# Estimation of Ask and Bid Prices for Geometric Asian Options

Authors: Tao Chen; Kaili Xiang; Xuemei Luo
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

Traditional derivative pricing theories usually focus on the risk-neutral price or the equilibrium price. However, in highly competitive financial markets, we observed two prices which are called bid and ask prices; then the unique risk-neutral price fails to hold. In this paper, within the framework of conic finance, we provide a useful approach to evaluate the ask and bid prices of geometric Asian options and obtain the explicit formulas for the ask and bid prices. Finally, numerical examples show that the higher the market liquidity parameterγ, the wider the spread and hence the less the liquidity.


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## Body

## 1. Introduction

Asian options give the holder a payoff that depends on the average price of the underlying over some prescribed period. This averaging of the underlying can bring about two significant advantages: one is to reduce the risk of manipulating the underlying asset and the other is that it costs less than standard American options and European options (see Wilmott [1], chap. 25). Asian options are actively traded in both exchanges and over-the-counter markets. In Black-Scholes framework, the study of exotic options has attracted the attention of many scholars.Although there are many methods, such as the PDE method, Martingale method, Monte Carlo simulation, and the binomial tree, to solve option price, an efficient method has not been found yet. The most important factor is that the real market has many uncertain factors. As we all know, in traditional financial mathematics, the foundations of the option pricing theory are built on the paradigm of frictionless and competitive markets. However, in the real market, the risk elimination is typically unattainable and not available. Furthermore, we observe two prices, one for buying from the market called the ask price and another for selling to the market called the bid price. Hence, in the real market, we can no longer depend on the unique risk-neutral price (or the law of one price or equilibrium price).There are diversity of theoretical approaches to estimating ask and bid prices. Barles and Soner [2], Cvitanić and Karatzas [3], Constantinides [4], Lo et al. [5], and Jouini and Kallal [6] attempted to study spreads which included transaction costs of trading in liquid markets. Easley and O’Hara [7] and Han and Shino [8] study price formation in securities markets. Copeland and Galai [9] discuss the effects of information on the bid-ask spread. Glosten and Milgrom [10] focus on the effects of heterogeneously informed traders on market makers. In [11–14] the researcher have carried out inventory costs and order processing of liquidity providers. In [15–18], statistical studies are used to model bid-ask spread. However, these models are not effective enough to explain the magnitude of the spreads observed in the markets. A new theory is built up by Cherny and Madan [19, 20], referred to as the conic finance theory. In the conic finance framework, the market acts as a passive counterparty to all transactions, buying at the ask price and selling at the bid price. The spread between bid-ask prices is a measure of illiquidity.Although there are a number of literatures based on conic finance theory, they focus on credit risk [21, 22], design of portfolio [23, 24], and hedging of financial and insurance risks [25, 26]. To the best of our knowledge, there is no literature research on valuation of ask and bid prices for geometric Asian option. In this paper, within the framework of conic finance, we lead to the explicit formulas for the ask and bid prices of geometric Asian option.The content of this paper is organized as follows. In Section2 we introduce the risk-neutral price for the geometric Asian option. Within the framework of conic finance, Section 3 is devoted to estimating the bid-ask prices for geometric Asian option. And we obtain the explicit formulas for the bid-ask prices. In Section 4, we present numerical results for the bid-ask prices. Finally, we finish our paper by concluding remarks in the last section.


## 2. Geometric Asian Option under the Law of One Price

In this section, we start with a brief description of the geometric Asian option model presented in [27].Under a probability space(Ω,F,P), Kemna and Vorst [27] set up the pricing model in which the underlying asset follows the geometric Brownian motion: (1)dSt=rStdt+σStdWt,0≤t≤T,S0=S0where r is the risk-free rate and σ is volatility. And these parameters are usually assumed to be constant. Let T be the maturity date and [T0,T] be the final time interval over which the average value of the stock is calculated. Let V(St,t) be the option price with the underlying price St, time to maturity T, and the strike price K. Then, the price of the geometric Asian option under the risk neutral measure P~, at time t, may be represented as follows: (2)VSt,t=e-rT-tE~GT-K+,where GT is the geometric average of the underlying asset prices during the time to the maturity T.Kemna and Vorst [27] introduce a process GT defined by(3)GT=exp⁡1T-T0∫T0Tln⁡Sudu,to represent the geometric average of underlying asset St until the time T. And the discrete case equation (3) can be written as follows:(4)GT=∏i=0nSTi1/n+1.In both continuous case (3) and discrete case (4), the variable GT is log-normally distributed so that its expectation and variance values may be calculated explicitly. For the continuous case (3) the log-normal distribution is(5)log⁡GT~N12r-12σ2T-T0+log⁡ST0,13σ2T-T0.Then, the price of geometric Asian call option at time T0 is given as(6)VST0,T0=ST0ed∗Nd1-Ke-rT-T0Nd2,where (7)d∗=-12r+16σ2T-T0,d1=log⁡ST0/K+1/2r+1/6σ2T-T0σ1/3T-T0,d2=d1-σT-T03.


## 3. Estimation of Bid-Ask Prices Formula

In this section, within the framework of conic finance, we derive the explicit formulas for the bid-ask prices of geometric Asian options. We first present a brief description of conic finance theory. Then we present our main conclusion in the next section.


### 3.1. Conic Finance Theory

Conic finance is a brand-new quantitative finance theory, which originates from the work by Cherny and Madan [20] and Madan and Cherny [19]. The key to the foundations of the conic finance is an underlying concept of acceptable risks in the economy. Markets are modeled as counterparty accepting at nonnegative stochastic cash flow X that have an acceptability level γ. The theory assumes that price depends on the direction of trade and there are two prices, one for buying from the market called the ask price aγ(X) and one for selling to the market called the bid price bγ(X). The difference between both prices gives rise to the bid-ask spread observed in financial markets.LetL∞≔L∞(Ω,F,P) be the space of all essentially bounded. Madan and Cherny [19] derive these bid and ask prices from the theory of acceptability indices (see [20]), which are functions α:L∞→[0,∞). In particular, they call a net cash flow, or trade, X~∈L∞ acceptable at an acceptability level γ if and only if α(X~)≥γ. Suppose that the market maker sell a cash flow X, for which driven by competition he charges a minimal price of a. Nevertheless, the emerging remaining cash flow a-X ought to be acceptable at level γ. Hence, this price a would be the ask price of X. So the minimal price is given by(8)aγX=inf⁡a:αa-X≥γ=inf⁡a:EQa-X≥γforanyQ∈Dγ=supQ∈Dγ⁡EQX,where a family of sets of probability measures (Dγ)γ≥0 is equivalent to the initial probability measure of P.When the market maker buysX for a price b, it is X-b that must be acceptable at level γ and the maximal price is(9)bγX=infQ∈Dγ⁡EQX.As proposed by Madan and Cherny [20], parameter family of distortion functions can be used to formulate an operational index of acceptability. The index α(X) is characterized as (10)αX=sup⁡γ≥0:∫-∞∞xdψγFXx,or(11)EQγX=∫-∞∞xdψγFXx,where X is an nonnegative stochastic variable, FX(x) is the distribution function of X, and (ψγ)γ≥0 is a pointwise increasing family of concave distortion functions.Because the distortion function plays a crucial part in the realization of explicit bid and ask prices, Cherny and Madan [20] conclude a series of potential distortion function that can be used. In the following definition, we first give some particular distortion function that has been used extensively in the literature.Definition 1 (distortion function).
A functionψ:[0,1]→[0,1] is a distortion function if and only if it is monotone, ψ(0)=0,ψ(1)=1.Definition 2 (Wang transform [28, 29]).
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(12)ψγu=ΦΦ-1u+γis called the Wang transform.Definition 3 (the Maxminvar distortion function [20]).
The concave distortion function is given by(13)ψγu=1-1-u1+γ1/1+γ,∀u∈0,1,γ≥0.Definition 4 (the Minmaxvar distortion function [20]).
The concave distortion function is given by(14)ψγu=1-1-u1/1+γ1+γ,∀u∈0,1,γ≥0.From a family of concave distortion functions(ψγ)γ≥0 and the properties of the distortion expectation (11), Cherny and Madan [19] lead to the following formulas of bid-ask prices. (15)αa-X≥γ⇔∫-∞∞xdψγFa-Xx≥0⇔a+∫-∞∞xdψγF-Xx≥0,so that the minimum value of a leads to the ask price:(16)aγX=-∫-∞∞xdψγF-Xx.Analogously, the maximum of b leads to the bid price: (17)αX-b≥γ⇔∫-∞∞xdψγFX-bx≥0⇔-b+∫-∞∞xdψγFXx≥0;we obtain(18)bγX=∫-∞∞xdψγFXx.Under a nonadditive probability using Choquet expectation which introduced by Choquet in [30], the bid-ask prices (16)-(18) may also be presented in the following definition.Definition 5 (single-period bid-ask prices [19]).
Let(ψγ)γ≥0 be a pointwise increasing family of concave distortion functions and γ is the market liquidity level. Then, the bid price of a discounted cash flow X∈L∞ is given by(19)bγX=EψγX=C∫Xdψγ∘P=-∫-∞0ψγPX≤xdx+∫0∞1-ψγPX≤xdxand its ask price is(20)aγX=-Eψγ-X=∫-∞0ψγPX>x-1dx+∫0∞ψγPX>xdxIn particular, forγ=0, the bid-ask prices are equivalent and they reduce to the regular Black-Scholes formula which is undistorted under the risk-neutral probability measure. In addition, we also have the relation as follows: (21)bγX≤EX≤aγX.


### 3.2. Bid-Ask Formulas for Geometric Asian Option

In this subsection, we give our main conclusion. For evaluation of the explicit formulas for the bid-ask prices of geometric Asian options, we first use the distortion function based on the Wang Transform from Definition2. Furthermore, by using Choquet expectation in Definition 5 we can derive the bid-ask price explicit formulas of the geometric Asian call and put options. The following theorem shows our main results.Theorem 6.
Assume that the distortion functionψγ(u) is the Wang Transform; then the bid-ask prices of the geometric Asian call option at time t is given by(22)bγC=StedbΦdb1-e-rT-tKΦdb2,(23)aγC=StedaΦda1-e-rT-tKΦda2,and the bid-ask prices of the geometric Asian put option at time t are given by(24)bγP=e-rT-tKΦd~b2-Sted~bΦd~b1,(25)aγP=e-rT-tKΦd~a2-Sted~aΦd~a1,where (26)τ=T-t,db1=ln⁡St/K+1/2r+1/6σ2τσ1/3τ-γ,db2=db1-στ3,da1=ln⁡St/K+1/2r+1/6σ2τσ1/3τ+γ,da2=da1-στ3,db=-12r+16σ2τ-γστ3,da=-12r+16σ2τ+γστ3,d~b1=ln⁡K/St-1/2r+1/6σ2τσ1/3τ-γ,d~b2=d~b1+στ3,d~a1=ln⁡K/St-1/2r+1/6σ2τσ1/3τ+γ,d~a2=d~a1+στ3,d~b=-12r+16σ2τ+γστ3,d~a=-12r+16σ2τ-γστ3.In particular, for γ=0, the bid-ask prices are equivalent and the bid-ask prices of geometric Asian call option reduce to formula (6).Proof.
Let the payoff of geometric Asian call option beCT=(GT-K)+, with GT being the geometric average of the underlying asset price during the time t to the maturity T. The continuous case of GT is defined by (3). And (5) shows that random variable GT has a lognormal distribution; it means that(27)FGTx=Φlog⁡x/St-1/2r-1/2σ2T-tσ1/3T-t,where Φ is the standard normal cumulative distribution function.
Now, by using Choquet expectation in Definition5, we can derive the bid price of the geometric Asian call option:(28)bγC=EψγCT=C∫CTdψγ∘P=∫0∞xdψγFCTx=∫K∞x-KdψγFGTx=∫K∞xdψγFGTx-∫K∞KdψγFGTx≔A1-B1.If we apply the Wang transform (12) to the distribution function FGT, we will get the following representation:(29)ψγFGTx=ΦΦ-1FGTx+γ=Φlog⁡x/St-1/2r-1/2σ2T-tσ1/3T-t+γ.And, if X~Lognormalμ,σ2 with cumulative distribution function (CDF) FX(x), then we can obtain(30)∫K∞xdFXx=∫K∞xdΦln⁡x-μσ=1σ2π∫ln⁡K∞eye-y-μ2/2σ2dy=1σ2π∫ln⁡K∞eμ+1/2σ2e-y-μ+σ22/2σ2dy=eμ+1/2σ2Φμ+σ2-ln⁡Kσ.
By using (29) and (30), we can calculate the integral A1 in (28). It is shown that(31)A1=∫K∞xdψγFGTx=∫ln⁡K∞eydΦy-ln⁡St-1/2r-1/2σ2T-tσT-t/3+γ=12π∫ln⁡K∞1σT-t/3·exp⁡-y-ln⁡St-1/2r-1/2σ2T-t/σT-t/3+γ22+ydy=exp⁡ln⁡St+12r-12σ2T-t-γσT-t3+16σ2T-t·Φln⁡St/K+1/2r-1/2σ2T-t+1/3σ2T-tσ1/3T-t-γ=exp⁡ln⁡St+12r-16σ2T-t-γσT-t3·Φln⁡St/K+1/2r+1/6σ2T-tσ1/3T-t-γ.And the second integral B1 in (28) can be calculated as(32)B1=∫K∞KdψγFGTx=∫K∞KdΦln⁡x-ln⁡St-1/2r-1/2σ2T-tσ1/3T-t+γ=K1-Φln⁡K/St-1/2r-1/2σ2T-tσ1/3T-t+γ=KΦln⁡St/K+1/2r-1/2σ2T-tσ1/3T-t-γ.Substituting (31) and (32) into (28) and multiplying by a discount factor e-r(T-t), we can get the following expression for the bid price:(33)bγC=Stexp⁡-12r+16σ2T-t-γσT-t3·Φln⁡St/K+1/2r+1/6σ2T-tσ1/3T-t-γ-e-rT-tKΦln⁡St/K+1/2r-1/2σ2T-tσ1/3T-t-γ.
Now, by using Choquet expectation in Definition5 we can derive the ask price of the geometric Asian call option:(34)aγC=-Eψγ-CT=-∫-∞0xdψγF-CTx=-∫-∞0xdψγ1-FGTK-x=-∫0∞xdψγ1-FGTK+x=-∫K∞x-Kdψγ1-FGTx=-∫K∞xdψγ1-FGTx+∫K∞Kdψγ1-FGTx≕A2+B2.
Similar to the way in which we obtain the bid price. From (27), (30), and Wang transform (12), we get the first integral in (34) as follows:(35)A2=-∫K∞xdψγ1-FGTx=-∫K∞xdψγln⁡St/x+1/2r-1/2σ2T-tσT-t/3=-∫K∞xdΦln⁡St/x+1/2r-1/2σ2T-tσT-t/3+γ=∫K∞xdΦln⁡x/St-1/2r-1/2σ2T-tσT-t/3-γ=exp⁡ln⁡St+12r-12σ2T-t+γσT-t3+16σ2T-t·Φln⁡St/K+1/2r-1/2σ2T-t+1/3σ2T-tσ1/3T-t+γ,and the second integral can be calculated as(36)B2=∫K∞Kdψγ1-FGTx=∫K∞KdΦln⁡St/x+1/2r-1/2σ2T-tσ1/3T-t+γ=-∫K∞KdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ=-K1-Φln⁡K/St-1/2r-1/2σ2T-tσ1/3T-t-γ=-KΦln⁡St/K+1/2r-1/2σ2T-tσ1/3T-t+γ.
By combining partsA2 and B2 and considering the continuous discount factor e-r(T-t), we have the ask price:(37)aγC=Stexp⁡-12r+16σ2T-t+γσT-t3·Φln⁡St/K+1/2r+1/6σ2T-tσ1/3T-t+γ-e-rT-tKΦln⁡St/K+1/2r-1/2σ2T-tσ1/3T-t+γ
In addition, we show the bid-ask prices of the geometric Asian put option. LetPT=(K-GT)+; by utilizing Choquet expectation in Definition 5 and transformations (29) and (30), we can derive the bid price of the geometric Asian put option as follows:(38)bγP=EψγPT=∫0∞xdψγFPTx=∫0∞xdψγ1-FGTK-x=-∫0KK-xdψγ1-FGTx=∫0KK-xdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ.
Conducting the same calculation steps that we used during the derivation of the bid price of the call option, we split the integral into two parts as follows:(39)A=∫0KKdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ=KΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ0K=KΦln⁡K/St-1/2r-1/2σ2T-tσ1/3T-t-γ,and (40)B=∫0KxdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ=∫0∞-∫K∞xdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ=exp⁡ln⁡St+12r-12σ2T-t+γσT-t3+16σ2T-t·1-Φln⁡St/K+1/2r-1/2σ2T-t+1/3σ2T-tσT-t/3+γ=Stexp⁡12r-12σ2T-t+γσT-t3+16σ2T-t·Φln⁡K/St-1/2r-1/2σ2T-t-1/3σ2T-tσ1/3T-t-γ.Substituting the results of the two integrals in (38) and multiplying by a discount factor e-r(T-t), we get the bid price of put option as follows:(41)bγP=e-rT-tKΦln⁡K/St-1/2r-1/2σ2T-tσ1/3T-t-γ-Stexp⁡-12r+16σ2T-t+γσT-t3·Φln⁡K/St-1/2r+1/6σ2T-tσ1/3T-t-γ
Finally, by the same methods that we used in the calculation process of ask price of the geometric Asian call option, we can get the formula of the ask price of the geometric Asian put option:(42)aγP=-Eψγ-PT=-∫-∞0xdψγF-PTx=-∫-∞0xdψγFGTK+x=-∫0∞xdψγFGTK-x=∫0KK-xdψγFGTx=∫0KK-xdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t+γ=e-rT-tKΦln⁡K/St-1/2r-1/2σ2T-tσ1/3T-t+γ-Stexp⁡-12r+16σ2T-t-γσT-t3·Φln⁡K/St-1/2r+1/6σ2T-tσ1/3T-t+γThis completes the proof of Theorem 6.


## 3.1. Conic Finance Theory

Conic finance is a brand-new quantitative finance theory, which originates from the work by Cherny and Madan [20] and Madan and Cherny [19]. The key to the foundations of the conic finance is an underlying concept of acceptable risks in the economy. Markets are modeled as counterparty accepting at nonnegative stochastic cash flow X that have an acceptability level γ. The theory assumes that price depends on the direction of trade and there are two prices, one for buying from the market called the ask price aγ(X) and one for selling to the market called the bid price bγ(X). The difference between both prices gives rise to the bid-ask spread observed in financial markets.LetL∞≔L∞(Ω,F,P) be the space of all essentially bounded. Madan and Cherny [19] derive these bid and ask prices from the theory of acceptability indices (see [20]), which are functions α:L∞→[0,∞). In particular, they call a net cash flow, or trade, X~∈L∞ acceptable at an acceptability level γ if and only if α(X~)≥γ. Suppose that the market maker sell a cash flow X, for which driven by competition he charges a minimal price of a. Nevertheless, the emerging remaining cash flow a-X ought to be acceptable at level γ. Hence, this price a would be the ask price of X. So the minimal price is given by(8)aγX=inf⁡a:αa-X≥γ=inf⁡a:EQa-X≥γforanyQ∈Dγ=supQ∈Dγ⁡EQX,where a family of sets of probability measures (Dγ)γ≥0 is equivalent to the initial probability measure of P.When the market maker buysX for a price b, it is X-b that must be acceptable at level γ and the maximal price is(9)bγX=infQ∈Dγ⁡EQX.As proposed by Madan and Cherny [20], parameter family of distortion functions can be used to formulate an operational index of acceptability. The index α(X) is characterized as (10)αX=sup⁡γ≥0:∫-∞∞xdψγFXx,or(11)EQγX=∫-∞∞xdψγFXx,where X is an nonnegative stochastic variable, FX(x) is the distribution function of X, and (ψγ)γ≥0 is a pointwise increasing family of concave distortion functions.Because the distortion function plays a crucial part in the realization of explicit bid and ask prices, Cherny and Madan [20] conclude a series of potential distortion function that can be used. In the following definition, we first give some particular distortion function that has been used extensively in the literature.Definition 1 (distortion function).
A functionψ:[0,1]→[0,1] is a distortion function if and only if it is monotone, ψ(0)=0,ψ(1)=1.Definition 2 (Wang transform [28, 29]).
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(12)ψγu=ΦΦ-1u+γis called the Wang transform.Definition 3 (the Maxminvar distortion function [20]).
The concave distortion function is given by(13)ψγu=1-1-u1+γ1/1+γ,∀u∈0,1,γ≥0.Definition 4 (the Minmaxvar distortion function [20]).
The concave distortion function is given by(14)ψγu=1-1-u1/1+γ1+γ,∀u∈0,1,γ≥0.From a family of concave distortion functions(ψγ)γ≥0 and the properties of the distortion expectation (11), Cherny and Madan [19] lead to the following formulas of bid-ask prices. (15)αa-X≥γ⇔∫-∞∞xdψγFa-Xx≥0⇔a+∫-∞∞xdψγF-Xx≥0,so that the minimum value of a leads to the ask price:(16)aγX=-∫-∞∞xdψγF-Xx.Analogously, the maximum of b leads to the bid price: (17)αX-b≥γ⇔∫-∞∞xdψγFX-bx≥0⇔-b+∫-∞∞xdψγFXx≥0;we obtain(18)bγX=∫-∞∞xdψγFXx.Under a nonadditive probability using Choquet expectation which introduced by Choquet in [30], the bid-ask prices (16)-(18) may also be presented in the following definition.Definition 5 (single-period bid-ask prices [19]).
Let(ψγ)γ≥0 be a pointwise increasing family of concave distortion functions and γ is the market liquidity level. Then, the bid price of a discounted cash flow X∈L∞ is given by(19)bγX=EψγX=C∫Xdψγ∘P=-∫-∞0ψγPX≤xdx+∫0∞1-ψγPX≤xdxand its ask price is(20)aγX=-Eψγ-X=∫-∞0ψγPX>x-1dx+∫0∞ψγPX>xdxIn particular, forγ=0, the bid-ask prices are equivalent and they reduce to the regular Black-Scholes formula which is undistorted under the risk-neutral probability measure. In addition, we also have the relation as follows: (21)bγX≤EX≤aγX.


## 3.2. Bid-Ask Formulas for Geometric Asian Option

In this subsection, we give our main conclusion. For evaluation of the explicit formulas for the bid-ask prices of geometric Asian options, we first use the distortion function based on the Wang Transform from Definition2. Furthermore, by using Choquet expectation in Definition 5 we can derive the bid-ask price explicit formulas of the geometric Asian call and put options. The following theorem shows our main results.Theorem 6.
Assume that the distortion functionψγ(u) is the Wang Transform; then the bid-ask prices of the geometric Asian call option at time t is given by(22)bγC=StedbΦdb1-e-rT-tKΦdb2,(23)aγC=StedaΦda1-e-rT-tKΦda2,and the bid-ask prices of the geometric Asian put option at time t are given by(24)bγP=e-rT-tKΦd~b2-Sted~bΦd~b1,(25)aγP=e-rT-tKΦd~a2-Sted~aΦd~a1,where (26)τ=T-t,db1=ln⁡St/K+1/2r+1/6σ2τσ1/3τ-γ,db2=db1-στ3,da1=ln⁡St/K+1/2r+1/6σ2τσ1/3τ+γ,da2=da1-στ3,db=-12r+16σ2τ-γστ3,da=-12r+16σ2τ+γστ3,d~b1=ln⁡K/St-1/2r+1/6σ2τσ1/3τ-γ,d~b2=d~b1+στ3,d~a1=ln⁡K/St-1/2r+1/6σ2τσ1/3τ+γ,d~a2=d~a1+στ3,d~b=-12r+16σ2τ+γστ3,d~a=-12r+16σ2τ-γστ3.In particular, for γ=0, the bid-ask prices are equivalent and the bid-ask prices of geometric Asian call option reduce to formula (6).Proof.
Let the payoff of geometric Asian call option beCT=(GT-K)+, with GT being the geometric average of the underlying asset price during the time t to the maturity T. The continuous case of GT is defined by (3). And (5) shows that random variable GT has a lognormal distribution; it means that(27)FGTx=Φlog⁡x/St-1/2r-1/2σ2T-tσ1/3T-t,where Φ is the standard normal cumulative distribution function.
Now, by using Choquet expectation in Definition5, we can derive the bid price of the geometric Asian call option:(28)bγC=EψγCT=C∫CTdψγ∘P=∫0∞xdψγFCTx=∫K∞x-KdψγFGTx=∫K∞xdψγFGTx-∫K∞KdψγFGTx≔A1-B1.If we apply the Wang transform (12) to the distribution function FGT, we will get the following representation:(29)ψγFGTx=ΦΦ-1FGTx+γ=Φlog⁡x/St-1/2r-1/2σ2T-tσ1/3T-t+γ.And, if X~Lognormalμ,σ2 with cumulative distribution function (CDF) FX(x), then we can obtain(30)∫K∞xdFXx=∫K∞xdΦln⁡x-μσ=1σ2π∫ln⁡K∞eye-y-μ2/2σ2dy=1σ2π∫ln⁡K∞eμ+1/2σ2e-y-μ+σ22/2σ2dy=eμ+1/2σ2Φμ+σ2-ln⁡Kσ.
By using (29) and (30), we can calculate the integral A1 in (28). It is shown that(31)A1=∫K∞xdψγFGTx=∫ln⁡K∞eydΦy-ln⁡St-1/2r-1/2σ2T-tσT-t/3+γ=12π∫ln⁡K∞1σT-t/3·exp⁡-y-ln⁡St-1/2r-1/2σ2T-t/σT-t/3+γ22+ydy=exp⁡ln⁡St+12r-12σ2T-t-γσT-t3+16σ2T-t·Φln⁡St/K+1/2r-1/2σ2T-t+1/3σ2T-tσ1/3T-t-γ=exp⁡ln⁡St+12r-16σ2T-t-γσT-t3·Φln⁡St/K+1/2r+1/6σ2T-tσ1/3T-t-γ.And the second integral B1 in (28) can be calculated as(32)B1=∫K∞KdψγFGTx=∫K∞KdΦln⁡x-ln⁡St-1/2r-1/2σ2T-tσ1/3T-t+γ=K1-Φln⁡K/St-1/2r-1/2σ2T-tσ1/3T-t+γ=KΦln⁡St/K+1/2r-1/2σ2T-tσ1/3T-t-γ.Substituting (31) and (32) into (28) and multiplying by a discount factor e-r(T-t), we can get the following expression for the bid price:(33)bγC=Stexp⁡-12r+16σ2T-t-γσT-t3·Φln⁡St/K+1/2r+1/6σ2T-tσ1/3T-t-γ-e-rT-tKΦln⁡St/K+1/2r-1/2σ2T-tσ1/3T-t-γ.
Now, by using Choquet expectation in Definition5 we can derive the ask price of the geometric Asian call option:(34)aγC=-Eψγ-CT=-∫-∞0xdψγF-CTx=-∫-∞0xdψγ1-FGTK-x=-∫0∞xdψγ1-FGTK+x=-∫K∞x-Kdψγ1-FGTx=-∫K∞xdψγ1-FGTx+∫K∞Kdψγ1-FGTx≕A2+B2.
Similar to the way in which we obtain the bid price. From (27), (30), and Wang transform (12), we get the first integral in (34) as follows:(35)A2=-∫K∞xdψγ1-FGTx=-∫K∞xdψγln⁡St/x+1/2r-1/2σ2T-tσT-t/3=-∫K∞xdΦln⁡St/x+1/2r-1/2σ2T-tσT-t/3+γ=∫K∞xdΦln⁡x/St-1/2r-1/2σ2T-tσT-t/3-γ=exp⁡ln⁡St+12r-12σ2T-t+γσT-t3+16σ2T-t·Φln⁡St/K+1/2r-1/2σ2T-t+1/3σ2T-tσ1/3T-t+γ,and the second integral can be calculated as(36)B2=∫K∞Kdψγ1-FGTx=∫K∞KdΦln⁡St/x+1/2r-1/2σ2T-tσ1/3T-t+γ=-∫K∞KdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ=-K1-Φln⁡K/St-1/2r-1/2σ2T-tσ1/3T-t-γ=-KΦln⁡St/K+1/2r-1/2σ2T-tσ1/3T-t+γ.
By combining partsA2 and B2 and considering the continuous discount factor e-r(T-t), we have the ask price:(37)aγC=Stexp⁡-12r+16σ2T-t+γσT-t3·Φln⁡St/K+1/2r+1/6σ2T-tσ1/3T-t+γ-e-rT-tKΦln⁡St/K+1/2r-1/2σ2T-tσ1/3T-t+γ
In addition, we show the bid-ask prices of the geometric Asian put option. LetPT=(K-GT)+; by utilizing Choquet expectation in Definition 5 and transformations (29) and (30), we can derive the bid price of the geometric Asian put option as follows:(38)bγP=EψγPT=∫0∞xdψγFPTx=∫0∞xdψγ1-FGTK-x=-∫0KK-xdψγ1-FGTx=∫0KK-xdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ.
Conducting the same calculation steps that we used during the derivation of the bid price of the call option, we split the integral into two parts as follows:(39)A=∫0KKdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ=KΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ0K=KΦln⁡K/St-1/2r-1/2σ2T-tσ1/3T-t-γ,and (40)B=∫0KxdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ=∫0∞-∫K∞xdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t-γ=exp⁡ln⁡St+12r-12σ2T-t+γσT-t3+16σ2T-t·1-Φln⁡St/K+1/2r-1/2σ2T-t+1/3σ2T-tσT-t/3+γ=Stexp⁡12r-12σ2T-t+γσT-t3+16σ2T-t·Φln⁡K/St-1/2r-1/2σ2T-t-1/3σ2T-tσ1/3T-t-γ.Substituting the results of the two integrals in (38) and multiplying by a discount factor e-r(T-t), we get the bid price of put option as follows:(41)bγP=e-rT-tKΦln⁡K/St-1/2r-1/2σ2T-tσ1/3T-t-γ-Stexp⁡-12r+16σ2T-t+γσT-t3·Φln⁡K/St-1/2r+1/6σ2T-tσ1/3T-t-γ
Finally, by the same methods that we used in the calculation process of ask price of the geometric Asian call option, we can get the formula of the ask price of the geometric Asian put option:(42)aγP=-Eψγ-PT=-∫-∞0xdψγF-PTx=-∫-∞0xdψγFGTK+x=-∫0∞xdψγFGTK-x=∫0KK-xdψγFGTx=∫0KK-xdΦln⁡x/St-1/2r-1/2σ2T-tσ1/3T-t+γ=e-rT-tKΦln⁡K/St-1/2r-1/2σ2T-tσ1/3T-t+γ-Stexp⁡-12r+16σ2T-t-γσT-t3·Φln⁡K/St-1/2r+1/6σ2T-tσ1/3T-t+γThis completes the proof of Theorem 6.


## 4. Numerical Examples

In this section, we present numerical results obtained for the geometric Asian option pricing model proposed in this paper. Assume the risk-free interest rater being 8% per annum, the stock price volatility σ being 20%, and the geometric Asian option with 3 months to expiry (i.e., T=3/12). We show the bid-ask prices for the geometric Asian option with the different market liquidity parameter γ, which are displayed in Table 1 and Figure 1.Table 1
The bid-ask prices for geometric Asian options at differentγ with r=0.02,σ=0.8,t=0,T=3/12.
Level S 0 K Call Put ask bid spread ask bid spread 0 100 90 10.752 10.752 0 0.048 0.048 0 100 2.748 2.748 0 1.845 1.845 0 110 0.178 0.178 0 9.077 9.077 0  0.02 100 90 10.864 10.641 0.223 0.050 0.045 0.005 100 2.814 2.682 0.131 1.894 1.797 0.097 110 0.186 0.170 0.016 9.183 8.971 0.212  0.4 100 90 13.031 8.575 4.456 0.129 0.016 0.113 100 4.250 1.629 2.621 2.985 1.037 1.948 110 0.420 0.066 0.354 11.224 7.009 4.216  0.9 100 90 15.984 6.068 9.916 0.372 0.003 0.369 100 6.613 0.737 5.875 4.844 0.434 4.410 110 1.045 0.016 1.029 13.924 4.669 9.256Figure 1
The price information for geometric Asian options at differentγ with S0=78, K=76, r=0.02, σ=0.8, t=0, T=3/12.Table1 provides the numerical results for the bid-ask prices of geometric Asian put and call options. For γ=0, the ask and bid prices are equivalent and they reduce to the analytic expression (6) presented by Kemna and Vorst [27]. Figure 1 plots bid-ask spread for the geometric Asian put and call options at different static market liquidity parameter γ. The spread between bid-ask prices is a measure of illiquidity. The nonnegative parameter γ gives an indication of the markets’ liquidity: the higher the γ, the wider the spread and hence the less the liquidity.


## 5. Conclusion

In this paper, within the framework of conic finance, we propose a useful approach to evaluate the ask and bid prices of geometric Asian options and obtain the explicit formulas for the ask and bid prices. Finally, by using the explicit formulas of geometric Asian options, we carry out the impacts of the static market liquidity parameterγ on bid-ask prices.



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