# Shadow Price Approximation for the Fractional Black Scholes Model

Authors: Dolemweogo Sibiri Narcisse; Béré Frédéric; Nitiéma Pierre Clovis
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## 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.


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## Body

## 1. Introduction

Let us consider a financial market without frictions and a portfolio consisting of a risky assetSt and a nonrisky asset St0 such that the dynamics of the evolution of the assets is respectively given by the equations dSt=μStdt+σStdWt and dSt0=rSt0dt where r, μ, and σ are real constants. Let ϕt0 be the proportion of the nonrisky asset in the portfolio and ϕt1 the proportion of the risky asset in the portfolio. For any pair Θt=ϕt0,ϕt10≤t≤TT∈ℝ, the value of the portfolio at time t is given by VΘt=ϕt0St0+ϕt1St. A portfolio optimization consists in determining an optimal allocation Θt=ϕt0,ϕt10≤t≤T of the portfolio which maximizes the expectation of the utility function under terminal wealth Xx,T=x+∫0Tϕt1dSt, i.e., find Θt=ϕt0,ϕt1 which maximizes(1)EUXx,t=EUx+∫0Tϕt1dSt.ϕt0 is given by the following relation:(2)ϕt0=x+∫0tϕt1dSt−ϕt1St,where Ux 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 μ/σ2. In [2], it is proved that this result remains valid when μt and σt are bounded predictable processes. The following relation holds,(3)ϕt1Stϕt0+ϕt1St=μtσt2,where μt and σt 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 λ (with λ∈0;1) proportional to the risky asset, i.e., the investor buys the asset at price St but receives the amount 1−λSt at the time of sale. In this case, the terminal wealth is replaced by Xx,Tλ=x+∫0Tϕu1dSu−λ∫0TSudϕ1u.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 functionEUx+∫0Tϕu1dSu−λ∫0TSudϕ1u 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 St^ 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 assetSt1 in the portfolio follows a fractional Brownian motion:(4)St1=S0expμt+σBtH,where(5)BtH=C∫−∞0t−sH−1/2−−sH−1/2dWs+∫0tt−sH−1/2dWs,and S0>0, the process value at t = 0. We extended the result to(6)St2=S0expμt+σBtH−12σ2t2H,and 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(7)dS˜t=μ˜S˜tdt+σ˜S˜tdWt.The paper is structured as follows: in Section1, we state some basic facts about the shadow price and its application to the case of a problem driven by St1=S0expμt+σBtH and by extension to St2=S0expμt+σBtH−1/2σ2t2H which is a generalization of the classical Black scholes model in the fractional case.Section2 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 assetBt=ert and a risky asset St defined on a filtered probability space (Ω, ℱ, ℱt, P) having the following dynamics:(8)dSt=μStdt+σStdBtH,where BtH denotes fractional Brownian motion and t∈0;T with T<∞; this equation is known as the fractional Black scholes equation. Portfolio optimization under transaction cost λ proportional to St consists to find an admissible and optimal strategy which maximizes the function EUXx,TΘt. The optimization problem can be presented in the following form: how to find Θt=ϕt0,ϕt1 which maximizes(9)EUXx,TΘt=EUx+∫0Tϕu1dSu−λ∫0TSudϕ1u.Definition 1.
LetSt be a continuous process on a filtered probability space (Ω, ℱ, ℱt, P); the process St^ is called shadow price for the problem (9) if:(1)
St^∈1−λSt;St.(2)
The solutionΨt of the problem of maximization without transaction cost of the utility expectation(10)EUVTΨ=EUx+∫0TΨsdSs^,exists and the optimal solutionΨt^ of (10) coincides with the solution of the equation (9) Θ^ under transaction transaction cost.Definition 2.
(Bender): letX=Xt0≤t≤T be a real-valued continuous stochastic process. For a finite stopping time τ,
τ+:=inft>τ:Xt−Xτ>0 and τ−:=inft>τ:Xt−Xτ<0.
X verifies the TWC (two-way crossing) condition of crossing if τ+=τ−, for all finite stopping times τ.
The existence of the shadow price (see [7]) is related to the following conditions:(1)
St is continuous and satisfies TWC for U:0;+∞⟶ℝ(2)
St is continuous and sticky for U:ℝ⟶ℝ
Czichowsky and Schachermayer (see [2, 11]) use duality results to prove the existence of the shadow price when St1=S0expμt+σBtH with Ux=lnx.
Guasoni in [9] shows that if Xt=X0expft+σBtH with ft:ℝ+⟶ℝ a continuous function and if small transaction costs λSt,λ∈0,1 are taken into account, then Xt is sticky and there is no arbitrage in the portfolio. The existence results can be extended St2=St1exp−1/2σ2t2H.


### 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 processBtH, t∈ℝ+ almost surely with continuous trajectories such that B0=0 and Bt+sH−BtH is independent of σBsH,s≤t,s∈T of normal distribution N(0, t) for all t≥0. In particular, Bt1/2 is the standard Brownian motion.
In [12], Benoit and John show that BtH=C∫ℝt−s+H−1/2−−s+H−1/2dWs with C=1/2H+∫0∞1+sH−1/2−sH−1/22ds−1/2, H ∈ ]0; 1[ et x+=maxx;0.
IfH≠0, then BtH is neither a Markov process, nor a semimartingale with respect to ℱt (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 1/2; we will only study the case H∈1/2;1. Benoit and John [13] show that BtH is not differentiable. Let B˜tH=C∫−∞tt+τ−sH−1/2−−s+H−1/2dWs, for τ>0.
B˜tH is infinitely differentiable (mean square) and we can give a meaning to the derivative of B˜tH.(11)d˜BtHdτ=d˜BtHt+τ,wdτ=CH−12∫−∞tt+τ−sH−3/2dWs.
LetSt be a process such that dSt=μStdt+σStdBtH and ft,x a function of class C1,2; in [15], we show that(12)fT,ST=ft,St+∫tT∂fu,Su∂udu+∫tTμSu∂fu,Su∂xdu+σ∫tTSu∂fu,Su∂xdBuH+Hσ2∫tTu2H−1Su2∂2fu,Su∂x2du.
The differential form of the Wick-Ito lemma for geometric Brownian motion can be written as(13)dft,St=∂ft,St∂t+μSt∂ft,St∂St+Hσ2t2H−1St2∂2ft,St∂St2dt+σSt∂ft,St∂StdBtH.
Using Tran Dung and Thao’s approximation of fractional Brownian motion [10], we have(14)ESt,ζ−St2≤Mζ2α,α=H−12,M>0,ζ∈0,1,and(15)EBt,ζH−BtH2≤Tζ2α,where(16)Bt,ζH=C∫−∞0t−sH−1/2−−sH−1/2dWs+∫0tt−s+ζH−1/2dWs,and(17)St,ζ=S0expμt+σBt,ζH−12σ2t2H.
We used a method of approximatingBtH by a semimartingale Bt,ζH to write dft,St,ζ in the form(18)dft,St,ζ=gt,ft,St,ζdt+ht,ft,St,ζdWt.


## 2.1. Existence of the Shadow Price for a Fractional Black Scholes Financial Model

Consider a financial portfolio consisting of a nonrisky assetBt=ert and a risky asset St defined on a filtered probability space (Ω, ℱ, ℱt, P) having the following dynamics:(8)dSt=μStdt+σStdBtH,where BtH denotes fractional Brownian motion and t∈0;T with T<∞; this equation is known as the fractional Black scholes equation. Portfolio optimization under transaction cost λ proportional to St consists to find an admissible and optimal strategy which maximizes the function EUXx,TΘt. The optimization problem can be presented in the following form: how to find Θt=ϕt0,ϕt1 which maximizes(9)EUXx,TΘt=EUx+∫0Tϕu1dSu−λ∫0TSudϕ1u.Definition 1.
LetSt be a continuous process on a filtered probability space (Ω, ℱ, ℱt, P); the process St^ is called shadow price for the problem (9) if:(1)
St^∈1−λSt;St.(2)
The solutionΨt of the problem of maximization without transaction cost of the utility expectation(10)EUVTΨ=EUx+∫0TΨsdSs^,exists and the optimal solutionΨt^ of (10) coincides with the solution of the equation (9) Θ^ under transaction transaction cost.Definition 2.
(Bender): letX=Xt0≤t≤T be a real-valued continuous stochastic process. For a finite stopping time τ,
τ+:=inft>τ:Xt−Xτ>0 and τ−:=inft>τ:Xt−Xτ<0.
X verifies the TWC (two-way crossing) condition of crossing if τ+=τ−, for all finite stopping times τ.
The existence of the shadow price (see [7]) is related to the following conditions:(1)
St is continuous and satisfies TWC for U:0;+∞⟶ℝ(2)
St is continuous and sticky for U:ℝ⟶ℝ
Czichowsky and Schachermayer (see [2, 11]) use duality results to prove the existence of the shadow price when St1=S0expμt+σBtH with Ux=lnx.
Guasoni in [9] shows that if Xt=X0expft+σBtH with ft:ℝ+⟶ℝ a continuous function and if small transaction costs λSt,λ∈0,1 are taken into account, then Xt is sticky and there is no arbitrage in the portfolio. The existence results can be extended St2=St1exp−1/2σ2t2H.


## 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 processBtH, t∈ℝ+ almost surely with continuous trajectories such that B0=0 and Bt+sH−BtH is independent of σBsH,s≤t,s∈T of normal distribution N(0, t) for all t≥0. In particular, Bt1/2 is the standard Brownian motion.
In [12], Benoit and John show that BtH=C∫ℝt−s+H−1/2−−s+H−1/2dWs with C=1/2H+∫0∞1+sH−1/2−sH−1/22ds−1/2, H ∈ ]0; 1[ et x+=maxx;0.
IfH≠0, then BtH is neither a Markov process, nor a semimartingale with respect to ℱt (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 1/2; we will only study the case H∈1/2;1. Benoit and John [13] show that BtH is not differentiable. Let B˜tH=C∫−∞tt+τ−sH−1/2−−s+H−1/2dWs, for τ>0.
B˜tH is infinitely differentiable (mean square) and we can give a meaning to the derivative of B˜tH.(11)d˜BtHdτ=d˜BtHt+τ,wdτ=CH−12∫−∞tt+τ−sH−3/2dWs.
LetSt be a process such that dSt=μStdt+σStdBtH and ft,x a function of class C1,2; in [15], we show that(12)fT,ST=ft,St+∫tT∂fu,Su∂udu+∫tTμSu∂fu,Su∂xdu+σ∫tTSu∂fu,Su∂xdBuH+Hσ2∫tTu2H−1Su2∂2fu,Su∂x2du.
The differential form of the Wick-Ito lemma for geometric Brownian motion can be written as(13)dft,St=∂ft,St∂t+μSt∂ft,St∂St+Hσ2t2H−1St2∂2ft,St∂St2dt+σSt∂ft,St∂StdBtH.
Using Tran Dung and Thao’s approximation of fractional Brownian motion [10], we have(14)ESt,ζ−St2≤Mζ2α,α=H−12,M>0,ζ∈0,1,and(15)EBt,ζH−BtH2≤Tζ2α,where(16)Bt,ζH=C∫−∞0t−sH−1/2−−sH−1/2dWs+∫0tt−s+ζH−1/2dWs,and(17)St,ζ=S0expμt+σBt,ζH−12σ2t2H.
We used a method of approximatingBtH by a semimartingale Bt,ζH to write dft,St,ζ in the form(18)dft,St,ζ=gt,ft,St,ζdt+ht,ft,St,ζdWt.


## 3. Approximation Results

We make the following assumptions:Hypothesis 1.
By definition, the shadow priceSt^ is a semimartingale process which takes its values in the interval 1−λSt;St; we will suppose that the shadow price St^ can be written as St^=ft,St and that S^t,ζ∈1−λSt,ζ;St,ζ with ζ∈0;1.Hypothesis 2.
Letf:0;T×ℝ+⟶ℝ+ be a function of class C1,3 such that ∀t∈0;T, x, y∈ℝ+∗,(H1)
∂ft,x/∂t−∂ft,y/∂t≤M1x−y(H2)
∂ft,x/∂x≤M2(H3)
∂2ft,x/∂x2≤M3(H4)
∂3ft,x/∂x3≤M4
whereM1,M2,M3,M4 are real positive constants.
The following lemma establishes the convergence result of the processft,St,ζ to the process ft,St.Lemma 1.
LetSt,ζ and St be the respective solutions of the equations dSt,ζ=μSt,ζdt+σSt,ζdBt,ζH and dSt=μStdt+σStdBtH, t∈ [0, T]. Let Vt,ft,St,ζ=ϕt0St0+ϕt1ft,St,ζ the wealth process and f a function defined on 0;T×ℝ+ and satisfying the assumptions (H1), (H2), (H3), (H4).(1)
ft,St,ζ is a semimartingale which converges uniformly to ft,St in L2Ω when ζ tends to 0(2)
The wealth processVt,ft,St,ζ is a semimartingale which converges uniformly to Vt,ft,St in L2Ω when ζ tends to 0Proof of Lemma 1.
Letf be a function of class C1,3, using the differential form of de Bs,ζH, in (11) and the Wick-Ito lemma, we have(19)dBt,ζH=CH−12∫−∞tt−s+ζH−3/2dWsdt+CζH−1/2dWt=Atdt+CζH−1/2dWt,with At=CH−1/2∫−∞tt−s+ζH−3/2dWs and(20)ft,St,ζ=f0,S0,ζ+∫0t∂f∂ss,Ss,ζds+∫0tμSs,ζ∂f∂xs,Ss,ζds+σ∫0tμSs,ζ∂f∂xs,Ss,ζdBs,ζH+Hσ2∫0ts2H−1Ss,ζ2∂2f∂x2s,Ss,ζds=f0,S0,ζ+∫0t∂f∂ss,Ss,ζ+μSs,ζ∂f∂xs,Ss,ζ+Hσ2s2H−1Ss,ζ2∂2f∂x2s,Ss,ζ+σμAtSs,ζ∂f∂xs,Ss,ζdt,+CζH−1/2σμSt,ζ∂f/∂xt,St,ζdWt, so ft,St,ζ is a semimartingale.
We have(21)ft,St,ζ−ft,St=f0,S0,ζ−f0,S0+∫0t∂f∂ss,Ss,ζ−∂f∂ss,Ssds+∫0tμSs,ζ∂f∂xs,Ss,ζ−μSs∂f∂xs,Ssds+Hσ2∫0ts2H−1Ss,ζ2∂2fs,Ss,ζ∂x2−s2H−1Ss2∂2fs,Ss∂x2ds+σ∫0tSs,ζ∂f∂xs,Ss,ζdBs,ζH−Ss∂f∂xs,SsdBsH.
In the relation (7), we have S0,ζ=S0expσB0,ζH=S0 so fS0,ζ=fS0.
Using the inequalitya+b+c+d2≤4a2+b2+c2+d2, we get(22)Eft,St,ζ−ft,St2≤4∫0tE∂f∂ss,Ss,ζ−∂f∂ss,Ss2ds+4∫0tEμSs,ζ∂f∂xs,Ss,ζ−μSs∂f∂xs,Ss2ds+4H2σ4∫0tEs2H−1Ss,ζ2∂2fs,Ss,ζ∂x2−s2H−1Ss2∂2fs,Ss∂x22ds+4σ2∫0tESs,ζ∂f∂xs,Ss,ζdBs,ζH−Ss∂f∂xs,SsdBsH2.
LetC1=∫0tE∂f/∂ss,Ss,ζ−∂f/∂ss,Ss2ds,(23)C2=∫0tEμSs,ζ∂f∂xs,Ss,ζ−μSs∂f∂xs,Ss2ds,C3=H2σ4∫0tEs2H−1Ss,ζ2∂2fs,Ss,ζ∂x2−s2H−1Ss2∂2fs,Ss∂x22ds,C4=σ2∫0tESs,ζ∂f∂xs,Ss,ζdBs,ζH−Ss∂f∂xs,SsdBsH2.
Using the relations (14), (15), and the Hypotheses (H1), (H2), (H3), (H4), we get(24)C1=∫0tE∂f∂ss,Ss,ζ−∂f∂ss,Ss2ds≤M12∫0tESs,ζ−Ss2ds≤M12∫0tMζ2αds≤tM12Mζ2α,C2=∫0tEμSs,ζ∂f∂xs,Ss,ζ−μSs∂f∂xs,Ss2ds=μ2∫0tESs,ζ∂f∂xs,Ss,ζ−Ss∂f∂xs,Ss2ds≤2μ2∫0tE∂f∂xs,Ss,ζ2ESs,ζ−Ss2ds+2μ2∫0tESs2E∂f∂xs,Ss,ζ−∂f∂xs,Ss2ds≤2μ2Mζ2αM22t+2μ2M32Mζ2α∫0tESs2ds,where ESs2=S0expμt−1/2σ2t2HEexpBtH2 which is finite.
In fact,∀t∈0,T, EBtH2=t2H thus ∫0tESs2ds is finite.
LetA=∫0tESs2ds,
We have(25)C2≤2μ2Mζ2αM22T+2μ2M32Mζ2αA≤2μ2MM22T+μ2M32MAζ2α,C3=H2σ4∫0tEs2H−1Ss,ζ2∂2fs,Ss,ζ∂x2−s2H−1Ss2∂2fs,Ss∂x22ds≤2H2σ4∫0ts4H−2E∂2fs,Ss,ζ∂x2Ss,ζ2−Ss22ds+2H2σ4∫0ts4H−2ESs2E∂2fs,Ss,ζ∂x2−∂2fs,Ss∂x22ds≤2H2σ4∫0ts4H−2E∂2fs,Ss,ζ∂x22ESs,ζ2−Ss22ds+2H2σ4M4∫0ts4H−2ESs,ζ−Ss2ESs2ds≤2H2σ4M32Mζ2α∫0ts4H−2ESs,ζ+Ss2ds+2M4H2σ4Mζ2α∫0ts4H−2ESs2ds.
AsESs2 is finite, ESs,ζ+Ss2 is also finite.
LetB=ESs2 and C=ESs,ζ+Ss2, then we have(26)C3≤2H2σ4M32Mζ2αC∫0ts4H−2ds+2M4H2σ4Mζ2αB∫0ts4H−2ds≤2H2σ4M32Mζ2αC+M4H2σ4Mζ2αBt4H−14H−1≤2H2σ4M32MC+M4MB4H−1T4H−1ζ2α,C4=σ2∫0tESs,ς∂f∂xs,Ss,ςdBs,ςH−Ss∂f∂xs,SsdBsH2≤2σ2∫0tE∂f∂xs,Ss,ςSs,ς−SsdBs,ςH2+2σ2∫0tESs2E∂f∂xs,Ss,ςdBs,ςH−∂f∂xs,SsdBsH2≤2σ2∫0tESs,ς−Ss2E∂f∂xs,Ss,ςdBs,ςH2+4σ2∫0tESs2E∂f∂xs,Ss,ςdBs,ςH−∂f∂xs,SsdBs,ςH2+4σ2∫0tESs2E∂f∂xs,SsdBs,ςH−∂f∂xs,SsdBsH2≤2σ2Mς2α∫0tE∂f∂xs,Ss,ς2EdBs,ςH2+4σ2M32B∫0tESs,ς−Ss2EdBs,ςH2+4σ2B∫0tE∂f∂xs,Ss2EdBs,ςH−dBsH2≤2σ2Mς2αM22EBt,ςH2+4σ2M32BMς2αEBt,ςH2+4σ2BM22EBt,ςH−BtH2≤ς2α2σ2MT2HM22+4σ2M32BMT2H+4σ2BM22T.
We conclude thatEft,St,ζ−ft,St2≤Dζ2α where D a constant. We deduce that ft,St,ζ converges in mean square to ft,St when ζ tends to 0.
Vt,ft,St,ζ=ϕt0St0+ϕt1ft,St,ζ is also a semimartingale because ft,St,ζ is a semimartingale and ϕt0St0 is a finite variation process.
Similarly,(27)Vt,ft,St,ζ−Vt,ft,St=ϕt1ft,St,ζ−ft,St,EVt,ft,St,ζ−Vt,ft,St2=ϕt12Eft,St,ζ−ft,St2≤ϕt12Dζ2α.Hence, the convergence of Vt,ft,St,ζ to Vt,ft,St in L2Ω.Theorem 1.
LetSt,ζ be a continuous semimartingale defined on a probability space (Ω, ℱ, P) and f a function of class C1,3 then under certain assumptions there exist two processes S^t,ζ=ft,St,ζ and Vt,ft,St,ζ such that(1)
S^t,ζ is the unique solution of the equation(28)dS^t,ζS^t,ζ=gt,ft,St,ζdt+ht,ft,St,ζdWt,and verifies the relation(29)σ2C2St,ζ2ζ2H−1∂ft,St,ζ/∂St,ζ2β+ft,St,ζ=∂ft,St,ζ∂t+∂ft,St,ζ∂St,ζμ+αtSt,ζ+Hσ2t2H−1∂2ft,St,ζ∂St,ζ2St,ζ2,with(30)gt,ft,St,ζ=1ft,St,ζ∂ft,St,ζ∂t+∂ft,St,ζ∂St,ζμ+αtSt,ζ+Hσ2t2H−1∂2ft,St,ζ∂St,ζ2St,ζ2,and(31)ht,ft,St,ζ=1ft,St,ζCσSt,ζ∂ft,St,ζ∂St,ζζH−1/2,two bounded predictable processes andαt=CσH−1/2∫−∞tt−s+ζH−3/2dWs.(2)
The processesft,St,ζ and Vt,ft,St,ζ are Markovian processes.Proof of Theorem 1.
(1)
Letf be a function of class C1,3 and the process St defined in the relation (5); according to the Wick-ito, we have:(32)dft,St=∂ft,St∂t+∂ft,St∂StμSt+Hσ2t2H−1∂2ft,St∂St2St2dt+σSt∂ft,St∂StdBtH.We approximateBtH by Bt,ζH, and we have(33)C−1BtH=∫−∞0t−sH−1/2−−sH−1/2dWs+∫0tt−sH−1/2dWs,and the integral(34)∫−∞0t−sH−1/2−−sH−1/2dWs,is a process with absolutely continuous trajectories, and by takingτ=0 in the relation (11), we can give a meaning to the derivative(35)ddt∫−∞0t−sH−1/2dWs.Using the fractional Brownian motion approximation of Tran Hung Thao, we have(36)C−1dBt,ζH=ddt∫−∞0t−s+ζH−1/2dWs+H−12∫0tt−s+ζH−3/2dWsdt+ζH−1/2dWt.Therefore,(37)dBt,ζH=CH−12∫−∞tt−s+ζH−3/2dWsdt+CζH−1/2dWt.ReplacingdBtH by dBt,ζH, St by St,ζ and ft,St by ft,St,ζ in (32), we obtain(38)dS^t,ζS^t,ζ=1ft,St,ζ∂ft,St,ζ∂t+∂ft,St,ζ∂St,ζμ+αtSt,ζ+Hσ2t2H−1∂2ft,St,ζ∂St,ζ2St,ζ2dt+1ft,St,ζCσSt,ζ∂ft,St,ζ∂St,ζζH−1/2dWs.Let(39)gt,ft,St,ζ=1ft,St,ζ∂ft,St,ζ∂t+∂ft,St,ζ∂St,ζμ+αtSt,ζ+Hσ2t2H−1∂2ft,St,ζ∂St,ζ2St,ζ2,and(40)ht,ft,St,ζ=1ft,St,ζCσSt,ζ∂ft,St,ζ∂St,ζζH−1/2.We have(41)dS^t,ζS^t,ζ=dft,St,ζft,St,ζ=gt,ft,St,ζdt+ht,ft,St,ζdWt.From hypothesis (H1) and (H2), we have(42)1−λSt,ζ≤ft,St,ζ≤St,ζ,ht,ft,St,ζ=1ft,St,ζCσSt,ζ∂ft,St,ζ∂xζH−1/2≤11−λCσM2ζH−1/2.So,∫0ths,fs,Ss,ζ2ds<∫0t1/1−λCσM2ζH−1/22ds<1/1−λCσM2ζH−1/22T<+∞.(43)gt,ft,St,ζ=1ft,St,ζ∂ft,St,ζ∂t+∂ft,St,ζ∂St,ζμ+αtSt,ζ+Hσ2t2H−1∂2ft,St,ζ∂St,ζ2St,ζ2.From Hypothesis (H1) 1 and the hypothesis(H2), (H3),(44)gt,ft,St,ζ≤11−λSt,ζ∂ft,St,ζ∂t+M2μ+αtSt,ζ+Hσ2t2H−1M3St,ζ2gt,ft,St,ζ≤Supt∈0,T11−λSt,ζ∂ft,St,ζ∂t+M2μ+αt1−λ+Hσ2T2H−1M31−λSupt∈0,TSt,ζ.Therefore,(45)∫0tgu,fu,Su,ζdu<+∞.From Assumption 1 and the hypothesis(H3), we have(46)ht,ft,x1−ht,ft,x2=1ft,x1Cσx1∂ft,x1∂xζH−1/2−1ft,x2Cσx2∂ft,x2∂xζH−1/2≤σCζH−1/21−λ∂ft,x1∂x−∂ft,x2∂x≤σCζH−1/21−λM3x1−x2,kt=μt,0+σt,0=1ft,0∂ft,0∂t+∂ft,0∂x+C.The functionsf(t, 0), ∂ft,0/∂x, ∂ft,0/∂t are continuous and bounded functions for t∈0;T so E∫0tku2du<+∞.Assume thatf is monotonic, i.e., ft,x2>ft,x1 and x2>x1 (where ft,x2>ft,x1 and x2<x1) and N=Supt∈0,T1/ft,x1,(47)gt,x1−gt,x2≤1ft,x1∂ft,x1∂t−∂ft,x2∂t+1ft,x1∂ft,x1∂xμ+αtx1−∂ft,x2∂xμ+αtx2+Hσ2t2H−11ft,x1∂2ft,x1∂x2x12−∂2ft,x2∂x2x22.Under the Hypotheses(H1), (H3), and (H4), we have(48)gt,x1−gt,x2≤NM1x1−x2+M3Nx1μ+αtx1−x2+Hσ2t2H−1M4Nx12x1−x2≤Supt∈0,TM1N+M3Nx1μ+αt+NHσ2t2H−1M4x12x1−x2.We have(49)ht,x1−ht,x2+gt,x1−gt,x2≤x1−x2Supt∈0,TM1N+M3Nx1μ+αt+Hσ2t2H−1M4Nx12+σCζH−1/21−λM3.Using Hypothesis1, we have S^t,ζ∈1−λSt,ζ;St,ζ,(50)gt,ft,St,ζ=1ft,St,ζ∂ft,St,ζ∂t+∂ft,St,ζ∂St,ζμ+αtSt,ζ+Hσ2t2H−1∂2ft,St,ζ∂St,ζ2St,ζ2,and(51)ht,ft,St,ζ=1ft,St,ζCσSt,ζ∂ft,St,ζ∂St,ζζH−1/2.According to the relation (3),(52)ϕt1S^t,ζϕt0+ϕt1S^t,ζ=μtσt2.We have(53)ϕt1ft,St,ζϕt0+ϕt1ft,St,ζ=1/ft,St,ζ∂ft,St,ζ/∂t+∂ft,St,ζ/∂St,ζμ+αtSt,ζ+Hσ2t2H−1∂2ft,St,ζ/∂St,ζ2St,ζ2C2/ft,St,ζ2σ2St,ζ2ζ2H−1∂ft,St,ζ/∂St,ζ2,ft,St,ζβ+ft,St,ζ=ft,St,ζ∂ft,St,ζ/∂t+∂ft,St,ζ/∂St,ζμ+αtSt,ζ+Hσ2t2H−1∂2ft,St,ζ/∂St,ζ2St,ζ2C2σ2St,ζ2ζ2H−1∂ft,St,ζ/∂St,ζ2,withβ=ϕt0/ϕt1,(54)1β+ft,St,ζ=∂ft,St,ζ/∂t+∂ft,St,ζ/∂St,ζμ+αtSt,ζ+Hσ2t2H−1∂2ft,St,ζ/∂St,ζ2St,ζ2σ2C2St,ζ2ζ2H−1∂ft,St,ζ/∂St,ζ2,σ2C2St,ζ2ζ2H−1∂ft,St,ζ/∂St,ζ2β+ft,St,ζ=∂ft,St,ζ∂t+∂ft,St,ζ∂St,ζμ+αtSt,ζ+Hσ2t2H−1∂2ft,St,ζ∂St,ζ2St,ζ2.(2)
The processft,St,ζ is a solution of (12); thus, it is Markovian process. We deduce that Vt,ft,St,ζ=ϕt0St0+ϕt1ft,St,ζ 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.



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*Source: 4719482-2022-08-29.xml*