geometric asian options pricing under the double heston...
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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 Β© 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 βaverage priceβ) and floatingstrike price options (sometimes called βaverage strikeβ). 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
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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 π. Let (Ξ©,F, {Fπ‘}0β€π‘β€π, π)be a complete probability space equipped with a filtration{Fπ‘} satisfying the usual conditions, whereπ is a risk-neutralprobability measure. Suppose πππ‘ and πππ‘ (π = 1, 2) are allstandard Brownian motions defined on the probability space,and the filtration {Fπ‘}π‘β₯0 is generated by these Brownianmotions. Moreover, ππ1π‘ ππ1π‘ = π1ππ‘, ππ2π‘ ππ2π‘ = π2ππ‘,and any other Brownian motions are pairwisely independent.Assume that the asset price process ππ‘, without payingany dividend, satisfies the following stochastic differentialequation under π:
πππ‘ππ‘ = ππ‘ππ‘ + βV1π‘ππ1π‘ + βV2π‘ππ2π‘ ,πV1π‘ = π 1 (π1 β V1π‘) ππ‘ + π1βV1π‘ππ1π‘ ,πV2π‘ = π 2 (π2 β V2π‘) ππ‘ + π2βV2π‘ππ2π‘ ,
(1)
where π π, ππ, ππ (π = 1, 2) are all nonnegative constants,which represent the mean-reverting rates, long-term meanlevels, and volatilities of variance processes Vππ‘, respectively.We suppose that 2π πππ β₯ π2π . The instantaneous interestrate, ππ‘, is assumed to be a linear combination of V1π‘ andV2π‘, i.e., ππ‘ = V1π‘ + V2π‘, 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 π forthese options are as follows:
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Complexity 3
β (ππ, πΊ[0,π]) ={{{{{{{{{{{{{{{{{
max {πΊ[0,π] β πΎ, 0} , a fixed strike geometric Asian calls,max {πΎ β πΊ[0,π], 0} , a fixed strike geometric Asian puts,max {πΊ[0,π] β ππ, 0} , a floating strike geometric Asian calls,max {ππ β πΊ[0,π], 0} , a floating strike geometric Asian puts,
(2)
where πΎ is a fixed strike price and πΊ[0,π] is the geometricaverage of the underlying asset price ππ‘ until time π; i.e.,πΊ[0,π] = exp((1/π) β«π0 ln ππ’ππ’). 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 ππ‘ = V1π‘ + V2π‘, onecan express the price at time π‘ of a zero-coupon bond withmaturate π as follows (see Cox et al. [37]):
π (π‘, π) = exp {π (π‘, π) β π (π‘, π) ππ‘} , (3)where
π (π‘, π) = 2βπ=1
[π πππ (π π β πΎπ) (π β π‘)π2π+ 2π ππππ2π ln
2πΎπ2πΎπ + (π π β πΎπ) (1 β πβπΎπ(πβπ‘))] ,
π (π‘, π) = 2βπ=1
[ 2 (1 β πβπΎπ(πβπ‘))2πΎπ + (π π β πΎπ) (1 β πβπΎπ(πβπ‘))] ,πΎπ = βπ 2π + 2π2π , for π = 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 ππ and the log-geometric mean value of theasset price over a certain time period.
Let ππ‘(π , π€) = πΈπ[πβ β«ππ‘ ππππ+π lnπΊ[π‘,π]+π€ ln ππ | Fπ‘] be thediscounted joint characteristic function of two-dimensionalrandom variable, (lnπΊ[π‘,π], ln ππ), conditioned on Fπ‘ underπ, whereπΊ[π‘,π] = exp{(1/π) β«ππ‘ ln ππ’ππ’} and πΈπ[β | Fπ‘] is theconditional expectation under π for π‘ β [0, π]. Denote π· ={(π , π€) β C2 : R(π ) β₯ 0,R(π€) β₯ 0, 0 β€ R(π ) +R(π€) β€ 1}.Proposition 1. Suppose that ππ‘, V1π‘, and V2π‘ follow thedynamics in (1). If (π , π€) β π· and π‘ β [0, π], thenπΈπ[|πβ β«ππ‘ ππππ+π lnπΊ[π‘,π]+π€ ln ππ |] < β and
ππ‘ (π , π€) = ππ0πΈπ[[exp{{{2βπ=1
[ππ1 β«ππ‘(π β π)2 Vππππ + ππ2 β«π
π‘(π β π) Vππππ + ππ3 β«π
π‘Vππππ + ππ4Vππ]}}} | V1π‘, V2π‘
]] , (5)
where
π0 = π0 (π , π€) = π [[π β π‘π ln ππ‘
β 1π2βπ=1
(πππ πππ2ππ (π β π‘)2 +ππ (π β π‘)ππ Vππ‘)]]
+ π€[[ln ππ‘ β2βπ=1
(πππ πππππ (π β π‘) +ππππ Vππ‘)]] ,
ππ1 = ππ1 (π , π€) = π 2 (1 β π2π )2π2 ,
ππ2 = ππ2 (π , π€) = π (2πππ π + ππ)2πππ +π π€ (1 β π2π )π ,
ππ3 = ππ3 (π , π€) = π πππππ +π€ (2πππ π + ππ)2ππ
+ π€2 (1 β π2π )2 β 1,ππ4 = ππ4 (π , π€) = π€ππππ .
(6)
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4 Complexity
Proof. (i) We first prove that the integrability conditionguarantees the existence of the cumulant function ππ‘(π , π€) inπ·. If (π , π€) β π· and π‘ β [0, π), then
πΈπ [πββ«π
π‘ππππ+π lnπΊ[π‘,π]+π€lnππ
]= π (π‘, π) πΈππ [ππ lnπΊ[π‘,π]+π€lnππ ]β€ π (π‘, π) πΈππ [ 1π β π‘ β«
π
π‘ππ’ππ’ + ππ]
= [ 1π β π‘ β«π
π‘
π (π‘, π)π (π‘, π’) ππ’ + 1] ππ‘ < β,
(7)
where ππ is the πβ forward measure given by the Radon-Nikodym derivative: (πππ/ππ)|Fπ = exp{β β«π0 ππ’ππ’}/π(0, π), and π(π‘, π) is given above in (3). In the case of π‘ = π,it is triviality.
(ii) In order to determine (5), we start from model(1) and develop ln ππ with the Brownian motions π1π‘ andπ2π‘ expressed as π1π‘ = π1π1π‘ + β1 β π12π1π‘ and π2π‘ =π1π2π‘ + β1 β π22π2π‘ , respectively, where (π1π‘ , π2π‘ , π1π‘ , π2π‘ ) are4-dimensional Brownian motion defined on the probabilityspace. For the process, ln ππ, we have
ln ππ = ln ππ‘ + 2βπ=1
[βπππ πππππ (π β π‘) βππππ Vππ‘
+ (πππ πππ +12)β«π
π‘Vππππ + ππππ Vππ
+ β1 β π2π β«ππ‘βVππππππ] = ln ππ‘ + 2β
π=1[ππ2 (π β π‘)
+ Vππ‘2π π βVππ2π π + (ππ +
ππ2π π)β«π
π‘βVππππππ
+ β1 β π2π β«ππ‘βVππππππ] .
(8)
On the other hand, we have
lnπΊ[π‘,π] = π β π‘π ln ππ‘ + 1π2βπ=1
[ππ4 (π β π‘)2
+ Vππ‘2π π (π β π‘) β12π π β«π
π‘Vππππ
+ (ππ + ππ2π π)β«π
π‘(π β π)βVππππππ
+ β1 β π2π β«ππ‘(π β π)βVππππππ] .
(9)
Using the fact
β«ππ‘(π β π)βVππππππ = 1ππ [π π β«
π
π‘(π β π) Vππππ
+ β«ππ‘Vππππ β (π β π‘) Vππ‘ β π πππ (π β π‘)22 ] ,
(10)
for π = 1, 2, thenlnπΊ[π‘,π] = π β π‘π ln ππ‘ + 1π
2βπ=1
[βπππ πππ2ππ (π β π‘)2
β ππVππ‘ππ (π β π‘) +ππππ β«π
π‘Vππππ
+ (πππ πππ +12)β«π
π‘(π β π) Vππππ
+ β1 β π2π β«ππ‘(π β π)βVππππππ] .
(11)
Let G be the πβfield generated byFπ‘ and {(π1π’, π2π’) : π‘
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Complexity 5
+ 2βπ=1
[π€(ππVππππ +2πππ π + ππ2ππ β«
π
π‘Vππππ)
β β«ππ‘Vππππ]}}} ,
π΄3 = exp{{{2βπ=1
β1 β π2π β«ππ‘[ π π (π β π) + π€]
β βVππππππ}}} .(13)
Since
πΈπ [π΄3 | G] = exp{{{122βπ=1
(1 β π2π )
β β«ππ‘[ π π (π β π) + π€]
2Vππππ}}}
= exp{{{2βπ=1
(1 β π2π )2β β«ππ‘[ π 2π2 (π β π)2 + 2π π€π (π β π) + π€2] Vππππ}}} ,
(14)
substituting (14) into (12) and applying the Markov propertyof {Vππ’ : 0 β€ π’ β€ π}, π = 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 V1π‘ and V2π‘ and the three different integrals of Vππ‘ (π =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 ππ‘(π , π€). Therefore,
two series of functions are introduced as follows. Define πΉππand πΉππ : C4 β C as
πΉππ (ππ1, ππ2, ππ3, ππ4) = ββπ=0
πππ , (15)πΉππ (ππ1, ππ2, ππ3, ππ4) = ββ
π=1
πππππ , (16)whereπππ , π = 0, 1, 2, β β β , are functions of ππ1, ππ2, ππ3, and ππ4defined as
ππβ2 = ππβ1 = 0,ππ0 = 1,ππ1 = (π π β ππ4π
2π ) π2 ,
πππ = β π2ππ22π (π β 1) (ππ1π2πππβ4 + ππ2ππππβ3
+ (ππ3 β π 2π2π2π )π
ππβ2) , π β₯ 2,
(17)
for π = 1, 2.For π = 1, 2, denoteπ·ππ = {(ππ1, ππ2, ππ3, ππ4) β C4 :
πΈπ [exp{R (ππ1) β«π0(π β π‘)2 Vππ‘ππ‘
+R (ππ2)β«π0(π β π‘) Vππ‘ππ‘ +R (ππ3)β«π
0Vππ‘ππ‘
+R (ππ4) Vππ}] < β} .
(18)
Proposition 2. (1) οΏ½e argument of πΉππ : argπΉππ(ππ1,ππ2, ππ3, ππ4) ΜΈ= 0 for every (ππ1, ππ2, ππ3, ππ4) β π·ππ (π = 1, 2).In particular, argπΉππ is continuous on π·ππ, andargπΉππ(ππ1, ππ2, ππ3, ππ4) = 0 for ππ1, ππ2, ππ3, and ππ4 areall real numbers inπ·ππ for π = 1, 2.
(2) For (ππ1, ππ2, ππ3, ππ4) β π·ππ, π = 1, 2, we have
πΈπ[[exp{{{2βπ=1
[ππ1 β«π0(π β π‘)2 Vππ‘ππ‘ + ππ2 β«π
0(π β π‘) Vππ‘ππ‘ + ππ3 β«π
0Vππ‘ππ‘ + ππ4Vππ]}}}
]]
= exp{{{2βπ=1
(π πVπ0 + π 2ππππππ β2Vπ0π2π
πΉππ (ππ1, ππ2, ππ3, ππ4)πΉππ (ππ1, ππ2, ππ3, ππ4) β2π ππππ2π lnπΉππ (ππ1, ππ2, ππ3, ππ4))
}}} .(19)
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6 Complexity
οΏ½e proof of Proposition 2 is similar to that of Kim andWee[18].
Using Proposition 2 to Proposition 1 leads to the explicitexpression of ππ‘(π , π€) in Proposition 3. To describe the simplic-ity of the result, we need to introduce new functions. Defineπ»ππ‘,πand οΏ½ΜοΏ½ππ‘,π: C2 β C asπ»ππ‘,π (π , π€)= πΉππβπ‘ (ππ1 (π , π€) , ππ2 (π , π€) , ππ3 (π , π€) , ππ4 (π , π€)) ,
(20)
οΏ½ΜοΏ½ππ‘,π (π , π€)= πΉππβπ‘ (ππ1 (π , π€) , ππ2 (π , π€) , ππ3 (π , π€) , ππ4 (π , π€)) ,
(21)
with π = 1, 2.Proposition 3. (1) If (π , π€) β π·, then π»ππ‘,π(π , π€) ΜΈ= 0 andargπ»ππ‘,π is continuous onπ·. In particular, argπ»ππ‘,π(π , π€) = 0 ifπ and π€ are real numbers for π = 1, 2.
(2) For ππ‘(π , π€) β π· with argπ»ππ‘,π defined as above, then
ππ‘ (π , π€) = exp{{{2βπ=1
(βππ1 οΏ½ΜοΏ½ππ‘,π (π , π€)π»ππ‘,π (π , π€)
β ππ2lnπ»ππ‘,π (π , π€) + ππ3π + ππ4π€ + ππ5)}}} .(22)
Here ππ1 = 2Vππ‘/π2π , ππ2 = 2π πππ/π2π ,
ππ3 = π β π‘2π ln ππ‘ βπππ πππ (π β π‘)22πππ β
ππ (π β π‘)πππ Vππ‘,ππ4 = ln ππ‘2 β
ππππ Vππ‘ βπππ πππ (π β π‘)ππ ,
ππ5 = π πVππ‘ + π 2πππ (π β π‘)π2π
(23)
for π = 1, 2.Proof. Assume that (π , π€) β π·, using the definitions of πππ =πππ(π , π€) and π·ππ for π = 1, 2, 3, 4 and π = 1, 2. Note that(ππ1, ππ2, ππ3, ππ4) β π·ππ β π·ππβπ‘ for every 0 β€ π‘ β€ π andπ = 1, 2. Therefore, (19) remains valid for any Vπ0 > 0. Thetime homogeneous Markov property of Vππ‘ implies that (22)holds when Propositions 1 and 2 are satisfied.
Now Substituting the expressions of ππ1, ππ2, ππ3, ππ4, π =1, 2 into (15) and (16),π»ππ‘,π and οΏ½ΜοΏ½ππ‘,π are expressed as follows,i.e., for (π , π€) β C2,
π»ππ‘,π (π , π€) = ββπ=0
βππ (π , π€) , (24)οΏ½ΜοΏ½ππ‘,π (π , π€) = ββ
π=1
ππ β π‘βππ (π , π€) , (25)where
βπβ2 (π , π€) = βπβ1 (π , π€) = 0,βπ0 (π , π€) = 1,βπ1 (π , π€) = (π β π‘) (π π β π€ππππ)2 ,βππ (π , π€) = (π β π‘)24π (π β 1) π2 {βπ 2π2π (1 β π2π ) (π β π‘)2
β βππβ4 (π , π€) β [π πππ(ππ + 2πππ π)+ 2π π€π2ππ (1 β π2π )] (π β π‘) βππβ3 (π , π€) + π [π 2ππβ 2π ππππ β π€ (2πππ π + ππ) πππ β π€2 (1 β π2π ) π2ππ+ 2π2ππ] βππβ2 (π , π€)} , π β₯ 2,
(26)
for π = 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 ππ‘, V1π‘, and V2π‘ follow the dynamicsin (1), then the price at time π‘ β [0, π] of the continuously mon-itored fixed strike geometric Asian call option with maturity πand the strike price πΎ is given by
πΊπ΄πΆππ (π‘, π, V1, V2, πΎ, π)= π(1/π) β«π‘0 lnππ’ππ’ππ‘ (1, 0) Ξ 1 β πΎπ (π‘, π)Ξ 2, (27)
where π(π‘, π) is given above in (3) andΞ π = 12 + 1π β«
+β
0R[πβππ’lnπΎπ‘,πππ (π’)ππ’ ]ππ’,
π = 1, 2,
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Complexity 7
πΎπ‘,π = πΎπβ(1/π) β«π‘0 lnππ’ππ’,π1 (π’) = ππ‘ (ππ’ + 1, 0)ππ‘ (1, 0) ,π2 (π’) = ππ‘ (ππ’, 0)ππ‘ (0, 0) ,
(28)
where π is the imaginary unit (π2 = β1).Proof. Since
πΊπ΄πΆππ (π‘, π, V1, V2, πΎ, π)= πΈπ [πβ β«ππ‘ ππ’ππ’ (πΊ[0,π] β πΎ)+ | Fπ‘]= π(1/π) β«π‘0 lnππ’ππ’πΈπ [πβ β«Tπ‘ ππ’ππ’πΊ[π‘,π]1{lnπΊ[π‘,π]β₯lnπΎπ‘,π} |Fπ‘] β πΎπΈπ [πβ β«ππ‘ ππ’ππ’1{lnπΊ[π‘,π]β₯lnπΎπ‘,π} | Fπ‘]= π(1/π) β«π‘0 lnππ’ππ’ππ‘ (1, 0)π1 (lnπΊ[π‘,π] β₯ lnπΎπ‘,π)β πΎπ (π‘, π)ππ (lnπΊ[π‘,π] β₯ lnπΎπ‘,π) ,
(29)
whereπ1 is defined by the following Radon-Nikodymderiva-tive:
ππ1ππFπ = πβ β«
π
π‘ππ’ππ’
πΊ[π‘,π]πΈπ [πβ β«ππ‘ ππ’ππ’πΊ[π‘,π] | Fπ‘] , (30)
and ππ is the πβ 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 π1 and ππ in (29) are given by
π1 (lnπΊ[π‘,π] β₯ lnπΎπ‘,π)= 12 + 1π β«
+β
0R [πβππ’lnπΎπ‘,ππ1 (π’)ππ’ ] ππ’ fl Ξ 1,
(31)
ππ (lnπΊ[π‘,π] β₯ lnπΎπ‘,π)= 12 + 1π β«
+β
0R [πβππ’lnπΎπ‘,ππ2 (π’)ππ’ ] ππ’ fl Ξ 2,
(32)
where
π1 (π’) = πΈπ1 [πππ’lnπΊ[π‘,π] | Fπ‘]= πΈπ [πβ β«ππ‘ ππ’ππ’+ππ’lnπΊ[π‘,π]πΊ[π‘,π] | Fπ‘]πΈπ [πβ β«ππ‘ ππ’ππ’πΊ[π‘,π] | Fπ‘]
= πΈπ [πβ β«ππ‘ ππ’ππ’+(ππ’+1)lnπΊ[π‘,π] | Fπ‘]πΈπ [πβ β«ππ‘ ππ’ππ’+lnπΊ[π‘,π] | Fπ‘]
= ππ‘ (ππ’ + 1, 0)ππ‘ (1, 0) ,(33)
π2 (π’) = πΈππ [πππ’lnπΊ[π‘,π] | Fπ‘]= πΈπ [πβ β«ππ‘ ππ’ππ’+ππ’lnπΊ[π‘,π] | Fπ‘]πΈπ [πβ β«ππ‘ ππ’ππ’ | Fπ‘] =
ππ‘ (ππ’, 0)π (π‘, π) .(34)
From (29), (31), and (32), we can obtain the requiredTheorem 4.
Theorem 5. Suppose that ππ‘, V1π‘, and V2π‘ follow the dynamicsin (1), then the price at time π‘ β [0, π] of the continuouslymonitored floating strike geometric Asian call option withmaturity π is given by
πΊπ΄πΆππ (π‘, π, V1, V2, π) = ππ‘Ξ Μ1β π(1/π) β«π‘0 lnππ’ππ’ππ‘ (1, 0) Ξ Μ2, (35)
where
Ξ Μ1= 12
β 1π β«+β
0R[[
ππ‘ (ππ’, 1 β ππ’) π(ππ’/π) β«π‘0 lnππ’ππ’ππ’ππ‘ ]]ππ’,Ξ Μ2= 12
β 1π β«+β
0R[[
ππ‘ (ππ’ + 1, βππ’) π(ππ’/π) β«π‘0 lnππ’ππ’ππ’ππ‘ (1, 0) ]]ππ’.
(36)
Proof. Since
πΊπ΄πΆππ (π‘, π, V1, V2, π) = πΈπ [πβ β«ππ‘ ππ’ππ’ (ππ β πΊ[0,π])+ | Fπ‘]= π(1/π) β«π‘0 lnππ’ππ’πΈπ [πβ β«ππ‘ ππ’ππ’ (πππβ(1/π) β«π‘0 lnππ’ππ’ β πΊ[π‘,π])+ |Fπ‘] = πΈπ [πβ β«ππ‘ ππ’ππ’ππ1{ln(πΊ[π‘,π]/ππ)β€β(1/π) β«π‘0 lnππ’ππ’} | Fπ‘]β π(1/π) β«π‘0 lnππ’ππ’πΈπ [πβ β«ππ‘ ππ’ππ’πΊ[π‘,π]1{ln(πΊπ‘,π/ππ)β€β(1/π) β«π‘0 lnππ’ππ’} |
-
8 Complexity
Fπ‘] = ππ‘π2 (ln(πΊ[π‘,π]ππ ) β€ β1π β«π‘
0ln ππ’ππ’)
β π(1/π) β«π‘0 lnππ’ππ’ππ‘ (1, 0)π1 (ln(πΊ[π‘,π]ππ ) β€ β1π β«π‘
0ln ππ’ππ’) ,
(37)
whereπ2 is defined by the following Radon-Nikodymderiva-tive:
ππ2πQFπ = πβ β«
π
π‘ππ’ππ’ ππππ‘ , (38)
under the two probability measures π1 and π2, the condi-tional characteristic functions of ln(πΊ[π‘,π]/ππ) are given by
πΈπ1 [πππ’ln(πΊ[π‘,π]/ππ) | Fπ‘]= πΈπ [πββ«ππ‘ ππ’ππ’+ππ’ln(πΊ[π‘,π]/Sπ)πΊ[π‘,π] | Fπ‘]
πΈπ [πββ«ππ‘ ππ’ππ’πΊ[π‘,π] | Fπ‘]= ππ‘ (ππ’ + 1, βππ’)ππ‘ (1, 0) ,
(39)
πΈπ2 [πππ’ln(πΊ[π‘,π]/ππ) | Fπ‘]= πΈπ [πββ«ππ‘ ππ’ππ’+ππ’ln(πΊ[π‘,π]/ππ)+lnππ | Fπ‘]
ππ‘= ππ‘ (ππ’, 1 β ππ’)ππ‘ .
(40)
Therefore
π1 (ln(πΊπ‘,πππ ) β€ β1π β«π‘
0ln ππ’ππ’)
= 12β 1π β«
+β
0R[[
ππ‘ (ππ’ + 1, βππ’) π(ππ’/π) β«π‘0 ln ππ’ππ’ππ’ππ‘ (1, 0) ]]ππ’,(41)
π2 (ln(πΊπ‘,πππ ) β€ β1π β«π‘
0ln ππ’ππ’)
= 12β 1π β«
+β
0R[[
ππ‘ (ππ’, 1 β ππ’) π(ππ’/π) β«π‘0 lnππ’ππ’ππ‘ β ππ’ ]] ππ’.(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 π 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 π»ππ‘,π and οΏ½ΜοΏ½ππ‘,π for π =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 π0 and π 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 βNIntegrate()β or βquadgk()β 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: π 1 = 1.15, π 2 =2.2, π1 = 0.2, π2 = 0.15, π1 = 0.5, π2 = 0.8, π1 = β0.4,π2 = β0.64, V10 = 0.09, V20 = 0.04, π0 = 100, and π‘ = 0. Theresults are shown in Table 1.
Table 1 investigates the influence of the number of termsπ 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 π =20 and π = 30 terms, at most. In addition, Table 1 providesoption prices with various maturities π, strike prices πΎ, andthe requiredCPU times (in seconds). FromTable 1, we can seethat some characteristics of the GACππ option are similar tothe plain European call option. For example, the values of theGACππ option are decreasing functions of the strike price. Inaddition, under the same parameter values setting in dbH-SImodel, longer maturating date π 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 π = 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: GACππ option prices with different maturities, strike prices, and varying n.
π πΎ π Option price CPU (sec.) π πΎ π 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 π0 and maturatingdate π with the strike price πΎ = 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 π0(orπ) is, the higher the GACππ option price is.
In Figure 2, we use the dbH-SI model to examine theeffects of the mean-reverting rates (π 1, π 2) and volatilities(π1, π2) of variance processes (V1π‘, V2π‘) on the GACππ optionprices (see Figures 2(a)β2(d)). We also examine the impactsof the correlation coefficients (π1, π2) (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 GACππ options.
π πΎ 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 π0
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 π
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 (π1, π2) have several effects depending on therelation between the strike price and the initial asset price.A negative (π1, π2) 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 (π1, π2) 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, π] 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
(οΌ₯1, οΌ₯2)=(1.5,1)
(οΌ₯1, οΌ₯2)=(1.5,2)
(οΌ₯1, οΌ₯2)=(2,1.5)
(οΌ₯1, οΌ₯2)=(1.5,1.5)
(οΌ₯1, οΌ₯2)=(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
(οΌ₯1, οΌ₯2)=(1.5,1)
(οΌ₯1, οΌ₯2)=(1.5,2)
(οΌ₯1, οΌ₯2)=(2,1.5)
(οΌ₯1, οΌ₯2)=(1.5,1.5)
(οΌ₯1, οΌ₯2)=(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 π0 and π 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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