Abstract
In this work, we used Tran Hung Thao’s approximation of fractional Brownian motion to approximate the shadow price of the fractional Black Scholes model. In the case to maximize expectation of the utility function in a portfolio optimization problem under transaction cost, the shadow price is approximated by a Markovian process and semimartingale.
1. Introduction
Let us consider a financial market without frictions and a portfolio consisting of a risky asset and a nonrisky asset such that the dynamics of the evolution of the assets is respectively given by the equations and where r, , and are real constants. Let be the proportion of the nonrisky asset in the portfolio and the proportion of the risky asset in the portfolio. For any pair , the value of the portfolio at time t is given by . A portfolio optimization consists in determining an optimal allocation of the portfolio which maximizes the expectation of the utility function under terminal wealth , i.e., find which maximizes is given by the following relation:where is the economic function which accounts for the risk aversion of an economic agent with initial wealth x. Robert Merton first dealt with this problem in the frictionless market case; in [1], optimal control methods is used to solve the (1). The utility function is assumed logarithmic and he proved that the optimal strategy consists to keep a constant portion of risky assets in the portfolio, which is also proportional to the sharp ratio . In [2], it is proved that this result remains valid when and are bounded predictable processes. The following relation holds,where and are bounded predictable processes.
In [3], Magill studies the case of the hedging with transaction cost. The hedging of the risky asset is done under transaction cost (with ) proportional to the risky asset, i.e., the investor buys the asset at price but receives the amount at the time of sale. In this case, the terminal wealth is replaced by .
To solve the problem (1), Micharl used the stochastic optimal control theory, which linked in particular the solution of partial differential equations of Hamilton Jacobi–Bellemann type in the Markovian framework, see [4, 5] for the details. An alternative approach called convex duality martingale method has been developed to take into account non-Markovian models, see [6, 7].
This method makes use of the results of convex analysis and martingales. If we consider the equation the maximization of the function as the primal of the optimization problem, the convex duality method allows to reduce the problem to the form of the problem (1). The method of convex duality is used to pass from a model with transaction cost to a model without transaction cost, in particular the existence of a new process which is a semimartingale called shadow price such that the optimal hedging strategy of the model with transaction cost coincides with the model without transaction cost.
The existence of the shadow is theoretically proved by the duality methods for portfolio optimization (see [6, 7]). Thanks to the work of Bender and Guasoni (see [8, 9]) on arbitrage, Christoph Czichowsky et al. proved in [2, 7] that the shadow price can exist for a non-semimartingale model under certain conditions.
Thus, the existence of the shadow price has been proved when the price of the risky asset in the portfolio follows a fractional Brownian motion:whereand , the process value at t = 0. We extended the result toand thanks to the work of Tran Dung and Thao [10] on the approximation of processes, it is proposed that an approximation of the shadow price which is a semimartingale process is of the form
The paper is structured as follows: in Section 1, we state some basic facts about the shadow price and its application to the case of a problem driven by and by extension to which is a generalization of the classical Black scholes model in the fractional case.
Section 2 is devoted to the recall of some results on fractional Brownian motion, and Section 3 is devoted to our main approximation results.
2. Preliminary
2.1. Existence of the Shadow Price for a Fractional Black Scholes Financial Model
Consider a financial portfolio consisting of a nonrisky asset and a risky asset defined on a filtered probability space (, , , P) having the following dynamics:where denotes fractional Brownian motion and with ; this equation is known as the fractional Black scholes equation. Portfolio optimization under transaction cost proportional to consists to find an admissible and optimal strategy which maximizes the function . The optimization problem can be presented in the following form: how to find which maximizes
Definition 1. Let be a continuous process on a filtered probability space (, , , P); the process is called shadow price for the problem (9) if:(1).(2)The solution of the problem of maximization without transaction cost of the utility expectation exists and the optimal solution of (10) coincides with the solution of the equation (9) under transaction transaction cost.
Definition 2. (Bender): let be a real-valued continuous stochastic process. For a finite stopping time ,
and .
X verifies the TWC (two-way crossing) condition of crossing if , for all finite stopping times .
The existence of the shadow price (see [7]) is related to the following conditions:(1) is continuous and satisfies for (2) is continuous and sticky for Czichowsky and Schachermayer (see [2, 11]) use duality results to prove the existence of the shadow price when with .
Guasoni in [9] shows that if with a continuous function and if small transaction costs are taken into account, then is sticky and there is no arbitrage in the portfolio. The existence results can be extended .
2.2. Stochastic Calculus for Fractional Brownian Motion and Application to the Fractional Black Scholes Model
In this section, we recall some results on fractional Brownian motion.
Definition 3. We call fractional Brownian motion a Gaussian process , almost surely with continuous trajectories such that and is independent of of normal distribution N(0, t) for all . In particular, is the standard Brownian motion.
In [12], Benoit and John show that with , H ]0; 1[ et .
If , then is neither a Markov process, nor a semimartingale with respect to (see [12, 13] for details and proofs). This process is not semimartingale, and the classical Itô lemma can not be applied, thus we make use of other types of integration theories (so-called Malliavin calculus, Wick-Itô calculus approach, and pathwise calculus), see [12, 14] for more details. In this article, we will use the Wick-Itô formulation which is closer to the Itô calculus and we will try to find some results of the classical Black Scholes model when H tends to ; we will only study the case . Benoit and John [13] show that is not differentiable. Let , for .
is infinitely differentiable (mean square) and we can give a meaning to the derivative of .Let be a process such that and a function of class ; in [15], we show thatThe differential form of the Wick-Ito lemma for geometric Brownian motion can be written asUsing Tran Dung and Thao’s approximation of fractional Brownian motion [10], we haveandwhereandWe used a method of approximating by a semimartingale to write in the form
3. Approximation Results
We make the following assumptions:
Hypothesis 1. By definition, the shadow price is a semimartingale process which takes its values in the interval ; we will suppose that the shadow price can be written as and that with .
Hypothesis 2. Let be a function of class such that , x, y,(H1)(H2)(H3)(H4)where are real positive constants.
The following lemma establishes the convergence result of the process to the process .
Lemma 1. Let and be the respective solutions of the equations and , t [0, T]. Let the wealth process and f a function defined on and satisfying the assumptions (H1), (H2), (H3), (H4).(1) is a semimartingale which converges uniformly to in when tends to 0(2)The wealth process is a semimartingale which converges uniformly to in when tends to 0
Proof of Lemma 1. Let f be a function of class , using the differential form of de , in (11) and the Wick-Ito lemma, we havewith and, so is a semimartingale.
We haveIn the relation (7), we have so .
Using the inequality , we getLet ,Using the relations (14), (15), and the Hypotheses (H1), (H2), (H3), (H4), we getwhere which is finite.
In fact, , thus is finite.
Let ,
We haveAs is finite, is also finite.
Let and , then we haveWe conclude that where D a constant. We deduce that converges in mean square to when tends to 0.
is also a semimartingale because is a semimartingale and is a finite variation process.
Similarly,Hence, the convergence of to in .
Theorem 1. Let be a continuous semimartingale defined on a probability space (, , P) and f a function of class then under certain assumptions there exist two processes and such that(1) is the unique solution of the equation and verifies the relation with and two bounded predictable processes and .(2)The processes and are Markovian processes.
Proof of Theorem 1. (1)Let f be a function of class and the process defined in the relation (5); according to the Wick-ito, we have: We approximate by , and we have and the integral is a process with absolutely continuous trajectories, and by taking in the relation (11), we can give a meaning to the derivative Using the fractional Brownian motion approximation of Tran Hung Thao, we have Therefore, Replacing by , by and by in (32), we obtain Let and We have From hypothesis (H1) and (H2), we have So, . From Hypothesis (H1) 1 and the hypothesis (H2), (H3), Therefore, From Assumption 1 and the hypothesis (H3), we have The functions f(t, 0), , are continuous and bounded functions for so . Assume that f is monotonic, i.e., and (where and ) and , Under the Hypotheses (H1), (H3), and (H4), we have We have Using Hypothesis 1, we have , and According to the relation (3), We have with ,(2)The process is a solution of (12); thus, it is Markovian process. We deduce that is also a Markovian process.
4. Conclusion
In this work, we have proposed a shadow price approximation method for the fractional Black model in the sense of Wick-Itô in the context of an optimization problem under transaction costs. We obtained (54) whose resolution gives a candidate process for the shadow price approximation. The problem is thus reduced to a frictionless optimization problem in the Markovian framework which could be solved by Hamilton–Jacobi type equations.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.