Valuation of bid and ask prices for European options under mixed fractional Brownian motion

1.   Introduction

The issue of market liquidity has drawn much attention among academic researchers, institutional professionals and financial regulators in various financial markets. However, there are few literatures on how to incorporate liquidity costs into option pricing. From the framework of acceptability indices proposed by ^[8], Madan and Cherny ^[19] developed the theory of conic finance which replaces the law of one price by the law of two prices, allowing for market participants sell to the market at the bid price and buy from the market at the higher ask price. The difference between the bid and ask prices is usually called the bid-ask spread which is an indication of the market liquidity.

The bid price of a cash flow $X$ is defined by its discounted distorted expectation and the ask price by minus the discounted distorted expectation of the cash flow $-X$.^* Madan and Cherny ^[19] proposed to model the liquidity of the market by a single market stress level $\gamma$ and then presented the closed-form expressions for bid and ask prices for European options. This conic option pricing model was further extended and taken to the real market data (S & P 500 option), see ^[1,9,11,20]. All of their empirical results indicated that the market-implied liquidity was far from being constant. That is to say, there is an implied liquidity risk premium. It is noteworthy that these papers are all studied on the premise of allowing explicit forms for the distribution function of the underlying asset price, see for example Black-Scholes and Variance Gamma models, such that the bid and ask prices can be further numerically calculated. Recently, Guillaume et al. ^[12] and Sonono and Mashele ^[26] derived the explicit formulas for the bid and ask prices of OTC interest rate options and European vanilla options, respectively, by using WANG-transform, which is a distortion function induced by a distribution function, in a Black-Scholes setting. Junike et al. ^[13] studied the convergence of bid and ask prices for various European and American possible path dependent options in a binomial model, where bid and ask prices are defined recursively using nonlinear expectations, which is closely related to discrete time conic finance models, see ^[15,17] for details. Based on the combination of Fourier cosine approximations and numerical integration, Li et al. ^[16] exhibited an efficient and fast numerical method to calculate the bid and ask prices for the European options.

^* One can refer to Section 2.2 of this paper for more related information on distorted expectation.

However, the long-range dependence of the underlying asset returns was not considered in the above mentioned researches. It is well known that the existence of long-range dependence in asset returns has been an interesting subject for both academics and market professionals for a long time. In addition, empirical evidence so far suggests that long-range dependence may be a characteristic of both exchange rates and stock markets, see ^{[3,14,25,27,28]}. In this case, it is possible to choose a fractional Brownian motion (hereafter fBm) to describe the dynamic of the financial asset price. Rogers ^[23] demonstrated that while a fBm could capture the long-range dependence between returns on different days, it also allowed arbitrage opportunities. To overcome this problem and to take into account the long memory property, El-Nouty ^[10] and Mishura ^[18] had shown that it was reasonable to use a mixed fractional Brownian motion(hereafter mfBm) to capture the fluctuations of the financial assets from time to time. Cheridito ^[6] had demonstrated that the mfBm was equivalent to a Brownian motion which means that no arbitrage was allowed. Whereafter, Xiao et al. ^[29] studied the problem of equity warrants pricing under a mfBm environment and employed a hybrid intelligent algorithm to solve the valuation of equity warrants. Sun ^[24] investigated pricing currency options when the driving force is a mixed fractional Brownian motion. Ballestra et al. ^[4] presented an integral representation for the pricing of the barrier options on an underlying asset driven by a mfBm. Both Prakasa Rao ^[22] and Zhang et al. ^[31] assumed that the price of the underlying stock followed a mfBm and derived the analytical pricing formula for the geometric Asian option. Recently, Zhang et al. ^[32] proposed a fuzzy mixed fractional Brownian motion model with jumps, which was to capture the features of both long memory and jump behaviour in financial assets under non-random uncertainty environment.

Motivated by the above mentioned insights, the main objective of this paper is to discuss the valuation of the bid and ask prices for European options under the mixed fractional Brownian motion, which is able to capture empirically observed patterns (the long range dependence of the underlying asset returns in real markets). Within the framework of conic finance, we then derive the explicit formulas for bid and ask prices of European call and put options by using WANG-transform as a distortion function. Finally, we perform numerical experiment for illustrating the effects of the Hurst index and market liquidity level on bid and ask prices.

The remainder of the paper proceeds as follows. In Section 2, we briefly introduce some basic concepts and properties of mixed fractional Brownian motion, distortion function and distorted expectation. Section 3 lays out the mixed fractional Brownian motion model for the underlying asset price. In Section 4, we present the analytical formulas for the bid and ask prices of European options within the framework of conic finance. Numerical experiment is performed in Section 5. Finally, some conclusions are stated in Section 6.

2.   Preliminaries

In this section, for better understanding the rest of this paper, we briefly review some basic concepts and the properties of mixed fractional Brownian motion and distorted expectation.

2.1.   Mixed fractional Brownian motion

Definition 1. A mixed fractional Brownian motion $M_{t}^{H}(\alpha, \beta)$ is a linear combination of Brownian motion and fractional Brownian motion(fBm), defined on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_{t}, t\geq0\}, \mathbb{P})$ by:

where $\alpha$ and $\beta$ are two real constants such that $(\alpha, \beta)\neq(0, 0)$, $\mathbb{P}$ is the physical probability measure, $B_{t}$ is a standard Brownian motion, $B_{t}^{H}$ is an independent standard fractional Brownian motion with the Hurst exponent $H \in (0, 1)$ and $\{\mathcal{F}_{t}\}_{t\geq0}$ denotes the $\mathbb{P}$-augmentation of the filtration generated by $(B_{\tau}, B_{\tau}^{H})$ for $\tau \leq t$.

In what follows, some properties of the mfBm are given in the form of proposition described below. For more detailed information about the properties of the mfBm, one can refer to ^[29,30].

Proposition 1. The mfBm $M_{t}^{H}(\alpha, \beta)$ for $t \in \mathbb{R^{+}}$ satisfies the following properties:

(i): $M_{t}^{H}(\alpha, \beta)$ is a centered Gaussian process and not a Markovian one for all $H\in (0, 1)\backslash \frac{1}{2}$;

(ii): $M_{0}^{H} = 0$ $\mathbb{P}$–almost surely;

(iii): the covariation function of $M_{t}^{H}(\alpha, \beta)$ and $M_{s}^{H}(\alpha, \beta)$ for any $t, s \in \mathbb{R}^{+}$ is given by

(iv): the increments of $M_{t}^{H}$ are stationary and mixed-self-similar for any $h > 0$

where $\triangleq$ denotes "to have the same law";

(v): the increments of $M_{t}^{H}$ are positively correlated if $\frac{1}{2} < H < 1$, uncorrelated if $H = \frac{1}{2}$ negatively correlated if $0 < H < \frac{1}{2}$;

(vi): the increments of $M_{t}^{H}$ are long-range dependent if and only if $H > \frac{1}{2}$.

2.2.   Distorted expectation

We fix a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and denote by $L^{\infty}(\Omega, \mathcal{F}, \mathbb{P})$ the space of all essentially bounded and $\mathbb{R}$-valued random variables on $(\Omega, \mathcal{F}, \mathbb{P})$.

Definition 2. (Distortion function). A function $\Psi :[0, 1]\rightarrow [0, 1]$ is a distortion function if and only if it is monotone, $\Psi(0) = 0$ and $\Psi(1) = 1$. The set function $\Psi\circ\mathbb{P}$ defined by

is called the distortion of the probability measure $\mathbb{P}$ with respect to the distortion function $\Psi$, i.e. the distorted probability measure.

With a probability distortion function $\Psi$ is associated the complementary distortion function $\hat{\Psi}$ given by

Given the probability distortion function $\Psi$, the Choquet integral $\mathbb{E}_{\Psi}[X]$ of $X\in L^{\infty}(\Omega, \mathcal{F}, \mathbb{P})$ can be defined by

Definition 3. (Distorted expectation). If we denote the distribution of random $X$ by $F_{X}$, then the Choquet integral $\mathbb{E}_{\Psi}[X]$ can be rewritten as

Here, $\mathbb{E}_{\Psi}[X]$ is generally referred to as the distorted expectation of a random $X$ with distribution function $F_{X}$ relative to the distortion function $\Psi$. Note that if $\Psi(u) = u$, and thereby $\mathbb{E}_{\Psi}[X]$ is the ordinary expectation.

In general, the distorted probability $\Psi\circ\mathbb{P}$ is no longer a probability measure. Nevertheless, it is still a finite monotone set function that is submodular, when the distortion function $\Psi$ is concave. Thus, it is possible to define a risk measure based on distorted probability using Choquet integral. Let $\Psi$ be a concave distortion function and a risk $X$. The function $\varrho^{\Psi}:L^{\infty}\rightarrow \mathbb{R}$ given by

is called a distortion risk measure induced by $\Psi$. By the properties of the Choquet integral, we can see that the function $\varrho^{\Psi}$ is a coherent risk measure.

3.   Models settings

Owing to a financial system is a complex system with great flexibility, investors do not make their decisions immediately after receiving the financial information, but rather wait until the information reaches to its threshold limit value. This behavior can lead to the features of "asymmetric leptokurtic" and "long memory". As mentioned above, the mixed fractional Brownian motion may be a useful tool to capture this phenomenon. Thus, in this section, we introduce a mixed fractional Brownian motion model for describing the dynamic of the underlying asset prices, which is a useful tool to capture the long memory of asset prices in real markets.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a complete probability space with information filtration $(\mathcal{F}_{t})_{t\geq0}$ satisfying the usual conditions, which is defined in Section 2.1. Suppose the underlying asset price $S_{t}$ takes the following stochastic differential equation:

where the drift parameters $\mu$ and the volatility $\sigma$ denote the expected return rate on the value of the underlying asset, the standard deviation of the return rate on the value of the underlying asset, respectively; $\bar{B}_{t}$ is a standard Brownian motion and $\bar{B}_{t}^{H}$ is an independent standard fractional Brownian motion with Hurst index $H > 3/4$. In fact, the Hurst exponent $H > 3/4$ ensures that the financial market does not allow arbitrage opportunity.

In addition, we also postulate that there are no transaction costs or taxes in purchasing or selling the financial assets, which means the financial market is frictionless. Thus, the market is complete and the risk-neutral martingale measure is unique under the aforementioned conditions, see ^[5,6,7] for details.

By using the fractional Girsanov theorem and the change of variables $B_{t}+B_{t}^{H} = \frac{\mu-r}{\sigma}t+\bar{B}_{t}+\bar{B}_{t}^{H}$, thus under the risk-neutral measure $\mathbb{Q}$ we have

where $r$ denotes the risk-free interest rate, $B_{t}$ is a standard Brownian motion and $B_{t}^{H}$ is an independent standard fractional Brownian motion under the risk-neutral measure $\mathbb{Q}$.

It follows from the mixed fractional pattern of Itô Lemma that the solution of Eq (3.2) can be expressed as

Obviously, the underlying asset price $S_{t}$ is log-normally distributed with

where $N(\tilde{\mu}, \tilde{\sigma}^{2})$ represents the Gaussian distribution with mean $\tilde{\mu}$ and variance $\tilde{\sigma}^{2}$.

4.   Bid and ask prices for European options

As mentioned above, the conic finance theory originates from the framework of acceptability indices developed by ^[8], where risk measures are defined in terms of distorted expectation of zero cash-flows $X$. Based on the framework of indices of acceptability, we say that a risk $X$ is acceptable or marketed if

where $\mathcal{M}$ is a convex set. This convex set $\mathcal{M}$ consists of test measures under which the expected cash-flow needs to have positive expectation in order for $X$ to be acceptable. Under a larger set $\mathcal{M}$, one has a smaller set of acceptable risks since there are more tests to be passed.

On the basis of this framework, cones of acceptability depend solely on the distribution function $F_{X}(x)$ of $X$ and on the parametric family of distortion functions $\{\Psi_{\gamma}, \gamma\geq0\}$, i.e., $X$ is acceptable if its distorted expectation relative to some distortion function $\Psi_{\gamma}$ is positive:

where the acceptability index $\gamma$ quantifies the degree of the distortion. The higher the $\gamma$, the higher the degree of distortion, i.e., the above obtained risk-adjusted distribution functions allocate more weight to the downside (losses) than the original distribution function.

For a given distortion function $\Psi_{\gamma}\in\{\Psi_{\gamma}, \gamma\geq0\}$ under which cash-flows are evaluated, and we can also interpret $\gamma$ as the market liquidity level. The higher the $\gamma$ (i.e., the more illiquid the market is), the higher the distorted probability measure. A liquidity level of zero (i.e., $\gamma = 0$) implies that there is no distortion at all, which corresponds to the perfect liquidity and hence to the complete market. In this case, the law of one price holds. As we know, the most prominent example of a family of distortion function induced by a distribution function is the WANG-transform, i.e.,

where $\Phi(\cdot)$ denotes the standard normal cumulate distribution function. In the following, we will focus on the valuation of the bid-ask prices for European options under the WANG-transform.^†

^† Moreover, there are many possible distortion functions, such as MINVAR, MAXVAR, MAXMINVAR and MINMAXVAR. For more related information one can refer to ^[8,21]. Unfortunately, there are basically no analytical expressionss for bid and ask prices in these distortion functions. Even so, Li et al. ^[16] presented a numerical method consisting of Fourier cosine approximations and numerical integration to calculate the bid and ask prices for European options as long as the characteristic function was known.

In conic finance theory, Madan and Cherny^[19] assumed that the market is taken as a counterparty willing to accept all stochastic cash-flows $X$ with an acceptability level $\gamma$. The ask price of a claim, $a_{\gamma}(X)$, is defined by the smallest price for which the cash-flow of selling the claim is acceptable at level $\gamma$ for the market. Similarly, the bid price of a claim, $a_{\gamma}(X)$, is defined by the highest price for which the cash-flow of buying the claim is acceptable at level $\gamma$ for the market. Let $X$ be the cash-flow generated by the claim at the future maturity data $T$. Then, the bid and ask prices of payoff $X$ are, respectively, determined by

and

We next consider a European call option $C = (S_{T}-K)^{+}$ and a put option $P = (K-S_{T})^{+}$ with strike price $K$ and maturity $T$ on the underlying $S_{T}$. Based on the above general expressions for bid and ask prices and the mixed fractional underlying asset price model (3.2), we can derive the analytical formulas of bid and ask prices for European can and put options by using WANG-transform. The results are given in the form of theorems stated below.

Theorem 1. If the underlying asset price $S_{t}$ satisfies the mfBm model (3.2) and the distortion function $\Psi_{\gamma}(u)$ is the WANG-transform, then the bid and ask prices of the European call option with strike price $K$ and maturity $T$ are, respectively, given by

and

where

and $\Phi(\cdot)$ denotes the standard normal cumulative distribution function.

Proof. By using the distorted expectation in Eq (4.4), we can derive the bid price of the European call option as follows:

It is easy to see that the distribution function of $S_{T}$ is

If we apply the WANG-transform to the distribution function $F_{S_{T}}(x)$, then the distorted distribution function $\Psi_{\gamma}\left(F_{S_{T}}(x)\right)$ has the following representation:

Thus, we can calculate the integrals $A_{1}$ and $B_{1}$ in Eq (4.8) as follows:^‡

^‡In the calculation of integral $A_{1}$, we have made the following transformation variables:

and

Substituting Eqs (4.11) and (4.12) to (4.8), we can obtain the bid price (4.6).

Applying similar arguments, we have the following expression for the ask price of the European call option:

Note from Eq (4.10) and WANG-transform that the distorted distribution function $\Psi_{\gamma}\left(1-F_{S_{T}}(x)\right)$ has the following representation:

Therefore, we also can calculate the integrals $A_{2}$ and $B_{2}$ in Eq (4.13) by

and

Finally, by combining Eqs (4.13), (4.15) and (4.16), we can get the ask price (4.7).

Theorem 2. If the underlying asset price $S_{t}$ satisfies the mfBm model (3.2) and the distortion function $\Psi_{\gamma}(u)$ is the WANG-transform, then the bid and ask prices of the European put option with strike price $K$ and maturity $T$ are, respectively, given by

and

where

and $\Phi(\cdot)$ denotes the standard normal cumulative distribution function.

Proof. Applying the arguments given in Theorem 1, we have the following expression for the bid price of the European put option:

Note from Eq (4.14) that the integrals $A_{3}$ and $B_{3}$ in Eq (28) can be, respectively, calculated by:

and

Substituting Eqs (4.20) and (4.21) to (4.19), we can obtain the bid price (4.17).

For the ask price of the European put option, we have

In a similar way, the integrals $A_{4}$ and $B_{4}$ in Eq (4.22) can be calculated by:

and

Finally, by combining Eqs (4.22)-(4.24), we can get the ask price (4.18).

Remark: Note that the resulting bid and ask prices depend on the parameter $\gamma$. In the special case, where $\gamma = 0$, bid price equals ask price and we are again in the classical one-price framework. In other words, a value of $\gamma$ equal to zero corresponds to a bid-ask spread of zero.

5.   Numerical analysis

In this section, we will present the numerical results for the conic option pricing. The model input parameters values for this numerical analysis are assumed to be $S_{0} = 100$, $T = 1.5$, $r = 0.05$, $\sigma = 0.2$.

Since that the main objective of this paper is to discuss the pricing problem of European option in a two-price economy, we further investigate the bid and ask prices of European call and put options with respect to Hurst index $H$ and market liquidity level $\gamma$. By utilizing the analytical formulas (4.6), (4.7), (4.17) and (4.18), Tables 1 and 2 reports the bid and ask prices of European call and put options with respect to different values of strike price $K$, Hurst index $H$ and market liquidity level $\gamma$.

From Tables 1 and 2, we can be clearly observed that the bid and ask prices of European call and put option are decreasing and increasing with respect to strike price $K$, respectively, which are consistent with the payment functions of the European options. We also can see that the bid-ask spread is increasing with market liquidity level $\gamma$. As mentioned above, the market liquidity goes hand in hand with bid and ask spreads and highly liquid asset have a small spreads. In other words, the higher $\gamma$, the wider the bid-ask spread and hence less the liquidity. We can also find that the bid-ask spread is decreasing with respect to strike price $K$ in theory, which is consistent with the empirical results of Leippold and Scharer ^[15].

In addition, it is worth noting that the bid-ask spread is increasing with respect to Hurst index $H$, which means that the stronger the long memory of underlying asset is, the higher bid-ask spread is. In spite of this, our numerical results presented here can at least illustrate that the valuation of bid and ask prices for European options considering the long-range dependence of underlying asset price may offer as a good competitor of the classical Black-Scholes ^[2] model, especially for some emerging markets.

6.   Conclusions

In a two-price economy, we study the valuation of the bid and ask prices for the European options in this paper. Considering the long range dependence of the underlying asset returns in real markets, we assume the dynamic of the underlying asset price follows a mixed fractional Brownian motion with Hurst index $H > 3/4$. In fact, the Hurst exponent $H > 3/4$ ensures that the financial market does not allow arbitrage opportunity. Within the framework of conic finance, we then derive the closed-form solutions of the bid and ask prices for European call and put options by using WANG-transform as a distortion function. Moreover, numerical experiment is performed to illustrate the effects of the Hurst index and market liquidity level on bid and ask prices.

Acknowledgments

The authors thank the referees for their comments and detailed suggestions. These have significantly improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 71901124) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20190695).

Conflict of interest

The authors declare no conflict of interest in this paper.