Evaluation of American strangles

Journal of Economic Dynamics & Control 29 (2005) 31–62www.elsevier.com/locate/econbase

Evaluation of American stranglesCarl Chiarella, Andrew Ziogas∗

School of Finance and Economics, University of Technology, P.O. Box 123 Sydney,Broadway NSW 2007, Australia

Abstract

This paper presents a generalisation of McKean’s free boundary value problem for Americanoptions by considering an American strangle position, where exercising one side of the payo/early knocks-out the remaining side. The Fourier transform technique is used to derive a coupledintegral equation system for the strangle’s free boundaries. A numerical algorithm is provided tosolve this system, and these free boundaries are then used to determine the price of the Americanstrangle position. Numerical comparisons between the strangle price and the price of a portfolioformed using a long American call and a long American put option are presented.? 2004 Elsevier B.V. All rights reserved.

JEL classi)cation: C61; D11

Keywords: American options; Coupled Volterra integral equation; Incomplete Fourier transform;Free-boundary problem

1. Introduction

American options are highly common derivative securities in today’s :nancial mar-kets. American calls and puts are frequently written on a range of underlying assets,including stocks, futures, and foreign exchange rates. Since the ground-breaking resultsof Merton (1973) and Black and Scholes (1973) regarding the analytic pricing of Eu-ropean call and put options, a great deal of research has been conducted into applyingthe Black–Scholes framework to American options. While McKean (1965) and Kim(1990) successfully extended the Black–Scholes European option pricing methodologyto American calls and puts, the method has never been generalised to allow a broaderrange of payo/-functions. This paper extends the results of McKean and Kim to a

∗ Corresponding author. Tel.: +61-2 9514 7757; fax: +61-2 9514 7711.E-mail address: [email protected] (A. Ziogas).

0165-1889/$ - see front matter ? 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jedc.2003.04.010

mailto:[email protected]

32 C. Chiarella, A. Ziogas / Journal of Economic Dynamics & Control 29 (2005) 31–62

special type of American strangle position, where the early exercise of one side of theposition will knock-out the remaining side. Through this example, a general Americanoption pricing framework for convex payo/s is provided. A numerical comparison isconducted between the two di/erent American strangle de:nitions, demonstrating thatMcKean’s method leads to useful representations for the price and free boundaries ofAmerican option portfolios.The fundamental di/erence between American and European options is that an Amer-

ican option can be exercised at any time up to and excluding the expiry date. Althoughthis di/erence is conceptually simple, it adds a large degree of mathematical complex-ity to the option pricing problem. Using mathematical results from Kolodoner (1956),McKean (1965) :rst derived the integral equation for both the price and early exerciseboundary of an American call option as the solution to a free boundary problem. Thiswas a natural extension of the Black–Scholes method for European options, and wasexplored further by Van Moerbeke (1974, 1976). A range of alternative methods basedon the Black–Scholes partial di/erential equation (PDE) were proposed, including thequadratic approximation of Barone-Adesi and Whaley (1987), and the compound op-tion approach of Geske and Johnson (1984). Ho et al. (1997) extended the Geske–Johnson technique to include stochastic interest rates. Other methods that have beenconsidered include the :nite di/erence method (Brennan and Schwartz, 1977) and thebinomial approximation (Parkinson, 1977).Karatzas (1988) was one of the :rst to re-visit the topic of pricing American options

as a solution to a free boundary problem using Martingale techniques. Kim (1990) re-produced McKean’s results, extending them to the American put case, and derivingthe exact behaviour of the early exercise boundary near expiry. By taking the limitof the Geske–Johnson method as the number of early exercise dates is increased toin:nity, he found a new set of integral equations for the free boundaries and prices ofAmerican calls and puts. These new integral equations were proven to be mathemati-cally equivalent to those found by McKean, and it was possible to express McKean’sintegral equations in Kim’s form using integration by parts. While McKean’s integralequations follow naturally from applying the Fourier transform technique to the Black–Scholes partial di/erential equation, they involve the derivative of the free boundary,which is diJcult to handle numerically. Furthermore there is no obvious economicinterpretation available for McKean’s integral equations. Kim’s integral equations arefree of the derivative of the free boundary, and their structure decomposes the valueof an American option into the sum of its European value and its early exercise pre-mium. Thus it is possible to apply McKean’s approach (which has the advantage ofbeing more straight forward to apply to complex payo/s, as we shall see below) toextend the Black–Scholes analysis to American options, while then transforming toKim’s representation to express the integral equations in a more numerically tractableand economically intuitive form. Several important papers on American options werepublished around the time of Kim’s :ndings, exploring American option prices in thecontext of free boundary problems. Kim’s results were con:rmed from a di/erent ap-proach by Elliott et al. (1990). Jacka (1991) proved both existence and uniquenessfor the American put option price and early exercise boundary. Carr et al. (1992)expressed the American option price as the sum of its early exercise and intrinsic

C. Chiarella, A. Ziogas / Journal of Economic Dynamics & Control 29 (2005) 31–62 33

value. Jamshidian (1989a, b, 1992) has also conducted extensive research into the freeboundary method for pricing American calls and puts.Despite the large amount of research conducted into the American option pricing

problem, there is still no universal framework with which one can derive the integralequations and free boundaries for a generic payo/ function, either monotonic, convexor concave. It should be noted that Elliott, Myneni and Viswanathan (henceforth EMV)(1990) considered the American straddle pricing problem, deriving the coupled integralequation system for the straddle’s free boundaries, however it is not clear how theirapproach could be extended to general convex/concave payo/ functions. While theyused probability theory in their analysis, in this paper we revisit McKean’s incompleteFourier transform method for American call options and derive the coupled integralequation system via this approach. We extend the :ndings of EMV by applying McK-ean’s method to a special kind of American strangle. If exercised early, the entirepayo/ is optimally realised, making this fundamentally di/erent to an American stran-gle formed using individual calls and puts. Thus the strangle under consideration is theanalogue of the straddle considered by EMV. Alobaidi and Mallier (2002) also con-sidered the American straddle problem, using incomplete Laplace transforms to deriveintegral equations for both its price and free boundary. The primary drawback in theirmethod is that while a solution is readily found in transform-space, inverting the solu-tion appears analytically impossible, making the :nal result very diJcult to implementnumerically. In this paper we demonstrate that the incomplete Fourier transform is apreferable solution technique, leading to solutions that are straight-forward to invert,and thereby more suitable for numerical solution methods.The American strangle is an example of a more general American option position

with a convex payo/ function and indeed the methodology developed in this papermay be applied to evaluate such positions. It should be emphasised that the Fouriertransform technique has the advantage that it can be readily applied to a broad classof market-relevant American option pricing problems beyond that of American port-folios. The integral equations for capped American call options presented by Broadieand Detemple (1995) can be readily derived using McKean’s analysis. Furthermore,Chiarella and Ziogas (2004) demonstrate how to apply McKean’s method to Americancall options under jump-di/usion dynamics. The application of McKean’s method inthis paper is based on the exposition in Kucera and Ziogas (2004), who amplify theanalysis of the incomplete Fourier transform techniques used by McKean. We alsotransform the results into Kim’s integral equation form, and then proceed to implementthese equations numerically to :nd :rstly the strangle’s early exercise boundaries, and:nally the strangle’s price.We use this American strangle contract as our illustrative example for several rea-

sons. The American strangle is a natural generalisation of EMV’s results. It is typicalfor option traders to deal in positions rather than single options, implying that thereexists a market for option portfolios comprised of American options. By applying ouranalysis to this alternative form of American strangle, we are given the opportunity toexplore the di/erences between the two American strangle de:nitions, both analyticallyand numerically. Such analysis has not been presented in the existing literature to ourknowledge.


The alternative strangle de:nition contains several implicit advantages over a ‘tra-ditional’ American strangle constructed using a long position in both an Americancall and an American put. The new strangle is self-closing, since exercising one sideof the position will knock-out the other. These implicit knock-out barriers will makethis strangle cheaper than a ‘traditional’ one, and may have market applications withinpure-volatility strategies. It is important to note that the proposed strangle loses theMexibility to be decomposed into its component options. An integral equation existsfor the delta of the strangle, and this can be solved in the same manner as the pricingintegral equation. Thus the strangle can be hedged in the same manner as any Americancall 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 American strangle option pricingproblem. Section 3 applies the incomplete Fourier transform to solve the PDE in termsof a transform variable. The transform is inverted in Section 4, to provide McKean’sintegral equations for the American strangle price, and a corresponding integral equationsystem for the strangle’s two early exercise boundaries. Section 5 outlines the numericalsolution method for both the free boundaries and strangle price, including the transformfrom McKean’s equations to Kim’s equations. A selection of numerical results areprovided in Section 6, with concluding remarks presented in Section 7.

2. Problem statement

Let Aa1 ;a2 (S; t) be the price of an American strangle position written on an underlyingasset with price S at time t, with time to expiry (T − t). This position is formed usinga long put with strike K1, and a long call with strike K2. Note that K1 ¡K2. Let theearly exercise boundary on the put side be denoted by a1(t), and the early exerciseboundary on the call side be denoted by a2(t). Fig. 1 demonstrates the payo/ andcontinuation region for Aa1 ;a2 (S; t).Under the assumption that the price of the underlying asset is driven by geometric

Brownian motion

dS = S dt + S dW

Continuation Regiona1(t) < S < a2 (t) max( S − K2, 0)min(K1 − S, 0)

Stopping RegionS > a2(t)

Stopping RegionS < a 1(t)

A a1,a2 (S, t)

0 K1 K2a1(t) a2(t) S

Fig. 1. Continuation region in S-space.


with drift , volatility and Wiener process increments dW , it is known that A satis:esthe Black–Scholes PDE:

9A9t +

122S2 92A

9S2 + (r − q)S9A9S − rA= 0; 06 t6T (1)

in the region a1(t)¡S¡a2(t) where r is the risk-free rate, and q is the dividendrate of S (continuously compounded), subject to the following :nal time and boundaryconditions:

Aa1 ;a2 (S; T ) = max(K1 − S; 0) + max(S − K2; 0); 0¡S¡∞; (2)

Aa1 ;a2 (a1(t); t) = K1 − a1(t); t¿ 0; (3)

Aa1 ;a2 (a2(t); t) = a2(t) − K; t¿ 0; (4)

limS→a1(t)

9A9S = −1; lim

S→a2(t)

9A9S = 1; t¿ 0: (5)

Condition (2) is the payo/ function for the strangle at expiry, while conditions (3)–(5) are collectively known as the ‘smooth-pasting’ conditions. These ensure that theprice, Aa1 ;a2 (S; t), and its :rst derivative with respect to S are both continuous. This isnecessary to maintain the Black–Scholes assumption of an arbitrage-free market.Firstly, we shall transform PDE (1) to a forward-in-time equation. Setting S = ex

and t = T − �, we de:ne the transformed function V by

Aa1 ;a2 (S; t) = Vc1 ;c2 (x; �): (6)

The transformed PDE for V is then

9V9� =

122 92V9x2 + k

9V9x − rV; 06 �6T; (7)

in the region ln c1(�)¡x¡ ln c2(�) where k = r − q − 12

2, and the transformed freeboundaries are given by c1(�) = a1(t) and c2(�) = a2(t).The transformed initial and boundary conditions are

Vc1 ;c2 (x; 0) = max(K1 − ex; 0) + max(ex − K2; 0); −∞¡x¡∞; (8)

Vc1 ;c2 (ln c1(�); �) = K1 − c1(�); �¿ 0; (9)

Vc1 ;c2 (ln c2(�); �) = c2(�) − K2; �¿ 0; (10)

limx→ln c1(�)

9V9x = −c1(�); (11)

limx→ln c2(�)

9V9x = c2(�): (12)

In what follows, we will use the notation c1 ≡ c1(�) and c2 ≡ c2(�) for simplicity.


In order to be able to apply transform methods to solve this PDE for Vc1 ;c2 (x; �), thex domain shall be extended to −∞¡x¡∞ by expressing the PDE as

H (ln c2 − x)H (x − ln c1)(9V9� − 1

22 92V9x2 − k

9V9x + rV

)= 0;

where H (x) is the Heaviside step function, de:ned as

H (x) =

1; x¿ 0;12 ; x = 0;

0; x¡ 0:

(13)

The reason for the appearance of the factor of 12 at the point of discontinuity is ex-

plained below. The initial and boundary conditions remain unchanged.

3. Applying the Fourier transform

We propose to solve the problem de:ned by Eqs. (7)–(12) by using the Fouriertransform technique to reduce the PDE to an ODE, whose solution is readily obtain-able. Note that in using the Fourier transform approach, we employ all the standardtechniques that apply when solving the Black–Scholes PDE for the price of Europeanoptions. This includes the assumption that for the purposes of the transform method, thefunction V and its :rst two derivatives with respect to x can be treated as zero whenx tends to ±∞. The assumption is subsequently justi:ed by virtue of the facts thatthe solution obtained satis:es the partial di/erential equation and associated boundaryconditions, and that the solution is unique.Since the x domain is now −∞¡x¡∞, the Fourier transform of the PDE can be

found. The Fourier transform of V; F{Vc1 ;c2 (x; �)}, is de:ned as

F{Vc1 ;c2 (x; �)} =∫ ∞

−∞ei� xVc1 ;c2 (x; �) dx:

Thus, the transformed PDE appears as

F

{H (ln c2 − x)H (x − ln c1)

9V9�

}=

122F

{H (ln c2 − x)H (x − ln c1)

92V9x2

}

+ kF{H (ln c2 − x)H (x − ln c1)

9V9x

}− rF{H (ln c2 − x)H (x − ln c1)V}:

By de:nition

F{H (ln c2 − x)H (x − ln c1)Vc1 ;c2 (x; �)}

=∫ ∞

−∞ei� xH (ln c2 − x)H (x − ln c1)Vc1 ;c2 (x; �) dx


=∫ ln c2

ln c1ei� xVc1 ;c2 (x; �) dx

≡ Fc{Vc1 ;c2 (x; �)} ≡ Vc1; c2(�; �); (14)

where for convenience we introduce the notation Vc1; c2(�; �) to also denote the trans-form. We note that, Fc is an incomplete Fourier transform, since it is equivalent to astandard Fourier transform applied to Vc1 ;c2 (x; �) in the x-domain of ln c1 ¡x¡ ln c2.In Appendix A we show how the incomplete Fourier transform may be derived asa consequence of the standard Fourier transform and there derive the correspondinginversion theorem. To apply the incomplete Fourier transform to PDE (7), we need toconsider three speci:c properties of Fc.

Proposition 1. Given the de)nition of Fc in Eq. (14), the following identities existfor Fc:

Fc{9V9x

}= (c2 − K2)ei� ln c2 − (K1 − c1)ei� ln c1 − i� V ; (15)

Fc{92V9x2

}= ei� ln c2 (c2 − i�(c2 − K2))

− ei� ln c1 (−c1 − i�(K1 − c1)) − �2V ; (16)

Fc{9V9�

}=9V9� − c′

2

c2ei� ln c2 (c2 − K2) +

c′1

c1ei� ln c1 (K1 − c1): (17)

Proof. Refer to Appendix B.1. Note that in deriving the above results, we make useof the so-called ‘smooth pasting’ conditions given in Eqs. (9)–(12).

The PDE can now be transformed, as required.

Proposition 2. The incomplete Fourier transform of PDE (7) with respect to x sat-is)es the ordinary di;erential equation:

dVd�

+(122�2 + ki�+ r

)V = F(�; �); (18)

where

F(�; �) = ei� ln c2[2c22

+(c′2

c2− 2i�

2+ k

)(c2 − K2)

]

− ei� ln c1[− 2c1

2+

(c′1

c1− 2i�

2+ k

)(K1 − c1)

]; (19)

with initial condition

F{Vc1 ;c2 (x; 0)} ≡ Vc1; c2(�; 0)

being calculated from Eq. (8).


Proof. Refer to Appendix B.2.

Instead of solving a PDE for the function Vc1 ;c2 (x; �), we are now faced with thesimpler task of solving ODE (18) for the function Vc1; c2(�; �). This can then beinverted via the Fourier inversion theorem (see Appendix A) to recover the desiredfunction Vc1 ;c2 (x; �). Before concluding this section, we obtain the solution to (18).

Proposition 3. The solution for Vc1; c2(�; �) is given by

Vc1; c2(�; �) = Vc1; c2(�; 0)e−

(12 2�2+k i�+r

)�+

∫ �

0e−

(2�2

2 +k i�+r

)(�−s)

F(�; s) ds:

(20)

Proof. Recalling that c1 and c2 are functions of �, ODE (18) is of the form

dVd�

+ b1(�)V = b2(�; �):

Using the integrating factor eb1(�)�, the solution to the ODE may be expressed as

Vc1; c2(�; �)eb1(�)� − Vc1; c2(�; 0) =∫ �

0b2(�; s)eb1(�)s ds:

Referring back to the original ODE, the solution for V (�; �) is found to be Eq. (20).

4. Inverting the Fourier transform

Having now found Vc1; c2(�; �), it is necessary to recover Vc1 ;c2 (x; �), the Americanstrangle price in the x-� plane. Taking the inverse (complete) Fourier transform of (20)gives

Vc1 ;c2 (x; �) =F−1

{V (�; 0)e

−(12 2�2+k i�+r

)�}

+F−1

∫ �

0e−

(2�2

2 +k i�+r

)(�−s)

F(�; s) ds

≡ V (1)

c1 ;c2 (x; �) + V (2)c1 ;c2 (x; �); ln c1(�)¡x¡ ln c2(�):

We must now determine V (1)c1 ;c2 (x; �) and V (2)

c1 ;c2 (x; �).

Proposition 4. The function V (1)c1 ;c2 (x; �) is given by

V (1)c1 ;c2 (x; �) = [K1e−r�N (−d2(x; �;K1)) − exe−q�N (−d1(x; �;K1))]

+ [exe−q�N (d1(x; �;K2)) − K2e−r�N (d2(x; �;K2))]

− [K1e−r�N (−d2(x; �; c1(0))) − exe−q�N (−d1(x; �; c1(0)))]

− [exe−q�N (d1(x; �; c2(0))) − K2e−r�N (d2(x; �; c2(0)))]; (21)


where

d1(x; �; �) =x − ln � + (r − q+ 2=2)�

√�

;

d2(x; �; �) = d1(x; �; �) − √�

and

N (y) =1√2�

∫ y

−∞e− 2=2 d :

Proof. Refer to Appendix C.1.

Proposition 5. The function V (2)c1 ;c2 (x; �) is given by

V (2)c1 ;c2 (x; �) =

∫ �

0

e−r(�−s)

√2�(� − s)

[e−h2(x; s)Q2(x; s) + e−h1(x; s)Q1(x; s)] ds; (22)

where

hj(x; s) =(x − ln cj(s) + k(� − s))2

22(� − s)(23)

and

Qj(x; s) =2cj(s)

2+

(c′j(s)

cj(s)+

12

[k − (x − ln cj(s))

(� − s)

])(cj(s) − Kj) (24)

for j = 1; 2 and ln c1(�)¡x¡ ln c2(�).

Proof. Refer to Appendix C.2.

Hence, the value of the American strangle is given in the x-� plane by

Vc1 ;c2 (x; �) = V (1)c1 ;c2 (x; �) + V (2)

c1 ;c2 (x; �);

06 �6T ; ln c1(�)¡x¡ ln c2(�): (25)

Eq. (25) expresses the value of the American strangle position in terms of theearly exercise boundaries c1(�) and c2(�). At this point these remain unknown, butwe are able to obtain an integral equation system that determines them by requiringthe expression for Vc1 ;c2 (x; �) to satisfy the early exercise boundary conditions (9) and(10). Recalling our de:nition for the Heaviside function, the following integral equationsystem is obtained:

c2(�) − K2

2= Vc1 ;c2 (ln c2(�); �); (26)

K1 − c1(�)2

= Vc1 ;c2 (ln c1(�); �); (27)

where Vc1 ;c2 (x; �) is given by Eq. (25) in conjunction with (21)–(24). The factor of 12

appears in the left-hand side of (26) and (27) due to properties of the Fourier transform.


Recall that the complete Fourier transform was applied to discontinuous functions ofthe form H (ln c1 − x)H (x − ln c2)f(x; �). As proved in Dettman (1965, p. 360), theinverted Fourier transform of a discontinuous function will converge to the midpoint ofthe discontinuity. Thus in Eqs. (26) and (27), when V is evaluated at either ln c1(�) orln c2(�), the factor of 1

2 must be introduced into the left-hand side. This is accountedfor by our Heaviside function de:nition in Eq. (13).The system of integral equations (26) and (27) must be solved simultaneously using

numerical methods to :nd c1(�) and c2(�) since analytical solutions seem impossi-ble. Once these are found, it is a simple matter to evaluate Vc1 ;c2 (x; �) via numericalintegration.

5. Numerical implementation

To make the task of numerical implementation less complicated, we will transformEqs. (21)–(25) into the form presented by Kim (1990). This essentially has the e/ectof removing the c′

1(�) and c′2(�) terms from the integral by using integration by parts.

The :rst step is to re-write the pricing equation in terms of the original underlyingasset S.

Proposition 6. The solution to the free boundary value problem (1)–(5) in terms ofS and � is given by

Ac1 ;c2 (S; �) = A(1)c1 ;c2 (S; �) + A(2)

c1 ;c2 (S; �); (28)

where

A(1)c1 ;c2 (S; �) = [K1e−r�N (−d2(A; �;K1)) − Se−q�N (−d1(S; �;K1))]

+[Se−q�N (d1(S; �;K2)) − K2e−r�N (d2(x; �;K2))]

−[K1e−r�N (−d2(S; �; c1(0))) − Se−q�N (−d1(S; �; c1(0)))]

−[Se−q�N (d1(S; �; c2(0))) − K2e−r�N (d2(S; �; c2(0)))] (29)

with

d1(S; �; �) =ln(S=�) + (r − q+ 2=2)�

√�

;

d2(S; �; �) = d1(S; �; �) − √�

and

A(2)c1 ;c2 (S; �) =

∫ �

0

e−r(�−%)

√2�(� − %)

[e−h2(S;%)Q2(S; %) + e−h1(S;%)Q1(S; %)] d% (30)

with

hj(S; %) =(ln(S=cj(%)) + k(� − %))2

22(� − %)(31)


and

Qj(S; %) =2cj(%)

2+

(c′j(%)

cj(%)+

12

[k − ln(S=cj(%))

(� − %)

])(cj(%) − Kj) (32)

for j = 1; 2 and c1(�)¡S¡c2(�).

Proof. Recall that x = ln S, and substitute this into Eqs. (21)–(25).

Proposition 7. Eqs. (28)–(32) can be expressed as follows:

Ac1 ;c2 (S; �) = AP(S; �) + AC(S; �); (33)

where

AP(S; �) =K1e−r�N (−d2(S; �;K1)) − Se−q�N (−d1(S; �;K1))

+∫ �

0[K1re−r(�−%)N (−d2(S; � − %; c1(%)))

−Sqe−q(�−%)N (−d1(S; � − %; c1(%)))] d% (34)

and

AC(S; �) = Se−q�N (d1(S; �;K2)) − K2e−r�N (d2(S; �;K2))

+∫ �

0[Sqe−q(�−%)N (d1(S; � − %; c2(%)))

−K2re−r(�−%)N (d2(S; � − %; c2(%)))] d%: (35)

Proof. The above is derived using integration by parts, as outlined in the appendixof Kim (1990). Note that Eqs. (34) and (35) do not involve the derivatives c′

1and c′

2.

The integral equation system for the free boundaries c1(�) and c2(�) is now

c2(�) − K2 = Ac1 ;c2 (c2(�); �); (36)

K1 − c1(�) = Ac1 ;c2 (c1(�); �): (37)

It is of value to note that Eq. (33) is simply the sum of the integral pricing equationsfor an American put and an American call option. The added complexity in pricingan American strangle therefore arises from the early exercise boundaries. Each freeboundary is dependent upon the other free boundary in system (36) and (37), andtherefore these boundaries are not equal to those found when valuing an American calland put option separately. Thus it is important to understand the nature of the earlyexercise boundaries for American option portfolios in order to obtain the correct earlyexercise boundary values.To solve this system, we propose using a numerical scheme similar to that usually

used for Volterra integral equations. Firstly, discretise the time variable � into n equally


spaced intervals of length h. Thus �i = ih for i = 0; 1; 2; : : : ; n, and h= T=n. Followingthe methods of Kim (1990), it can be readily shown that the initial values are givenby (see Appendix D)

c1(0) = min(K1; K1

rq

); and c2(0) = max

(K2; K2

rq

): (38)

Thus by starting at i=1, there are only two unknown values in system (36) and (37)for each i, namely c1(�i) and c2(�i). We use Simpson’s rule to evaluate the integralterms.For each i beginning with i = 1, the bisection method is applied to the following

non-linear equation to :nd c1(�i):

c1(ih) = K1 − AP(c1(ih); ih; c1) − AC(c1(ih); ih; c2); (39)

where

AP(c1(ih); ih; c1) =K1e−rihN (−d2(c1(ih); ih;K1))

−c1(ih)e−qihN (−d1(c1(ih); ih;K1))

+ hi∑

j=0

wj[K1re−rh(i−j)N (−d2(c1(ih); h(i − j); c1(jh)))

−c1(ih)qe−qh(i−j)N (−d1(c1(ih); h(i − j); c1(jh)))] (40)

and

AC(c1(ih); ih; c2) = c1(ih)e−qihN (d1(c1(ih); ih;K2))

−K2e−rihN (−d2(c1(ih); ih;K2))

+ hi∑

j=0

wj[c1(ih)qe−qh(i−j)N (d1(c1(ih); h(i − j); c2(jh)))

−K2re−rh(i−j)N (d2(c1(ih); h(i − j); c2(jh)))]: (41)

The summation weights wj are those dictated by the numerical integration scheme, inthis case Simpson’s rule, and the extended Simpson’s rule (used on the end furthestfrom expiry whenever i is odd). The bisection method was chosen over more com-plex techniques, such as Newton’s method, because the monotonic nature of the freeboundaries enables us to eJciently reduce our search region for the unknown root as iincreases. The bisection method also saves us from having to evaluate the derivativesof the free boundary integral equations.Similarly, we apply the bisection method to the following to :nd c2(�i):

c2(ih) = K2 + AP(c2(ih); ih; c1) + AC(c2(ih); ih; c2); (42)


where

AP(c2(ih); ih; c1) =K1e−rihN (−d2(c2(ih); ih;K1))

− c2(ih)e−qihN (−d1(c2(ih); ih;K1))

+ hi∑

j=0

wj[K1re−rh(i−j)N (−d2(c2(ih); h(i − j); c1(jh)))

− c2(ih)qe−qh(i−j)N (−d1(c2(ih); h(i − j); c1(jh)))] (43)

and

AC(c2(ih); ih; c2) = c2(ih)e−qihN (d1(c2(ih); ih;K2))

−K2e−rihN (−d2(c2(ih); ih;K2))

+ hi∑

j=0

wj[c2(ih)qe−qh(i−j)N (d1(c2(ih); h(i − j); c2(jh)))

−K2re−rh(i−j)N (d2(c2(ih); h(i − j); c2(jh)))]: (44)

It is important to note that at j = i, both d1 and d2 are singular. This can be handledby using the following:

lim%→�

N (d1(c1(�); � − %; c2(%))) = lim%→�

N (−d1(c2(�); � − %; c1(%))) = 0; (45)

lim%→�

N (d1(c2(�); � − %; c2(%))) = lim%→�

N (−d1(c1(�); � − %; c1(%))) = 0:5: (46)

These limits are the same for d2. It is also important to see that while, for example,(39)–(41) depends upon c2(�i) it does not explicitly require c2(�i), due to the needto use limits (45) and (46) at j = i. Hence, the simultaneous integral equation system(36) and (37) can be solved by :nding c1(�i) using all known values of c1(�j) andc2(�j); j=0; 1; 2; : : : ; i−1. That is, the interdependence at the ith time point is removeddue to the need to consider the limits of d1 and d2 when evaluating (39)–(41) and(42)–(44).The above numerical scheme is :rstly carried out using a time-step size of h, and

then repeated using h=2. In each case, since it is necessary to alternate between twodi/erent numerical integration schemes (for odd and even values of i) it turns out thatthe free boundaries have non-monotonic gradients. This is recti:ed by combining thetwo estimates using Richardson’s extrapolation. Pricing the American strangle is thenachieved via numerical integration using Simpson’s rule, combined with the estimatesof c1(�i) and c2(�i).The algorithm American Strangle Price presented below demonstrates how the

Richardson’s extrapolation was implemented when solving the integral equation systemfor the free boundaries. Note that the algorithm allows for a :ner grid to be used closeto the expiry date of the strangle, since this is the region where the free boundariesexperience the most rapid change. The :nal American strangle price is readily obtainedusing numerical integration once the free boundaries have been estimated.


Algorithm. American Strangle Price Input: S, r, q, , K1, K2, T (time to expiry), n(number of time intervals), nsml (number of starting time intervals for :ner grid), ns(number of time intervals within the :ne-grid region).Output: AS (American strangle price).

1. h= T=n; hs = h ∗ nsml=ns2. c1;0 = K1 ∗ min(r=q; 1); c2;0 = K2 ∗ max(r=q; 1)3. a1;0 = c1;0; a2;0 = c2;0; a21;0 = c1;0; a22;0 = c2;04. b1;0 = c1;0; b2;0 = c2;0; b21;0 = c1;0; b22;0 = c2;05. for i = 1 to ns6. do solve the integral equation system for b1; i and b2; i using time step-size hs7. for i = 1 to ns ∗ 28. do solve the integral equation system for b21; i and b22; i using time step-size hs=29. for i = 1 to nsml ∗ 2

10. do j = i ∗ ns=(2 ∗ nsml)11. a21; i = (24 ∗ b21;2∗j − b1; j)=(24 − 1)12. a22; i = (24 ∗ b22;2∗j − b2; j)=(24 − 1)13. for i = 1 to nsml14. do a1; i = a21; i∗215. a2; i = a22; i∗216. for i = 1 + nsml to n17. do solve the integral equation system for a1; i and a2; i using time step-size h18. for i = 1 + nsml to n ∗ 219. do solve the integral equation system for a21; i and a22; i using time step-size h=220. for i = 1 + nsml to n21. do c1; i = (24 ∗ a21;2∗i − a1; i)=(24 − 1)22. c2; i = (24 ∗ a22;2∗i − a2; i)=(24 − 1)23. calculate AS using c1, c2 and S

6. Results

Firstly we examined the accuracy of the numerical scheme being implemented. Themethod was applied to a 1-year American strangle position, using n = 100; 200; 400and 800. This has been compared against a standard Crank–Nicolson scheme using4 time steps per day. The r and q parameters were chosen to be non-zero and un-equal. The results are summarised in Table 1 for :ve spot prices, chosen to representat-the-money, in-the-money and out-of-the-money prices for the strangle within thecontinuation region.From Table 1, it can be seen that for the strangle at-the-money on the call side, the

Crank–Nicolson scheme has converged to four decimal places, while it has convergedto :ve decimal places at the other spot values. Thus we take the Crank–Nicolsonresults as being the true solution to an accuracy of around four decimal places.For all the values of n used, the American strangle prices found using Kim’s integral

equation system match those found using Crank–Nicolson to four decimal places. We


Table 1American strangle price found numerically

S CN 60 000 CN 120 000 Kim 100 Kim 200 Kim 400 Kim 800

0.75 0.275648 0.275648 0.275647 0.275647 0.275647 0.2756471.00 0.100322 0.100319 0.100332 0.100332 0.100332 0.1003321.25 0.038560 0.038560 0.038563 0.038562 0.038561 0.0385611.50 0.092316 0.092314 0.092344 0.092341 0.092341 0.0923401.75 0.255619 0.255619 0.255631 0.255632 0.255633 0.255633

The Crank–Nicolson :nite di/erence scheme involved four time steps per day, and involved 60 000 and120 000 space-nodes, as indicated in the table. The numerical scheme for solving Kim’s integral equationsused n = 100; 200; 400 and 800, respectively, as indicated in the table. The parameter values were r = 5%,q = 10%, T − t = 1, K1 = 1, K2 = 1:5 and = 20%.

conclude from these results that the numerical method employed in solving the integralequations has an accuracy of four decimal places. It can also be seen that the numericalscheme for Kim’s integral equations has converged to :ve decimal places for n aslow as 100. We therefore select n = 200 for the purposes of generating all furtherresults. The algorithm was implemented using LAHEYTMFORTRAN 95 on a PC witha Pentium III 500 MHz processor, 128 MB of RAM, and running the Windows 98operating system. With n=200, the code takes approximately 3 min and 15 s to solvethe integral equation system for the American strangle’s free boundaries.By considering the number of calculations required by the code for a given n, we

can provide some insight into how the value of n a/ects the code’s run-time. Assumethat when using the bisection method to solve the integral equation system, the meannumber of iterations required for each application is m. When :nding the ith pair offree boundary values for i=1; 2; : : : ; n, we must evaluate the integrand at i+1 points.Therefore to estimate each free boundary using the bisection method, we are requiredto make m[(n+ 1)(n+ 2)=2] integrand evaluations. Thus it can be concluded that thenumber of operations required by our code is of order mn2.

To demonstrate the early exercise boundaries and price properties of the Americanstrangle, we implemented the method using n = 200 time nodes. To improve the ac-curacy of the method where the free boundaries change rapidly, a :ner grid was usedbetween the :rst three nodes (speci:cally, 40 nodes between i=0 and 2). The methodwas also applied in the same manner to the American call and put contracts whichde:ne the components of the strangle’s payo/ function. By comparing the results forthe strangle against those of the independent call and put, we can demonstrate howthe American strangle’s free boundaries and price are a/ected by the interdependencebetween c1(�) and c2(�).Firstly, consider an American strangle with 1 year until maturity. Let the put-side

strike be 1 and the call-side strike be 1.1, with the volatility of the underlying at 20%.In Figs. 2–5, we present the call- and put-side boundaries for the American strangle,with r ¿q (Figs. 2 and 5), r ¡q (Figs. 3 and 4), and :nally r = q (Figs. 4 and 5).In all cases, we include the free boundary for the corresponding American call or put.The same results are presented in Figs. 2, 3, 6 and 7, but the call-side strike has beenreduced to 1.001, moving the strangle position closer to a straddle.


2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

2.4

2.45

2.5

0 0.2 0.4 0.6 0.8 1

c 2(T

-t)

(T-t)

Free Boundary: American Call and Strangle

Call: K2 = 1.1, r > qStrangle: K2 = 1.1, r > q

Call: K2 = 1.001, r > qStrangle: K2 = 1.001, r > q

Fig. 2. K1 = 1, = 20%, r = 10% and q = 5%.

0.44

0.45

0.46

0.47

0.48

0.49

0.5

0 0.2 0.4 0.6 0.8 1

c 1(T

-t)

(T-t)

Free Boundary: American Put and Strangle

Put: K2 = 1.1, r < qStrangle: K2 = 1.1, r < q

Put: K2 = 1.001, r < qStrangle: K2 = 1.001, r < q

Fig. 3. K1 = 1, = 20%, r = 5% and q = 10%.

There are several distinct features that can be ascertained from these free bound-ary plots. The :rst is that the relative values of r and q directly a/ect whether ornot the American strangle free boundaries will show signi:cant divergence from thecorresponding American call and put boundaries. In particular, when r ¿q, only the


1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

0 0.2 0.4 0.6 0.8 1

c 2(T

-t)

(T-t)

Free Boundary: American Call and Strangle K2 = 1.1

Call: r < qStrangle: r < q

Call: r = qStrangle: r = q

Fig. 4. K1 = 1, K2 = 1:1, = 20%, q = 10% and r = 5% when r ¡q.

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.2 0.4 0.6 0.8 1

c 1(T

-t)

(T-t)

Free Boundary: American Put and Strangle K2 = 1.1

Put: r > qStrangle: r > q

Put: r = qStrangle: r = q

Fig. 5. K1 = 1, K2 = 1:1, = 20%, r = 10% and q = 5% when r ¿q.

put-side boundaries diverge, and when r ¡q only the call-side boundaries diverge.When r=q, there is divergence in both boundaries, but it is smaller than in the other twocases.


1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

0 0.2 0.4 0.6 0.8 1

c 2(T

-t)

(T-t)

Free Boundary: American Call and Strangle K2 = 1.001

Call: r < qStrangle: r < q

Call: r = qStrangle: r = q

Fig. 6. K1 = 1, K2 = 1:001, = 20%, q = 10% and r = 5% when r ¡q.

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.2 0.4 0.6 0.8 1

c 1(T

-t)

(T-t)

Free Boundary: American Put and Strangle K2 = 1.001

Put: r > qStrangle: r > q

Put: r = qStrangle: r = q

Fig. 7. K1 = 1, K2 = 1:001, = 20%, r = 10% and q = 5% when r ¿q.


1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0 0.2 0.4 0.6 0.8 1

c 2(T

-t)

(T-t)

Free Boundaries by Volatility: American Call and Strangle K2=1.001, r=5%. q=10%

Call: 20% volStrangle: 20% vol



Fig. 8. Changes in the call-side free boundaries for di/erent values of .

Since the early exercise of the in-the-money side of the strangle will knock-out theother side, it is expected that the strangle will have to be deeper in-the-money towarrant early exercise than one formed using independent American calls and puts. Inall cases, the call-side free boundary for the strangle is always greater than or equalto that of the corresponding American call free boundary, while the put-side is alwaysless than or equal to that of the corresponding American put free boundary. This is inkeeping with the economic intuition behind the American strangle position.In all three cases of r and q values, moving the call-side’s strike closer to the

put-side’s strike increases any divergence between the American strangle free bound-aries and those of the corresponding American call and put. This is again as one wouldexpect, since the closer the strangle is to being a straddle, the more intrinsic value theout-of-the-money strangle component will contribute to the early exercise decision. Itcan also be seen that as the time to maturity increases, the divergence between thestrangle free boundaries and the corresponding call and put boundaries increases. Whenthe strangle has a very short time to maturity, say 2 weeks or less, then the divergencebetween the two free boundaries becomes minimal.Fig. 8 demonstrates how the early exercise boundary of the American call and the

call-side free boundary of the strangle vary with changes in the volatility of S. Wefocus on the case where r ¡q, since this is when the call-side di/erences are mostpronounced. As the volatility increases, the divergence between the corresponding freeboundaries becomes larger. A similar result can be seen for the put side of the stran-gle, and it’s corresponding American put option, as displayed in Fig. 9, although theincreased divergence is far less obvious. We used r ¿q, again so that we could focus


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

c 1(T

-t)

(T-t)

Free Boundaries by Volatility: American Put and Strangle K2=1.001, r=10%. q=5%

Put: 20% volStrangle: 20% vol



0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

c 1(T

-t)

(T-t)

Free Boundaries by Volatility: American Put and Strangle K2=1.001, r=10%. q=5%




Fig. 9. Changes in the put-side free boundaries for di/erent values of .

on the case where the put-side di/erences were most extreme. Smaller values of rand q were also explored, and this produced similar behaviour in the location of thefree boundaries as presented for changes in the volatility. These results have not beenpresented for the sake of brevity.While it is clear that the early exercise boundaries for the strangle are not always

equivalent to those of the component American call and put in the examples provided,the di/erence never exceeds 0.1, which in relative terms is no more than 10% of theput-side’s strike price. Past research into American options, such as Ju (1998) andAitSahalia and Lai (2001), has found that the price of American call and put optionsis not greatly a/ected by the free boundary estimate used. While a 10% di/erence inthe free boundary has obvious early exercise timing repercussions, it remains to beseen whether the price of the strangle using these free boundaries is far removed fromthat of a strangle priced simply using the sum of an American call and an Americanput. To explore the e/ect of these free boundary di/erences on the strangle’s price,we compare the price of the American strangle against the ‘traditional’ American callplus American put approach. The prices were found using Simpson’s rule with 100nodes (implying no need for interpolation when using our c1(�) and c2(�) estimates),and were compiled for a range of volatilities (Tables 2 and 3) and call-side strikes(Tables 4 and 5). In all cases, the prices were found for a range of underlying assetvalues, S, between 0 and 300 000. Thus, these results are indicative of a position ina contract involving several thousand American strangle contracts. The tables presentonly the prices for which the di/erence between the strangle and the call-put sum wasgreatest. The time to maturity is always set at 1 year.


Table 2Maximum price di/erences between the American Strangle and the same position formed using an AmericanCall and an American Put for a range of values

(%) Max price di/erence S (000s) Relative di/erence (%)

20 1201.13 80 5.8440 1156.51 60 2.8560 1078.79 40 1.8080 1137.30 30 1.62

r = 10%, q= 5%, T − t = 1; K1 = 100 000; K2 = 100 100 and prices of underlying range from S = 0 toS = 300 000 in steps of 10 000.

Table 3Maximum price di/erences between the American Strangle and the same position formed using an AmericanCall and an American Put for a range of values

(%) Max price di/erence S (000s) Relative di/erence (%)

20 1478.68 130 4.9340 1996.79 170 2.8360 2776.88 240 1.9880 3458.06 300 1.71

r = 5%, q= 10%, T − t = 1, K1 = 100 000, K2 = 100 100 and prices of underlying range from S = 0 toS = 300 000 in steps of 10 000.

Table 4Maximum price di/erences between the American Strangle and the same position formed using an AmericanCall and an American Put for a range of K2 values

K2 (000s) Max price di/erence S (000s) Relative di/erence (%)

100.01 1207.49 80 5.87101.00 1138.72 80 5.56110.00 630.92 80 3.14150.00 11.23 80 0.06

r = 10%, q = 5%, T − t = 1, K1 = 100 000, = 20% and prices of underlying range from S = 0 toS = 300 000 in steps of 10 000.

It should be noted that in all cases, the American strangle price is always less thanor equal to the sum of the corresponding American call and put prices. This is asexpected, since the American strangle is equivalent to combining a long knock-outAmerican call and a long knock-out American put, where the knock-out barriers arec1(�) and c2(�) for the call and put, respectively. The decrease in the strangle’s pricereMects the presence of these implicit knock-out barriers, and hence the inability toseparate the call and put sides in this new strangle position.From Table 2, we see that when r ¿q, the largest di/erence appears on the put-side,

as one would expect. The di/erence remains around 1000 for all the volatilities, but


Table 5Maximum price di/erences between the American Strangle and the same position formed using an AmericanCall and an American Put for a range of K2 values

K2 (000s) Max price di/erence S (000s) Relative di/erence (%)

100.01 1482.61 130 4.93101.00 1433.60 130 4.92110.00 748.99 140 2.50150.00 15.16 180 0.05

r = 5%, q = 10%, T − t = 1, K1 = 100 000, = 20% and prices of underlying range from S = 0 toS = 300 000 in steps of 10 000.

as the volatility increases, the relative di/erence decreases, and is at most between 5%and 6%. The greatest di/erences occur when the put is deep in-the-money, and thismaximum occurs deeper in-the-money as the volatility increases.When r ¡q, the maximum di/erence occurs on the call-side. Table 3 shows that this

can exceed 3000 for a large enough volatility, but as is the case in Table 2, the smallerthe volatility, the greater the relative price di/erence is. This di/erence never exceeds5%, and the greatest di/erences arise when the strangle is deep in-the-money on thecall side. Thus the largest relative price deviations will occur for low volatilities. Whilethis result appears counter-intuitive, it is important to note that ‘realistic’ volatilities(e.g. 20%) produce the greatest relative price di/erences.Table 4 considers the maximum price di/erences for a range of call-side strikes, with

r ¿q. As in Table 2, the greatest di/erences occur deep in-the-money on the put-side,and become smaller as K2 −K1 increases. A similar result is shown in Table 5, wherer ¡q, and the di/erence is now greatest deep in-the-money on the call side. Once thecall-side strike reaches 150 000, the relative price di/erence is at most less than 0.1%,while the largest relative di/erences, when the strangle is e/ectively a straddle, arestill no more than 6%. Overall, it appears that a 10% di/erence in one of the earlyexercise boundaries will produce at most a 6% di/erence in the price, when comparingthis American strangle with a position formed by going long in both an American calland an American put. From a market perspective, it appears that the reduction in thestrangle’s premium by foregoing the Mexibility to separate the strangle’s componentsis extremely small. There appears little premium advantage in creating an Americanstrangle with early exercise triggered knock-out features for the out-of-the-money sideof the position. This clearly demonstrates that American strangles would generallybe of little value to investors and traders on the premium side. To what extent thisalternate de:nition would impact on transaction and investment costs to the holderremains unknown, since the greatest di/erences arise in the timing of early exercise,and the volume of transactions required to close-out the strangle position.

7. Conclusion

In this paper we have presented a generalisation of McKean’s free boundary valueproblem for pricing American options. We have considered the example of an American


strangle position, where exercising one side of the position early will knock-out the re-maining side. McKean’s integral equations for this strangle’s price were derived, alongwith the integral equation system for its two free boundaries. The integral equationswere re-expressed in a more economically intuitive form using Kim’s simpli:cations.It was shown that analytically, the free boundaries for the American strangle are notequal to those found when valuing independent American calls and puts.Kim’s form of the integral equation system was solved using a scheme typically

applied to non-linear Volterra integral equations. It was found that numerically, theearly exercise boundary of this strangle only di/ered signi:cantly from the bound-aries of corresponding American calls and puts for certain values of the risk-free rateand continuous dividend yield parameters. The di/erences became larger as the dis-tance 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 longAmerican call and a long American put, we showed that for several call-side strikesand volatilities, our strangle was cheaper than the ‘traditional’ one by no more than 6%,and that these di/erences were most apparent when the strangle was deep in-the-money.Economically, this pricing di/erence can be viewed as the reduction in value causedby introducing the knock-out e/ect into the new strangle, and foregoing the freedomto separate the call and put sides.The early exercise boundaries for our strangle required that the position be deeper

in-the-money than a ‘traditional’ strangle, to compensate the intrinsic value forgone onthe out-of-the-money side. If one does not calculate these free boundaries correctly,there is the potential to exercise the American strangle presented in this paper tooearly. Despite these early exercise di/erences, the prices of the two strangles wereusually very close, and an important contribution of this paper has been to quantify thisdi/erence. An investor interested in an American strangle position may be indi/erentwhen choosing between this proposal and a ‘traditional’ American strangle, since onlya small premium is required for the added Mexibility of the latter. Whether or not thereduced transaction costs from the self-closing strangle would bene:t the investor is amatter we leave to future study.The methodology developed here is applicable to American positions with quite

general convex or concave payo/s. One avenue for future research would be to considerother complex payo/ types, such as an American butterMy (i.e. concave payo/), or anAmerican bear/bull spread (i.e. monotonic payo/). These positions can be constructedwith similar early exercise conditions to our American strangle, and can be evaluatedusing our generalisation of McKean’s framework. The numerical method presentedshould be rigourously tested against existing techniques, such as binomial trees and:nite di/erences. We are also exploring the potential to numerically solve McKean’sintegral equation in its original form.

Appendix A. The incomplete Fourier transform

Our aim is to prove that if f(x; �) = H (b − x)H (x − a)g(x; �); a ≡ a(�); b ≡b(�); a¡b ∀ �∈ [0; T ], and H (x) ≡ Heaviside Function, then application of the


standard Fourier inversion theorem

f(x; �) =12�

∫ ∞

−∞

[∫ ∞

−∞f(x; �)ei� xdx

]e−i� x d�; −∞¡x¡∞;

yields

g(x; �) =12�

∫ ∞

−∞

[∫ b

ag(x; �)ei� xdx

]e−i� x d�; a¡x¡b;

which may be regarded as an inversion theorem for the incomplete Fourier transform.Firstly,

RHS =12�

∫ ∞

−∞

[∫ ∞

−∞H (b − x)H (x − a)g(x; �)ei� xdx

]e−i� x d�

=12�

∫ ∞

−∞

[∫ b

ag(x; �)ei� xdx

]e−i� x d�:

Next consider

LHS = H (b − x)H (x − a)g(x; �) =

{g(x; �); a¡x¡b;

0 otherwise:

Hence

H (b − x)H (x − a)g(x; �) =12�

∫ ∞

−∞

[∫ b

ag(x; �)ei� x dx

]e−i� x d�;

−∞¡x¡∞or alternatively,

g(x; �) =12�

∫ ∞

−∞

[∫ b


]e−i� x d�; a¡x¡b

and

g(x; �)2

=12�

∫ ∞

−∞

[∫ b


]e−i� x d�; x = a; b:

(See Section 4 for an explanation regarding the factor of 12 on the left-hand side.)

Appendix B. Properties of the incomplete Fourier transform

B.1. Proof of Proposition 1

Firstly consider

Fc{9V9x

}= Vc1 ;c2 (ln c2; �)e

i� ln c2 − Vc1 ;c2 (ln c1; �)ei� ln c1 − i� Vc1; c2(�; �):


Finally by use of boundary conditions (9) and (10),

Fc{9V9x

}= (c2 − K2)ei� ln c2 − (K1 − c1)ei� ln c1 − i� V :

Next consider

Fc{92V9x2

}=9Vc1 ;c2 (x; �)

9x

∣∣∣∣x=ln c2

· ei� ln c2 − 9Vc1 ;c2 (x; �)9x

∣∣∣∣x=ln c1

·ei� ln c1 − i�Fc{9V9x

}= c2ei� ln c2 + c1ei� ln c1 − i�[(c2 − K2)ei� ln(c2)

−(K1 − c1)ei� ln(c1) − i� V ];

where the last equality follows by use of the boundary conditions (11) and (12), andthe transform result (15). The last equation simpli:es to

Fc{92V9x2

}= ei� ln c2 (c2 − i�(c2 − K2)) − e−i� ln c1 (−c1 − i�(K1 − c1)) − �2V :

Finally consider

Fc{9V9�

}=99�

[∫ ln c2

ln c1ei� xVc1 ;c2 (x; �) dx

]− c′

2

c2ei� ln c2Vc1 ;c2 (ln c2; �)

+c′1

c1ei� ln c1Vc1 ;c2 (ln c1; �)

=99� [F

c{V}] − c′2

c2ei� ln c2Vc1 ;c2 (ln c2; �) +

c′1

c1ei� ln c1Vc1 ;c2 (ln c1; �);

where c′j ≡ dcj(�)=d�, j=1; 2. Applying the boundary conditions (9) and (10) we have

Fc{9V9�

}=9V9� − c′

2

c2ei� ln c2 (c2 − K2) +

c′1

c1ei� ln c1 (K1 − c1):

B.2. Proof of Proposition 2

Taking the incomplete Fourier transform of Eq. (7) with respect to x and using(15)–(17), we obtain

9V9� +

(122�2 + ki�+ r

)V

=ei� ln c2[c′2

c2(c2 − K2) +

122(c2 − i�(c2 − K2)) + k(c2 − K2)

]

− ei� ln c1[c′1

c1(K1 − c1) +

122(−c1 − i�(K1 − c1)) + k(K1 − c1)

]:


It is a simple matter to rewrite this in terms of F(�; �) to produce Eqs. (18) and (19),and the initial condition is obtained by de:nition.

Appendix C. Derivation of the American strangle integral equations

C.1. Proof of Proposition 4

We shall evaluate the inverse Fourier transform, V (1)c1 ;c2 (x; �), using the standard Fourier

convolution result:

F−1{F(�; �1)G(�; �2)} =∫ ∞

−∞f(x − u; �1)g(u; �2) du:

Let

F(�; �1) = e−( 12 2�2+k i�+r)�:

Hence

f(x; �1) =e−r�

2�

∫ ∞

−∞e− 1

2 2�2�−i�(x+k�) d�=e−r�

√2��

e−(x+k�)2=22�;

where we have used the result that∫ ∞

−∞ e−p�2−q� d�= eq2=4p

√�=p; Re(p)¿ 0.

Next let G(�; �2) = Vc1; c2(�; 0). Hence we have

g(x; �2) =H (ln c2(0) − x)H (x − ln c1(0))Vc1 ;c2 (x; 0)

=H (ln c2(0) − x)H (x − ln c1(0))[H (lnK1 − x)(K1 − ex)

+H (x − lnK2)(ex − K2)]:

Thus

V (1)c1 ;c2 (x; �) =

∫ ∞

−∞

e−r�

√2��

e− (x−u+k�)2

22� H (ln c2(0) − u)H (u − ln c1(0))

× [H (lnK1 − u)(K1 − eu) + H (u − lnK2)(eu − K2)] du:

It is known that c1(0)6K1 and c2(0)¿K2. Hence

H (lnK1 − u)H (ln c2(0) − u)H (u − ln c1(0)) = H (lnK1 − u)H (u − ln c1(0))

and

H (u − lnK2)H (ln c2(0) − u)H (u − ln c1(0)) = H (u − lnK2)H (ln c2(0) − u):

Therefore

V (1)c1 ;c2 (x; �) =

∫ ln K1

ln c1(0)

K1e−r�

√2��

e−(x−u+k�)2=22� du

−∫ ln K1

ln c1(0)

eue−r�

√2��

e−(x−u+k�)2=22� du


+∫ ln c2(0)

ln K2

eue−r�

√2��

e−(x−u+k�)2=22� du

−∫ ln c2(0)

ln K2

K2e−r�

√2��

e−(x−u+k�)2=22� du

= I1 − I2 + I3 − I4:

To simplify V (1)c1 ;c2 (x; �) further, we shall re-express it in terms of the cumulative standard

normal distribution, N (y). For the :rst term, I1:

I1 =K1e−r�

√2��

∫ ln K1

ln c1(0)e−(x−u+k�)2=22� du:

By de:ning d2(x; �; �) ≡ (x − ln � + k�)=√�, the integral I1 then becomes

I1 = K1e−r�[N (−d2(x; �;K1)) − N (−d2(x; �; c1(0)))]:

For the second term, I2:

I2 =e−r�

√2��

∫ ln K1

ln c1(0)e−(u−[x+(k+2)�])2=22�ex+k�+2�=2 du:

Recall that k = r − q− 12

2, and by de:ning d1(x; �; �) ≡ (x − ln �+ (k + 2)�)=√�,

the integral I2 then becomes

I2 = −exe−q�[N (−d1(x; �;K1)) − N (−d1(x; �; c1(0)))]:

Similarly it can be shown that

I3 = exe−q�[N (d1(x; �;K2)) − N (d1(x; �; c2(0)))]

and

I4 = K2e−r�[N (d2(x; �;K2)) − N (d2(x; �; c2(0)))]:

Thus it is concluded that

V (1)c1 ;c2 (x; �) = [K1e−r�N (−d2(x; �;K1)) − exe−q�N (−d1(x; �;K1))]

+ [exe−q�N (d1(x; �;K2)) − K2e−r�N (d2(x; �;K2))]

− [K1e−r�N (−d2(x; �; c1(0))) − exe−q�N (−d1(x; �; c1(0)))]

− [exe−q�N (d1(x; �; c2(0))) − K2e−r�N (d2(x; �; c2(0)))]:

C.2. Proof of Proposition 5

We begin by noting that

V (2)c1 ;c2 (x; �) =F

−1{∫ �

0F2(�; s)e−((2�2=2)+k i�+r)(�−s) ds

}

−F−1{∫ �

0F1(�; s)e−((2�2=2)+k i�+r)(�−s) ds

};


where

F2(�; s) = ei� ln c2(s)[2c2(s)

2+

(c′2(s)c2(s)

− 2 i�2

+ k)(c2(s) − K2)

]and

F1(�; s) = ei� ln c1(s)[2c1(s)

2+

(c′1(s)c1(s)

− 2 i�2

+ k)(K1 − c1(s))

]:

Following the approach outlined in Kucera and Ziogas (2004), V (2)c1 ;c2 (x; �) evaluates to

V (2)c1 ;c2 (x; �) =

∫ �

0

[2c2(s)

2+

(c′2(s)c2(s)

+12

[k − (x − ln c2(s))

(� − s)

])(c2(s) − K2)

]

× e−g2(x; s)

√2�(� − s)

ds −∫ �

0

[− 2c1(s)

2+

(c′1(s)c1(s)

+12

×[k − (x − ln c1(s))

(� − s)

])(K1 − c1(s))

]e−g1(x; s)

√2�(� − s)

ds; (47)

where we set

g2(x; s) =(x − ln c2(s) + k(� − s))2

22(� − s)+ r(� − s) (48)

and

g1(x; s) =(x − ln c1(s) + k(� − s))2

22(� − s)+ r(� − s): (49)

With a simple change of notation, Eq. (47) may be written as it is appears in Eqs.(22)–(24).

Appendix D. Value of the American strangle free boundaries at expiry

In deriving Eq. (38), it is necessary to analyse the limit of Eqs. (36) and (37) as� tends to zero. Using the method outlined by Kim (1990), we begin by consideringEq. (37):

K1 − c1(�) = c1(�)e−q�[N (d1(c1(�); �;K2)) − N (−d1(c1(�); �;K1))]

− e−r�[K2N (d2(c1(�); �;K2)) − K1N (−d2(c1(�); �;K1))]

+∫ �

0qc1(%)e−q(�−%)[N (d1(c1(�); � − %; c2(%)))

−N (−d1(c1(�); � − %; c1(%)))] d%


−∫ �

0re−r(�−%)[K2N (d2(c1(�); � − %; c2(%)))

−K1N (−d2(c1(�); � − %; c1(%)))] d%:

This equation can be factorised to produce

c1(�){1 + e−q�[N (d1(c1(�); �;K2)) − N (−d1(c1(�); �;K1))]

+∫ �

0qe−q(�−%)[N (d1(c1(�); � − %; c2(%))) − N (−d1(c1(�); � − %; c1(%)))] d%

}=K1 + e−r�[K2N (d2(c1(�); �;K2)) − K1N (−d2(c1(�); �;K1))]

+∫ �

0re−r(�−%)[K2N (d2(c1(�); � − %; c2(%)))

−K1N (−d2(c1(�); � − %; c1(%)))] d%;

which then yields the following implicit equation for c1(�):

c1(�) =(K1 + e−r�[K2N (d2(c1(�); �;K2)) − K1N (−d2(c1(�); �;K1))]

+∫ �

0re−r(�−%)[K2N (d2(c1(�); � − %; c2(%)))

−K1N (−d2(c1(�); � − %; c1(%)))] d%)

×(1 + e−q�[N (d1(c1(�); �;K2)) − N (−d1(c1(�); �;K1))]

+∫ �

0qe−q(�−%)[N (d1(c1(�); � − %; c2(%)))

−N (−d1(c1(�); � − %; c1(%)))] d%)−1

: (50)

Before proceeding further, it should be noted that K1 ¡K2, c1(�)6K1, and c2(�)¿K2.To :nd the value of c1(0), we take the limit of Eq. (50) as � tends to zero. In order toevaluate this limit, we need to :nd 4 limits involving d1 and d2. The :rst to consideris

lim�→0

d2(c1(�); �;K2) = lim�→0

ln(c1(�)=K2)√�

= −∞; since c1(�)¡K2: (51)

Secondly, we have

lim�→0

d2(c1(�); �;K1) = lim�→0

ln(c1(�)=K1)√�

=

{0; c1(0) = K1;

−∞; c1(0)¡K1:(52)


Similarly the following limits can be shown to be

lim�→0

d1(c1(�); �;K2) = −∞; since c1(�)¡K2 (53)

and

lim�→0

d1(c1(�); �;K1) =

{0; c1(0) = K1;

−∞; c1(0)¡K1:(54)

Note also that N (−∞) = 0; N (0) = 0:5, and N (∞) = 1. Given that limits (52) and(54) depend on the value of c1(0) relative to K1, there are two cases to consider when:nding the limit of Eq. (50). Consider the :rst case where c1(0)=K1. Taking the limitof Eq. (50) as � tends to zero, and using the results from Eqs. (51)–(54), we obtain

lim�→0

c1(�) = K1: (55)

Now consider the second case, where c1(0)¡K1. The limit as � tends to zero of Eq.(50) is now of the form 0

0 , and L’Hopital’s rule can therefore be applied. Firstly, let

lim�→0

c1(�) = lim�→0

N 1(�)

D1(�);

where

N 1(�)≡K1 + e−r�[K2N (d2(c1(�); �;K2)) − K1N (−d2(c1(�); �;K1))]

+∫ �

0re−r(�−%)[K2N (d2(c1(�); � − %; c2(%)))

−K1N (−d2(c1(�); � − %; c1(%)))] d%

and

D1(�)≡ 1 + e−q�[N (d1(c1(�); �;K2)) − N (−d1(c1(�); �;K1))]

+∫ �

0qe−q(�−%)[N (d1(c1(�); � − %; c2(%)))

−N (−d1(c1(�); � − %; c1(%)))] d%:

To apply L’Hopital’s rule, we must di/erentiate both N 1(�) and D1(�), and take theirlimit as � tends to zero. For N 1(�) we have

N ′1(�) =− re−r�[K2N (d2(c1(�); �;K2)) − K1N (−d2(c1(�); �;K1))]

+ e−r�[K2N ′(d2(c1(�); �;K2))

9d2(c1(�); �;K2)9�

+K1N ′(−d2(c1(�); �;K1))9d2(c1(�); �;K1)

9�

]+ r[K2N (d2(c1(�); 0; c2(�))) − K1N (−d2(c1(�); 0; c1(�)))]


+ r∫ �

0{−re−r(�−%)[K2N (d2(c1(�); � − %; c2(%)))

−K1N (−d2(c1(�); � − %; c1(%)))]

+ e−r(�−%)[K2N ′(d2(c1(�); � − %; c2(%)))61

+K1N ′(−d2(c1(�); � − %; c1(%)))62]} d%;where

61 =9d2(c1(�); � − %; c2(%))

9� and 62 =9d2(c1(�); � − %; c1(%))

9� :

Note that N ′(x)= e−x2=2=√2�, and that as x → ∞, N ′(x) → 0 at a faster rate than any

other terms observed in N ′1(�) (see Kim, 1990). We also note that

lim%→�

d2(c1(�); � − %; c1(�)) = 0

and

lim%→�

d2(c1(�); � − %; c2(�)) = −∞:

Combining all these limit results, it is concluded that

lim�→0

N ′1(�) =

r2K1: (56)

Similarly for D′1(�) it can be shown that

lim�→0

D′1(�) =

q2: (57)

Thus it is concluded that

lim�→0

c1(�) =rqK1: (58)

Recalling that this result only holds when c1(0)¡K1, it follows that we must haver ¡q. Combining the results from Eqs. (55) and (58) we :nd that

lim�→0

c1(�) = min(K1;

rqK1

)which is the :rst part of Eq. (38). Similarly the process can be repeated for Eq. (37),yielding

lim�→0

c2(�) = max(K2;

rqK2

)which is the second part of Eq. (38).

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Evaluation of American strangles