An Interval of No-Arbitrage Prices in Financial Markets with Volatility Uncertainty

Abstract

In financial markets with volatility uncertainty, we assume that their risks are caused by uncertain volatilities and their assets are effectively allocated in the risk-free asset and a risky stock, whose price process is supposed to follow a geometric -Brownian motion rather than a classical Brownian motion. The concept of arbitrage is used to deal with this complex situation and we consider stock price dynamics with no-arbitrage opportunities. For general European contingent claims, we deduce the interval of no-arbitrage price and the clear results are derived in the Markovian case.

1. Introduction

Though many choice situations show uncertainty, owing to the Ellsberg Parasox, the impacts of ambiguity aversion on economic decisions are established and Beissner [1] considered general equilibrium economies with a primitive uncertainty model that features ambiguity about continuous-time volatility. Under uncertainty, multiple priors can be used to model decisions. Recently, these multiple priors models have attracted much attention. The decision theoretical setting of multiple priors was introduced by Gilboa and Schmeidler [2] and Artzner et al. [3] adapted it to monetary risk measures. Afterwards, Maccheroni et al. [4] generalized multiple priors to preferences. In diffusion models, Girsanov’s theorem was employed to consider stochastic processes by Chen and Epstein [5], but these multiple priors can only lead to uncertainty. When these multiple priors are used in finance areas, they result in drift uncertainty for stock prices. In the risk-neutral world, when we assess financial claims, the uncertainty of this drift will disappear.

Under the assumption of no arbitrage and volatility uncertainty, Fernholz and Karatzas [6] considered to outperform the market. Compared with this, our paper is to model volatility uncertain financial markets which have no arbitrage. Epstein and Ji [7] or Vorbrink [8] used a specific example to illustrate an uncertain volatility model. On the basis of our predecessors, our paper solves a few basic problems of the volatility uncertainty in finance markets. Our aim is to analyze the volatility uncertain financial markets and we take advantage of the framework of sublinear expectation and -Brownian motion which is introduced by Peng [9] to deal with the model in financial markets. The -Brownian motion is no longer a classical Brownian motion. The construction of stochastic integration, ’s lemma, and martingale theory is utilized to the framework of -Brownian motion. In order to control the model risk, the -Brownian motion is employed to concern the model and evaluate claims by means of -expectation which is a sublinear expectation.

In our financial markets with volatility uncertainty, the wealth is invested in risk-free asset and risky asset, in which the risky asset, that is, stock and its price process , is given by the following geometric -Brownian motion:where constant interest rate is an expected instantaneous return of the stock and is the volatility of which is associated with . The canonical process is a -Brownian motion relating to a sublinear expectation , called -expectation (see [9, 10] for a detailed construction). The stochastic calculus with respect to -Brownian motion can also be established, especially integral [9]. The ordinary martingales are replaced by -martingales. Denis et al. [11] developed the -framework of Peng [10] (see [12]) in the framework of quasi-sure analysis. An upper expectation of classical expectations is used to represent the sublinear expectation established by Denis et al. [11]; that is to say, there exists a set of probability measures such that .

In this paper, we prove that the considered financial market does not admit any arbitrage opportunity, but it allows for uncertain volatility. In our analysis, the notion of -martingale which replaces the notion of martingale in classical probability theory plays a major role.

One of our aims is to solvewhere denotes the payoff of contingent claims at maturity and is a discounting. presents a series of different probability measures.

The stochastic environment can bring about a set of probability measures that are not equivalent but even mutually singular. To illustrate this, let be a Brownian motion under a measure and think about the processes and . Using and , we describe the distributions over continuous trajectories which are induced by the two processes. These measures describe two possible hypotheses of real probability measure which drives the volatility uncertainty by (1). Therefore, we havewhere both priors are mutually singular.

The definition of trading strategy and portfolio process is applied to obtain the wealth equation. Defining the concept of no-arbitrage in financial markets and the hedging classes, we gain the interval of no-arbitrage price for general European contingent claims. Finally, the connection of the lower and upper arbitrage prices is presented.

In such an ambiguous financial market, our subject is to analyze the European contingent claim concerning pricing and hedging. The asset pricing is extended to the financial markets with volatility uncertainty. The notion of no-arbitrage plays an important role in our analysis. Owing to the fact that the volatility uncertainty leads to additional source of risk, the classical definition of arbitrage will no longer be adequate. For this reason, a new arbitrage definition is presented to adjust our multiple priors model with mutually singular priors which are shown in (3). In this modified sense, we confirm that our volatility uncertain financial markets do not admit any arbitrage opportunity.

Utilizing the notion of no-arbitrage, we have obtained several results, which provide us with a better economic understanding of financial markets under volatility uncertainty. For general contingent claims, we determine an interval of no-arbitrage prices. The bounds of this interval are the upper and lower arbitrage prices and , which are obtained as the expected value of the claim’s discounted payoff with respect to -expectation (see (2)). They specify the lowest initial capital. We use the capital to hedge a short position in the claim or long position, respectively. Generally speaking, because is a sublinear expectation, we have . This verifies the market’s incompleteness. In a few words, no arbitrage will be generated when price is in the interval for a European contingent claim. In Section 4, when the contingent claim’s payoff is only determined by the current stock price, we deduce a more clear structure about the upper and lower arbitrage prices by a partial differential equation (PDE for short). We derive an explicit representation for the corresponding supper-hedging strategies and consumption plans. Given the special situation when the payoff function shows convexity (concavity), the upper arbitrage price solves the classical Black-Scholes PDE with a volatility equal to , and vice versa concerning the lower arbitrage price.

The novelties of this paper are that the volatility of in our model is a variable which is related to . This is different from works of Vorbrink [8] in which the volatility of is a constant. We employ the -framework including -expectation, -Brownian motion, and the concept of arbitrage to study the financial markets with volatility uncertainty; we gain the interval of no arbitrage, which is different from that in Denis and Martini [12].

This paper is organized as follows. Section 2 introduces the financial markets. We focus on and extend the terminology from mathematical finance. Section 3 applies a series of definitions and lemmas to derive the interval of no arbitrage. Section 4 restricts us to the Markovian case and derives results which are analogy to those in Avellaneda et al. [13] or Vorbrink [8]. Conclusions are given in Section 5.

2. The Market Model and the Mathematical Setting

2.1. -Brownian Motion and the Multiple Priors Setting

In the whole paper, the one-dimensional case is considered and we fix an interval with . This interval describes the volatility uncertainty. and denote a lower and upper bound for volatility, respectively.

Definition 1 (see [9]). Let be a given set. Let be a linear space of real valued functions defined on with for all constants , and if . ( is considered as the space of random variables.) A sublinear expectation on is a functional satisfying the following properties: for any , it has(1)Monotonicity: if then .(2)Constant preserving: .(3)Subadditivity: .(4)Positive homogeneity: .The triple is called a sublinear expectation space.

Definition 2 (see [10] (-normal distribution)). In a sublinear expectation space , a random variable is called (centralized) -normal distributed if for any where is an independent copy of . Here the letter denotes the function

Note that has no mean-uncertainty; that is, it has . Moreover, the following important identity holds: with and . We write that is distributed. Therefore, we say that -normal distribution is characterized by the parameters .

Remark 3 (see [10]). The random variable which is defined in (4) is generated by the following parabolic PDE defined in .
For any , define ; then is the unique (viscosity) solution ofEquation (7) is called a -equation.

Definition 4 (see [10] (-Brownian motion)). A process in a sublinear expectation space is called a -Brownian motion if the following properties are satisfied: (i).(ii)For each the increment is distributed and independent from for each , .Condition (ii) can be replaced by the following three conditions giving a characterization of -Brownian motion: (i)For each and as .(ii)The increment is independent from for each , .(iii).

For each , it has that is a -Brownian motion.

Let us briefly depict the construction of -expectation and its corresponding -Brownian motion. As in the previous sections, we fix a time horizon and set -the space of all real valued continuous paths starting at zero. Considering the canonical process , we define A -Brownian motion is firstly constructed in . For this purpose, let be a sequence of random variables in a sublinear expectation space such that is -normal distributed and is independent of for each integer . Then a sublinear expectation in is constructed by the following procedure: for each with for some , , set

The related conditional expectation of as above under , is defined by where [(, )]. One checks that consistently defines a sublinear expectation in and the canonical process represents a -Brownian motion.

Let and be the collection of all -valued -adapted processes on . We write and as the law of ; that is, is distribution over trajectories. Let the set of multiple priors be the closure of under the topology of weak convergence.

Theorem 5 (see [8]). For any , , it holds that Furthermore,

We use the set of priors to define the -expectation . It is given by where is any random variable. So the -expectation can be defined. Relative to the -expectation, the space of random variable is denoted by .

In this paper, we consider the tuple as and the canonical process is a -expectation motion with respect to as given in the previous. The -framework enables the analysis of stochastic processes for all priors of . The terminology of “” (q.s.) is proved to be very useful.

Unless there are special instructions, all equations should also be understood as “quasi-sure.” This means a property almost surely for all conceivable scenarios.

As mentioned in the preceding, -expectation can be defined in the space . It is the completion of , the set of bounded continuous functions on under the norm . Because the stochastic integrals are required to define trading strategies in the next sections, we briefly introduce the basic concepts about stochastic calculus and the construction of Itô integral with respect to -Brownian motion.

For , let be the collection of simple processes of the following form: for a given partition , , for any the process is defined bywhere . For each , let We denote by the completion of under the norm .

Definition 6 (see [14]). For with the presentation in (15), the integral mapping is defined by and

We consider the quadratic variation process of -Brownian motion. It has It is a continuous, increasing process, absolutely continuous with respect to . It contains all the statistical uncertainty of the -Brownian motion. For we have and it is independent of .

Definition 7 (see [9]). Let , and . Then the process is a -martingale.

Specially, the nonsymmetric part , is a -martingale, which is quite a surprising result because is continuous, nonincreasing with quadratic variation equal to zero.

Remark 8 (see [15]). is a symmetric -martingale if and only if .

Theorem 9 (see [15] (martingale representation)). Let and . Then the -martingale with , has the following unique representation: where is a continuous, increasing process with , , , and a -martingale.

If and bounded from above, and (see [14]).

A construction of the stochastic integral for the domain is established by Song [15]. Although the structure of these spaces is similar as before, the norm for completion is different and the random variables in (15) are elements of a subset of . We will also use the domain which is necessary for the martingale representation in the -framework (see Theorem 9). For , both domains coincide (see Song [15]). As a consequence, we can define the stochastic integral since is contained in . In financial fields, more trading strategies will be feasible.

2.2. The Financial Market Model

We consider the following financial market which includes a risk-free asset and a single risky asset and two assets are traded continuously over . Assume that the risk-free asset is a bond and its interest rate is . So the discount process can be defined to satisfy the following formula:where constant is the interest rate of the riskless bond as in the classical theory.

Assume that the risky asset is a stock with price at time , whose price process is given by the following equation:where denotes the canonical process which is a -Brownian motion under or , respectively, with parameters .

Since is a -Brownian motion, the volatility of is related to which is different from that of Vorbrink [8], where the volatility of stock price is a constant . Consequently, the stock price evolution involves not only risk modeled by the noise part but also ambiguity about the risk due to the unknown deviation of the process from its mean. According to financial fields, this ambiguity is called volatility uncertainty.

Compared with the classical stock price process, (22) does not contain any volatility parameter . This is due to the characteristics of the -Brownian motion . Apparently, if we choose , then we will be in the classical Black-Scholes model.

Remark 10. Take notice of the discounted stock price process which is a symmetric -martingale relative to the corresponding -expectation . As everyone knows, both the pricing and hedging of contingent claims are treated under a risk-neutral measure. This leads to a favorable situation in which the discounted stock price process is a (local) martingale [16]. In our ambiguous setting, this is also allowed. In order to model as a symmetric -martingale (see Definition 7), we do not need to change the sublinear expectation. A symmetric -martingale is required to make sure that the stock is the same for all participants, whether they sell or buy.

Definition 11 (see [17]). In the market , a trading strategy is an -adapted vector process , a member of such that , and for all .
A cumulative consumption process is a nonnegative -adapted process with values in , and with increasing, right-continuous, and , q.s.

A basic assumption in the market is that the stock price process defined by (22) is an element of . We impose the so-called self-financing condition. In other words, consumption and trading in satisfywhere denotes the value of the trading strategy at time . The meaning of (23) is that, starting with an initial amount of wealth, all changes in wealth are due to capital gains (appreciation of stocks and interest from the bond), minus the amount consumed. The means quasi-surely, which is the same as before.

For economic and mathematical considerations, it is more appropriate to introduce wealth and a portfolio process which presents the proportions of wealth invested in the risky stock.

Remark 12 (see [17]). A portfolio process represents proportions of a wealth which is invested in the stock. If we define then we have . As long as constitutes a portfolio process with corresponding wealth process , the is a trading strategy in the sense of (23).

Definition 13 (see [8]). A portfolio process is an -adapted real valued process if with values in .

Definition 14. For a given initial capital , a portfolio process , and a cumulative consumption process , consider the wealth equation with initial wealth . Or equivalently,If this equation has a unique solution , then it is called the wealth process corresponding to the triple .

In the setup of Definition 14, notice that the must hold quasi-surely. Thus, we need to impose requirements , or .

Definition 15. A portfolio/consumption process pair is called admissible for an initial capital if (i)the pair obeys the conditions of Definitions 11, 13, and 14,(ii),(iii)the solution satisfies where is a nonnegative random variable in .

We then have .

In the above Definitions 11 and 13–15, it is necessary to guarantee that the financial fields and related stochastic analysis can be well defined. In particular, condition (ii) of Definition 15 makes sure that the mathematical framework does not collapse by allowing for many portfolio processes.

3. Arbitrage and Contingent Claims

Definition 16 (see [8] (arbitrage in )). We say that there is an arbitrage opportunity in if there exist an initial wealth and an admissible pair with such that, at some time ,

Lemma 17 (no arbitrage). In the financial market , there does not exist any arbitrage opportunity.

Proof. Assume that there exists an arbitrage opportunity; that is to say, there exist and a pair with such that quasi-surely for some . Then we have . By definition of the wealth process, it has Since the -expectation of an integral with respect to -Brownian motion is zero, we have . This implies q.s. Therefore, cannot constitute an arbitrage.

In the financial market , we consider a European contingent claim and assume that its payoff at maturity time is . Here, represents a nonnegative, -adapted random variable. Regardless of any time, we impose the assumption . The price of the claim at time 0 is denoted by . For the sake of finding reasonable prices for , we need to utilize the concept of arbitrage. Considering that the financial market contains the original market and the contingent claim . Similar to the above, an arbitrage opportunity needs to be defined in the financial market .

Definition 18 (see [17] (arbitrage in )). We say that there is an arbitrage opportunity in () if there exist an initial wealth (, resp.), an admissible pair , and a constant (, resp.), such that at time 0, and at time .

The values in Definition 18 indicate short or long positions in the claims , respectively. This definition of arbitrage is standard in the literature [17]. For the same reasons as before, we again require quasi-sure dominance for the wealth at time and again with positive probability for only one possible scenario.

In the following, we show that there exist no-arbitrage prices for a claim . Under these prices, there is no-arbitrage opportunity. Because the uncertainty caused by the quadratic variation cannot be dispelled, generally speaking, there is no self-financing portfolio strategy which replicates the European claim or a risk-free hedge for the claim in our ambiguous market .

Roughly stated, since there is only one kind of situation where stocks will be traded, the measures induced by the -framework result in market’s incompleteness.

Definition 19 (see [17]). Given a European contingent claim , the upper hedging class is defined by and the lower hedging class is defined by In addition, the upper arbitrage price is defined by and the lower arbitrage price is defined by

Lemma 20 (see [17]). and implies . Analogously, and implies .

The proof uses the idea that one “just consumes immediately the difference between the two initial wealth” (see [17] for the complete proof process).

For any , we define the Black-Scholes price of a European contingent claim as follows:

Similar to the constrained circumstances [17], we prove the next three lemmas which are related to the European contingent claim .

Lemma 21. For any , it holds that belongs to the interval .

Proof. Let . From the definition of , we know that there exists a pair such that q.s. Employing the properties of -expectation as stated in Definition 1, we obtain for any that Therefore, . We know ; hence, .
Analogously, let . By definition of , there exists a pair such that q.s. For the same reason, we obtain for any that Therefore, . We know ; hence, .

Lemma 22. For any price , there exists an arbitrage opportunity. Also for any price , there exists an arbitrage opportunity.

Proof. The idea of proving this lemma comes from [8]. We only consider the case since the argument is similar. Assume , and let . By definition of and Lemma 20, we deduce that . Hence, there exists a pair with This implies the existence of arbitrage in the sense of Definition 18. If with , then and . Let ; without loss of generality, we may assume . Due to we deduce Assume and we deduce , q.s. This contradicts q.s. Hence, . Hence, constitutes an arbitrage.

Lemma 23. For any , there is no arbitrage in the financial market .

Proof. The idea of proving this lemma also comes from [8]. We prove it by contradiction. Assume , and that there exists an arbitrage opportunity in . We suppose that it satisfies Definition 18 for . The case works similarly.
By definition of arbitrage, there exists , with Therefore, . By Lemma 21, it has . This contradicts our assumption.

Theorem 24. For the financial market , the following identities hold:

Proof. Firstly, let us begin with the identity . As seen in the proof of Lemma 22, for any we have . Therefore, .
To show the opposite inequality we need to define the -martingale by By the martingale representation theorem [15] (see Theorem 9), we know there exists and continuous, increasing processes with such that for any For any , we set , , and . Then the wealth process satisfies The properties of and obey the conditions of a cumulative consumption process in the sense of Definition 11. Due to , for , the wealth process is bounded from below, where is admissible for .
As quasi-surely, we have . Due to the definition of , we conclude that .
The proof for the second identity is analogous. Again, using the proof of Lemma 22, we obtain for any . Hence, .
In order to obtain , we define a -martingale by The remaining part is almost a copy of the above. By the martingale representation theorem [15], there exist , and a continuous, increasing process with such that, for any , As the above, for any , let and . Then the wealth process satisfies where obeys the condition of a cumulative consumption process due to the properties of . Moreover, for any , it haswhich is bounded from below in the sense of item (iii) in Definition 15 because . Therefore, the wealth process is bounded from below. Consequently, is admissible for .
Since q.s., it has .
Due to the definition of , we conclude . So far, we have completed the proof of Theorem 24.

The proof of Theorem 24 is different from that of Vorbrink [8], because the volatility of stock price is related to rather than a constant.

Remark 25. Because of sublinear expectation , by Theorem 24 we have . This means that the market is not complete implying that not all claims can be hedged perfectly. Therefore, there are many no-arbitrage prices for . As long as is not a symmetric -martingale, it has . Under other circumstances, the process is identically equal to zero (see Remark 8), meaning that is symmetric and can be hedged perfectly owing to Remark 8 and Theorem 9.

Theorem 26. For any price of a European contingent claim at time zero, there does not exist any arbitrage opportunity in . For any price there exists an arbitrage in the market.

Proof. The first part directly follows from Lemma 23. From Lemma 22, we know that implies the existence of an arbitrage opportunity. Thus, we only need to show that and admit an arbitrage opportunity.
We only treat the case . Then ; that is, . The second case is similar. Comparing the proof of Theorem 24 and letting , for , there exists a pair such that Then , where is an increasing, continuous process with . So we can select such that (see Remark 25). Then the pair satisfiesThus, and we conclude that constitutes an arbitrage.

On account of Theorem 26, we call the arbitrage free interval. Particularly, in the Markovian case where for some Lipschitz function , we can give more structural details about the bounds and . We investigate this issue in Section 4.

4. The Markovian Case

For the European contingent claims , we have the form for some Lipschitz function . We use a nonlinear Feynman-Kac formula which is established in Peng [9]. Let us rewrite the dynamics of in (22) as Analogy to the lower and upper arbitrage prices at time 0, at time , the lower and upper arbitrage prices are noted by and , respectively. At a considered time , the stock price is replaced by the variable . That is, .

Theorem 27. Given a European contingent claim , its upper arbitrage price is given by , where is the unique solution of the following PDE:A precise representation for the corresponding trading strategy in the stock and the cumulative consumption process is given by

Analogously, is the lower arbitrage price , where . solves (55) but with terminal condition , .

Proof. Firstly, we consider the Backward Stochastic Differential Equation (in short BSDE): where is a given Lipschitz function. Peng [9] showed that the BSDE has a unique solution. So we can define a function by . In the light of the knowledge of the nonlinear Feynman-Kac formula [9], the function is a viscosity solution of the following PDE:We define the function According to the above definition, for , solves (58). Since the function is nondegenerate, turns into a classical -solution (see page 19 in [9]). Consequently, together with Itô’s formula (Theorem5.4 in [18]), it hasNow, consider the function For , based on Theorem 24, it has Moreover, can be used as a solution of (55). In addition, the function defined by solves (55) owing to the nonlinear Feynman-Kac formula since . By uniqueness of the solution in (55) (see [19]; is obviously bounded in ), we have . Thus, and it uniquely solves (55).
In combination with the proof of Theorem 24, by using its notions and Remark 12, we obtain the precise expressions for the trading strategy and the cumulative consumption process . That is, it has . Therefore, .
Analogously, we derive

Due to Theorem 27, the functions and can be characterized as the unique solutions of (55). Under the circumstances of being a convex or concave function, respectively, (55) simplifies greatly.

Lemma 28. (1) If is concave, then is convex for any .
(2) If is concave, then is concave for any . Similarly, if is convex, then is concave for any . If is concave, then is convex for any .

Proof. We only need to take into account the upper arbitrage price which is determined by the function First of all, let be convex, , and . For any , it has where we used the convexity of , the monotonicity of , and, in the second inequality, the sublinearity of . Therefore, is convex for all .
Secondly, let be concave. For any , we define where Since is a classical Brownian motion under , solves the Black-Scholes PDE (7) with replaced by .
Because is linear, this is straightforward to mean that is concave for any . Consequently, also solves (55). By uniqueness, we conclude that . Therefore, is concave for any .

5. Conclusion

In order to analyse the financial markets with volatility uncertainty, we consider a stock price modeled by a geometric -Brownian motion which features volatility uncertainty. This is all based on the structure of a -Brownian motion. The “-framework” is summarized by Peng [9] which gives us a useful mathematical setting. A little new arbitrage free concept is utilized to obtain the detailed results which give us an economically better understanding of financial markets under volatility uncertainty. We establish the connection of the lower and supper arbitrage prices by means of partial differential equations. The outcomes in this paper are only applied to European contingent claims. For other cases, we would extend these results to American contingent claims in our forthcoming paper.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.