the solution to a free boundary problem. This was a natural extension of the BlackScholes method for European options, and was explored further by Van Moerbeke (94, 9). range of alternative methods based on the BlackScholes partial differential equation (PDE) were proposed, including the quadratic approximation of Baronedesi and Whaley (98), and the compound option approach of Geske and Johnson (984). Ho, Stapleton and Subrahmanyam (99) extended the GeskeJohnson technique to include stochastic interest rates. Other methods that have been considered include the finite difference method (Brennan and Schwarts, 9) and the binomial approximation (Parkinson, 9). Karatzas (988) was one of the first to revisit the topic of pricing merican options as a solution to a free boundary problem. Kim (990) reproduced McKean s results, extending them to the merican put case, and deriving the exact behaviour of the early exercise boundary near expiry. He also proved that taking the limit of the GeskeJohnson method as the number of early exercise dates is increased to infinity, generated integral equations that are mathematically equivalent to those of McKean. Furthermore, Kim derived a new representation for these integral equations using the cumulative normal distribution, simultaneously removing their dependence on the derivative of the free boundary. Several important papers on merican options were published at this time, exploring merican option prices in the context of free boundary problems. Kim s results were confirmed by Elliot, Myneni and Viswanathan (990). Jacka (99) proved both existence and uniqueness for the merican put option price and early exercise boundary. Carr, Jarrow and Myneni (992) expressed the merican option price as the sum of its early exercise and intrinsic value. Jamshidian (989a,

3 EVLUTION OF MERICN STRNGLES 989b, 992) has also conducted extensive research into the free boundary method for pricing merican calls and puts. Though Kim s integral equations for merican call and put prices could be interpreted as the sum of their European values and their early exercise premiums, the early exercise boundary could not be found explicitly. Numerical solutions for the early exercise boundary have been considered by many authors. Underwood and Wang (2000) implemented Kim s equations using a fixedpoint iterative scheme, which was prone to slow convergence. Huang, Subrahmanyam and Yu (99) used a recursive method. They estimated the free boundary at a small number of time points, combining the GeskeJohnson compound option technique with analytic valuation formulae. The complete early exercise boundary was then developed using Richardson extrapolation. This approximation was used in the analytic integral equations to find the merican option s price. Ju (998) approximated the free boundary using a piecewise exponential function,finding that the prices obtained using these estimates were highly accurate. it Sahalia and Lai (200) used linear splines with very few knots to approximate the free boundary. Solving a transformed free boundary integral equation at several time points, they estimated the entire free boundary using linear interpolation, and were able to obtain very accurate merican option prices using these early exercise boundary estimates. It was postulated that the price of an merican option was not overly sensitive to the early exercise boundary approximation. Despite the large amount of research conducted into the merican option pricing problem, there is still no clear framework with which one can derive the integral equations and free boundaries for a generic payoff function, either monotonic, convex or concave. In this paper, we revisit McKean s incomplete Fourier transform method for merican call options, and apply it to a special kind of merican strangle. If exercised early, the entire payoff is optimally realised, making this fundamentally different to an merican strangle formed using individual calls and puts. This is an example of a more general merican option position with a convex payoff function. The application of McKean s method is based on Kucera and Ziogas (2002). We also transform the results into Kim s integral equation form, and then proceed to implement these equations numerically to find firstly the strangle s early exercise boundaries, and finally the strangle s price. Such a strangle has several implicit advantages over a traditional merican strangle constructed using a long position in both an merican call and an merican put. The new strangle is selfclosing, since exercising one side of the position will knockout the other. These implicit knockout barriers will make this strangle cheaper than a traditional one, and may have market applications within purevolatility strategies. It is important to note that the proposed strangle loses the flexibility to be decomposed into its component options. n analytic delta exists for the strangle, and the strangle can be hedged in the same manner as any merican call or put. The remainder of this paper is structured as follows. Section 2 outlines McKean s free boundary problem that arises from this paper s merican strangle option pricing

4 4 CRL CHIRELL ND NDREW ZIOGS problem. Section applies the incomplete Fourier transform to solve the PDE in terms of a transform variable. The transform is inverted in Section 4, to provide McKean s integral equations for the merican strangle price, and a corresponding integral equation system for the strangle s two early exercise boundaries. Section outlines the numerical solution method for both the free boundaries and strangle price, including the transform from McKean s equations to Kim s equations. selection of numerical results are provided in Section, with concluding remarks presented in Section. 2. PROBLEM STTEMENT Let be the price of an merican strangle position written on at time, with time to expiry. This position is formed using a long put with strike, and a long call with strike. Note that. Let the early exercise boundary on the put side be denoted by!, and the early exercise boundary on the call side be denoted by ". Diagram demonstrates the payoff and continuation region for. Diagram : Continuation Region in Sspace &,. +% Continuation Region $ % &')( * +% Stopping Region 0 / Stopping Region % 0 % It is known that in the region2! H <H satisfies the BlackScholes PDE: 8 0 % :9;=<>, where volatility of riskfree rate, and! 8?9@BC+D+FE$GE$ dividend rate of (continuously compounded) ()

5 EVLUTION OF MERICN STRNGLES subject to the following final time and boundary conditions: ; &')( +% &')( * + + % %!,.& 8 +!,.& 8 + +% + (2) Condition (2) is the payoff function for the strangle at expiry, while conditions () () are collectively known as the smoothpasting conditions. These ensure that the price,, and its first derivative with respect to are both continuous. This is necessary to maintain the BlackScholes assumption of an arbitragefree market. Firstly, we shall transform the PDE () to a forwardintime equation with constant coefficients. Let where The transformed PDE is then in the region $ 8 =% 0$?9! D+FE" E$ $ where B9;=<; 8 and the transformed free boundaries are given by The new initial and boundary conditions are &% & % +% &')( ' +% &')( ( +% $ )* + $ )* + +,.&, & +,.&, & & In what follows, we will use the notation /. and. () (4) () () () (8) (9) (0) for simplicity. () (2) () (4)

6 CRL CHIRELL ND NDREW ZIOGS In order to solve this PDE for, the domain shall be extended to by expressing the PDE as 8 B 9!BB+ D+FE E$ where is the Heaviside step function, defined as /0+ C+ + + The initial and boundary conditions remain unchanged.. PPLYING THE FOURIER TRNSFORM We will now find a solution to the problem defined by equations (9)(4). Specifically, we propose the use of the Fourier transform technique to reduce the PDE to an ODE, whose solution is readily obtainable. Note that in using the Fourier transform approach, we employ all the standard techniques that apply when solving the Black Scholes PDE for the price of European options. This includes the assumption that for the purposes of the transform method, the function and its first two derivatives with respect to can be treated as zero when tends to. Since the domain is now, the Fourier transform of the PDE can be found. Define the Fourier transform of,, to be! Thus, the transformed PDE appears as: By definition $ $ $ 9 $! +,! +, ". ()

7 . EVLUTION OF MERICN STRNGLES and for convenience, let us introduce the notation We note that,. is an incomplete Fourier transform, since it is equivalent to a standard Fourier transform applied to in the domain of $ $. In ppendix we show how the incomplete Fourier transform may be derived as a consequence of the standard Fourier transform and there derive the corresponding inversion theorem. To apply the incomplete Fourier transform to the PDE (9), we need to consider three specific properties of. Proposition : Given the definition of for : Proof: Refer to ppendix B.. in equation (), the following identities exist +, B +, % +, +, % +, +, & () () (8) The PDE can now be transformed, as required. Proposition 2: The incomplete Fourier transform of the PDE (9) with respect to satisfies the ordinary differential equation! 9 (9) where +, +, (20) with initial condition +% +%&

8 8 CRL CHIRELL ND NDREW ZIOGS Proof: Refer to ppendix B.2. Instead of solving a PDE for the function, we are now faced with the simpler task of solving the ODE (9) for the function. This can then be inverted via the Fourier inversion theorem (see ppendix ) to recover the desired function. Before concluding this section, we shall find the solution to (9). Proposition : The solution for +% is given by )! (2) Proof: Recalling that and are functions of, the ODE (9) is of the form! & Using the integrating factor!, the ODE becomes whose solution may be expressed as +% ) Referring back to the original ODE, the solution for (2).! is found to be equation 4. INVERTING THE FOURIER TRNSFORM Having now found, it is necessary to recover, the merican strangle price in the plane. Taking the inverse (complete) Fourier transform of (2) gives +% ) &% $ $ & We must now determine and. Proposition 4:

9 The function where EVLUTION OF MERICN STRNGLES 9 % % and is given by 2 Proof: Refer to ppendix C.. % ' % % +% ' % +% % 9;?< / % % % +% % +% (22) Proposition : The function ) where and ) ) ) ) ) for and $ Proof: Refer to ppendix C.2. is given by $ ) ) ) ) $ ) ) ) > ) (2) (24) (2) Hence, the value of the merican strangle is given in the plane by +FE E$ % $ & Equation (2) expresses the value of the merican strangle position in terms of the. t this point these remain unknown, but early exercise boundaries and we are able to obtain an integral equation system that determines them by requiring the expression for to satisfy the early exercise boundary conditions () (2)

10 0 CRL CHIRELL ND NDREW ZIOGS and (2). Recalling our definition for the Heaviside function, the following integral equation system is obtained: $ $ where is given by equation (2) in conjunction with (22)(2). For a more detailed explanation of why the factor of appears in the left hand side of (2) and (28), see Kucera and Ziogas (2002). This system of integral equations must be solved simultaneously using numerical methods to find and. Once these are found, it is simple to evaluate via numerical integration. It is of value to note that equation (2) is simply the sum of the integral pricing equations for an merican put and an merican call option. The added complexity in pricing an merican strangle therefore arises from the early exercise boundaries. Each boundary is dependent upon all other free boundaries in the system (2)(28), and therefore these boundaries are not equal to those found when valuing an merican call and put option separately. Thus it is important to understand the nature of the early exercise boundaries for merican option portfolios in order to obtain the correct early exercise boundary values. (2) (28). NUMERICL IMPLEMENTTION equations (22)(2) into the form presented by Kim (990). This will remove the To make the task of numerical implementation less complicated, we will transform and terms from the integral using integrations by parts. The first step is to rewrite the pricing equation in terms of the original underlying asset. Proposition : The solution to the boundary value problem ()() in terms of and is given by B (29) where : % % % % % +% % +% % +% % +% (0)

11 EVLUTION OF MERICN STRNGLES with and with % % and " " for and Proof: Recall that " " " "G %. 9;?< " " " " " " " " 2, and substitute this into equations (22)(2). () (2) () Proposition : Equations (29)() can be expressed as follows: where and % % 9 % " ; < % " 2 % % < % " 9 % "& Proof: The above is derived using integration by parts, as outlined in the appendix of Kim (990). (4) ()

12 % % 9 < % % 2 CRL CHIRELL ND NDREW ZIOGS The integral equation system for the free boundaries & and is now To solve this system, we propose using a numerical scheme similar to those used for Volterra integral equations. Firstly, discretise the time variable into equally spaced intervals of length. Thus for + B, and. Following the methods of Kim (990), it can be readily shown that +% &,. +% &')( Thus by starting at, there are only two unknown values in the system ()() for each, namely and. We use Simpson s rule to evaluate the integral terms. For each beginning with find : where and 9 < = () () (8) (9), the bisection method is applied to the following to 9 < < 9 &% &% &% &% (40) (4) The summation weights are those dictated by the numerical integration scheme, in this case Simpson s rule, and the extended Simpson s rule (used on the end furthest from expiry whenever is odd). Similarly, we apply the bisection method to the following to find : (4) (42)

13 % % % % EVLUTION OF MERICN STRNGLES where 9 < &% &% (44) and It is important to note that at by using the following:,.&,.&,.&,.& < 9 &% &% (4), both and are singular. This can be handled % " + % " + % " + % " + These limits are the same for. It is also important to see that while, for example, (40)(42) depends upon it does not explicitly require, due to the need to use the limits (4)(49). Hence, the simultaneous integral equation system () () can be solved by finding using all known values of and +, (4) (4) (48) (49). That is, the interdependence at the th time point is removed due to the need to consider the limits of and when evaluating (40)(42) and (4)(4). The above numerical scheme is firstly carried out using using a timestep size of, and then repeated using. In each case, alternating between two different numerical integration schemes (for odd and even values of ) leads to free boundaries that have nonmonotonic gradients. This is rectified by combining the two estimates using Richardson s extrapolation. Pricing the merican strangle is then achieved via numerical integration using Simpson s rule, combined with the estimates of and.

14 < 4 CRL CHIRELL ND NDREW ZIOGS. RESULTS +>+ To demonstrate the early exercise boundaries and price properties of the merican strangle, we implemented the method using time nodes. To improve the accuracy of the method where the free boundaries change rapidly, a finer + grid was used between the first three nodes (specifically, 40 nodes between and ). The method was also applied in the same manner to the merican call and put contracts which define the components of the strangle s payoff function. By comparing the results for the strangle against those of the independent call and put, we can demonstrate how the merican strangle s free boundaries and price are affected by the interdependence between and. Firstly, consider an merican strangle with one year until maturity. Let the putside strike be and the callside strike be., with the volatility of the underlying at 20%. In figures 9*/, we present the call and putside boundaries 9 for the < merican strangle, with (figures 4), and finally (figures ). In all < (figures 2), 9 cases, we include the free boundary for the corresponding merican call or put. The same results are presented in figures 2, but the callside strike has been reduced to.00, moving the strangle position closer to a straddle. 2. Free Boundary: merican Call and Strangle r > q merican Call merican Strangle c 2 (T t) FIGURE.,, (T t) +, 9 +, < There are several distinct features that can be ascertained from these free boundary plots. The first is that the relative values of 9 and < directly affect whether or not the merican strangle free boundaries will show significant divergence 9$/ from < the corresponding merican call and put boundaries. In particular, when, only the

15 EVLUTION OF MERICN STRNGLES Free Boundary: merican Put and merican Strangle r > q merican Put merican Strangle c (T t) (T t) FIGURE 2.,, +, 9 +, <.4 Free Boundary: merican Call and merican Strangle r < q merican Call merican Strangle.4. c 2 (T t) (T t) FIGURE.,, +, 9, < +

16 CRL CHIRELL ND NDREW ZIOGS 0. Free Boundary: merican Put and merican Strangle r < q merican Put merican Strangle c (T t) (T t) FIGURE 4.,, +, 9, < +.. Free Boundary: merican Call and merican Strangle r = q merican Call merican Strangle.4.4 c 2 (T t) (T t) FIGURE.,, +, 9 +, < +

17 EVLUTION OF MERICN STRNGLES Free Boundary: merican Put and merican Strangle r = q merican Put merican Strangle c (T t) (T t) FIGURE.,, +, 9 +, < + 2. Free Boundary: merican Call and Strangle r > q merican Call merican Strangle c 2 (T t) (T t) FIGURE., +>+, +, 9 +, <

18 8 CRL CHIRELL ND NDREW ZIOGS Free Boundary: merican Put and merican Strangle r > q merican Put merican Strangle c (T t) (T t) FIGURE 8., +>+, +, 9 +, <. Free Boundary: merican Call and merican Strangle r < q merican Call merican Strangle..2 c 2 (T t) (T t) FIGURE 9., +>+, +, 9, < +

19 EVLUTION OF MERICN STRNGLES 9 0. Free Boundary: merican Put and merican Strangle r < q merican Put merican Strangle c (T t) (T t) FIGURE 0., +>+, +, 9, < Free Boundary: merican Call and merican Strangle r = q merican Call merican Strangle.. c 2 (T t) (T t) FIGURE., +>+, +, 9 +, < +

20 < 20 CRL CHIRELL ND NDREW ZIOGS Free Boundary: merican Put and merican Strangle r = q merican Put merican Strangle c (T t) FIGURE 2., +>+ (T t), +, 9 +, < + putside 9F boundaries < 9 diverge, and when only the callside boundaries diverge. When, there is divergence in both boundaries, but it is smaller than in the other two cases. Since the early exercise of the inthemoney side of the strangle will knockout the other side, it is expected that the strangle will have to be deeper inthemoney to warrant early exercise than one formed using independent merican calls and puts. In all cases, the callside free boundary for the strangle is always greater than or equal to that of the corresponding merican call free boundary, while the putside is always less than or equal to that of the corresponding merican put free boundary. This is in keeping with the economic 9 intuition < behind this merican strangle problem. In all three cases of and values, moving the callside s strike closer to the putside s strike increases any divergence between the merican strangle free boundaries and those of the corresponding merican call and put. This is again as one would expect, since the closer the strangle is to being a straddle, the more intrinsic value the outofthemoney strangle component will contribute to the early exercise decision. It can also be seen that as the time to maturity increases, the divergence between the strangle free boundaries and the corresponding call and put boundaries increases. When the strangle has a very short time to maturity, say 2 weeks or less, then the divergence between the two free boundaries becomes minimal. While it is clear that the early exercise boundaries for the strangle are not always equivalent to those of the component merican call and put in the examples provided,

21 EVLUTION OF MERICN STRNGLES 2 the difference never exceeds 0., which in relative terms is no more than 0% of the putside s strike price. Past research into merican options, such as Ju (998) and itsahalia and Lai (200), has found that the price of merican call and put options is not greatly affected by the free boundary estimate used. While a 0% difference in the free boundary has obvious early exercise timing repercussions, it remains to be seen whether the price of the strangle using these free boundaries is far removed from that of a strangle priced simply using the sum of an merican call and an merican put. To explore the effect of these free boundary differences on the strangle s price, we compare the price of the merican strangle against the traditional merican call plus merican put approach. The prices were found using Simpson s rule with 00 nodes (implying no need for interpolation when using our and estimates), and were compiled for a range of volatilities (tables 2) and callside strikes (tables 4). In all cases, the prices were found for a range of underlying asset values,, between 0 and 00,000. The tables present only the prices for which the difference between the strangle and the callput sum was greatest. The time to maturity is always set at year. (%) Max Price Difference (000 s) Relative Difference 20, % 40, % 0, % 80, % TBLE. Maximum price differences between the new merican Strangle and the same position formed using an merican Call and an merican Put for a range of 9 values. + <, B, ; 00,000;?? 00,00; prices found from 0 to 00,000 in steps of 0,000. (%) Max Price Difference (000 s) Relative Difference 20, % 40, % 0 2, % 80, % TBLE 2. Maximum price differences between the new merican Strangle and the same position formed using an merican Call and an merican Put for a range of 9 values. <, + B, ; 00,000;?? 00,00; prices found from 0 to 00,000 in steps of 0,000.

22 < 22 CRL CHIRELL ND NDREW ZIOGS (000 s) Max Price Difference (000 s) Relative Difference 00.0, % 0.00, % % % TBLE. Maximum price differences between the new merican Strangle and the same position formed using an merican Call and an merican Put for a range of 9 values. + <, 0?, ; + C B ; prices found from 0 to 00,000 in 00,000; steps of 0,000. (000 s) Max Price Difference (000 s) Relative Difference 00.0, % 0.00, % % % TBLE 4. Maximum price differences between the new merican Strangle and the same position formed using an merican Call and an merican Put for a range of 9 values. <, + 0?, ; + C B ; prices found from 0 to 00,000 in 00,000; steps of 0,000. It should be noted that in all cases, the merican strangle price is always less than or equal to the sum of the corresponding merican call and put prices. This is as expected, since the merican strangle is equivalent to combining a long knockout merican call and a long knockout merican put, where the knockout barriers are and for the call and put respectively. The decrease in the strangle s price reflects the presence of these implicit knockout barriers, and hence the inability to separate the call and put sides in this 9F/ new < strangle position. From table, we see that when, the largest difference appears on the putside, as one would expect. The difference remains around,000 for all the volatilities, but as the volatility increases, the relative difference decreases, and is at most between %. The greatest differences occur when the put is deep inthemoney, and this maximum occurs deeper 9 inthemoney as the volatility increases. When, the maximum difference occurs on the callside. Table 2 shows that this can exceed,000 for a large enough volatility, but as is the case in table, the smaller the volatility, the greater the relative price difference is. This difference never exceeds %, and the greatest differences arrive when the strangle is deep inthemoney

23 < EVLUTION OF MERICN STRNGLES 2 on the call side. Thus the largest relative price deviations will occur for low volatilities. While this result appears counterintuitive, it is important to note that realistic volatilities (e.g. 20%) produce the greatest relative price differences. Table 9/ < considers the maximum price differences for a range of callside strikes, with. s in table, the greatest differences occur deep inthemoney on the putside, and 9 become smaller as increases. similar result is shown in table 4, where, and the difference is now greatest deep inthemoney on the call side. Once the callside strike reaches 0,000, the relative price difference is at most less than 0.%, while the largest relative differences, when the strangle is effectively a straddle, are still no more than %. Overall, it appears that a 0% difference in one of the early exercise boundaries will produce at most a % difference in the price, when comparing this merican strangle with a position formed by going long in both an merican call and an merican put.. CONCLUSION In this paper we have presented a generalisation of McKean s free boundary value problem for pricing merican options. We have considered the example of an merican strangle position, where exercising one side of the position early will knockout the remaining side. McKean s integral equations for this strangle s price were derived, along with the integral equation system for its two free boundaries. It was shown that analytically, the free boundaries for the merican strangle are not exactly equal to those found when valuing independent merican calls and puts. Kim s form of the integral equation system was solved using a scheme typically applied to Volterra integral equations. It was found that numerically, the early exercise boundary of this strangle only differed significantly from the boundaries of corresponding merican calls and puts for certain values of the riskfree rate and continuous dividend yield parameters. The differences became larger as the distance between the strangle s strikes was reduced, and as the time to expiry increased. Comparing the prices of this new strangle to those of a strangle formed using a long merican call and a long merican put, we showed that for several callside strikes and volatilities, our strangle was cheaper than the traditional one by no more than %, and that these differences were most apparent when the strangle was deep inthemoney. Economically, this pricing difference can be viewed as the reduction in value caused by introducing the knockout effect into the new strangle, and foregoing the freedom to separate the call and put sides. The early exercise boundaries for our strangle required that the position be deeper inthemoney than a traditional strangle, to compensate the intrinsic value forgone on the outofthemoney side. If one does not calculate these free boundaries correctly, there is the potential to exercise the merican strangle presented in this paper too early. Despite these early exercise differences, the prices of the two strangles were usually very close. n investor interested in an merican strangle position may be indifferent when choosing between this proposal and a traditional merican strangle, since only

24 .. 24 CRL CHIRELL ND NDREW ZIOGS a small premium is required for the added flexibility of the latter. Whether or not the reduced transaction costs from our selfclosing strangle would benefit the investor is a matter we leave to future study. One avenue for future research would be to consider other complex payoff types, such as an merican butterfly (i.e. concave payoff), or an merican bear/bull spread (i.e. monotonic payoff). These positions can be constructed with similar early exercise conditions to our merican strangle, and can be evaluated using our generalisation of McKean s framework. The numerical method presented should be rigourously tested against existing techniques, such as binomial trees and finite differences. We are also exploring the potential to solve McKean s integral equation in its original form. PPENDIX. THE INCOMPLETE FOURIER TRNSFORM Our aim is to prove that if + then the standard Fourier inversion theorem yields Firstly, and. Heaviside Function!!!!!!!!

25 EVLUTION OF MERICN STRNGLES 2 Next consider Hence or alternatively, and + otherwise.!!! PPENDIX B. PROPERTIES OF THE INCOMPLETE FOURIER TRNSFORM B.. Proof of Proposition. Firstly consider +, +,! +, +, +,! +, $ +, $ +, & Finally by use of boundary conditions () and (2), +, B Next consider +, +,! +, +, +, +, +, +, +, +,! +, +, +, +, +,

26 . 2 CRL CHIRELL ND NDREW ZIOGS where the last equality follows by use of the boundary conditions () and (4), and the transform result (). The last equation simplifies to +, ' +, Finally consider where +, +, +, +,!! +, $ +, +, +, $! pplying the boundary conditions () and (2) we have Hence, finally +, +, +, +, & $ B.2. Proof of Proposition 2. Taking the incomplete Fourier transform of equation (9) with respect to and using () (8), we obtain: +, +, +, +, +, +,!?9 which may be rearranged to B 8 +, +, 9

27 EVLUTION OF MERICN STRNGLES 2 It is a simple matter to rewrite this in terms of and the initial condition is obtained by definition. to produce equations (9)(20), PPENDIX C. DERIVTION OF THE MERICN STRNGLE INTEGRL EQUTIONS C.. Proof of Proposition 4. In ppendix D we derive the convolution result that: Let Hence Using the result that we have $ & " + Next let +%. Hence we have +% &')( "+% &')( ( +% '@ ( & B $ ' (; $ +% $ +% Thus Consider the following Heaviside functions: i)!

28 28 CRL CHIRELL ND NDREW ZIOGS ii) / iii) +%! $ +% iv) $ +%! / $ +% v) B! / $ ; $ vi) $ $ ; $ It is known that Hence and Therefore +% E and +% B $ +% $ +% $ +% $ $ B $ +% $ +% $ +%& $ +% > +% ( +, +, +, +, +, +, +, +, To simplify 2 2 / further, we shall reexpress it in terms of the cumulative standard normal distribution,, where For the first term, : Let +, +, %.

29 .. therefore By defining the integral then becomes For the second term, : Recall that and let which in turn implies that By defining EVLUTION OF MERICN STRNGLES 29 the integral then becomes and % % % Similarly it can be shown that G +, +, +, +, B9;=<; 8 ( % % % % % % +% +% % +% +%

30 0 CRL CHIRELL ND NDREW ZIOGS Thus it is concluded that % ' % % % % +% ' % +% % +% % +% C.2. Proof of Proposition. We begin by noting that where and ) +, ) ) ) ) ) ) +, ) ) ) ) Following the same approach outlined in Kucera and Ziogas (2002), evaluates to: where we set and ) ) ) ) ) ) ) > ) ) ) $ ) ) ) $ ) ) ) ) ) ) ) $ ) ) 92 ) 92 )& With a simple change of notation, equation (0) may be written as it is appears in equations (2)(2). ) () (2) (0)

31 and EVLUTION OF MERICN STRNGLES PPENDIX D. CONVOLUTION INTEGRL FOR THE INCOMPLETE FOURIER TRNSFORM Using the definition for the incomplete Fourier transform of a function, let 8 Consider Let F! therefore B B!!

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