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Research ArticleGeometric Asian Options Pricing under the Double
HestonStochastic Volatility Model with Stochastic Interest Rate
Yanhong Zhong and Guohe Deng
College of Mathematics and Statistics, Guangxi Normal
University, Guilin 541004, China
Correspondence should be addressed to Guohe Deng;
[email protected]
Received 16 August 2018; Revised 6 December 2018; Accepted 24
December 2018; Published 10 January 2019
Academic Editor: Hassan Zargarzadeh
Copyright © 2019 Yanhong Zhong and Guohe Deng. This is an open
access article distributed under the Creative CommonsAttribution
License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work isproperly
cited.
This paper presents an extension of double Heston stochastic
volatility model by incorporating stochastic interest rates and
derivesexplicit solutions for the prices of the continuously
monitored fixed and floating strike geometric Asian options. The
discountedjoint characteristic function of the log-asset price and
its log-geometric mean value is computed by using the change of
numeraireand the Fourier inversion transform technique. We also
provide efficient approximated approach and analyze several effects
onoption prices under the proposed model. Numerical examples show
that both stochastic volatility and stochastic interest rate havea
significant impact on option values, particularly on the values of
longer term options.The proposedmodel is suitable formodelingthe
longer time real-market changes and managing the credit risks.
1. Introduction
Asian option is a special type of option contract in which
thepayoff depends on the average of the underlying asset priceover
somepredetermined time interval.The averaging featureallows Asian
options to reduce the volatility inherent in theoption. There are
some advantages to trading Asian optionsin a financial market. One
is that these decrease the riskof market manipulation of the
financial derivative at expiry.Another is that Asian options have
lower relative chargethan European or American options. In general,
the averageconsidered can be a arithmetic or geometric one and it
canbe calculated either discretely, for which the average is
takenover the underlying asset prices at discrete monitoring
timepoints, or continuously, for which the average is calculatedvia
the integration of the underlying asset price over themonitoring
time period. Asian options can be differentiatedinto two main
classes according to their payoff: fixed strikeprice options
(sometimes called “average price”) and floatingstrike price options
(sometimes called “average strike”). Allthese details are specified
by the contracts stipulated bytwo counterparts, as Asian options
are traded actively on
the OTC market among investors or traders for hedgingthe average
price of a commodity. For a brief introductionto the development of
Asian options, see Boyle and Boyle[1].
As the probability distribution of the average prices of
theunderlying asset generally does not have a simple
analyticalexpression, it is difficult to obtain the analytical
pricingformula for Asian option. Since the best-known
closed-formpricing formula for the European vanilla option
derivedby Black and Scholes [2]), many researchers have
devotedthemselves to developing the Asian options pricing based
onthe Black-Scholes assumptions; see, e.g., Kemna and Vorst(1990),
Turnbull andWakeman [3], Ritchken et al. [4], Gemanand Yor [5],
Rogers and Shi [6], Boyle et al. [7], Angus [8],Linetsky [9], Cui
et al. [10], and the references therein. For arecent review, one
can refer to Fusai and Roncoroni [11] andSun et al. [12].
In practice, the Black-Scholes assumptions are hardly
sat-isfied, especially the constant volatility and constant
interestrate hypothesis. As the empirical behaviors of the
impliedvolatility smile and heavy tailed in the distribution of
log-returns are commonly observed in financial markets. For
HindawiComplexityVolume 2019, Article ID 4316272, 13
pageshttps://doi.org/10.1155/2019/4316272
http://orcid.org/0000-0002-9344-5193https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/4316272
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2 Complexity
this reason, stochastic volatility (hereafter SV) models
havebeen proposed in finance (see Hull and White [13], Steinand
Stein [14], Heston [15], and others). These models havebeen applied
to value the Asian options (see, e.g., Wongand Cheung [16], Hubalek
and Sgarra [17], Kim and Wee[18], and Shi and Yang [19]). In
addition, interest rates arestochastic and stock returns are
negatively correlated withinterest rate changes, which have been
examined in previousresearch.
Although these models mentioned above are able toaccount for the
empirical behaviors, they are still based on asingle-factor for
volatility dynamics that is inconsistent withthe long range memory
characteristic of the volatility correc-tions and the stiff
volatility skews. See Alizadeh et al. [20],Fiorentini et al. [21],
Chernov et al. [22], Gourieroux [23],Christoffersen et al. [24],
Romo [25], and Nagashima et al.[26] for the empirical results. To
address this issue, mul-tifactor SV models have recently generated
attention inthe option pricing literature. For instance, Duffie et
al.[27] proposed multifactor affine stochastic volatility
models.Based upon the Black-Scholes framework, Fouque et al.
[28]introduced a multiscale SV model, in which the
volatilityprocesses are driven by two mean-reverting diffusion
pro-cesses. Gourieroux [23] proposed a multivariate model inwhich
the volatility-covolatility matrix follows a Wishartprocess.
On the basis of the findings of Christoffersen et al. [24],a
double Heston (dbH) model, which consists of two inde-pendent
variance processes, has recently been reported betterthan the plain
Heston [15] model in the performances ofhedging (see Sun [29]) and
has also been applied to arithmeticAsian option under discrete
monitoring (Mehrdoust andSaber [30]) and forward starting option
(Zhang and Sun [31]).However, its extension to continuously
monitored geometricAsian option is yet to be considered. On the
other hand,many of Asian options often have long-dated maturities
sincethey are used as part of the structured notes which has along
maturity. The movement of interest rates becomes anissue in such
cases and constant interest rate assumptionshould be replaced by an
appropriate dynamic interest ratemodel. Several results are
available on the Asian option inthe stochastic interest rate
framework; see, e.g., Nielsen andSandmann [32, 33], Zhang et al.
[34], Donnelly et al. [35],and He and Zhu [36]. In the above
stochastic interest rateframework, the short-term interest rate is
assumed to followa specific parametric one-factor model (see, e.g.,
Cox et al.[37], Hull and White [13], and Vasicek [38]), which
tendsto oversimplify the true behavior of interest rate
movement.However, empirical tests reported in Lonstaff and
Schwartz[39] and Pearson and Sun [40] show that the term struc-ture
for the interest rate should involve several sources ofuncertainty,
and introducing additional state variables (suchas the rate of
inflation, GDP, etc.) significantly improves thefit.
In this paper, we study the pricing of the continuouslymonitored
geometric Asian options under dbH stochasticvolatility model with
stochastic interest rate framework(hereafter, dbH-SI model). The
contribution of the present
paper is twofold. Firstly, this paper extends the dbH modelby
introducing stochastic interest rate, which is assumed tofollow
two-factor model with two state variables. Secondly,this paper
provides a semiexplicit valuation formula for thegeometric Asian
options with fixed or floating strike price,which is extremely
useful also for the arithmetic averageoption valuation via Monte
Carlo methods with controlvariables.
The rest of the paper is organized as follows. Section 2develops
the underlying pricing model and describes thegeometric Asian
option. Section 3 derives the joint char-acteristic function of a
log-return of the underlying assetand its geometric average.
Section 4 obtains the analyticexpressions for the prices of the
fixed strike geometric Asiancall option and the floating strike
Asian call option undercontinuous monitoring. Section 5 provides
some numericalexamples for the proposed approach. Section 6
concludes thepaper.
2. Model Formulation
We consider an arbitrage-free, frictionless financial
marketwhere only riskless asset and risky asset are traded
continu-ously up to a fixed horizon date 𝑇. Let (Ω,F, {F𝑡}0≤𝑡≤𝑇,
𝑄)be a complete probability space equipped with a filtration{F𝑡}
satisfying the usual conditions, where𝑄 is a
risk-neutralprobability measure. Suppose 𝑊𝑗𝑡 and 𝑍𝑗𝑡 (𝑗 = 1, 2) are
allstandard Brownian motions defined on the probability space,and
the filtration {F𝑡}𝑡≥0 is generated by these Brownianmotions.
Moreover, 𝑑𝑊1𝑡 𝑑𝑍1𝑡 = 𝜌1𝑑𝑡, 𝑑𝑊2𝑡 𝑑𝑍2𝑡 = 𝜌2𝑑𝑡,and any other Brownian
motions are pairwisely independent.Assume that the asset price
process 𝑆𝑡, without payingany dividend, satisfies the following
stochastic differentialequation under 𝑄:
𝑑𝑆𝑡𝑆𝑡 = 𝑟𝑡𝑑𝑡 + √V1𝑡𝑑𝑊1𝑡 + √V2𝑡𝑑𝑊2𝑡 ,𝑑V1𝑡 = 𝜅1 (𝜃1 − V1𝑡) 𝑑𝑡 +
𝜎1√V1𝑡𝑑𝑍1𝑡 ,𝑑V2𝑡 = 𝜅2 (𝜃2 − V2𝑡) 𝑑𝑡 + 𝜎2√V2𝑡𝑑𝑍2𝑡 ,
(1)
where 𝜅𝑗, 𝜃𝑗, 𝜎𝑗 (𝑗 = 1, 2) are all nonnegative constants,which
represent the mean-reverting rates, long-term meanlevels, and
volatilities of variance processes V𝑗𝑡, respectively.We suppose
that 2𝜅𝑗𝜃𝑗 ≥ 𝜎2𝑗 . The instantaneous interestrate, 𝑟𝑡, is assumed
to be a linear combination of V1𝑡 andV2𝑡, i.e., 𝑟𝑡 = V1𝑡 + V2𝑡,
which designates the interest rateas an affine function of
two-factor economic variables V1and V2 and offers the analytic
tractability (see Duffie et al.[27]).
In financial market, there are four types of Europeanstyle
continuously monitoring geometric Asian options: fixedstrike
geometric Asian calls, fixed strike geometric Asianputs, floating
strike geometric Asian calls, and floating strikegeometric Asian
puts.The payoffs at the expiration date 𝑇 forthese options are as
follows:
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Complexity 3
ℎ (𝑆𝑇, 𝐺[0,𝑇]) ={{{{{{{{{{{{{{{{{
max {𝐺[0,𝑇] − 𝐾, 0} , a fixed strike geometric Asian calls,max
{𝐾 − 𝐺[0,𝑇], 0} , a fixed strike geometric Asian puts,max {𝐺[0,𝑇] −
𝑆𝑇, 0} , a floating strike geometric Asian calls,max {𝑆𝑇 − 𝐺[0,𝑇],
0} , a floating strike geometric Asian puts,
(2)
where 𝐾 is a fixed strike price and 𝐺[0,𝑇] is the
geometricaverage of the underlying asset price 𝑆𝑡 until time 𝑇;
i.e.,𝐺[0,𝑇] = exp((1/𝑇) ∫𝑇0 ln 𝑆𝑢𝑑𝑢). In the following, we
consideronly the pricing problem of the geometric Asian call
options(hereafter, GAC), while the put options can be dealt
withsimilarly.
For the instantaneous interest rate 𝑟𝑡 = V1𝑡 + V2𝑡, onecan
express the price at time 𝑡 of a zero-coupon bond withmaturate 𝑇 as
follows (see Cox et al. [37]):
𝑃 (𝑡, 𝑇) = exp {𝑎 (𝑡, 𝑇) − 𝑏 (𝑡, 𝑇) 𝑟𝑡} , (3)where
𝑎 (𝑡, 𝑇) = 2∑𝑗=1
[𝜅𝑗𝜃𝑗 (𝜅𝑗 − 𝛾𝑗) (𝑇 − 𝑡)𝜎2𝑗+ 2𝜅𝑗𝜃𝑗𝜎2𝑗 ln
2𝛾𝑗2𝛾𝑗 + (𝜅𝑗 − 𝛾𝑗) (1 − 𝑒−𝛾𝑗(𝑇−𝑡))] ,
𝑏 (𝑡, 𝑇) = 2∑𝑗=1
[ 2 (1 − 𝑒−𝛾𝑗(𝑇−𝑡))2𝛾𝑗 + (𝜅𝑗 − 𝛾𝑗) (1 − 𝑒−𝛾𝑗(𝑇−𝑡))] ,𝛾𝑗 = √𝜅2𝑗 +
2𝜎2𝑗 , for 𝑗 = 1, 2.
(4)
3. The Joint Characteristic Function
Given the dynamic of the underlying asset price, it is
possibleto obtain the discounted joint characteristic function for
thelog-asset value 𝑆𝑇 and the log-geometric mean value of theasset
price over a certain time period.
Let 𝜓𝑡(𝑠, 𝑤) = 𝐸𝑄[𝑒− ∫𝑇𝑡 𝑟𝜏𝑑𝜏+𝑠 ln𝐺[𝑡,𝑇]+𝑤 ln 𝑆𝑇 | F𝑡] be
thediscounted joint characteristic function of
two-dimensionalrandom variable, (ln𝐺[𝑡,𝑇], ln 𝑆𝑇), conditioned on
F𝑡 under𝑄, where𝐺[𝑡,𝑇] = exp{(1/𝑇) ∫𝑇𝑡 ln 𝑆𝑢𝑑𝑢} and 𝐸𝑄[⋅ | F𝑡] is
theconditional expectation under 𝑄 for 𝑡 ∈ [0, 𝑇]. Denote 𝐷 ={(𝑠,
𝑤) ∈ C2 : R(𝑠) ≥ 0,R(𝑤) ≥ 0, 0 ≤ R(𝑠) +R(𝑤) ≤ 1}.Proposition 1.
Suppose that 𝑆𝑡, V1𝑡, and V2𝑡 follow thedynamics in (1). If (𝑠, 𝑤)
∈ 𝐷 and 𝑡 ∈ [0, 𝑇], then𝐸𝑄[|𝑒− ∫𝑇𝑡 𝑟𝜏𝑑𝜏+𝑠 ln𝐺[𝑡,𝑇]+𝑤 ln 𝑆𝑇 |] <
∞ and
𝜓𝑡 (𝑠, 𝑤) = 𝑒𝑔0𝐸𝑄[[exp{{{2∑𝑗=1
[𝑔𝑗1 ∫𝑇𝑡(𝑇 − 𝜏)2 V𝑗𝜏𝑑𝜏 + 𝑔𝑗2 ∫𝑇
𝑡(𝑇 − 𝜏) V𝑗𝜏𝑑𝜏 + 𝑔𝑗3 ∫𝑇
𝑡V𝑗𝜏𝑑𝜏 + 𝑔𝑗4V𝑗𝑇]}}} | V1𝑡, V2𝑡
]] , (5)
where
𝑔0 = 𝑔0 (𝑠, 𝑤) = 𝑠[[𝑇 − 𝑡𝑇 ln 𝑆𝑡
− 1𝑇2∑𝑗=1
(𝜌𝑗𝜅𝑗𝜃𝑗2𝜎𝑗 (𝑇 − 𝑡)2 +𝜌𝑗 (𝑇 − 𝑡)𝜎𝑗 V𝑗𝑡)]]
+ 𝑤[[ln 𝑆𝑡 −2∑𝑗=1
(𝜌𝑗𝜅𝑗𝜃𝑗𝜎𝑗 (𝑇 − 𝑡) +𝜌𝑗𝜎𝑗 V𝑗𝑡)]] ,
𝑔𝑗1 = 𝑔𝑗1 (𝑠, 𝑤) = 𝑠2 (1 − 𝜌2𝑗 )2𝑇2 ,
𝑔𝑗2 = 𝑔𝑗2 (𝑠, 𝑤) = 𝑠 (2𝜌𝑗𝜅𝑗 + 𝜎𝑗)2𝜎𝑗𝑇 +𝑠𝑤 (1 − 𝜌2𝑗 )𝑇 ,
𝑔𝑗3 = 𝑔𝑗3 (𝑠, 𝑤) = 𝑠𝜌𝑗𝑇𝜎𝑗 +𝑤 (2𝜌𝑗𝜅𝑗 + 𝜎𝑗)2𝜎𝑗
+ 𝑤2 (1 − 𝜌2𝑗 )2 − 1,𝑔𝑗4 = 𝑔𝑗4 (𝑠, 𝑤) = 𝑤𝜌𝑗𝜎𝑗 .
(6)
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4 Complexity
Proof. (i) We first prove that the integrability
conditionguarantees the existence of the cumulant function 𝜓𝑡(𝑠, 𝑤)
in𝐷. If (𝑠, 𝑤) ∈ 𝐷 and 𝑡 ∈ [0, 𝑇), then
𝐸𝑄 [𝑒−∫𝑇
𝑡𝑟𝜏𝑑𝜏+𝑠ln𝐺[𝑡,𝑇]+𝑤ln𝑆𝑇
]= 𝑃 (𝑡, 𝑇) 𝐸𝑄𝑇 [𝑒𝑠ln𝐺[𝑡,𝑇]+𝑤ln𝑆𝑇 ]≤ 𝑃 (𝑡, 𝑇) 𝐸𝑄𝑇 [ 1𝑇 − 𝑡 ∫
𝑇
𝑡𝑆𝑢𝑑𝑢 + 𝑆𝑇]
= [ 1𝑇 − 𝑡 ∫𝑇
𝑡
𝑃 (𝑡, 𝑇)𝑃 (𝑡, 𝑢) 𝑑𝑢 + 1] 𝑆𝑡 < ∞,
(7)
where 𝑄𝑇 is the 𝑇− forward measure given by the Radon-Nikodym
derivative: (𝑑𝑄𝑇/𝑑𝑄)|F𝑇 = exp{− ∫𝑇0 𝑟𝑢𝑑𝑢}/𝑃(0, 𝑇), and 𝑃(𝑡, 𝑇) is
given above in (3). In the case of 𝑡 = 𝑇,it is triviality.
(ii) In order to determine (5), we start from model(1) and
develop ln 𝑆𝑇 with the Brownian motions 𝑊1𝑡 and𝑊2𝑡 expressed as 𝑊1𝑡
= 𝜌1𝑍1𝑡 + √1 − 𝜌12𝑍1𝑡 and 𝑊2𝑡 =𝜌1𝑍2𝑡 + √1 − 𝜌22𝑍2𝑡 , respectively,
where (𝑍1𝑡 , 𝑍2𝑡 , 𝑍1𝑡 , 𝑍2𝑡 ) are4-dimensional Brownian motion
defined on the probabilityspace. For the process, ln 𝑆𝑇, we
have
ln 𝑆𝑇 = ln 𝑆𝑡 + 2∑𝑗=1
[−𝜌𝑗𝜅𝑗𝜃𝑗𝜎𝑗 (𝑇 − 𝑡) −𝜌𝑗𝜎𝑗 V𝑗𝑡
+ (𝜌𝑗𝜅𝑗𝜎𝑗 +12)∫𝑇
𝑡V𝑗𝜏𝑑𝜏 + 𝜌𝑗𝜎𝑗 V𝑗𝑇
+ √1 − 𝜌2𝑗 ∫𝑇𝑡√V𝑗𝜏𝑑𝑍𝑗𝜏] = ln 𝑆𝑡 + 2∑
𝑗=1[𝜃𝑗2 (𝑇 − 𝑡)
+ V𝑗𝑡2𝜅𝑗 −V𝑗𝑇2𝜅𝑗 + (𝜌𝑗 +
𝜎𝑗2𝜅𝑗)∫𝑇
𝑡√V𝑗𝜏𝑑𝑍𝑗𝜏
+ √1 − 𝜌2𝑗 ∫𝑇𝑡√V𝑗𝜏𝑑𝑍𝑗𝜏] .
(8)
On the other hand, we have
ln𝐺[𝑡,𝑇] = 𝑇 − 𝑡𝑇 ln 𝑆𝑡 + 1𝑇2∑𝑗=1
[𝜃𝑗4 (𝑇 − 𝑡)2
+ V𝑗𝑡2𝜅𝑗 (𝑇 − 𝑡) −12𝜅𝑗 ∫𝑇
𝑡V𝑗𝜏𝑑𝜏
+ (𝜌𝑗 + 𝜎𝑗2𝜅𝑗)∫𝑇
𝑡(𝑇 − 𝜏)√V𝑗𝜏𝑑𝑍𝑗𝜏
+ √1 − 𝜌2𝑗 ∫𝑇𝑡(𝑇 − 𝜏)√V𝑗𝜏𝑑𝑍𝑗𝜏] .
(9)
Using the fact
∫𝑇𝑡(𝑇 − 𝜏)√V𝑗𝜏𝑑𝑍𝑗𝜏 = 1𝜎𝑗 [𝜅𝑗 ∫
𝑇
𝑡(𝑇 − 𝜏) V𝑗𝜏𝑑𝜏
+ ∫𝑇𝑡V𝑗𝜏𝑑𝜏 − (𝑇 − 𝑡) V𝑗𝑡 − 𝜅𝑗𝜃𝑗 (𝑇 − 𝑡)22 ] ,
(10)
for 𝑗 = 1, 2, thenln𝐺[𝑡,𝑇] = 𝑇 − 𝑡𝑇 ln 𝑆𝑡 + 1𝑇
2∑𝑗=1
[−𝜌𝑗𝜅𝑗𝜃𝑗2𝜎𝑗 (𝑇 − 𝑡)2
− 𝜌𝑗V𝑗𝑡𝜎𝑗 (𝑇 − 𝑡) +𝜌𝑗𝜎𝑗 ∫𝑇
𝑡V𝑗𝜏𝑑𝜏
+ (𝜌𝑗𝜅𝑗𝜎𝑗 +12)∫𝑇
𝑡(𝑇 − 𝜏) V𝑗𝜏𝑑𝜏
+ √1 − 𝜌2𝑗 ∫𝑇𝑡(𝑇 − 𝜏)√V𝑗𝜏𝑑𝑍𝑗𝜏] .
(11)
Let G be the 𝜎−field generated byF𝑡 and {(𝑍1𝑢, 𝑍2𝑢) : 𝑡
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Complexity 5
+ 2∑𝑗=1
[𝑤(𝜌𝑗V𝑗𝑇𝜎𝑗 +2𝜌𝑗𝜅𝑗 + 𝜎𝑗2𝜎𝑗 ∫
𝑇
𝑡V𝑗𝜏𝑑𝜏)
− ∫𝑇𝑡V𝑗𝜏𝑑𝜏]}}} ,
𝐴3 = exp{{{2∑𝑗=1
√1 − 𝜌2𝑗 ∫𝑇𝑡[ 𝑠𝑇 (𝑇 − 𝜏) + 𝑤]
⋅ √V𝑗𝜏𝑑𝑍𝑗𝜏}}} .(13)
Since
𝐸𝑄 [𝐴3 | G] = exp{{{122∑𝑗=1
(1 − 𝜌2𝑗 )
⋅ ∫𝑇𝑡[ 𝑠𝑇 (𝑇 − 𝜏) + 𝑤]
2V𝑗𝜏𝑑𝜏}}}
= exp{{{2∑𝑗=1
(1 − 𝜌2𝑗 )2⋅ ∫𝑇𝑡[ 𝑠2𝑇2 (𝑇 − 𝜏)2 + 2𝑠𝑤𝑇 (𝑇 − 𝜏) + 𝑤2] V𝑗𝜏𝑑𝜏}}}
,
(14)
substituting (14) into (12) and applying the Markov propertyof
{V𝑗𝑢 : 0 ≤ 𝑢 ≤ 𝑇}, 𝑗 = 1, 2, lead to (5), which completes
theproof.
FromProposition 1, it is clear thatweneed to search for anexact
formula for the discounted joint characteristic functionof V1𝑡 and
V2𝑡 and the three different integrals of V𝑗𝑡 (𝑗 =1, 2) appearing in
(5). We use the same approach introducedby Kim and Wee [18] to
obtain an explicit formula for thediscounted joint characteristic
function 𝜓𝑡(𝑠, 𝑤). Therefore,
two series of functions are introduced as follows. Define 𝐹𝑗𝜏and
𝐹𝑗𝜏 : C4 → C as
𝐹𝑗𝜏 (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) = ∞∑𝑛=0
𝑓𝑗𝑛 , (15)𝐹𝑗𝜏 (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) = ∞∑
𝑛=1
𝑛𝜏𝑓𝑗𝑛 , (16)where𝑓𝑗𝑛 , 𝑛 = 0, 1, 2, ⋅ ⋅ ⋅ , are functions of
𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, and 𝑔𝑗4defined as
𝑓𝑗−2 = 𝑓𝑗−1 = 0,𝑓𝑗0 = 1,𝑓𝑗1 = (𝜅𝑗 − 𝑔𝑗4𝜎
2𝑗 ) 𝜏2 ,
𝑓𝑗𝑛 = − 𝜎2𝑗𝜏22𝑛 (𝑛 − 1) (𝑔𝑗1𝜏2𝑓𝑗𝑛−4 + 𝑔𝑗2𝜏𝑓𝑗𝑛−3
+ (𝑔𝑗3 − 𝜅2𝑗2𝜎2𝑗 )𝑓
𝑗𝑛−2) , 𝑛 ≥ 2,
(17)
for 𝑗 = 1, 2.For 𝑗 = 1, 2, denote𝐷𝑗𝜏 = {(𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) ∈
C4 :
𝐸𝑄 [exp{R (𝑔𝑗1) ∫𝜏0(𝜏 − 𝑡)2 V𝑗𝑡𝑑𝑡
+R (𝑔𝑗2)∫𝜏0(𝜏 − 𝑡) V𝑗𝑡𝑑𝑡 +R (𝑔𝑗3)∫𝜏
0V𝑗𝑡𝑑𝑡
+R (𝑔𝑗4) V𝑗𝜏}] < ∞} .
(18)
Proposition 2. (1) �e argument of 𝐹𝑗𝜏 : arg𝐹𝑗𝜏(𝑔𝑗1,𝑔𝑗2, 𝑔𝑗3,
𝑔𝑗4) ̸= 0 for every (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) ∈ 𝐷𝑗𝜏 (𝑗 = 1, 2).In
particular, arg𝐹𝑗𝜏 is continuous on 𝐷𝑗𝜏, andarg𝐹𝑗𝜏(𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3,
𝑔𝑗4) = 0 for 𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, and 𝑔𝑗4 areall real numbers in𝐷𝑗𝜏 for 𝑗
= 1, 2.
(2) For (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) ∈ 𝐷𝑗𝜏, 𝑗 = 1, 2, we have
𝐸𝑄[[exp{{{2∑𝑗=1
[𝑔𝑗1 ∫𝜏0(𝜏 − 𝑡)2 V𝑗𝑡𝑑𝑡 + 𝑔𝑗2 ∫𝜏
0(𝜏 − 𝑡) V𝑗𝑡𝑑𝑡 + 𝑔𝑗3 ∫𝜏
0V𝑗𝑡𝑑𝑡 + 𝑔𝑗4V𝑗𝜏]}}}
]]
= exp{{{2∑𝑗=1
(𝜅𝑗V𝑗0 + 𝜅2𝑗𝜃𝑗𝜏𝜎𝑗 −2V𝑗0𝜎2𝑗
𝐹𝑗𝜏 (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4)𝐹𝑗𝜏 (𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) −2𝜅𝑗𝜃𝑗𝜎2𝑗 ln𝐹𝑗𝜏
(𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4))
}}} .(19)
-
6 Complexity
�e proof of Proposition 2 is similar to that of Kim
andWee[18].
Using Proposition 2 to Proposition 1 leads to the
explicitexpression of 𝜓𝑡(𝑠, 𝑤) in Proposition 3. To describe the
simplic-ity of the result, we need to introduce new functions.
Define𝐻𝑗𝑡,𝑇and �̃�𝑗𝑡,𝑇: C2 → C as𝐻𝑗𝑡,𝑇 (𝑠, 𝑤)= 𝐹𝑗𝑇−𝑡 (𝑔𝑗1 (𝑠, 𝑤) ,
𝑔𝑗2 (𝑠, 𝑤) , 𝑔𝑗3 (𝑠, 𝑤) , 𝑔𝑗4 (𝑠, 𝑤)) ,
(20)
�̃�𝑗𝑡,𝑇 (𝑠, 𝑤)= 𝐹𝑗𝑇−𝑡 (𝑔𝑗1 (𝑠, 𝑤) , 𝑔𝑗2 (𝑠, 𝑤) , 𝑔𝑗3 (𝑠, 𝑤) ,
𝑔𝑗4 (𝑠, 𝑤)) ,
(21)
with 𝑗 = 1, 2.Proposition 3. (1) If (𝑠, 𝑤) ∈ 𝐷, then 𝐻𝑗𝑡,𝑇(𝑠, 𝑤)
̸= 0 andarg𝐻𝑗𝑡,𝑇 is continuous on𝐷. In particular, arg𝐻𝑗𝑡,𝑇(𝑠, 𝑤) =
0 if𝑠 and 𝑤 are real numbers for 𝑗 = 1, 2.
(2) For 𝜓𝑡(𝑠, 𝑤) ∈ 𝐷 with arg𝐻𝑗𝑡,𝑇 defined as above, then
𝜓𝑡 (𝑠, 𝑤) = exp{{{2∑𝑗=1
(−𝑐𝑗1 �̃�𝑗𝑡,𝑇 (𝑠, 𝑤)𝐻𝑗𝑡,𝑇 (𝑠, 𝑤)
− 𝑐𝑗2ln𝐻𝑗𝑡,𝑇 (𝑠, 𝑤) + 𝑐𝑗3𝑠 + 𝑐𝑗4𝑤 + 𝑐𝑗5)}}} .(22)
Here 𝑐𝑗1 = 2V𝑗𝑡/𝜎2𝑗 , 𝑐𝑗2 = 2𝜅𝑗𝜃𝑗/𝜎2𝑗 ,
𝑐𝑗3 = 𝑇 − 𝑡2𝑇 ln 𝑆𝑡 −𝜌𝑗𝜅𝑗𝜃𝑗 (𝑇 − 𝑡)22𝜎𝑗𝑇 −
𝜌𝑗 (𝑇 − 𝑡)𝜎𝑗𝑇 V𝑗𝑡,𝑐𝑗4 = ln 𝑆𝑡2 −
𝜌𝑗𝜎𝑗 V𝑗𝑡 −𝜌𝑗𝜅𝑗𝜃𝑗 (𝑇 − 𝑡)𝜎𝑗 ,
𝑐𝑗5 = 𝜅𝑗V𝑗𝑡 + 𝜅2𝑗𝜃𝑗 (𝑇 − 𝑡)𝜎2𝑗
(23)
for 𝑗 = 1, 2.Proof. Assume that (𝑠, 𝑤) ∈ 𝐷, using the
definitions of 𝑔𝑗𝑘 =𝑔𝑗𝑘(𝑠, 𝑤) and 𝐷𝑗𝜏 for 𝑘 = 1, 2, 3, 4 and 𝑗 = 1,
2. Note that(𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4) ∈ 𝐷𝑗𝜏 ⊂ 𝐷𝑗𝑇−𝑡 for every 0 ≤ 𝑡 ≤ 𝑇
and𝑗 = 1, 2. Therefore, (19) remains valid for any V𝑗0 > 0.
Thetime homogeneous Markov property of V𝑗𝑡 implies that (22)holds
when Propositions 1 and 2 are satisfied.
Now Substituting the expressions of 𝑔𝑗1, 𝑔𝑗2, 𝑔𝑗3, 𝑔𝑗4, 𝑗 =1, 2
into (15) and (16),𝐻𝑗𝑡,𝑇 and �̃�𝑗𝑡,𝑇 are expressed as follows,i.e.,
for (𝑠, 𝑤) ∈ C2,
𝐻𝑗𝑡,𝑇 (𝑠, 𝑤) = ∞∑𝑛=0
ℎ𝑗𝑛 (𝑠, 𝑤) , (24)�̃�𝑗𝑡,𝑇 (𝑠, 𝑤) = ∞∑
𝑛=1
𝑛𝑇 − 𝑡ℎ𝑗𝑛 (𝑠, 𝑤) , (25)where
ℎ𝑗−2 (𝑠, 𝑤) = ℎ𝑗−1 (𝑠, 𝑤) = 0,ℎ𝑗0 (𝑠, 𝑤) = 1,ℎ𝑗1 (𝑠, 𝑤) = (𝑇 −
𝑡) (𝜅𝑗 − 𝑤𝜌𝑗𝜎𝑗)2 ,ℎ𝑗𝑛 (𝑠, 𝑤) = (𝑇 − 𝑡)24𝑛 (𝑛 − 1) 𝑇2 {−𝑠2𝜎2𝑗 (1 −
𝜌2𝑗 ) (𝑇 − 𝑡)2
⋅ ℎ𝑗𝑛−4 (𝑠, 𝑤) − [𝑠𝜎𝑗𝑇(𝜎𝑗 + 2𝜌𝑗𝜅𝑗)+ 2𝑠𝑤𝜎2𝑗𝑇 (1 − 𝜌2𝑗 )] (𝑇 − 𝑡)
ℎ𝑗𝑛−3 (𝑠, 𝑤) + 𝑇 [𝜅2𝑗𝑇− 2𝑠𝜌𝑗𝜎𝑗 − 𝑤 (2𝜌𝑗𝜅𝑗 + 𝜎𝑗) 𝜎𝑗𝑇 − 𝑤2 (1 − 𝜌2𝑗 )
𝜎2𝑗𝑇+ 2𝜎2𝑗𝑇] ℎ𝑗𝑛−2 (𝑠, 𝑤)} , 𝑛 ≥ 2,
(26)
for 𝑗 = 1, 2.4. Pricing Geometric Asian Option
Once the discounted joint characteristic function is found,the
European continuously monitored geometric Asianoption can be valued
using numeraire change technique andthe inverse Fourier transform
approach which are appliedin many research works (see, e.g., Geman
et al. [41] andDeng [42]). This section derives the pricing
formulas for thecontinuously monitored fixed and floating strike
geometricAsian call options.
Theorem 4. Suppose that 𝑆𝑡, V1𝑡, and V2𝑡 follow the dynamicsin
(1), then the price at time 𝑡 ∈ [0, 𝑇] of the continuously
mon-itored fixed strike geometric Asian call option with maturity
𝑇and the strike price 𝐾 is given by
𝐺𝐴𝐶𝑓𝑖 (𝑡, 𝑆, V1, V2, 𝐾, 𝑇)= 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝜓𝑡 (1, 0) Π1 − 𝐾𝑃
(𝑡, 𝑇)Π2, (27)
where 𝑃(𝑡, 𝑇) is given above in (3) andΠ𝑗 = 12 + 1𝜋 ∫
+∞
0R[𝑒−𝑖𝑢ln𝐾𝑡,𝑇𝜙𝑗 (𝑢)𝑖𝑢 ]𝑑𝑢,
𝑗 = 1, 2,
-
Complexity 7
𝐾𝑡,𝑇 = 𝐾𝑒−(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢,𝜙1 (𝑢) = 𝜓𝑡 (𝑖𝑢 + 1, 0)𝜓𝑡 (1, 0) ,𝜙2
(𝑢) = 𝜓𝑡 (𝑖𝑢, 0)𝜓𝑡 (0, 0) ,
(28)
where 𝑖 is the imaginary unit (𝑖2 = −1).Proof. Since
𝐺𝐴𝐶𝑓𝑖 (𝑡, 𝑆, V1, V2, 𝐾, 𝑇)= 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢 (𝐺[0,𝑇] − 𝐾)+ | F𝑡]=
𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝐸𝑄 [𝑒− ∫T𝑡 𝑟𝑢𝑑𝑢𝐺[𝑡,𝑇]1{ln𝐺[𝑡,𝑇]≥ln𝐾𝑡,𝑇} |F𝑡] − 𝐾𝐸𝑄
[𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢1{ln𝐺[𝑡,𝑇]≥ln𝐾𝑡,𝑇} | F𝑡]= 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝜓𝑡 (1, 0)𝑄1
(ln𝐺[𝑡,𝑇] ≥ ln𝐾𝑡,𝑇)− 𝐾𝑃 (𝑡, 𝑇)𝑄𝑇 (ln𝐺[𝑡,𝑇] ≥ ln𝐾𝑡,𝑇) ,
(29)
where𝑄1 is defined by the following
Radon-Nikodymderiva-tive:
𝑑𝑄1𝑑𝑄F𝑇 = 𝑒− ∫
𝑇
𝑡𝑟𝑢𝑑𝑢
𝐺[𝑡,𝑇]𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢𝐺[𝑡,𝑇] | F𝑡] , (30)
and 𝑄𝑇 is the 𝑇− forward measure given above, it is wellknown
that the probability distribution functions can becalculated by
using the Fourier inversion transform, and thenthe above two
probabilities 𝑄1 and 𝑄𝑇 in (29) are given by
𝑄1 (ln𝐺[𝑡,𝑇] ≥ ln𝐾𝑡,𝑇)= 12 + 1𝜋 ∫
+∞
0R [𝑒−𝑖𝑢ln𝐾𝑡,𝑇𝜙1 (𝑢)𝑖𝑢 ] 𝑑𝑢 fl Π1,
(31)
𝑄𝑇 (ln𝐺[𝑡,𝑇] ≥ ln𝐾𝑡,𝑇)= 12 + 1𝜋 ∫
+∞
0R [𝑒−𝑖𝑢ln𝐾𝑡,𝑇𝜙2 (𝑢)𝑖𝑢 ] 𝑑𝑢 fl Π2,
(32)
where
𝜙1 (𝑢) = 𝐸𝑄1 [𝑒𝑖𝑢ln𝐺[𝑡,𝑇] | F𝑡]= 𝐸𝑄 [𝑒− ∫𝑇𝑡
𝑟𝑢𝑑𝑢+𝑖𝑢ln𝐺[𝑡,𝑇]𝐺[𝑡,𝑇] | F𝑡]𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢𝐺[𝑡,𝑇] | F𝑡]
= 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢+(𝑖𝑢+1)ln𝐺[𝑡,𝑇] | F𝑡]𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢+ln𝐺[𝑡,𝑇] |
F𝑡]
= 𝜓𝑡 (𝑖𝑢 + 1, 0)𝜓𝑡 (1, 0) ,(33)
𝜙2 (𝑢) = 𝐸𝑄𝑇 [𝑒𝑖𝑢ln𝐺[𝑡,𝑇] | F𝑡]= 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢+𝑖𝑢ln𝐺[𝑡,𝑇] |
F𝑡]𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢 | F𝑡] =
𝜓𝑡 (𝑖𝑢, 0)𝑃 (𝑡, 𝑇) .(34)
From (29), (31), and (32), we can obtain the requiredTheorem
4.
Theorem 5. Suppose that 𝑆𝑡, V1𝑡, and V2𝑡 follow the dynamicsin
(1), then the price at time 𝑡 ∈ [0, 𝑇] of the continuouslymonitored
floating strike geometric Asian call option withmaturity 𝑇 is given
by
𝐺𝐴𝐶𝑓𝑙 (𝑡, 𝑆, V1, V2, 𝑇) = 𝑆𝑡Π̂1− 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝜓𝑡 (1, 0) Π̂2,
(35)
where
Π̂1= 12
− 1𝜋 ∫+∞
0R[[
𝜓𝑡 (𝑖𝑢, 1 − 𝑖𝑢) 𝑒(𝑖𝑢/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝑖𝑢𝑆𝑡 ]]𝑑𝑢,Π̂2= 12
− 1𝜋 ∫+∞
0R[[
𝜓𝑡 (𝑖𝑢 + 1, −𝑖𝑢) 𝑒(𝑖𝑢/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝑖𝑢𝜓𝑡 (1, 0) ]]𝑑𝑢.
(36)
Proof. Since
𝐺𝐴𝐶𝑓𝑙 (𝑡, 𝑆, V1, V2, 𝑇) = 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢 (𝑆𝑇 − 𝐺[0,𝑇])+ | F𝑡]=
𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢 (𝑆𝑇𝑒−(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢 − 𝐺[𝑡,𝑇])+
|F𝑡] = 𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢𝑆𝑇1{ln(𝐺[𝑡,𝑇]/𝑆𝑇)≤−(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢} | F𝑡]−
𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝐸𝑄 [𝑒− ∫𝑇𝑡 𝑟𝑢𝑑𝑢𝐺[𝑡,𝑇]1{ln(𝐺𝑡,𝑇/𝑆𝑇)≤−(1/𝑇) ∫𝑡0
ln𝑆𝑢𝑑𝑢} |
-
8 Complexity
F𝑡] = 𝑆𝑡𝑄2 (ln(𝐺[𝑡,𝑇]𝑆𝑇 ) ≤ −1𝑇 ∫𝑡
0ln 𝑆𝑢𝑑𝑢)
− 𝑒(1/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝜓𝑡 (1, 0)𝑄1 (ln(𝐺[𝑡,𝑇]𝑆𝑇 ) ≤ −1𝑇 ∫𝑡
0ln 𝑆𝑢𝑑𝑢) ,
(37)
where𝑄2 is defined by the following
Radon-Nikodymderiva-tive:
𝑑𝑄2𝑑QF𝑇 = 𝑒− ∫
𝑇
𝑡𝑟𝑢𝑑𝑢 𝑆𝑇𝑆𝑡 , (38)
under the two probability measures 𝑄1 and 𝑄2, the condi-tional
characteristic functions of ln(𝐺[𝑡,𝑇]/𝑆𝑇) are given by
𝐸𝑄1 [𝑒𝑖𝑢ln(𝐺[𝑡,𝑇]/𝑆𝑇) | F𝑡]= 𝐸𝑄 [𝑒−∫𝑇𝑡
𝑟𝑢𝑑𝑢+𝑖𝑢ln(𝐺[𝑡,𝑇]/S𝑇)𝐺[𝑡,𝑇] | F𝑡]
𝐸𝑄 [𝑒−∫𝑇𝑡 𝑟𝑢𝑑𝑢𝐺[𝑡,𝑇] | F𝑡]= 𝜓𝑡 (𝑖𝑢 + 1, −𝑖𝑢)𝜓𝑡 (1, 0) ,
(39)
𝐸𝑄2 [𝑒𝑖𝑢ln(𝐺[𝑡,𝑇]/𝑆𝑇) | F𝑡]= 𝐸𝑄 [𝑒−∫𝑇𝑡 𝑟𝑢𝑑𝑢+𝑖𝑢ln(𝐺[𝑡,𝑇]/𝑆𝑇)+ln𝑆𝑇
| F𝑡]
𝑆𝑡= 𝜓𝑡 (𝑖𝑢, 1 − 𝑖𝑢)𝑆𝑡 .
(40)
Therefore
𝑄1 (ln(𝐺𝑡,𝑇𝑆𝑇 ) ≤ −1𝑇 ∫𝑡
0ln 𝑆𝑢𝑑𝑢)
= 12− 1𝜋 ∫
+∞
0R[[
𝜓𝑡 (𝑖𝑢 + 1, −𝑖𝑢) 𝑒(𝑖𝑢/𝑇) ∫𝑡0 ln 𝑆𝑢𝑑𝑢𝑖𝑢𝜓𝑡 (1, 0) ]]𝑑𝑢,(41)
𝑄2 (ln(𝐺𝑡,𝑇𝑆𝑇 ) ≤ −1𝑇 ∫𝑡
0ln 𝑆𝑢𝑑𝑢)
= 12− 1𝜋 ∫
+∞
0R[[
𝜓𝑡 (𝑖𝑢, 1 − 𝑖𝑢) 𝑒(𝑖𝑢/𝑇) ∫𝑡0 ln𝑆𝑢𝑑𝑢𝑆𝑡 ⋅ 𝑖𝑢 ]] 𝑑𝑢.(42)
Plugging (41) and (42) in (37) yields (35).
In addition to option prices, one can compute derivativesto
hedge against changes in the underlying asset price 𝑆
andvolatilities V1 and V2. We omit it due to its triviality.
5. Numerical Examples
In this section, we use the dbH-SI model to analyze the
val-uation of the continuously monitored fixed strike
geometricAsian call option using some numerical examples. To
imple-ment the analytic formula given in (27) numerically, we
firstdetermine the number of terms taken for the computationof the
infinite series expansions for 𝐻𝑗𝑡,𝑇 and �̃�𝑗𝑡,𝑇 for 𝑗 =1, 2 in
(24) and (25). Second, we investigate the accuracyand efficiency of
the approximated analytic formula givenin (27). In the end, we
compare the option prices varyingby 𝑆0 and 𝑇 under different models
including the dbH-SI,H-SI (i.e., Heston stochastic volatility model
with stochasticinterest rate), dbH, and Heston models. Furthermore,
we usethis proposed model (1), i.e., the dbH-SI model, to
examinethe impacts of the model parameters on option prices. Herewe
implement the integral formulas given in (27)
withouttruncation,modification, and approximation inMathematicaand
Matlab software. Numerical integration was performedusing the
“NIntegrate()” or “quadgk()” commands whichcan handle infinite
ranges, oscillatory integrands, and sin-gularities in their default
version, which employs automaticadaptive integration.
Model (1) parameters are set as follows: 𝜅1 = 1.15, 𝜅2 =2.2, 𝜃1
= 0.2, 𝜃2 = 0.15, 𝜎1 = 0.5, 𝜎2 = 0.8, 𝜌1 = −0.4,𝜌2 = −0.64, V10 =
0.09, V20 = 0.04, 𝑆0 = 100, and 𝑡 = 0. Theresults are shown in
Table 1.
Table 1 investigates the influence of the number of terms𝑛 taken
in (24) and (25). It is shown that the numerical valuestend to be
quite stable as the number of terms increases. Inparticular, the
option prices stay unchanged after taking 𝑛 =20 and 𝑛 = 30 terms,
at most. In addition, Table 1 providesoption prices with various
maturities 𝑇, strike prices 𝐾, andthe requiredCPU times (in
seconds). FromTable 1, we can seethat some characteristics of the
GAC𝑓𝑖 option are similar tothe plain European call option. For
example, the values of theGAC𝑓𝑖 option are decreasing functions of
the strike price. Inaddition, under the same parameter values
setting in dbH-SImodel, longer maturating date 𝑇 will result in
higher optionvalues.
It would be interesting to see the performance of
ourapproximated analytic formula approach implemented to thefixed
strike GAC option under dbH-SImodel. Our numericalexample uses the
number of terms 𝑛 = 20 taken in (24) and(25); other parameter
values are similar to those settings ofTable 1. We compute the
0.5-,1.5-, and 3-year maturity GACoptions by our approximated
analytic formula approach andcompare the results with Monte Carlo
simulation with 10000sample paths and 100 points for time axis. The
numericalresults are displayed in Table 2.
Table 2 shows that the approximated analytic formulaapproach is
considerably faster than those of the MonteCarlo simulation (MC).
For a given set of parameters,the approximated analytic formula
approach calculates theoption prices for 5 different strikes and 3
different maturatesin approximately 5 seconds. The Monte Carlo
simulationtakes around 110 seconds for each option price. Table 2
alsocompares their pricing accuracy. It can be seen that
therelative error (RE) in prices is less than 0.4% for all cases.
If
-
Complexity 9
Table 1: GAC𝑓𝑖 option prices with different maturities, strike
prices, and varying n.
𝑇 𝐾 𝑛 Option price CPU (sec.) 𝑇 𝐾 𝑛 Option price CPU (sec.)0.5
90 5 17.8597 0.287 1.5 90 5 26.4805 0.291
10 17.7859 0.291 10 25.2393 0.29415 17.7849 0.294 15 25.2254
0.29920 17.7849 0.296 20 25.2251 0.30225 17.7849 0.301 25 25.2251
0.30730 17.7849 0.312 30 25.2251 0.315
100 5 9.7733 0.279 100 5 19.8642 0.27610 9.8624 0.288 10 19.1846
0.28515 9.8623 0.290 15 19.1758 0.29020 9.8623 0.295 20 19.1755
0.30225 9.8623 0.298 25 19.1755 0.30930 9.8623 0.302 30 19.1755
0.311
110 5 4.2061 0.283 110 5 14.1405 0.28310 4.5395 0.291 10 14.0976
0.29115 4.5415 0.304 15 14.1003 0.29720 4.5415 0.306 20 14.1002
0.30425 4.5415 0.312 25 14.1002 0.31030 4.5415 0.315 30 14.1002
0.315
1 90 5 22.3192 0.268 3 90 5 30.5831 0.28910 22.0988 0.283 10
29.6250 0.29315 22.0938 0.287 15 29.2870 0.29920 22.0938 0.296 20
29.2835 0.30525 22.0938 0.313 25 29.2835 0.31430 22.0938 0.319 30
29.2835 0.317
100 5 15.0971 0.281 100 5 26.7286 0.26810 15.1874 0.287 10
25.6231 0.28415 15.1850 0.294 15 25.3192 0.29120 15.1849 0.298 20
25.3151 0.29525 15.1849 0.301 25 25.3151 0.29830 15.1849 0.307 30
25.3151 0.307
110 5 9.2791 0.275 110 5 22.5318 0.28110 9.7604 0.278 10 21.9684
0.28415 9.7650 0.281 15 21.7105 0.29720 9.7651 0.292 20 21.7064
0.30125 9.7651 0.294 25 21.7064 0.30530 9.7651 0.301 30 21.7064
0.310
we regard the Monte Carlo price as the benchmark, then
thisnumerical example confirms that the approximated
analyticformula approach is accurate and efficient.
After examining accuracy and efficiency. We shall turnto
investigate the comparison of the option prices underdifferent
models. Figure 1 compares the dbH-SI model withHeston, dbH, and
H-SI models in pricing the fixed strikeGAC option against the
initial asset price 𝑆0 and maturatingdate 𝑇 with the strike price 𝐾
= 100. From Figure 1, it isinteresting to notice that the option
prices of the fixed strikeGAC option calculated from the dbH-SI
model are muchhigher than those of the dbHmodel, H-SImodel, and
Hestonmodel. One possible reason for this phenomena is that
thevolatilities are always bigger for the dbH-SI model than
those
for other models. This implies that the proposed model, i.e.,the
dbH-SI model, has a more significant influence than thedbHmodel,
H-SImodel, and Hestonmodel on option prices.We can also clearly
observe that the bigger the value of 𝑆0(or𝑇) is, the higher the
GAC𝑓𝑖 option price is.
In Figure 2, we use the dbH-SI model to examine theeffects of
the mean-reverting rates (𝜅1, 𝜅2) and volatilities(𝜎1, 𝜎2) of
variance processes (V1𝑡, V2𝑡) on the GAC𝑓𝑖 optionprices (see
Figures 2(a)–2(d)). We also examine the impactsof the correlation
coefficients (𝜌1, 𝜌2) (see Figures 2(e)and 2(f)), which determine
both the correlation betweenthe underlying asset and its
volatilities, and the correla-tion between the underlying asset and
the short interestrates.
-
10 Complexity
Table 2: Comparison of the approximated approach and MC for
GAC𝑓𝑖 options.
𝑇 𝐾 Approximated approach CPU(sec.) Monte Carlo CPU(sec.)
RE(%)0.5 90 17.7849 0.284 17.7892 110.302 -0.0242
95 13.5305 0.291 13.5246 110.384 0.0436100 9.8623 0.278 9.8602
110.403 0.0213105 6.8598 0.296 6.8624 110.329 -0.3810110 4.5415
0.287 4.5407 110.376 0.0176
1.5 90 25.2251 0.265 25.2206 110.376 0.017895 22.0851 0.281
22.0891 110.391 -0.0181100 19.1755 0.294 19.1787 110.417 -0.0167105
16.5109 0.312 16.5045 110.485 0.3888110 14.1002 0.293 14.1039
110.392 -0.0262
3 90 29.2835 0.271 29.2894 110.428 -0.020195 27.2560 0.289
27.2513 110.397 0.0173100 25.3151 0.307 25.3106 110.405 0.0178105
23.4644 0.295 23.4595 110.471 0.0209110 21.7065 0.292 21.7128
110.396 -0.0290
80 85 90 95 100 105 110 115 120S0
0
5
10
15
20
25
30
35
40
45
optio
n pr
ice
dbH-SI modeldbH modelH-SI modelHeston model
(a) Option price against 𝑆0
0.5 1 1.5 2 2.5 3 3.5 4 4.5T
5
10
15
20
25
30
35
40
optio
n pr
ice
dbH-SI modeldbH modelH-SI modelHeston model
(b) Option price against 𝑇
Figure 1: Comparison of option prices under Heston, dbH, H-SI,
and dbH-SI models.
From Figures 2(a) and 2(b), we can observe that theoption prices
increase as the mean-reverting rate increases.Particularly, it is
quite remarkable that the mean-revertingrates have a significant
effect on the longer termoption values.The option price decreases
as the volatilities of varianceprocesses increase (see Figures 2(c)
and 2(d)).The correlationcoefficients (𝜌1, 𝜌2) have several effects
depending on therelation between the strike price and the initial
asset price.A negative (𝜌1, 𝜌2) tends to produce higher values for
ITM(in-the-money) options and lower values for OTM
(out-of-the-money) options (see Figure 2(e)). A similar result
holdswith respect to the expiration date: a negative (𝜌1, 𝜌2) tends
to
produce higher value for the longer term options and lowervalues
for the shorter options (see Figure 2(f)).
6. Conclusions
Theproposedmodel (the dbH-SImodel) incorporates severalimportant
features of the underlying asset returns variability.We derive the
discounted joint characteristic function ofthe log-asset price and
its log-geometric mean value overtime period [0, 𝑇] and obtain
approximated analytic solutionsto the continuously monitored fixed
and floating strikegeometric Asian call options using the change of
numeraire
-
Complexity 11
S0
5
10
15
20
25
30
35
optio
n pr
ice
80 85 90 95 100 105 110 115 120
(E1, E2)=(1.5,1)
(E1, E2)=(1.5,2)
(E1, E2)=(2,1.5)
(E1, E2)=(1.5,1.5)
(E1, E2)=(1,1.5)
(a)
T
5
10
15
20
25
30
35
optio
n pr
ice
0.5 1 1.5 2 2.5 3 3.5 4
(E1, E2)=(1.5,1)
(E1, E2)=(1.5,2)
(E1, E2)=(2,1.5)
(E1, E2)=(1.5,1.5)
(E1, E2)=(1,1.5)
(b)
S0
05
101520253035404550
optio
n pr
ice
80 85 90 95 100 105 110 115 120
(1, 2)=(0.2,0.5)
(1, 2)=(0.2,0.7)
(1, 2)=(0.2,0.9)
(1, 2)=(0.6,0.5)
(1, 2)=(0.9,0.5)
(c)
T
10
15
20
25
30
35
40
optio
n pr
ice
0.5 1 1.5 2 2.5 3 3.5 45
(1, 2)=(0.2,0.5)
(1, 2)=(0.2,0.7)
(1, 2)=(0.2,0.9)
(1, 2)=(0.6,0.5)
(1, 2)=(0.9,0.5)
(d)
S0
0
10
20
30
40
50
60
70
80
optio
n pr
ice
80 85 90 95 100 105 110 115 120
(1, 2)=(0.6,-0.6)
(1, 2)=(0.6,0)
(1, 2)=(0.6,0.6)
(1, 2)=(-0.6,0.6)
(1, 2)=(0,0.6)
(1, 2)=(-0.6,-0.6)
(e)
T
510152025303540455055
optio
n pr
ice
0.5 1 1.5 2 2.5 3 3.5 4
(1, 2)=(0.6,-0.6)
(1, 2)=(0.6,0)
(1, 2)=(0.6,0.6)
(1, 2)=(-0.6,0.6)
(1, 2)=(0,0.6)
(1, 2)=(-0.6,-0.6)
(f)
Figure 2: Option prices with respect to 𝑆0 and 𝑇 in the dbH-SI
model.
-
12 Complexity
technique and the Fourier inversion transform approach.Some
numerical examples are provided to examine theeffects of the
proposed model, which reveals some additionalfeatures having a
significant impact on option values, espe-cially long-term options.
The proposed model can be testedempirically by using the option
price data from the optionmarket.
Data Availability
All data used to support of the findings for this study
areincluded within this article.
Conflicts of Interest
The authors declare that there are no conflicts of
interestregarding the publication of this paper.
Acknowledgments
The authors acknowledge the financial support providedby the
Natural Sciences Foundation of China (11461008),the Guangxi Natural
Science Foundation (2018JJA110001),and the Guangxi Graduate
Education Innovation Programproject (XYCSZ2018059).
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