# Estimation of the Bid-Ask Prices for the European Discrete Geometric Average and Arithmetic Average Asian Options

Authors: Xiankang Luo; Tao Chen
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Conic finance is a new and exciting development in quantitative finance, which is widely applied to several topics in finance. The theory of conic finance extends the law of one price to the law of two prices, which yields closed forms for bid-ask prices of European options. In this paper, within the framework of conic finance, we derive effective, explicit, approximate formulas to estimate the bid-ask prices for the European discrete geometric average and arithmetic average Asian options. Finally, we give two examples to demonstrate and validate that the approximate closed-form solutions are efficient and accurate.


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## Body

## 1. Introduction

In the OTC markets, the Asian option is one of the most popular exotic options. Asian options are options whose payoffs depend on some average price of the underlying asset over some prescribed period; Asian options are also called average price options. There are two main advantages of Asian options: one for reducing the risk of market manipulation of the underlying instrument at maturity and the other one for reducing the volatility inherent in the option. Because of the path-dependent characteristic, Asian options have a lower volatility and a lower cost than American options [1]. Therefore, Asian options have attracted much attention, and their volume has grown rapidly in the OTC markets.In the Black–Scholes–Merton framework, many scholars have sufficient research on exotic options. They develop many methods such as chaos theory, fractal Brownian motion, time-changed Brownian, stochastic volatility, and models based on Lévy processes to incorporate two empirical features: the asymmetric leptokurtic features and the volatility smile [2, 3]. In a parallel development, many mathematical methods are used to solve the value of options, such as the numerical method (PDE numerical calculation, binomial lattice, Monte Carlo simulation, and Fourier transform), partial differential equation method, and deep learning method [4–6]. These methods are well known and can be found in many option pricing textbooks. However, there are many uncertain factors in the real financial market, which profoundly affect the option pricing, so there is no effective method to solve the option value. As we all know, classical option pricing theories are usually built on the law of one price or the risk-neutral price which assumes implicitly that market participants can buy or sell any amount of assets with the market taking the opposite position. However, empirical studies showed that even in highly liquid financial markets, we observe two prices, namely, the price at which the market is willing to buy (bid) and a price at which the market is willing to sell (ask). When liquidity is insufficient, it can lead to a higher difference between the bid price and ask price, and we can no longer rely on the law of one price.There are a variety of theoretical approaches that try to model bid-ask spreads (see Madan and Cherny [7] for general reviews). A different approach was taken by Madan and Cherny [7] which is based on the theory of conic finance, originating from the framework of acceptability indices presented in Cherny and Madan [8]. In the conic finance framework, the price depends on the direction of trade, one for buying from the market at the ask price and another for selling to the market at the bid price. Conic finance is a new and exciting development in quantitative finance, which is widely applied to several topics in finance. One of the applications is option pricing with liquidity. In [7], Madan and Cherny established the conic finance theory by using the acceptability indices’ theory and the nonlinear expectation theory and derived the general closed-form formulas for bid and ask prices for European options. In 2010, Madan [9] studied the discrete-time constant liquidity model. Leippold and Schärer [10] first proved that multiperiod bid-ask prices with stochastic liquidity are time-consistent and dynamically translation-invariant nonlinear expectations. The results extend the model of Madan [9] to develop a discrete-time option pricing model with stochastic liquidity and open ample ground for future research. Furthermore, Leippold and Schärer [11] developed the conic finance framework for optimal execution strategies with stochastic liquidity. In [12–15], the scholars made an empirical analysis of the option pricing model under the conic finance framework. The empirical results showed the huge advantage of conic finance theory in modeling liquidity.At the same time, many research topics including conic CVA and DVA, conic portfolio theory, conic hedging of financial and insurance risks, and conic trading are presented in the recently published book by Madan and Schoutens [16]. Recently, the researchers derived the explicit formulas for the bid and ask prices of European vanilla options [15, 16], interest rate options [17], and continuous geometric average Asian options [18], respectively, by using the conic finance theory. To the best of our knowledge, there is no literature research to estimate the bid-ask prices of the discrete Asian options. In this paper, motivated by the conic finance theory, we derive an effective, explicit, approximate formula to calculate the bid and ask prices for the discrete Asian options including discrete geometric average and arithmetic average Asian options.The content of this paper is organized as follows. In Section2, we review the conic finance theory. Section 3 is devoted to derive an effective, explicit, approximate formula to calculate the bid and ask prices for the discrete Asian options. In Section 4, we present two numerical examples which show that the approximate closed forms are efficient and accurate. Section 5 concludes the paper.


## 2. Conic Finance Theory

Conic finance is a new and exciting development in quantitative finance, which extends the law of one price to the law of two prices. The conic finance theory originates from the framework of acceptability developed by Cherny and Madan [8]. In the conic finance framework, there are two prices for financial assets in the market. If a nonnegative stochastic cash flow X is acceptable at liquidity level γ, the market maker buys at the ask price aγX and sells at the bid price bγX. The difference of bid and ask prices gives rise to the bid-ask spread which can measure market liquidity.The foundation of the conic finance theory is an underlying concept of acceptable risk in the economy. The theory of acceptability indices is developed in [8], and the index of acceptability is a new measure for performance evaluation which is a function α:L∞⟶0,∞, where L∞:=L∞Ω,ℱ,ℙ is the space of all essentially bounded. For the acceptability index αX and for each liquidity level γ>0, there is a family of sets of probability measures Dγγ≥0 equivalent to the initial probability measure of ℙ. A risk X is acceptable at level γ if and only if EℚX≥0, for all ℚ∈Dγ. That is to say, the acceptability index αX˜ will satisfy the following operational acceptability level:(1)αX=supγ≥0:EℚX≥0,for anyℚ∈Dγ.Suppose the trader, due to competition, will sell a net cash flowX; it charges a minimal price a. The residual cash flow a−X ought to be α-acceptable at level γ. For a−X to be acceptable at level γ, the price a must exceed EℚX≥0 for any ℚ∈Dγ, so the ask price is derived as follows:(2)aγX=infa:αa−X≥γ=infa:Eℚa−X≥γfor anyℚ∈Dγ=supℚ∈DγEℚX.For the buy-side, when the market maker buys a net cash flowX, again driven by the competition, he charges a maximum price of b. The residual cash flow X−b must be α-acceptable at level γ. Hence, the maximal price is(3)bγX=infℚ∈DγEℚX.As proposed by Cherny and Madan [8], Choquet expectation (see [19]) based on the concave distortion function can be used to formulate an operational index of acceptability. The distorted expectation of a risk X with distribution function FXx relative to the concave distortion function ψγγ≥0 is defined as(4)EψγX=∫−∞∞xdψγFXx,and the acceptability index αX is characterized as(5)αX=supγ≥0:∫−∞∞xdψγFXx,where X is a nonnegative stochastic variable, FXx is the distribution function of X, and ψγγ≥0 is a pointwise increasing family of concave distortion functions. Because the distortion function is an important bridge between Choquet expectation (nonlinear expectation) and linear expectation, Cherny and Madan [8] concluded a series of distortion functions ψγγ≥0, such as MINVAR, MAXVAR, MAXMINVAR, and MINMAXVAR. In this paper, we need to use the Choquet expectation on Wang transform, so we first give related definitions as follows.Definition 1 (distortion function).
A functionψ:0,1⟶0,1 is a distortion function if and only if it is monotone, ψ0=0andψ1=1.Definition 2 (Wang transform; see [20, 21]).
LetΦ denote the standard normal cumulative distribution function, and let γ be a nonnegative constant. Then, a distortion function ψγ:0,1⟶0,1 defined by(6)ψγu=ΦΦ−1u+γ,is called the Wang transform.By using the distortion expectation (4) and a family of concave distortion functions ψγγ≥0, the ask price is derived as follows:(7)αa−X≥γ⇒∫−∞∞xdψγFa−Xx≥0⇒a+∫−∞∞xdψγF−Xx≥0.so that the minimum price of a leads to the ask price:(8)aγX=−∫−∞∞xdψγF−Xx.Similarly, the maximum price ofb leads to the bid price:(9)αX−b≥γ⇒∫−∞∞xdψγFX−bx≥0⇒−b+∫−∞∞xdψγFXx≥0.We obtain(10)bγX=∫−∞∞xdψγFXx.In order to present the relationship between the distorted expectation and classical integral, Madan and Cherny [7] led to the following definition of bid and ask prices.Definition 3.
(single-period bid-ask prices; see [7]). Let ψγγ≥0 be a pointwise increasing family of concave distortion functions and γ be the market liquidity level. Then, the bid price of a discounted cash flow X∈L∞ is given by(11)bγX=EψγX=C∫Xdψγ∘ℙ=−∫−∞0ψγℙX≤xdx+∫0∞1−ψγℙX≤xdx,and its ask price is(12)aγX=−Eψγ−X=∫−∞0ψγℙX>x−1dx+∫0∞ψγℙX>xdx.In particular, the Choquet expectation, introduced by Choquet in [19], is a nonlinear expectation when the market liquidity level γ=0, the distortion expectation reduces to a linear expectation on the reference pricing measure ℙ, and bid-ask prices (8)–(10) are equivalent. In addition, the concavity of the distortion ensures that the ask price is greater than the bid price, i.e.,(13)bγX≤EX≤aγX,where EX is a linear expectation on the reference pricing measure ℙ, which means that distortion is not applied at all.


## 3. The Bid-Ask Prices of Discrete Average Asian Options

In this section, we first introduce the pricing of discrete average Asian options under the law of one price. Second, within the framework of conic finance, we derive effective, explicit, approximate formulas to calculate the bid and ask prices for the discrete Asian options including discrete geometric average and arithmetic average Asian options.


### 3.1. Discrete Average Asian Option under the Law of One Price

In this section, we start with a brief description of the discrete geometric and arithmetic Asian option under the law of one price.Under a probability spaceΩ,ℱ,ℙ, the reference probability measure ℙ is assumed to be a risk-neutral measure. It is assumed that the underlying asset follows the geometric Brownian motion, with a stochastic differential equation having the following expression:(14)dSt=rStdt+σStdWt,0≤t≤T,S0=S0,where r is the risk-free rate and σ is volatility. And these parameters are usually assumed to be constant. The solution of equation (14) can be calculated to the following result:(15)ST=Stexpr−σ22τ+σZτ,where τ=T−t is the time to maturity and Zτ=WT−Wt is the Brownian motion.Suppose thatn prices are taken from equation (15) with observation frequency h, i.e.,(16)ai=Sτ−n−ih=S0expr−σ22τ−n−ih+σZτ−n−ih,where i=1,2,3,⋅⋅⋅,n, and τ=T−t is the time to maturity. We can see that the averaging period starts with the first observation at T=τ−n−1h and stops at the last observation i=n at T=τ. The averaging time period is thus from τ−n−1h to τ or n−1h.The standard geometric average (GA) ofn positive prices a1,a2,⋅⋅⋅,an is defined as(17)GAn=∏i=1nai1/n,and the arithmetic average (AA) of n positive prices a1,a2,…,an is defined as(18)AAn=1n∑i=1nai,i=1,2,3,…,n,where n is the number of observations and ai is the ith observation.LetVSt,t be the option price with the underlying price St, the time to maturity T, and the strike price K. Then, the price of the discrete geometric Asian call option under the risk-neutral measure ℙ, at time t, may be represented as(19)VSt,t=e−rT−tEGAn−K+,and the price of the discrete arithmetic Asian put option is given by(20)VSt,t=e−rT−tEK−AAn+,where GAn is the discrete geometric average and AAn is the arithmetic average of the underlying asset prices during the time to maturity T.In order to price the discrete geometric and arithmetic Asian option, we have to know the distribution of the geometric averageGAn and arithmetic average AAn. In both continuous case and discrete case of the geometric Asian option, the variable GAn is log-normally distributed so that its expectation and variance values may be calculated explicitly. Therefore, Kemma and Vorst [22], Turnbull and Wakeman, [23], and Zhang [1] obtained similar B-S pricing formulas of geometric average Asian options by different approximation methods.Asian option trading in the OTC markets is usually based on the arithmetic average prices of the underlying assets. However, even if all price observations follow a log-normal process, the arithmetic average is not a log-normal process. It is difficult to solve the closed-form solutions for arithmetic average Asian options. A representative method was proposed in Zhang [1]. Zhang [1] tried to express the difference in terms of the distribution of the given observations using the Taylor series expansion and then obtained an approximation formula for arithmetic means in terms of their corresponding geometric means. The following lemmas come from Zhang [1], which give the distribution of GAn and AAn.Lemma 1 (see Zhang [1]).
Let the averaging numbers be defined by equation (16); then, the standard geometric average GAn is a log-normal distribution, i.e.,(21)logGAn∼Nr−12σ2Tμ,n−jsa+lnBjsa+lnSt,σ2Tn−jsa,where(22)B0sa=1,Bjsa=∏i=1jSτ−n−ihS01/n,1≤j≤n,Tμ,n−jsa=n−jnτ−n−j−1h2,Tn−jsa=τn−j22−n−jn−j−14n−4j+1h6n2,and n is the number of observations specified in the contract, h is the observation frequency or the time interval between two consecutive observations, j is the number of observations already passed, Bj is the geometric average of the gross returns of those observations that have already passed, τ is the time to maturity of the option, and the two functions Tμ,n−j and Tn−j may be interpreted as the effective mean and volatility time functions, respectively, because they largely determine the effective mean and variance of the geometric average.Lemma 2 (see Zhang [1]).
The arithmetic average ofn price observations, each following the Brownian motion specified in (16), can be approximated with the corresponding geometric mean:(23)AAa≅κGAa,where(24)κ=1+E˜v+v2=1+12E˜Vlna+14VarVlna+E˜Vlna2,E˜Vlna=n2−1h612r−12σ22h+1nσ2,VarVlna=n2−13n2−215n3r−12σ22σ2h3.According to Lemmas1 and 2, we can easily derive the closed-form solution of the discrete average Asian option under the law of one price. These pricing formulas are well known and can be found in many option pricing textbooks.


### 3.2. The Bid-Ask Prices of Discrete Geometric Average Asian Options

In this section, within the framework of conic finance, we first derive the explicit approximate formulas for the bid-ask prices of the discrete geometric average Asian options. Furthermore, by using Lemma2 and a similar proof method of Theorem 1, we can readily derive explicit approximate formulas for the bid-ask price of the discrete geometric average Asian options. We will give the explicit approximate formulas in the following theorems.Theorem 1.
Assume that the distortion functionψγu is the Wang transform; then, the bid-ask prices of the European discrete geometric average Asian call option at time t are given by(25)bγC≅StAjbΦdn−jb+σTn−jsa−e−rτKΦdn−jb,(26)aγC≅StAjaΦdn−ja+σTn−jsa−e−rτKΦdn−ja,and the bid-ask prices of the European discrete geometric average Asian put option at time t are given by(27)bγP≅e−rτKΦ−dn−ja−StAjaΦ−dn−ja−σTn−jsa,(28)aγP≅e−rτKΦ−dn−jb−StAjbΦ−dn−jb−σTn−jsa,where τ=T−t is maturity,(29)dn−jb=lnBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa−γ,dn−ja=lnBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa+γ,Ajb=Bjsaexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa−γσTn−jsa,Aja=Bjsaexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+γσTn−jsa.
Bjsa, Tn−jsa, and Tμ,n−jsa are the same as in Lemma 1. In particular, for γ=0, the bid-ask prices are equivalent, and the bid-ask prices of the European discrete geometric average Asian option reduce to the B-S-type formulas in [1].Proof.
Consider the discrete geometric averageGAn defined by equation (17). And Lemma 1 shows that random variable GAn has a log-normal distribution under risk-neutral measure ℙ, which means that(30)FGAx=Φlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa,where Φ⋅ is the standard normal cumulative distribution function. If we apply Wang transform (6) to the distribution function FGA, we will get the following representation:(31)ψγFGAx=Φlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ.
Let the payoff of the European discrete geometric average Asian call option beCT=GAn−K+; we can derive the bid price for a European discrete geometric average Asian call option by Choquet integral:(32)bγC=e−rτEψγCT=e−rτC∫CTdψγ∘ℙ=e−rτ∫0∞xdψγFCTx=e−rτ∫K∞x−KdψγFGTx=e−rτ∫K∞xdψγFGTx−∫K∞KdψγFGTx≔I1−I2.
Neglecting the discount factore−rτ temporarily, we can calculate the integral I1 in equation (32). It is shown that(33)I1=∫K∞xdψγFGAx=∫K∞xdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ=exprTμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+lnBjsaSt−γσTn−jsa⋅ΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsa+σ2Tn−jsaσTn−jsa−γ.
And the second integralI2 in equation (32) can be calculated as(34)I2=∫K∞KdψγFGTx=∫K∞KdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ=K1−ΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ=KΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa−γ.
Substituting equations (33) and (34) into (32) and multiplying by a discount factor e−rτ, we can get the following expression for the bid price:(35)bγC=BjsaStexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa−γσTn−jsa⋅ΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsa+σ2Tn−jsaσTn−jsa−γ−e−rτKΦlnBsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa−γ.
Now, by using Choquet expectation in Definition3, we can derive the ask price of the European discrete geometric average Asian call option:(36)aγC=−Eψγ−CT=−∫−∞0xdψγF−CTx=−∫−∞0xdψγ1−FGAK−x=−∫0∞xdψγ1−FGAK+x=−∫K∞x−Kdψγ1−FGAx=−∫K∞xdψγ1−FGAx+∫K∞Kdψγ1−FGAx≕I¯1+I¯2.
Similar to the way in which we obtained the bid price, we get the first integral in equation (36) as follows:(37)I¯1=−∫K∞xdψγ1−FGAx=−∫K∞xdΦlnBjsaSt/x+r−1/2σ2Tμ,n−jsaσTn−jsa+γ=∫K∞xdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa−γ=exprTμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+lnBjsaSt+γσTn−jsa⋅ΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsa+σ2Tn−jsaσTn−jsa+γ,and the second integral can be calculated as(38)I¯2=∫K∞Kdψγ1−FGAx=∫K∞KdΦlnBjsaSt/x+r−1/2σ2Tμ,n−jsaσTn−jsa+γ=−∫K∞KdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa−γ=−K1−ΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa−γ=−KΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa+γ.
By combining partsI¯1 and I¯2 and considering the continuous discount factor e−rτ, we have the ask price:(39)aγC=BjsaStexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+σTn−jsa⋅ΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsa+σ2Tn−jsaσTn−jsa+γ−e−rτKΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa+γ.
In addition, let the put option payoff bePT=K−GAn+. Similar to the way in which we derived the bid-ask prices of the call option, we can derive the bid-ask prices of the European discrete geometric average Asian put option. We first gave the bid price of the put option as follows:(40)bγP=e−rτEψγPT=e−rτ∫0∞xdψγFPTx=e−rτ∫0∞xdψγ1−FGAK−x=−e−rτ∫0KK−xdψγ1−FGAx=e−rτ∫0KK−xdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa−γ=e−rτKΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa−γ−BjsaStexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+γσTn−jsa⋅ΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsa−σ2Tn−jsaσTn−jsa−γ.
Furthermore, we can also get the formula of the ask price of the European discrete geometric average Asian put option:(41)aγP=−e−rτEψγ−PT=−e−rτ∫−∞0xdψγF−PTx=−e−rτ∫−∞0xdψγFGAK+x=−e−rτ∫0∞xdψγFGAK−x=e−rτ∫0KK−xdψγFGAx=e−rτ∫0KK−xdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ=KΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ+BjsaStexprTμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa−γσTn−jsa⋅ΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsa−σ2Tn−jsaσTn−jsa+γ.
Finally, by designing some simple variables, we will lead to the bid-ask prices in equations (25)(27).
This completes the proof of Theorem1.Let the payoff of the European discrete arithmetic average Asian option beCT=AAn−K+ or PT=K−AAn+. By using Lemma 2 and applying the method of calculating Choquet expectation in Theorem 1, we can easily obtain the explicit approximate formulas of the discrete arithmetic average Asian options.Theorem 2.
Assume that the distortion functionψγu is the Wang transform; then, the bid-ask prices of the European discrete arithmetic average Asian call option at time t are given by(42)bγC≅StA¯jbΦd¯n−jb+σTn−jsa−e−rτKΦd¯n−jb,(43)aγC≅StA¯jaΦd¯n−ja+σTn−jsa−e−rτKΦd¯n−ja,and the bid-ask prices of the European discrete arithmetic average Asian put option at time t are given by(44)bγP≅e−rτKΦ−d¯n−ja−StA¯jaΦ−d¯n−ja−σTn−jsa,(45)aγP≅e−rτKΦ−d¯n−jb−StA¯jbΦ−d¯n−jb−σTn−jsa,where(46)d¯n−jb=lnκBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa−γ,d¯n−ja=lnκBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa+γ,A¯jb=κBjsaexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa−γσTn−jsa,A¯ja=κBjsaexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+γσTn−jsa,and Bjsa, Tn−jsa, Tμ,n−jsa, and κ are the same as in Lemmas 1 and 2. In particular, for γ=0, the bid-ask prices are equivalent, and the bid-ask prices of the European discrete arithmetic average Asian option reduce to the B-S-type formulas in [1].Remark 1.
If the observation frequencyh is continuous, the number of observations n approaches to infinity, and the observation frequency h tends to zero. Substituting n⟶∞ and h⟶0, n−1h⟶τ, into Tn−jsa and Tμ,n−jsa yields Tn−jsa⟶τ/2 and Tμ,n−jsa⟶τ/3. Furthermore, substituting these two limiting values into (25) and (27), we obtain the bid-ask prices of a continuous geometric average Asian option in [18]. On the contrary, substituting these two limiting values into (42) and (43), we can also derive the bid-ask prices of a continuous arithmetic average Asian option.


## 3.1. Discrete Average Asian Option under the Law of One Price

In this section, we start with a brief description of the discrete geometric and arithmetic Asian option under the law of one price.Under a probability spaceΩ,ℱ,ℙ, the reference probability measure ℙ is assumed to be a risk-neutral measure. It is assumed that the underlying asset follows the geometric Brownian motion, with a stochastic differential equation having the following expression:(14)dSt=rStdt+σStdWt,0≤t≤T,S0=S0,where r is the risk-free rate and σ is volatility. And these parameters are usually assumed to be constant. The solution of equation (14) can be calculated to the following result:(15)ST=Stexpr−σ22τ+σZτ,where τ=T−t is the time to maturity and Zτ=WT−Wt is the Brownian motion.Suppose thatn prices are taken from equation (15) with observation frequency h, i.e.,(16)ai=Sτ−n−ih=S0expr−σ22τ−n−ih+σZτ−n−ih,where i=1,2,3,⋅⋅⋅,n, and τ=T−t is the time to maturity. We can see that the averaging period starts with the first observation at T=τ−n−1h and stops at the last observation i=n at T=τ. The averaging time period is thus from τ−n−1h to τ or n−1h.The standard geometric average (GA) ofn positive prices a1,a2,⋅⋅⋅,an is defined as(17)GAn=∏i=1nai1/n,and the arithmetic average (AA) of n positive prices a1,a2,…,an is defined as(18)AAn=1n∑i=1nai,i=1,2,3,…,n,where n is the number of observations and ai is the ith observation.LetVSt,t be the option price with the underlying price St, the time to maturity T, and the strike price K. Then, the price of the discrete geometric Asian call option under the risk-neutral measure ℙ, at time t, may be represented as(19)VSt,t=e−rT−tEGAn−K+,and the price of the discrete arithmetic Asian put option is given by(20)VSt,t=e−rT−tEK−AAn+,where GAn is the discrete geometric average and AAn is the arithmetic average of the underlying asset prices during the time to maturity T.In order to price the discrete geometric and arithmetic Asian option, we have to know the distribution of the geometric averageGAn and arithmetic average AAn. In both continuous case and discrete case of the geometric Asian option, the variable GAn is log-normally distributed so that its expectation and variance values may be calculated explicitly. Therefore, Kemma and Vorst [22], Turnbull and Wakeman, [23], and Zhang [1] obtained similar B-S pricing formulas of geometric average Asian options by different approximation methods.Asian option trading in the OTC markets is usually based on the arithmetic average prices of the underlying assets. However, even if all price observations follow a log-normal process, the arithmetic average is not a log-normal process. It is difficult to solve the closed-form solutions for arithmetic average Asian options. A representative method was proposed in Zhang [1]. Zhang [1] tried to express the difference in terms of the distribution of the given observations using the Taylor series expansion and then obtained an approximation formula for arithmetic means in terms of their corresponding geometric means. The following lemmas come from Zhang [1], which give the distribution of GAn and AAn.Lemma 1 (see Zhang [1]).
Let the averaging numbers be defined by equation (16); then, the standard geometric average GAn is a log-normal distribution, i.e.,(21)logGAn∼Nr−12σ2Tμ,n−jsa+lnBjsa+lnSt,σ2Tn−jsa,where(22)B0sa=1,Bjsa=∏i=1jSτ−n−ihS01/n,1≤j≤n,Tμ,n−jsa=n−jnτ−n−j−1h2,Tn−jsa=τn−j22−n−jn−j−14n−4j+1h6n2,and n is the number of observations specified in the contract, h is the observation frequency or the time interval between two consecutive observations, j is the number of observations already passed, Bj is the geometric average of the gross returns of those observations that have already passed, τ is the time to maturity of the option, and the two functions Tμ,n−j and Tn−j may be interpreted as the effective mean and volatility time functions, respectively, because they largely determine the effective mean and variance of the geometric average.Lemma 2 (see Zhang [1]).
The arithmetic average ofn price observations, each following the Brownian motion specified in (16), can be approximated with the corresponding geometric mean:(23)AAa≅κGAa,where(24)κ=1+E˜v+v2=1+12E˜Vlna+14VarVlna+E˜Vlna2,E˜Vlna=n2−1h612r−12σ22h+1nσ2,VarVlna=n2−13n2−215n3r−12σ22σ2h3.According to Lemmas1 and 2, we can easily derive the closed-form solution of the discrete average Asian option under the law of one price. These pricing formulas are well known and can be found in many option pricing textbooks.


## 3.2. The Bid-Ask Prices of Discrete Geometric Average Asian Options

In this section, within the framework of conic finance, we first derive the explicit approximate formulas for the bid-ask prices of the discrete geometric average Asian options. Furthermore, by using Lemma2 and a similar proof method of Theorem 1, we can readily derive explicit approximate formulas for the bid-ask price of the discrete geometric average Asian options. We will give the explicit approximate formulas in the following theorems.Theorem 1.
Assume that the distortion functionψγu is the Wang transform; then, the bid-ask prices of the European discrete geometric average Asian call option at time t are given by(25)bγC≅StAjbΦdn−jb+σTn−jsa−e−rτKΦdn−jb,(26)aγC≅StAjaΦdn−ja+σTn−jsa−e−rτKΦdn−ja,and the bid-ask prices of the European discrete geometric average Asian put option at time t are given by(27)bγP≅e−rτKΦ−dn−ja−StAjaΦ−dn−ja−σTn−jsa,(28)aγP≅e−rτKΦ−dn−jb−StAjbΦ−dn−jb−σTn−jsa,where τ=T−t is maturity,(29)dn−jb=lnBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa−γ,dn−ja=lnBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa+γ,Ajb=Bjsaexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa−γσTn−jsa,Aja=Bjsaexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+γσTn−jsa.
Bjsa, Tn−jsa, and Tμ,n−jsa are the same as in Lemma 1. In particular, for γ=0, the bid-ask prices are equivalent, and the bid-ask prices of the European discrete geometric average Asian option reduce to the B-S-type formulas in [1].Proof.
Consider the discrete geometric averageGAn defined by equation (17). And Lemma 1 shows that random variable GAn has a log-normal distribution under risk-neutral measure ℙ, which means that(30)FGAx=Φlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa,where Φ⋅ is the standard normal cumulative distribution function. If we apply Wang transform (6) to the distribution function FGA, we will get the following representation:(31)ψγFGAx=Φlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ.
Let the payoff of the European discrete geometric average Asian call option beCT=GAn−K+; we can derive the bid price for a European discrete geometric average Asian call option by Choquet integral:(32)bγC=e−rτEψγCT=e−rτC∫CTdψγ∘ℙ=e−rτ∫0∞xdψγFCTx=e−rτ∫K∞x−KdψγFGTx=e−rτ∫K∞xdψγFGTx−∫K∞KdψγFGTx≔I1−I2.
Neglecting the discount factore−rτ temporarily, we can calculate the integral I1 in equation (32). It is shown that(33)I1=∫K∞xdψγFGAx=∫K∞xdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ=exprTμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+lnBjsaSt−γσTn−jsa⋅ΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsa+σ2Tn−jsaσTn−jsa−γ.
And the second integralI2 in equation (32) can be calculated as(34)I2=∫K∞KdψγFGTx=∫K∞KdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ=K1−ΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ=KΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa−γ.
Substituting equations (33) and (34) into (32) and multiplying by a discount factor e−rτ, we can get the following expression for the bid price:(35)bγC=BjsaStexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa−γσTn−jsa⋅ΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsa+σ2Tn−jsaσTn−jsa−γ−e−rτKΦlnBsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa−γ.
Now, by using Choquet expectation in Definition3, we can derive the ask price of the European discrete geometric average Asian call option:(36)aγC=−Eψγ−CT=−∫−∞0xdψγF−CTx=−∫−∞0xdψγ1−FGAK−x=−∫0∞xdψγ1−FGAK+x=−∫K∞x−Kdψγ1−FGAx=−∫K∞xdψγ1−FGAx+∫K∞Kdψγ1−FGAx≕I¯1+I¯2.
Similar to the way in which we obtained the bid price, we get the first integral in equation (36) as follows:(37)I¯1=−∫K∞xdψγ1−FGAx=−∫K∞xdΦlnBjsaSt/x+r−1/2σ2Tμ,n−jsaσTn−jsa+γ=∫K∞xdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa−γ=exprTμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+lnBjsaSt+γσTn−jsa⋅ΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsa+σ2Tn−jsaσTn−jsa+γ,and the second integral can be calculated as(38)I¯2=∫K∞Kdψγ1−FGAx=∫K∞KdΦlnBjsaSt/x+r−1/2σ2Tμ,n−jsaσTn−jsa+γ=−∫K∞KdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa−γ=−K1−ΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa−γ=−KΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa+γ.
By combining partsI¯1 and I¯2 and considering the continuous discount factor e−rτ, we have the ask price:(39)aγC=BjsaStexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+σTn−jsa⋅ΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsa+σ2Tn−jsaσTn−jsa+γ−e−rτKΦlnBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa+γ.
In addition, let the put option payoff bePT=K−GAn+. Similar to the way in which we derived the bid-ask prices of the call option, we can derive the bid-ask prices of the European discrete geometric average Asian put option. We first gave the bid price of the put option as follows:(40)bγP=e−rτEψγPT=e−rτ∫0∞xdψγFPTx=e−rτ∫0∞xdψγ1−FGAK−x=−e−rτ∫0KK−xdψγ1−FGAx=e−rτ∫0KK−xdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa−γ=e−rτKΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa−γ−BjsaStexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+γσTn−jsa⋅ΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsa−σ2Tn−jsaσTn−jsa−γ.
Furthermore, we can also get the formula of the ask price of the European discrete geometric average Asian put option:(41)aγP=−e−rτEψγ−PT=−e−rτ∫−∞0xdψγF−PTx=−e−rτ∫−∞0xdψγFGAK+x=−e−rτ∫0∞xdψγFGAK−x=e−rτ∫0KK−xdψγFGAx=e−rτ∫0KK−xdΦlnx/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ=KΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsaσTn−jsa+γ+BjsaStexprTμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa−γσTn−jsa⋅ΦlnK/BjsaSt−r−1/2σ2Tμ,n−jsa−σ2Tn−jsaσTn−jsa+γ.
Finally, by designing some simple variables, we will lead to the bid-ask prices in equations (25)(27).
This completes the proof of Theorem1.Let the payoff of the European discrete arithmetic average Asian option beCT=AAn−K+ or PT=K−AAn+. By using Lemma 2 and applying the method of calculating Choquet expectation in Theorem 1, we can easily obtain the explicit approximate formulas of the discrete arithmetic average Asian options.Theorem 2.
Assume that the distortion functionψγu is the Wang transform; then, the bid-ask prices of the European discrete arithmetic average Asian call option at time t are given by(42)bγC≅StA¯jbΦd¯n−jb+σTn−jsa−e−rτKΦd¯n−jb,(43)aγC≅StA¯jaΦd¯n−ja+σTn−jsa−e−rτKΦd¯n−ja,and the bid-ask prices of the European discrete arithmetic average Asian put option at time t are given by(44)bγP≅e−rτKΦ−d¯n−ja−StA¯jaΦ−d¯n−ja−σTn−jsa,(45)aγP≅e−rτKΦ−d¯n−jb−StA¯jbΦ−d¯n−jb−σTn−jsa,where(46)d¯n−jb=lnκBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa−γ,d¯n−ja=lnκBjsaSt/K+r−1/2σ2Tμ,n−jsaσTn−jsa+γ,A¯jb=κBjsaexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa−γσTn−jsa,A¯ja=κBjsaexp−rτ−Tμ,n−jsa−σ22Tμ,n−jsa−Tn−jsa+γσTn−jsa,and Bjsa, Tn−jsa, Tμ,n−jsa, and κ are the same as in Lemmas 1 and 2. In particular, for γ=0, the bid-ask prices are equivalent, and the bid-ask prices of the European discrete arithmetic average Asian option reduce to the B-S-type formulas in [1].Remark 1.
If the observation frequencyh is continuous, the number of observations n approaches to infinity, and the observation frequency h tends to zero. Substituting n⟶∞ and h⟶0, n−1h⟶τ, into Tn−jsa and Tμ,n−jsa yields Tn−jsa⟶τ/2 and Tμ,n−jsa⟶τ/3. Furthermore, substituting these two limiting values into (25) and (27), we obtain the bid-ask prices of a continuous geometric average Asian option in [18]. On the contrary, substituting these two limiting values into (42) and (43), we can also derive the bid-ask prices of a continuous arithmetic average Asian option.


## 4. Numerical Examples

In this section, we give two examples to demonstrate and validate that the approximate closed forms are efficient and accurate. Suppose there are 12 observations in the European discrete geometric average or arithmetic average Asian option, observation frequency in monthly, the averaging period has not started, and the time to maturity of the option is one year. That is to say,n=12, h=1/12, j=0, and τ=1. In addition, assume that the risk-free interest rate r=0.07, initial stock price S0=100, and the stock price volatility σ=0.2. By substituting these parameters into the approximate closed forms with the different market liquidity parameter γ and strike price K, we will obtain the numerical results which are displayed in Tables 1 and 2.Table 1
The bid-ask prices of the European discrete geometric Asian options at differentγ with r=0.07,σ=0.2,τ=1, n=12,h=1/12,andj=0.
γS0KCallPutAskBidSpreadAskBidSpread01009013.30413.30400.6970.69701006.4656.46503.1823.18201102.4312.43108.4728.47200.031009013.62012.9910.6290.7390.6570.0811006.6936.2420.4513.3133.0540.2601102.5512.3150.2368.7108.2360.4740.31009016.60110.3306.2721.2100.3760.8341008.9574.4534.5044.6582.0562.6021103.8391.4522.38710.9806.2624.7180.91009023.9625.54518.4173.0270.0892.93810015.2641.80213.4628.6080.7157.8931108.1720.4287.74416.5572.94713.610Table 2
The bid-ask prices of the European discrete arithmetic Asian options at differentγ with r=0.07,σ=0.2,τ=1, n=12,h=1/12,andj=0.
γS0KCallPutAskBidSpreadAskBidSpread01009013.59813.59800.6600.66001006.6766.67603.0633.06301102.5432.54308.2538.25300.031009013.91713.2820.6350.7000.6220.0781006.9096.4490.4603.1912.9380.2531102.6672.4230.2448.4898.0200.4690.31009016.92210.5916.3311.1530.3540.7991009.2134.6204.5924.5061.9682.5371103.9941.5272.46710.7366.0744.6630.91009024.3255.73018.5952.9150.0822.83310015.5951.88913.7068.3990.6767.7221108.4250.4557.96916.2892.83013.459In Table1, we provide the numerical results for the bid-ask prices of the discrete geometric average Asian options. And Table 2 shows the numerical results for the bid-ask prices of the discrete arithmetic average Asian options. For γ=0, the ask and bid prices are equivalent, and they reduce to the B-S-type formulas presented by Zhang [1]. From Tables 1 and 2, we find that the higher the liquidity level γ, the wider the bid-ask spread, which means that liquidity is insufficient.


## 5. Conclusion

In this paper, within the framework of conic finance, we derive effective, explicit, approximate formulas to estimate the bid-ask prices for the European discrete geometric average and arithmetic average Asian options. Finally, numerical examples show that the theory of conic finance has significant contributions to financial derivative pricing.



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