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Asymptotic Solutions for AustralianOptions with Low VolatilitySai Hung Marten Tinga & Christian-Oliver Ewaldb

a School of Mathematics and Statistics, University of Sydney,Sydney, Australiab Adam Smith Business School-Economics, University of Glasgow,Glasgow G12 8QQ, UKPublished online: 01 Sep 2014.

To cite this article: Sai Hung Marten Ting & Christian-Oliver Ewald (2014) Asymptotic Solutionsfor Australian Options with Low Volatility, Applied Mathematical Finance, 21:6, 595-613, DOI:10.1080/1350486X.2014.906973

To link to this article: http://dx.doi.org/10.1080/1350486X.2014.906973

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Asymptotic Solutions for AustralianOptions with Low Volatility

SAI HUNG MARTEN TING* & CHRISTIAN-OLIVER EWALD**

*School of Mathematics and Statistics, University of Sydney, Sydney, Australia, **Adam Smith BusinessSchool-Economics, University of Glasgow, Glasgow G12 8QQ, UK

(Received 17 November 2010; accepted 10 February 2014)

ABSTRACT In this paper we derive asymptotic expansions for Australian options in the case oflow volatility using the method of matched asymptotics. The expansion is performed on avolatility scaled parameter. We obtain a solution that is of up to the third order. In case thatthere is no drift in the underlying, the solution provided is in closed form, for a non-zero drift, allexcept one of the components of the solutions are in closed form. Additionally, we show that insome non-zero drift cases, the solution can be further simplified and in fact written in closed formas well. Numerical experiments show that the asymptotic solutions derived here are quite accuratefor low volatility.

KEY WORDS: Asian options, Australian options, stochastic volatility, asymptotic expansions

1. Introduction

Australian options are a class of exotic derivatives which so far have received verylittle attention from academia. The most notable work in this area is that of Handley(2000, 2003), Moreno and Navas (2008) and Ewald, Menkens, and Ting (2013). Thestate underlying of an Australian option can be simply defined as the ratio of theaverage stock price to its terminal stock price. There are various ways in which theaveraging can be defined: discrete or continuous and geometric or arithmetic. Thesubject of this paper will be that of the continuously sampled arithmetic Australianoptions, henceforth referred to as Australian options.

The averaging nature of an Australian option indicates that it can be considered asa special type of Asian option. In particular, fresh Australian options, i.e., options forwhich the start date of the averaging period is equal to the pricing date, can be shownto be equivalent to a floating and fixed strike (classical) Asian option. Details on thiscan be found in Ewald et al. (2013); also, compare Henderson and Wojakowski(2002) for the equivalence of fixed and floating strike Asian options. In this way,the pricing problem corresponding to a fresh Australian option can thus be regardedas a pricing problem corresponding to a fresh fixed strike Asian option. The latter has

Correspondence Address: Christian-Oliver Ewald, Adam Smith Business School-Economics, University ofGlasgow, Glasgow G12 8QQ, UK. Email: christian.ewald@glasgow.ac.uk

Applied Mathematical Finance, 2014Vol. 21, No. 6, 595–613, http://dx.doi.org/10.1080/1350486X.2014.906973

© 2014 Taylor & Francis

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received a lot of attention in the literature. Only to mention the most prominentresults, there are the Geman and Yor (1993) Laplace transform approach and theLinetsky (2004) spectral expansion approach, in addition to PDE approaches invol-ving dimension reduction techniques (see Vecer and Xu, 2004). In general, an inprogress Australian option, i.e., an Australian option for which the averaging startedbefore the pricing date, can be shown to be equivalent to that of an Asian option witha general payoff function. The pricing of general payoff Asian options has beenstudied by Vecer (2001) using a PDE approach.

Despite the various equivalences, some pricing techniques used for Asian optionsdo not directly carry over to Australian options. In some cases the conversionbetween Australian and Asian options hides potential problems that can occur witheither of them but also opportunities that may be missed out, when focusing on thecase of classical Asian options only. This applies particularly to the various approx-imations that have been obtained in the past. The focus in this paper is on the lowvolatility asymptotic expansion for Asian options as presented in Dewynne and Shaw(2008). Their results have been extended in Siyanko (2012) to cover all Taylorexpansion coefficients as well as the case of floating strikes. In the case of anAustralian option the resulting PDE will have two terms involving the volatility,and thus any expansion of the solution in terms of the (scaled) volatility must takeinto account an additional term. In this paper we investigate the application of theDewynne and Shaw (2008) methodology, but for Australian options. The asymptoticexpansion is performed on the pricing PDE corresponding to an Australian under-lying and is not derived through equivalence arguments.

The remainder of the paper is organized as follows: In Section 2, we will provide abrief revision, illustrating the model, the PDE corresponding to an Australian optionand the equivalence of Australian and Asian options. Furthermore, using the equiva-lence theorems, we will point out why the Dewynne and Shaw asymptotic solutioncannot be directly applied to the Australian case. Section 3 will contain our derivationof the asymptotic expansion. Here, a solution is derived for both an outer and aninner region, although in practice, only the solution for the inner region will be ofimportance. In our derivation, we also need to distinguish two cases, that of zero andnon-zero drift in the underlying. We investigate these two cases in two differentsubsections. In Section 4, we will focus on the numerical performance of our solu-tions. Numerical comparisons of the asymptotic solution with respect to otherestablished methods are made. Due to the lack of published results with numericalcontent, many of the comparisons will in fact be made with respect to the solutionobtained via the finite difference method applied to the Australian option PDE. Wewill emphasize the test cases involving low volatility but also present results for thecase of higher volatilities.

2. A Brief Review on Australian Options

2.1 Model Assumptions

Assume the stock price Xt follows a geometric Brownian motion described bythe SDE,

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dXt ¼ r � qð ÞXtdt þ σXtdWt; (1)

where r is the risk-free interest rate, q the continuous dividend yield and σ thevolatility. Let t0 ¼ 0 be the start date of the averaging period, T be the end date ofthe averaging period and t be the current time. Define the Australian state variable as

AUt ¼R t0 Xudu

TXt:

Note that inclusion of the factor 1T differs from article to article. We find it con-

venient to include it here, because the resulting PDE will then have a similar structureas the Asian option PDE derived in Dewynne and Shaw (2008). Using Itô’s lemma,the Australian state variable can be shown to satisfy the following SDE.

dAUt ¼ σ2 � r � qð Þ� �AUt þ 1

T

� �dt � σAUtdWt: (2)

The value of interest is the price of an Australian call option defined as

f η; tð Þ ¼ e�r T�tð ÞE AUT � kð Þþ jAUt ¼ η� �

:

The price of the corresponding put option can be calculated using a put-call parityrelation. Consider that

e�r T�tð ÞE k � AUTð Þþ jAUt ¼ η� � ¼ f η; tð Þ þ ke�r T�tð Þ � e�r T�tð ÞE AUT jAUt ¼ η½ �;

where parts of the last term can be calculated1 using,

E AUT1 AUtj ¼ η½ � ¼ηe σ2 � r þ qð Þ T1 � tð Þþ 1

T σ2�rþqð Þ e σ2�rþqð Þ T1�tð Þ � 1�

if σ2 � r þ q�0;1T T1 � tð Þ þ η if σ2 � r þ q ¼ 0;

8><>:

for t � T1 � T . The latter represents the Australian forward price. In Ewald et al.(2013), these results are used to help show that an Australian call option is mono-tonically increasing with volatility.

2.2 Pricing PDE

By defining η; tð Þ ¼ E AUT � kð Þþ jAUt ¼ η� �

and using the Feynman–Kac theorem,one can show that η; tð Þ satisfies the following PDE,

@

@tþ 1

2σ2η2

@2

@η2þ 1

Tþ σ2 � r � qð Þ� �

η

� �@

@η¼ 0;

η; Tð Þ ¼ max η� k; 0ð Þ:

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This PDE is then non-dimensionalized by introducing the following variables:

τ ¼ 1� t

T;

�2 ¼ σ2T ;

θ ¼ r � qð ÞT ;which results in,

@

@τ¼ 1

2�2η2

@2

@η2þ ð1þ ð�2 � θÞηÞ @

@η;

ðη; 0Þ ¼ maxðη� k; 0Þ:

The variables τ; � and θ can be regarded as the scaled time, volatility and drift ratevariable, respectively. The form of this PDE is similar to the one studied in Dewynneand Shaw (2008), but with the notable difference of having an additional �2η @ @η termand the one-half factor in the diffusion term. The additional � related advection termin the PDE needs to be taken care of, since the solution is expanded in terms of �. Infact, it prevents a direct application of the Dewynne and Shaw (2008) result.2

2.3 Australian and Asian Equivalence

Let P be the probability measure under which Equation (1) holds. We define anequivalent probability measure Q via the Radon–Nikodym derivative.

dQ

dPjF T

¼ exp r � q� σ2ð ÞTð ÞXT

;

¼ exp � 1

2σ2T � σWP

T

� �:

Girsanov’s theorem implies

WQt ¼ WP

t þ σt;

is a Brownian motion under Q and Bayes rule for conditional expectations impliesthat the price of an Australian option can then be computed as,

f η; tð Þ ¼ exp σ2 þ q� 2rð Þ T � tð Þð ÞTXt

EQ

Z T

0Xudu� kT XT

� �þjF t

" #: (3)

Under the probability measure Q , Xt satisfies the SDE,

dXt ¼ r � q� σ2� �

Xtdt þ σXtdWQt :

The expectation in Equation (3) is related to a floating strike Asian option, withdynamics under Q . By considering the PDE satisfied by the value of this expectation,

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it can be shown that the advection term has a coefficient which depends on σ. Similarto the Australian PDE with a volatility dependent coefficient in the advection term,this once more indicates that the results of Dewynne and Shaw (2008) thus cannot bedirectly applied, even after the equivalence relationship.

3. Asymptotic Expansion for Australian Option

The approach taken here to derive asymptotic expansions for Australian options willbe that of the method of matched asymptotic expansion. This methodology has beenused to great effect by Dewynne and Shaw (2008) for Asian options in findinganalytical approximations to Asian fixed strike call options in a low volatility regime.The derivation for the Australian case follows similarly; however, due to the addi-tional volatility related advection term in the Australian PDE, additional steps mustbe taken.

The method of matched asymptotic expansion assumes a solution for an outerregion and an inner region. The outer region can be simply thought of as the region inwhich the volatility parameter is negligible and the inner region where its effectcannot be ignored. Intuitively, the outer region corresponds to deeply in or deeplyout-of-the-money options. Obviously, once the outer region has been determined, theinner region is identified as well.

Given that the derivation is largely similar to the one in Dewynne and Shaw (2008),much of the details will be omitted with only the differences and results beingpresented. Interested readers are directed to Dewynne and Shaw (2008) to get abetter feel for the derivation techniques used.

3.1 Outer Region

The derivation of the solution in the outer region in the Australian option casefollows similarly to that of the Asian option case. The critical characteristic of thisperturbation expansion is given by

η�ðτÞ ¼ keθτþ1θ 1�eθτð Þ θ� 0;

k�τ θ¼ 0:

(4)

which differs slightly to Equation 17 in Dewynne and Shaw (2008), for the Asianaverage-rate call case.

3.2 Inner Region for r = q

When solving for the solution in the inner region, there are two cases to consider:θ ¼ 0 and θ � 0; corresponds to r ¼ q and r � q respectively. This section reportsthe asymptotic solution for the inner region when θ ¼ 0. As before, the criticalcharacteristic is given by Equation (4) and the resulting PDE of interest is

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@

@τ¼ 1

2�2η2

@2

@η2þ 1þ �2η� � @

@η;

η; 0ð Þ ¼ max η� k; 0ð Þ:

We define an inner variable via,

ζ ¼ 1

�η� η� τð Þð Þ;

and by applying the chain rule, the resulting PDE becomes,

@

@τ¼ 1

2�ζ þ η� τð Þð Þ @

2

@ζ 2þ � �ζ þ η� τð Þð Þ @

@ζ;

ζ ; 0ð Þ ¼ � max ζ ; 0ð Þ:

Assume that the solution for ζ ; τð Þ in the inner region can be written in terms of thepower series

ζ ; τð Þ ¼X1j¼1

� j j ζ ; τð Þ:

This solution gives rise to a sequence of PDEs which the j’s must satisfy. ThesePDEs are given as follows:

@ 1

@τ� 1

2η� τð Þ2 @

2 1

@ζ 2¼ 0; j ¼ 1;

@ 2

@τ� 1

2η� τð Þ2 @

2 2

@ζ 2¼ ζ η� τð Þ @

2 1

@ζ 2þ η� τð Þ @ 1

@ζ; j ¼ 2;

@ j

@τ� 1

2η� τð Þ2 @

2 j

@ζ 2¼ ζ η� τð Þ @

2 j�1

@ζ 2þ η� τð Þ @ j�1

@ζþ 1

2ζ 2@2 j�2

@ζ 2þ ζ

@ j�2

@ζ; j � 3;

with initial conditions,

j ζ ; 0ð Þ ¼ max ζ ;0ð Þ0

if j¼ 1;if j> 1:

nNote that for j > 1, each of the above PDEs feature an additional source term, whencompared to the corresponding sequence of PDEs in Dewynne and Shaw (2008). Thisis caused by the additional advection term in the original PDE and the maindifference in the PDEs to be solved.

3.2.1 First Order Solution. We are now aiming to solve the above sequence ofPDEs. For j ¼ 1, the corresponding PDE is given by

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@ 1

@τ¼ 1

2η� τð Þ2 @

2 1

@ζ 2;

ζ ; 0ð Þ ¼ max ζ ; 0ð Þ:

The solution to this is

1ðζ ; tÞ ¼ ζN ζffiffiffiffi2t

p� �

þffiffiffit

π

rexp � ζ 2

4t

� �;

with N xð Þ denoting the cumulative standard normal distribution function and t τð Þgiven by

t τð Þ ¼ k2τ2

� kτ2

2þ τ3

3:

The following partial derivatives will be required in determining the solution for thehigher orders.

G1 ζ ; tð Þ :¼ @2 1

@ζ 2¼

ffiffiffiffiffiffiffi1

4πt

rexp � ζ 2

4t

� �;

G2 ζ ; tð Þ :¼ @ 1

@ζ¼ N ζffiffiffiffi

2tp� �

:

3.2.2 Second Order Solution. The second order PDE is given by

@ 2

@τ� 1

2η� τð Þ2 @

2 2

@ζ 2¼ ζ η� τð Þ @

2 1

@ζ 2þ η� τð Þ @ 1

@ζ:

Using the functions G1 and G2 from above and transforming to the ζ ; tð Þ variables, thePDE becomes

@ 2

@ t� @2 2

@ζ 2¼ ζ η� τð ÞG1 ζ ; tð Þ þ η� τð ÞG2 ζ ; tð Þ

12 η

� τ2ð Þ ;

2 ζ ; 0ð Þ ¼ 0:

The solution to 2 ζ ; tð Þ is

2 ζ ; tð Þ ¼ ζg1 tð ÞG1 þ g2 tð ÞG2;

where the functions g1 and g2, written with τ as the variable can be calculated as

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g1 τð Þ ¼ τ 15k3 � 20k2τ þ 10kτ2 � 2τ3ð Þ10 3k2 � 3kτ þ τ2ð Þ

g2 τð Þ ¼ kτ � τ2

2:

3.2.3 Third Order Solution. For the third order solution, the following partialderivatives are required.

@ 2

@ζ¼ g1 þ g2 � g1

2tζ 2

� �G1;

@2 2

@ζ 2¼ � 3g1

2tþ g2

2t

� �ζ þ g1

4t2 ζ

3

� �G1:

Using these partial derivatives, the third order PDE in terms of the ζ ; tð Þ variablesbecomes

@ 3

@ t� @2 3

@ 2¼ H ζ ; tð ÞG1 þ ζG2

12 η

� τð Þ2 ;

where

H ζ ; tð Þ ¼ η� τð Þ g1 þ g2ð Þ þ 1

2� g2η� τð Þ

2t� 2g1η� τð Þ

t

� �ζ 2 þ g1η� τð Þ

4t2 ζ 4:

The solution for 3 ζ ; tð Þ is,

3 ζ ; tð Þ ¼ f1 tð Þ þ f2 tð Þζ 2 þ f3 tð Þζ 4� �G1 þ f4 tð ÞζG2;

where the functions f1, f2, f3 and f4, written in terms of τ are

f1 τð Þ ¼τ2P6

j¼0 f j1k6�jτj

� 4200 3k2 � 3kτ þ τ2ð Þ2 ;

f2 τð Þ ¼τP6

j¼0 f j2k6�jτj

� 4200 ð3k2 � 3kτ þ τ2Þ3 ;

f3 τð Þ ¼ 9 15k3 � 20k2τ þ 10kτ2 � 2τ3ð Þ2200 3k2 � 3kτ þ τ2ð Þ4 ;

f4 τð Þ ¼ τ;

and the f j1 and f j2 constants are given in the table below (Table 1).

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3.2.4 Higher Order. In principle, it is possible to obtain higher order solutions byconsidering solutions whose forms follows a similar pattern to that of 2 and 3.However, it will be shown that for the case of low volatility, the solution up to thethird order provides a very good approximation to the true solution.

3.3 Inner Region for r ≠ q

In this section, we discuss the solutions of the asymptotic expansion when θ � 0, i.e.,r � q. The corresponding PDE of interest is given as

@

@τ¼ 1

2�2η2

@2

@η2þ 1þ e2 � θ

� �η

� � @ @η;

η; 0ð Þ ¼ max η� k; 0ð Þ:

Again, we define an inner variable by

ζ ¼ 1

�η� η� τð Þð Þ;

where η� τð Þ is defined as in Section 3.1. Through the application of the chain rule, thePDE becomes

@

@τ¼ 1

2�ζ þ η� τð Þð Þ2 @

2

@ζ 2� θζ

@

@ þ � �ζ þ η� τð Þð Þ @

@ζ;

ζ ; 0ð Þ ¼ emax ζ ; 0ð Þ:

We assume that the solution can be written as a power series in the form of

ζ ; τð Þ ¼X1j¼1

� j j ζ ; τð Þ:

This solution gives rise to the following sequence of PDEs, which the j must satisfy:

Table 1. f j1 and f j2 constants.

j 0 1 2 3 4 5 6

f j1 36,225 −111,825 151,620 −115,185 51,645 −12,970 1437f j2 22,050 −61,425 74,445 −50,715 20,460 −4640 464

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@ 1

@τ� 1

2η� τð Þ2 @

2 1

@ζ 2þ θζ

@ 1

@ζ¼ 0; j ¼ 1;

@ 2

@τ¼ 1

2η� τð Þ2 @

2 2

@ζ 2þ θζ

@ 2

@ζ¼ ζ η� τð Þ @

2 1

@ζ 2þ η� τð Þ @ 1

@ζ; j ¼ 2;

@ j

@τ� 1

2η� τð Þ2 @

2 j

@ζ 2þ θζ

@ j

@ζ¼ ζ η� τð Þ @

2 j�1

@ζ 2þ η� τð Þ @ j�1

@ζ

þ 1

2ζ 2@2 j�2

@ζ 2þ ζ

@ j�2

@ζ; j � 3;

with initial conditions

j ζ ; 0ð Þ ¼ max ζ ; 0ð Þ if j ¼ 1;0 if j > 1:

Note that the additional source terms as compared to the Dewynne and Shaw (2008)PDEs are due to the additional advection term in the original PDE. Again, thedifferences in the PDEs lead to different solutions as compared to the Asian aver-age-rate call case.

3.3.1 First Order Solution. Define the new variables t ¼ t τð Þ and x ¼ f τð Þζ , suchthat,

@ f

@τþ θf ¼ 0;

dt

dτ� 1

2η � τð Þ2 f τð Þ2 ¼ 0:

(5)

Under this transformation, the PDE for j ¼ 1 then becomes

@ 1

@ t¼ @2 1

@x2;

1 x; 0ð Þ ¼ max x; 0ð Þ:

The solutions for f τð Þ and t τð Þ can be easily calculated as

f τð Þ ¼ e�θτ;

t τð Þ ¼ e�2θτ

4θ34eθτ 1� kθð Þ � 1þ e2θτ �3þ 2θτ þ 2k2θ3τ � 4kθ θτ � 1ð Þ� �� �

;

while the solution for 1 x; tð Þ is given by

1 x; tð Þ ¼ xN xffiffiffiffi2t

p� �

þffiffiffit

π

rexp � x2

4t

� �

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Further, we define the functions

G1 x; tð Þ :¼ @2 1

@x2¼

ffiffiffiffiffiffiffi1

4π t

rexp � x2

4t

� �;

G2 x; tð Þ :¼ @ 1

@x¼ N xffiffiffi

2p

t

� �;

which will be required to solve the higher order solutions.

3.3.2 Second Order Solution. The PDE for the second order is

@ 2

@τ� 1

2η� τð Þ2 @

2 2

@ζ 2þ θζ

@ 2

@ζ¼ ζ η� τð Þ @

2 1

@ζ 2þ η� τð Þ @ 1

@ζ:

Transforming this PDE to the x; tð Þ variables results in

@ 2

@ t� @2 2

@x2¼

ζ η� τð Þ @2 1

@ζ 2þ η� τð Þ @ 1

@ζ

12 η

� τð Þ2 f τð Þ2 ;

¼ 2

η� τð Þf τð Þ xG1 x; tð Þ þ 2

η� τð Þf τð Þ G2 x; tð Þ:

The solution for 2 x; tð Þ is in the form of

2 x; tð Þ ¼ g1 tð ÞxG1 x; tð Þ þ g2 tð ÞG2 x; tð Þ;

where the functions g1 and g2 are defined via the ODEs,

g1 tð Þt

þ dg1 tð Þdt

¼ 2

η� τð Þf τð Þ ;

dg2 tð Þdt

¼ 2

η� τð Þf τð Þ :

By further defining h τð Þ ¼ t τð Þg1 τð Þ (with g1 written in terms of τ) and showing,

dh

dτ¼ t τð Þη� τð Þf τð Þ; (6)

h τð Þ can be calculated as

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h τð Þ ¼ e�3τθ

24θ52þ h11 τð Þ þ h12 τð Þ þ h13 τð Þð Þ;

h11 τð Þ ¼ 15eτθ kθ � 1ð Þ;h12 τð Þ ¼ � 6e2τθ �5þ 2τθ � 4kθ τθ � 2ð Þ þ 2k2θ2 τθ � 1ð Þ� �

;

h13ðτÞ ¼ e3τθð�17þ 18τθ � 6τ2θ2 þ 6k3τ2θ5 � 6k2θ2ð2� 4τθ þ 3τ2θ2Þþ 3kθ 11� 14τθ þ 6τ2θ2

� ��:

This is then used to obtain g1 τð Þ. For g2 τð Þ (with g2 written in terms of τ), the solutionis calculated as

g2 τð Þ ¼ 1� e�τθ þ τθ kθ � 1ð Þθ2

3.3.3 Third Order Solution. The third order PDE is given as

@ 3

@τ� 1

2η� τð Þ2 @

2 3

@zþ θz

@ 3

@z¼ zη� τð Þ @

2 2

@z2þ η� τð Þ @ 2

@zþ 1

2z2@2 1

@z2þ z

@ 1

@z

In transforming to the x; tð Þ variable, the PDE becomes,

@ 3

@ t� @ 2

@x2¼ H x; tð ÞG1 x; tð Þ þ xG2 x; tð Þ

12 η

� τð Þ2 f τð Þ2 ;

where

H x; tð Þ ¼ η� f g1 þ g2ð Þ þ 1

2� η� f

t2g1 þ g2

2

� !xþ g1η� f

4t2 x4:

The solution to 3 x; tð Þ is of the form

3 x; tð Þ ¼ f1 tð Þ þ f2 tð Þx2 þ f3 tð Þx4� �G1 þ xf4 tð ÞG2:

In terms of the τ variable, the functions f1, f2, f3 and f4 can be determined using thefollowing ODEs:

df4dτ

¼ 1;

d t4f3

� dτ

¼ h

4

dh

dτ;

(7)

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d t2f2

� dτ

¼ 1

2t2 þ 6f3 η� f

� 2t2 � η� f

� 2hþ g2 t

2

� �;

df1dτ

¼ 1

2η� f� 2

2f2 þ 2f4ð Þ þ η� f�

g1 þ g2ð Þ;(8)

The solutions to f4 and f3 can be calculated in closed form and is given as

f4 τð Þ ¼ τ;

f3 τð Þ ¼ h2 τð Þ8t

4 ;

while f2 and f1 needs to be calculated numerically. The next subsection containsdetails and suggestions on the implementation of these solutions.

3.3.4 Notes on Implementation. Implementation of the asymptotic expansion pre-sented in the last subsection can be quite difficult. Compared to the Dewynne andShaw (2008) asymptotic solution for Asian options, the Australian option asymptoticsolution has an additional parameter in its functions. More specifically, the strikeprice of a fixed strike Asian option is absorbed in the space variable of the corre-sponding PDE, but this is not the case for an Australian option. As such, many of thederived functions in the solution are dependent on k (the Australian strike price) aswell. Even though, technically, software with symbolic capabilities are able to makemany of the above calculations, it may end up being quite time consuming andinefficient to do so. Below are some suggestions on how to implement the asymptoticsolution in Mathematica.

So far, the functions η�, f , t, f1, g1, g2, G1, G2, h, f4 and f3 have all been presented inclosed form. The functions f1 and f2 have been left in their differential form becausethey are in fact quite difficult to compute symbolically. In terms of implementation, itis simply easier to input the actual values for the parameters θ and k, occurring in thepricing problem, into the ODEs rather than trying to symbolically calculate thesolution for a general θ and k.

A method for obtaining the solution for f2ðτÞÞ is now presented. Define F2ðτÞ to bet2ðτÞf2 t τð Þð Þ and integrating Equation (8) gives the solution to F2ðτÞ as

F2 τð Þ ¼Z τ

0

1

2t τ0ð Þ2 þ 6f3 t τ0ð Þð Þ η� τ0ð Þf τ0ð Þ

� 2t τ0ð Þ2dτ0

�Z τ

0η� τ0ð Þf τ0ð Þ�

2h τ0ð Þ þ g2t τ0ð ÞÞt τ0ð Þ

2

� �dτ0:

� (9)

Observe that

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I1 τð Þ ¼Z τ

06f3 t τ0ð Þð Þ η� τ0ð Þf τ0ð Þ

� 2t τ0ð Þ2dτ0

¼Z τ

0

3h τ0ð Þ22t τ0ð Þ2

dt τ0ð Þdτ0

dτ0

¼ 3h τ0ð Þ22t τ0ð Þ

" #τ0

� 3

2

Z τ

0t τ0ð Þ d

dτ0h τ0ð Þ2t τ0ð Þ2

!dτ0

¼ 3h τð Þ22t τð Þ � 3

Z τ

0

h τ0ð Þt τ0ð Þ

dh τ0ð Þdτ0

dτ0 þ 2I1 τð Þ

¼ 3

Z τ

0h τ0ð Þη� τ0ð Þf τ0ð Þdτ0 � 3h τð Þ2

2t τð Þ ;

by using Equations (5) and (7), integration by parts, some elementary calculations,and Equation (6), for the second, third, fourth and fifth equalities, respectively.Furthermore, note that,

I2 τð Þ ¼Z τ

0η� τ0ð Þf τ0ð Þg2 t τ0ð Þð Þt τ0ð Þdτ0

¼Z τ

0g2 t τ0ð Þð Þ dh τ0ð Þ

dτ0dτ0

¼ g2 t τð Þð Þh τð Þ �Z τ

0h τ0ð Þ dg2 t τ0ð Þð Þ

dτ0dτ0

¼ g2 t τð Þð Þh τð Þ �Z τ

0h τ0ð Þη� τ0ð Þf τ0ð Þdτ0;

also by integration by parts. Thus, F2 τð Þ can be simplified as

F2 τð Þ ¼Z τ

0

1

2t τ0ð Þ2 � 2h τ0ð Þη� τ0ð Þf τ0ð Þ

� �dτ0 þ I1 τð Þ � 1

2I2 τð Þ

¼Z τ

0

1

2t τ0ð Þ2 þ 3

2h τ0ð Þη� τ0ð Þf τ0ð Þ

� �dτ0 � 3

2

h τð Þ2t τð Þ � 1

2g2 t τð Þð Þh τð Þ:

(10)

It is evident that the integral part in Equation (10) is much simpler to evaluate thanthe complete integral in Equation (9).

Up until now, the aforementioned integrals in the derivations could be computedsymbolically. However, the solution for f1 is even harder to obtain than the solutionfor f2. In calculating f1, we begin by approximating the solution to f2 using Taylorseries expansion in τ around the point zero, whilst keeping terms up to powers of 10 inτ. This helps immensely in the computation since the approximate solution is easier tohandle. Numerical integration is then performed to obtain the value for f1. Choosethe upper limits of the integral as the value of τ in the pricing problem, while thelower limit is chosen as something slightly bigger than zero, say 0.0001. This choice of

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lower limit helps create stability in the integral since the integrand may be singular atzero.

It must be said that, by opting for numerical integration, higher order solutionscannot be obtained. This is due to the fact that any higher order solution will nodoubt involve integrations whose integrands involve the functions fi’s; similarly,determining the fi’s require the functions gi’s.

3.3.5 Low Volatility Solution. Under specific circumstances, the asymptotic solu-tion for low volatility can be further reduced. For some parameters, the value of G1

can be effectively treated as zero. This is the case, for example, when the absolutevalue of x is very large or t τð Þ is very small. Whilst a general parameter space forwhich the value G1 can be ignored is hard to obtain in terms of all the parameters, ifonly varying σ is considered while keeping the others fixed, then a smaller σ implies asmaller �, which means a greater absolute value of x. Of course, it is prudent to checkthat G1 can be ignored before proceeding. If G1 can be ignored, then the calculationsfor g1, f1, f2 and f3 can be omitted, thus making the asymptotic solution much easierto obtain because closed form solutions for the other functions are available.

4. Numerical Results

This section compares the numerical results of the asymptotic solution by consideringmany test cases. Test cases involve those from published literature and low volatilityregimes.

4.1 Comparisons to Literature

In this section, we compare the numerical results obtained from the asymptoticsolution to that of published results. Literature on Australian options, in general, issparse; the only notable publications with numerical results being Moreno and Navas(2008) as well as Ewald et al. (2013). The parameter set used in Moreno and Navas(2008) does not really meet the low volatility assumption (σ ¼ 0.2 and 0.4); however,the comparison is still made. Furthermore, the solutions will be compared to thoseobtained using the Crank–Nicholson finite difference method to solve the PDE aswell as the authors’ Monte Carlo simulation. The latter differs from Moreno andNavas in the sense that it uses more monitoring dates and simulation paths.

The first four test cases use parameters taken from Moreno and Navas3. Theparameters are X0 ¼ 1; t ¼ 0; r ¼ 0:1 and q ¼ 0:03; with further parameters givenas in Table 2. MN indicates these results are taken from Moreno and Navas, withMC, W and GD representing their Monte Carlo, Wilkinson approximation and theGamma distribution solutions, respectively (Table 3). CN-FD and MC refers to theauthors’ Crank–Nicholson and Monte Carlo solution. Asymp-10 denotes the asymp-totic solution using power series expansions of up to power 10 in τ in calculating f1and f2. These results show that even for relatively large volatility, the asymptoticsolution does a reasonable job at approximating the true solution; however, highaccuracy is not achieved.

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4.2 Low Volatility

In the subsequent section, we compare the numerical solutions for low volatility testcases. We generally speak of a low volatility problem when the volatility is less thanor equal to 0.1.

4.2.1 Case: θ ≠ 0. The following test cases are for the θ � 0 case. The parameters arechosen in order to meet the low volatility requirement. Test cases 5 to 8 have parametersX0 ¼ 1; t ¼ 0; r ¼ 0:1; q ¼ 0:03; T ¼ 1 and k ¼ 0:8 while volatility varies in values of0.1, 0.05, 0.02 and 0.01. The low volatility solution that is computed by assuming G1 isnegligible, and is also presented and denoted as Asymp-LV (Table 4).4

Both the Asymp-10 and Asymp-LV solution match the Crank–Nicholson finitedifference method to a very high degree of accuracy. In particular, for lower volati-lity, the asymptotic solutions are accurate to at least 6 significant figures. For theparameters tested, it can be concluded that if G1 is indeed negligible the differencebetween the two asymptotic solutions are quite small.

Test case 9 involves looking at in-progress options. The parameters areXt ¼ 100; t ¼ 0:5; T ¼ 1;

R t0 Xudu ¼ 50; r ¼ 0:1; q ¼ 0:03; σ ¼ 0:05; while the strike

price k; varies from 0.9 to 0.1 in intervals of 0.1. In this case, only the Asymp-LV solutionis presented, since G1 being negligible in this parameter set. The percentage errors arecalculated assuming that the Crank–Nicholson solution is the true solution (Table 5).

Table 3. Results for different tests.

Test Asymp-10 CN-FD MC MN-MC MN-W MN-GD

1 0.215196 0.215666 0.215618 0.21535 0.21620 0.215352 0.004158 0.004146 0.004129 0.00413 0.00401 0.004243 0.183157 0.183217 0.183206 0.18319 0.18324 0.183214 0.168574 0.168665 0.168640 0.16899 0.16883 0.16857

Table 2. Further parameters for tests.

Test σ T k

1 0.4 0.5 0.82 0.2 0.5 1.13 0.2 0.5 0.84 0.2 1.0 0.8

Table 4. Results for different tests.

Test Asymp-10 Asymp-LV CN-FD MC

5 0.154345 0.154343 0.154360 0.1543616 0.151104 0.151104 0.151105 0.1511127 0.150197 0.150197 0.150197 0.1502078 0.150068 0.150068 0.150068 0.150079

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The percentage errors show that the Asymp-LV solution is able to produce a highlevel of accuracy within this parameter set.

4.2.2 Case: θ = 0. The following test cases are for θ ¼ 0. The same parameter setsare used as in test case 5 to 8, but with r ¼ q ¼ 0:03; and will be referred to as tests10–13. The closed form asymptotic solution up to the third order will be listedunder the Asymp column. Although the asymptotic solution does not perform aswell as its θ � 0 counterpart, there is still quite a high level of agreement betweenthe three solutions, up to 4 significant figures, in fact. Test case 14 is set up similarlyas test case 9, but with r = q = 0.03. The results are given in the following table(Tables 6 and 7).

Table 5. Percentage errors.

k Asymp-LV CN-FD Percentage error %

0.9 0.071406333 0.071408436 −0.00290.8 0.166524802 0.166525282 −0.00030.7 0.261647744 0.261648224 −0.00020.6 0.356770687 0.356771166 −0.00010.5 0.451893629 0.451894109 −0.00010.4 0.547016572 0.547017051 −0.00010.3 0.642139514 0.642139994 −0.00010.2 0.737262457 0.737262936 −0.00010.1 0.832385399 0.832385879 −0.0001

Table 6. Comparison with Crank–Nicholson and Monte Carlo.

Test Asymp CN-FD MC

10 0.198941 0.198958 0.19894711 0.195302 0.195303 0.19530012 0.194283 0.194283 0.19428313 0.194138 0.194145 0.194139

Table 7. Percentage errors.

k Asymp CN-FD Percentage error %

0.9 0.099434925 0.099435274 −0.000350.8 0.197945930 0.197946444 −0.000260.7 0.296457124 0.296457638 −0.000170.6 0.394968318 0.394968832 −0.000130.5 0.493479512 0.493480026 −0.000100.4 0.591990706 0.591991219 −0.000090.3 0.690501900 0.690502413 −0.000070.2 0.789013094 0.789013607 −0.000070.1 0.887524288 0.887524801 −0.00006

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The low percentage errors show that the asymptotic solution produces results ofhigh accuracy much like its θ � 0 counterpart.

5. Conclusion

We derived asymptotic solutions for Australian options in the case of low volatility.The methodology presented is an adaptation of the Dewynne and Shaw (2008)approach to compute asymptotic solutions for Asian options. The solution is writtenas a power series in � which can be regarded as a scaled volatility parameter.Although the pricing of Australian options becomes equivalent to that of pricing anAsian option under an appropriate transformation (see Ewald et al. (2013), a directapplications of the Dewynne and Shaw (2008) result is not possible.

Asymptotic solutions for the cases when r = q and r � q are derived up to the thirdorder, with the former case having an easy to implement closed form solution. In thelatter, we showed how the asymptotic solution can be simplified to have a closed formsolution, if the value of the function G1 is negligible. This is generally true in the lowvolatility regime.

Numerical solutions in published literature is sparse, although for those that exist,e.g., Moreno and Navas (2008), the numerics of the asymptotic solution performsmoderately well. In the low volatility regime, the asymptotic solution performs verywell with high level of accuracy. Test cases involved fresh and in-progress options,varying the strike, and for reasonable values of the interest and dividend rates.

Acknowledgement

Both authors acknowledge support from the Australian Research Council GrantDP1095969.

Notes1This calculation can be done by integrating the SDE in Equation (2) and then taking conditionalexpectations.

2An alternative approach is to change the drift term in Dewynne and Shaw (2008) to θ � �2; replacing θ byθ � �2 in the individual terms of the corresponding equation in Dewynne and Shaw (2008), expandingthese as powers of �2 and then collecting powers of �.

3Test cases 1 and 3 indicate that the Australian call price is increasing with volatility. This fact is establishedanalytically in Ewald et al. (2013) by using the Pontryagin maximum principle. Test cases 3 and 4 indicatethat the Australian call price may be decreasing with maturity. This relationship is slightly more subtle.The following relationship between the partial derivatives of the Australian call price function f withrespect to volatility σ, time of maturity T and interest rate r can be established: fT ¼ 1

2T fσσ þ 2frrð Þ.While

4fσ is always positive as shown in Ewald et al. (2013), it is not difficult to see that fr is always negative,causing the ambiguity in monotonicity of the Australian call price as a function of time-of-maturity.

References

Dewynne, J. N., & Shaw, W. T. (2008). Differential equations and asymptotic solutions for arithmeticAsian options: ‘Black-Scholes Formulae’ for Asian Rate Calls. European Journal of AppliedMathematics, 19, 353–391.

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Ewald, C.-O., Menkens, O., & Ting, S. H. M. (2013). Asian and Australian options: A common perspec-tive. Journal of Economic Dynamics and Control, 37, 5.

Geman, H., & Yor, M. (1993). Bessel processes, Asian options and perpetuities. Mathematical Finance, 3(4), 349–375. doi:10.1111/j.1467-9965.1993.tb00092.x

Handley, J. C. (2000). Variable purchase options. Review of Derivatives Research, 4, 219–230. doi:10.1023/A:1011331329906

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Henderson, V., & Wojakowski, R. (2002). On the equivalence of floating and fixed-strike Asian options.Journal of Applied Probability, 39(2), 391–394. doi:10.1239/jap/1025131434

Linetsky, V. (2004). Spectral expansions for Asian (average price) options. Operations Research, 52(6), 856–867. doi:10.1287/opre.1040.0113

Moreno, M., & Navas, J. F. (2008). Australian options. Australian Journal of Management, 33(1), 69–93.doi:10.1177/031289620803300105

Siyanko, S. (2012). Essentially exact asymptotic solutions for Asian derivatives. European Journal ofApplied Mathematics, 23, 395–415. doi:10.1017/S0956792511000441

Vecer, J. (2001). Unified Asian pricing. Risk, 15(6), 113–116.Vecer, J., & Xu, M. (2004). Pricing Asian options in a semimartingale model. Quantitative Finance, 4(2),

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