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

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 G-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.


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 continuoustime 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, ô'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  ⩾ 0 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 ô 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 P 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 solve where   denotes the payoff of contingent claims at maturity  and   is a discounting.P 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   fl (  ) and   fl (  ).Using   =  ∘ (  ) −1 and   =  ∘ (  ) −1 , 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 have where 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 V up and V low , 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 V low ̸ = V up .This verifies the market's incompleteness.In a few words, no arbitrage will be generated when price is in the interval (V low , V up ) 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.

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  > 0. This interval describes the volatility uncertainty. and  denote a lower and upper bound for volatility, respectively.
Definition 1 (see [9]).Let Ω ̸ = 0 be a given set.Let H be a linear space of real valued functions defined on Ω with  ∈ H for all constants , and || ∈ H if  ∈ H. (H is considered as the space of random variables.)A sublinear expectation Ê on H is a functional Ê : H → R satisfying the following properties: for any ,  ∈ H, it has (1) Monotonicity: if  ⩾  then Ê() ⩾ Ê().
Definition 4 (see [10] (-Brownian motion)).A process (  ) ⩾0 in a sublinear expectation space (Ω, H, Ê) is called a -Brownian motion if the following properties are satisfied: Condition (ii) can be replaced by the following three conditions giving a characterization of -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  > 0 and set Ω  =  0 ([0, ], R)-the space of all real valued continuous paths starting at zero.Considering the canonical process   () fl   ,  ⩽ ,  ∈ Ω, we define A -Brownian motion is firstly constructed in   (Ω  ).For this purpose, let (  ) ∈N be a sequence of random variables in a sublinear expectation space ( Ω, H, Ẽ) such that   is normal distributed and  +1 is independent of ( 1 , . . .,   ) for each integer  ⩾ 1.Then a sublinear expectation in   (Ω  ) is constructed by the following procedure: The related conditional expectation of  ∈   (Ω  ) as above under Ω   ,  ∈ N, is defined by where One checks that   consistently defines a sublinear expectation in   (Ω  ) and the canonical process  represents a -Brownian motion.
Let Θ fl [, ] and A Θ 0, be the collection of all Θ-valued (F  )-adapted processes on [0, ].We write and   as the law of  0, = ∫ ⋅ 0     ; that is,   =  0 ∘ ( 0, ) −1 is distribution over trajectories.Let the set of multiple priors P be the closure of {  |  ∈ A Θ 0, } under the topology of weak convergence.
Theorem 5 (see [8]).For any Furthermore, We use the set of priors P 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  1  (Ω  ).In this paper, we consider the tuple as (Ω  , F, (F  ), P) and the canonical process  = (  ) is a -expectation motion with respect to P as given in the previous.The -framework enables the analysis of stochastic processes for all priors of P. 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 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.
If  = 2 and  bounded from above,  ∈  2  (0, ) and   ∈  2  (Ω  ) (see [14]).A construction of the stochastic integral for the domain    (0, ),  ⩾ 1 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    (Ω   ).We will also use the domain  1  (0, ) which is necessary for the martingale representation in the -framework (see Theorem 9).For  = 2, both domains coincide (see Song [15]).As a consequence, we can define the stochastic integral since  2  (0, ) is contained in  1  (0, ).In financial fields, more trading strategies will be feasible.

The Financial Market Model.
We consider the following financial market M which includes a risk-free asset and a single risky asset and two assets are traded continuously over [0, ].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  ⩾ 0 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 P, respectively, with parameters  >  > 0.
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 1.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.
A basic assumption in the market M is that the stock price process   defined by ( 22) is an element of  2  (0, ) =  2  (0, ).We impose the so-called self-financing condition.In other words, consumption and trading in M satisfy where   denotes the value of the trading strategy at time .
The meaning of (23) is that, starting with an initial amount  −1 0  0 + 0  0 of wealth, all changes in wealth are due to capital gains (appreciation of stocks and interest from the bond), minus the amount consumed.The q.s.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 14.For a given initial capital , a portfolio process , and a cumulative consumption process , consider the wealth equation with initial wealth  0 = .Or equivalently, If this equation has a unique solution  = (  ) fl  ,, , then it is called the wealth process corresponding to the triple (, , ).
In the setup of Definition where  is a nonnegative random variable in  2  (Ω  ).
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.
(28) Lemma 17 (no arbitrage).In the financial market M, there does not exist any arbitrage opportunity.
Proof.Assume that there exists an arbitrage opportunity; that is to say, there exist  ⩽ 0 and a pair (, ) ∈ A() with  ≡ 0 such that  ,,0  ⩾ 0 quasi-surely for some  > 0. Then we have   ( ,,0  ) ⩾ 0. By definition of the wealth process, it has Since the -expectation of an integral with respect to -Brownian motion is zero, we have   (   ,,0  ) = 0.This implies    ,,0  = 0 q.s.Therefore, (, , 0) cannot constitute an arbitrage.
In the financial market M, we consider a European contingent claim  and assume that its payoff at maturity time  is   .Here,   represents a nonnegative, F  -adapted random variable.Regardless of any time, we impose the assumption   ∈  2  (Ω  ).The price of the claim at time 0 is denoted by  0 .For the sake of finding reasonable prices for , we need to utilize the concept of arbitrage.Considering that the financial market (M, ) contains the original market M and the contingent claim .Similar to the above, an arbitrage opportunity needs to be defined in the financial market (M, ).
Definition 18 (see [17] (arbitrage in (M, ))).We say that there is an arbitrage opportunity in (M, ) if there exist an initial wealth  ⩾ 0 ( ⩽ 0, resp.), an admissible pair (, ) ∈ A(), and a constant  = −1 ( = 1, resp.), such that The values  = ±1 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 noarbitrage 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 M.
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]).
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 𝜎 ∈ [𝜎, 𝜎], it holds that
Proof.Let  ∈ U. From the definition of U, we know that there exists a pair (, ) ∈ A() such that  ,,  ⩾   q.s.Employing the properties of -expectation as stated in Definition 1, we obtain for any  ∈ [, ] that Therefore,   0 ⩽ .We know Analogously, let  ∈ L. By definition of L, there exists a pair (, ) ∈ A(−) such that  −,,  ⩾ −  q.s.For the same reason, we obtain for any  ∈ [, ] that Therefore,  ⩽   0 .We know V low fl sup{ |  ∈ L}; hence, V low ⩽   0 .
Lemma 22.For any price  0 > V  , there exists an arbitrage opportunity.Also for any price  0 < V  , there exists an arbitrage opportunity.
Proof.The idea of proving this lemma comes from [8].We only consider the case  0 < V low since the argument is similar.Assume  0 < V low ,  ⩽ 0 and let − ∈ ( 0 , V low ).By definition of V low and Lemma 20, we deduce that − ∈ L.
Hence, there exists a pair (, ) ∈ A(−) with Proof.The idea of proving this lemma also comes from [8].We prove it by contradiction.Assume  0 ̸ ⊆ U,  0 ̸ ⊆ L and that there exists an arbitrage opportunity in (M, ).We suppose that it satisfies Definition 18 for  = 1.The case  = −1 works similarly.
Theorem 24.For the financial market (M, ), the following identities hold: Proof.Firstly, let us begin with the identity V up =   (    ).As seen in the proof of Lemma 22, for any  ∈ U we have  ⩾   (    ).Therefore, V up = inf{ |  ∈ U} ⩾   (    ).
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  ∈  1  (0, ) and continuous, increasing processes  = (  ) with   ∈  1  (Ω  ) such that for any  ⩽ For any  ⩽ , we set  =   (    ) ⩾ 0,     ]  =    −1  ∈  1  (0, ), 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    ,,  =   ⩾ 0, for ∀ ⩽ , the wealth process is bounded from below, where (, ) is admissible for .
Proof.Firstly, we consider the Backward Stochastic Differential Equation (in short BSDE): where  : R × R → R is a given Lipschitz function.
Peng [9] showed that the BSDE has a unique solution.So we can define a function  : [0, ] × R + → R by (, ) fl  ,  , (, ) ∈ [0, ] × R + .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  0, û solves (58).
Analogously, we derive Due to Theorem 27, the functions (, ) = V  up () and (, ) = −V  low () can be characterized as the unique solutions of (55).Under the circumstances of Φ being a convex or concave function, respectively, (55) simplifies greatly.
where we used the convexity of Φ, the monotonicity of   , and, in the second inequality, the sublinearity of   .Therefore, (, ⋅) is convex for all  ∈ [0, ].
Since  = (  ) is a classical Brownian motion under  0 ,  solves the Black-Scholes PDE (7) with  replaced by .

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.