European Option Pricing with Transaction Costs and Stochastic Volatility: an Asymptotic Analysis

# European Option Pricing with Transaction Costs and Stochastic Volatility: an Asymptotic Analysis

R. E. Caflisch111Department of Mathematics, UCLA, USA, rcaflisch@ipam.ucla.edu G. Gambino222Department of Mathematics, University of Palermo, Italy, gaetana@math.unipa.it M. Sammartino333Department of Mathematics, University of Palermo, Italy, marco@math.unipa.it C. Sgarra444Department of Mathematics, Politecnico di Milano, Italy, carlo.sgarra@polimi.it
###### Abstract

In this paper the valuation problem of a European call option in presence of both stochastic volatility and transaction costs is considered. In the limit of small transaction costs and fast mean reversion, an asymptotic expression for the option price is obtained. While the dominant term in the expansion it is shown to be the classical Black and Scholes solution, the correction terms appear at and . The optimal hedging strategy is then explicitly obtained for the Scott’s model.

## 1 Introduction

The intrinsic limitations of the Black-Scholes model in describing real markets behavior are very well known. Among the main assumptions underlying that model the assumptions of constant volatility and no transaction costs are probably the most relevant. In this paper we are going to consider the pricing problem of a European option in a model in which both proportional transaction costs are taken into account and the volatility is assumed to evolve according to a stochastic process of the Ornstein–Uhlenbeck type. To analyze this situation we shall follows a utility maximization procedure, following the seminal paper of M.H.A.Davis, V.G.Panas and T.Zariphopoulou [2].

If one uses the following utility function :

 U(x)=1−exp(−γx),

where expresses the risk aversion of the investor, one gets, as the result of this analysis, a non linear PDE for the expected value of the utility–maximized wealth held in the underlying asset of the option.

At this point we shall make two assumptions. First, that the process driving the volatility is fast mean–reverting with speed . Second, that the transaction costs are very small, .

The pricing of a European option in presence of small transaction costs was considered in [7], where a correction term to the Black&Scholes pricing formula was derived. This correction term was found to be order in the pricing cost. Moreover in [7] was found that the optimal hedging strategy consisted in not transacting when the process driving the stock price is in a strip around the classical Black&Scholes delta–hedging formula and in rebalancing the portfolio (selling or buying stocks) to keep the process inside the strip of no transaction. The width of the no transaction strip was found to be order in the transaction costs. In [7] the volatility was supposed to be constant. More Recently V.I.Zakamouline studied optimal hedging of European options with transaction costs via a utility optimization approach in ([12]).

The pricing of a European option with fast mean–reverting stochastic volatility was considered in in a series of papers (see e.g. [3], [5] and [4]). In the above mentioned papers the authors found the pricing formula whose leading order term is the classical Black&Scholes formula with averaged volatility. The correction term was order the square root of the characteristic time scale of the process driving the volatility. An optimal consumption-investment problem has been investigated in a paper by M.Bardi, A.Cesaroni and L.Manca ([8]) where a rigorous asymptotic analysis is performed and where the solution is characterized in the limit of fast volatility dynamics.

More recently, M.C.Mariani, I.SenGupta and P.Bezdek ([9]) proposed a numerical approximation scheme for European option prices in stochastic volatility models including transaction costs based on a finite-difference method. The stochastic volatility dynamics considered there is a slight generalization of that proposed by Hull and White [10], since they consider a drift coefficient which is a general (regular) deterministic function of both the time and the underlying asset price, while their diffusion coefficient is linear in the instantaneous volatility.

In the present paper we propose a different approximation method for European option pricing in stochastic volatility models with transaction costs, based on an asymptotic analysis which follows the approach pioneered by J.-P.Fouque, G.Sircar and R.Papanicolaou ([3]). The model we consider for the stochastic volatility dynamics is of Ornstein–Uhlenbeck type with a constant diffusion coefficient. This model has been originally proposed by Stein and Stein [11]. We provide closed-formulas for European option prices in the limit of fast volatility and small transaction costs.

The plan of the paper is the following: in Section 2 we introduce the multidimensional stochastic process for the dynamic of a portfolio of the writer of a European option. By solving a stochastic control problem, we obtain the Hamilton-Jacobi-Bellman equation. In Section 3, the asymptotic analysis is done assuming small transaction costs and fast mean reverting volatility. In section 4 the price of the option is computed and in section 5 the numerical results are provided and some conclusions outlined.

For the reader’s convenience, in Appendix A the source term of the equation obtained through the asymptotic analysis at is calculated; in Appendix B the averages with respect to the Ornstein-Uhlenbeck invariant measure using the model introduced by L.Scott ([1]) for the stochastic volatility are given; finally, in Appendix C the derivatives, which appears in the obtained corrected pricing formula, with respect to the stock price of the classical Black and Scholes model are recalled and collected.

## 2 Option pricing via utility maximization

We suppose to have the following multidimensional stochastic process:

 dB = rBdt−(1+λ)SdL+(1−μ)SdM (2.1) dy = dL−dM (2.2) dS = S(αdt+f(z)dW) (2.3) dz = ξ(m−z)dt+β(ρdW+√1−ρ2dZ). (2.4)

In the above equations and are the risk-free (the ”Bond”) and the risky asset (the ”Stock”) respectively, is the risk–free interest rate, is the drift rate of the stock, and are the (proportional) cost of buying and selling a stock, is the volatility function, which we shall suppose to depend on the stochastic variable , which is sometimes called the volatility driving process. and are the cumulative number of shares bought or sold, respectively, in the time interval considered . We keep the notations introduced in [2] and [7], where the reader can find a detailed justification for the model for transaction costs just introduced. In what follows we shall always suppose to be a function bounded away from :

 0

The process followed by the stochastic variable is a Ornstein–Uhlenbeck process with average . The parameter is the rate of mean reversion volatility.

The Brownian motions and are uncorrelated and is the instantaneous correlation coefficient between asset price and the volatility shocks. Usually one considers , i.e. the two processes are anti–correlated (e.g. when the prices go down the investors tend to be nervous and the volatility raises). For more details see [3] and [6].

We will suppose to deal with trading strategies absolutely continuous with respect to time, so that:

 L=∫t0lds,M=∫t0mds.

Therefore the process we are dealing with can be written in the form:

 dB = [rB−(1+λ)Sl+(1−μ)Sm]dt (2.5) dy = (l−m)dt (2.6) dS = S(αdt+f(z)dW) (2.7) dz = ξ(m−z)dt+β(ρdW+√1−ρ2dZ). (2.8)

The final value of a portfolio of the writer of a European option with strike price , after following the strategy is:

 Φw(T,Bπ(T),yπ(T),S(T),z(T)) = Bπ(T)+I(S(T)K)[c(yπ(T)−1,S(T))+K].

On the other hand the final value of a portfolio which does not include the option is simply:

 Φ1(T,Bπ(T),yπ(T),S(T),z(T))=Bπ(T)+c(yπ(T),S(T)). (2.10)

We can now define the following value functions:

 Vj(B)=supπ∈TE(U(Φj(T,Bπ(T),yπ(T),S(T),z(T)))) (2.11)

for . Notice how this value functions depend on the initial endowment .

Following [2] we now define:

 Bj=inf{B:Vj(B)≥0}.

The fair price of the option to avoid arbitrage, i.e. the amount of money that the writer has to receive to accept the obligation implicit in writing the option, will therefore be:

 C=Bw−B1. (2.12)

For this price the investor would in fact be indifferent between the two possibilities of going into the market to hedge the option, or of going into the market without the option.

We can define the following function that will be useful in the sequel:

 Ψj(T,Bπ(T),yπ(T),S(T),z(T))=Φj(T,Bπ(T),yπ(T),S(T),z(T))−Bπ(T). (2.13)

We have to find an equation for . In what follows we shall suppress the index and denote with . The problem we are dealing with is a stochastic control problem, where the control is the trading strategy and . The equation for is the Hamilton–Jacobi–Bellman equation.

The Hamilton–Jacobi–Bellman equation: Suppose we have the following -dimensional controlled stochastic process:

 dX=b(t,X,y)dt+σ(t,X,y)dW.

Let us define the performance functional:

 Jy(s,x)=E(K(T,X(T))).

In general one can define a performance functional that depends on the whole trajectory. In our case we do not need this generality.

The infinitesimal operator associated with the stochastic process is:

 Lyf=∂tf+bi∂xif+aij∂xixjf,

where

 aij=12(σσT)ij.

If one defines:

 V=sup{Jy:y=y(x)is a Markov control},

then we have the following result according to the Hamilton-Jacobi-Bellman Theorem:

In the present case this turns out to be equivalent to the following otimization problem:

 max0≤l,m≤k{(∂yVj−(1+λ)S∂BVj)l−(∂yVj−(1−μ)S∂BVj)m+ ∂tVj+rB∂BVj+αS∂SVj+ξ(m−z)∂zVj+ 12[f(z)]2S2∂SSVj+12β2∂zzVj+βfSρ∂SzVj}=0. (2.14)

With some analysis, as in [2], one gets that the above equation is equivalent to the following equation:

 max{(∂yVj−(1+λ)S∂BVj),−(∂yVj−(1−μ)S∂BVj), ∂tVj+rB∂BVj+αS∂SVj+ξ(m−z)∂zVj+ 12[f(z)]2S2∂SSVj+12β2∂zzVj+βfSρ∂SzVj}=0. (2.15)

We now consider the case of the exponential utility function . We note, just in passing, that this gives for the following expression:

 Vj=1−inf{E[exp(−γB(T))exp(−γΨj)]},

where has been previously introduced.

In the above maximization problem let us change the variables passing :

 Vj=1−exp(−γδ(B+Wj)),

where

 δ≡exp[−r(T−t)].

Notice that with the above expression for the price of the option , as given in (2.12), now becomes:

 C=W1−Ww. (2.16)

The maximization problem for is equivalent to the following minimization problem for :

 min{(∂yWj−(1+λ)S),(−∂yWj+(1−μ)S), ∂tWj−rWj+αS∂SWj+ξ(m−z)∂zWj+ 12[f(z)]2S2[∂SSWj−γδ(∂SWj)2]+12β2[∂zzWj−γδ(∂zWj)2]+ βfSρ[∂SzWj−γδ∂SWj∂zWj]}=0.

## 3 Small transaction costs and fast mean reverting volatility: the asymptotic analysis

We now suppose small transaction costs and fast mean reverting volatility. Moreover we will assume that the transaction costs are much smaller than the rate of mean reversion.

 λ=μ=ε2,ξ=1ε,β=√2ν√ε.

Buying and selling costs are assumed to be the same for simplicity.

We believe that our asymptotic assumptions are consistent with a situation where a large investor, facing very small transaction costs, is involved. In fact, in the empirical study [4] it is found that . For large investor, typically .

In absence of transaction costs and with a deterministic volatility , the investor would continuously trade and get a perfect hedge staying at , the “B&S” hedging strategy. When transaction costs are present there is a strip of small thickness around where he does not transact. To resolve this strip we introduce the inner rescaled coordinate :

 y=y∗+εaYand∂y⟶ε−a∂Y. (3.1)

The unknown boundaries between the no-transaction region and the buy and sell regions are located at:

 y=y∗+εaY+andy=y∗−εaY−.

It is very important from the practical hedger point of view to determine and .

We impose the following matching conditions (see e.g.[7]):

 WNT(Y=Y±)=W(y=y∗±εaY±) continuity∂YWNT(Y=Y±)=εa∂yW(y=y∗±εaY±) continuity of the  first derivative∂YYWNT(Y=Y±)=ε2a∂yyW(y=y∗±εaY±) smooth pasting boundary condition

These boundary conditions will force, in the asymptotic analysis below, . Therefore the strip of no-transaction will have a thickness .

In the buy region () we have the equation:

 (∂yWj−(1+λ)S)=0, (3.2)

which solves to:

 W=(1+λ)Sy+H−(t,S,λ). (3.3)

In the sell region () we have the equation:

 (∂yWj−(1−λ)S)=0,

which solves to:

 W=(1−λ)Sy+H+(t,S,λ). (3.4)

In the no-transaction region we have the equation:

 ∂tWj−rWj+αS∂SWj+1ε(m−z)∂zWj +12[f(z)]2S2[∂SSWj−γδ(∂SWj)2]+1εν2[∂zzWj−γδ(∂zWj)2] +1√εν√2fSρ[∂SzWj−γδ∂SWj∂zWj] = 0, (3.5)

whose solution will be find in what follows.

### 3.1 The solution in the no transaction region

As we said, in the no transaction region we use the rescaled variable defined by (3.1). The change of variable leads to the following transformation rules for the derivatives:

 ∂y ⟶ ε−1/3∂Y ∂S ⟶ ∂S−ε−1/3y∗S∂Y ∂t ⟶ ∂t−ε−1/3y∗t∂Y ∂z ⟶ ∂z−ε−1/3y∗z∂Y

We write the solution in the no transaction region in the form:

 WNT=S(y∗+ε1/3Y)+U0(S,t,z)+13∑i=2εi/6Ui(S,t,z)+ε14/6U14(S,t,z,Y)+... (3.6)

The derivative with respect to writes as:

 ∂tWNT=U0t+11∑i=2εi/6Uit+ε12/6(U12t−y∗tU14Y)+...

The derivatives with respect to writes as:

 ∂SWNT = y∗+U0S+ε2/6(Y+U2S)+11∑i=3εi/6UiS+ε12/6(U12S−y∗SU14Y)+... ∂SSWNT = U0SS+9∑i=2εi/6UiSS+ε10/6(U10S+(y∗S)2U14YY)+...

The derivatives with respect to writes as:

 ∂zWNT = U0z+11∑i=2εi/6Uiz+ε12/6(U12z−y∗zU14Y)+... ∂zzWNT = U0zz+9∑i=2εi/6Uizz+ε10/6(U10zz+(y∗z)2U14YY)+... (3.7)

The derivative with respect to and writes as:

 ∂SzWNT = U0Sz+9∑i=2εi/6UiSz+ε10/6(U10Sz+y∗Sy∗zU14YY)+...

### 3.2 The O(ε−1) up to O(ε−1/6) order equations

To simplify the notation, and following the use in [5] and [6], we define the linear operators and the non linear operator :

 L0U = (m−z)Uz+ν2Uzz (3.8) L1U = −ν√2ρ(α−r)fUz+ν√2fSρUSz (3.9) L2U = Ut+12f2S2USS−rU+rSUS (3.10) NLU = −ν2γδ(Uz)2 (3.11)

The equation is simply:

 L0U0+NLU0=0. (3.12)

The above equation can be considered an ODE in for :

 ν2U0zz+(m−z)U0z−ν2γδ(U0z)2=0. (3.13)

In [6] it is proved that the only solution of an equation of this form is a which does not depend on . The conclusion we therefore draw is that:

 U0does not depend on z.

The equations, for are:

 L0Ui=0.

The conclusion is:

 Uii=2,⋯,5does not depend on z.

### 3.3 The O(1) equation

The equation writes:

 U0t−r(Sy∗+U0)+αS(y∗+U0S)+L0U6 +12[f(z)]2S2[U0SS−γδ(y∗+U0S)2] = 0. (3.14)

The above equation will be analyzed in the subsection 3.5.

### 3.4 The O(ε1/6) equation

The equation is:

 L0U7=0.

We therefore get:

 U7does not depend on z.

### 3.5 The O(ε2/6) equation

The equation is:

 L0U8+U2t−r(SY+U2)+αS(Y+U2S)+ +12[f(z)]2S2[U2SS−2γδ(y∗+U0S)(Y+U2S)] = 0. (3.15)

In the above equation there are terms that do not depend on , and terms linear in . They must be equal to zero separately. ¿From the terms linear in one gets:

 y∗=−U0S+(α−r)δf2Sγ. (3.16)

The above expression gives the leading order (in absence of transaction costs) optimal hedging strategy. One recognizes the Black&Scholes -hedging strategy.

If one inserts the above expression into the equation (3.14), one gets:

 L0U6+∂tU0+12f2S2∂SSU0+rS∂SU0−rU0+12δγ(α−r)2f2=0. (3.17)

The above equation, considered as an ODE for , is of the form:

 L0U=χ. (3.18)

In [5] it is shown that the solvability condition for the equation (3.18) is:

 ⟨χ⟩=0, (3.19)

where the average is taken with respect to the Ornstein–Uhlenbeck process invariant measure:

 ⟨χ⟩=1ν√2π∫Rχ(z)e−(m−z)2/2ν2dz. (3.20)

Therefore the solvability condition for (3.17) is:

 ∂tU0+12¯σ2S2∂SSU0−rU0+rS∂SU0=−δ(α−r)22γ1¯τ2, (3.21)

where is the effective constant volatility:

 ¯σ2=⟨f2⟩,

and is defined as:

 1¯τ2=⟨1f2⟩.

Once one imposes to (3.21) the appropriate final condition, which will be different for the investor with option liability and the investor without it, then is determined. One can go back to equation (3.17) and solve it for . The solution can be written in the form:

 U6=U(z)6(S,z,t)+~U6(S,t), (3.22)

where, , the part of which depends on , has the following expression:

 U(z)6(S,z,t)=−L−10[12S2(f2−¯σ2)U0SS+12δγ(α−r)2(1f2−1¯τ2)];

on the other hand is a function that does not depend on and that will be determined by the equation in the asymptotic procedure.

We can get a more explicit representation for , that will be useful in the next subsection. We first define the functions and as the solutions of the following problems:

 L0φ = f2−⟨f2⟩, (3.23) L0ψ = 1f2−⟨1f2⟩. (3.24)

Therefore the above expression for can be written as:

 U(z)6=−[12S2U0SSφ+12δγ(α−r)2ψ]. (3.25)

One can now go back to equation (3.15), collect all terms independent of and get the following equation:

 L0U8+L2U2=0

The above equation is a Poisson problem of the type (3.18). The solvability condition is:

 ⟨L2⟩U2=0. (3.26)

Notice that the above equation is homogeneous in . Given that the final condition, both for the investor with option liability and for the investor without it, is , one gets the following conclusions:

 U2 ≡ 0 U8 = ~U8(S,t)is independent of z.

Therefore the correction to the Black and Scholes value comes up to order.

### 3.6 The O(ε3/6) equation

Using the expression (3.16) for , the equation can be written as:

 L2U3+L1U6+L0U9=0. (3.27)

Notice that in the above equation appears which, until now we have derived only up to the function , to be determined by to a higher order asymptotics. However, in (3.27) is hit by the operator , which cancels .

Therefore one can consider (3.27) as a Poisson problem for , whose solvability condition is:

 ⟨L2⟩U3=−⟨L1U6⟩. (3.28)

The above equation is a Black and Scholes equation for with final condition and with a source term. We now want to rewrite the source term.

By using for the expression (3.22) the operator cancels the part not depending on , and taking into account the expression (3.25), one can express the source term in the equation (3.28) as:

 −⟨L1U6⟩ = ⟨L1[12S2U0SSφ+12δγ(α−r)2ψ]⟩ =

Therefore solves the following Black and Scholes equation:

 U3t+12¯σ2S2U3SS−rU3+rSU3S =

with zero final data.

### 3.7 The O(ε2/3) equation

Since does not depend on , the equation can be written as:

 L2U4+L0U10+ν2(y∗z)2U14YY−γδf2S2Y2=0. (3.30)

The equation (3.30) can be considered as an ODE in for . It writes as:

 U14YY=AY2+B, (3.31)

where we have defined the following quantities:

 A = γδf2S2ν2(y∗z)2, B = −L2U4+L0U10ν2(y∗z)2.

The above equation (3.31) solves to:

 U14=A12Y4+12BY2+CY+D, (3.32)

with and independent of . Now we have to impose the matching conditions.

Being:

and imposing the continuity of the gradient at the two boundaries:

one gets, at the :

 ∂YU14(Y=−Y−)=S, (3.34) ∂YU14(Y=Y+)=−S. (3.35)

Therefore, using (3.32), one gets:

 −A3(Y−)3−BY−+C = S, (3.36) A3(Y+)3+BY++C = −S. (3.37)

Moreover, being in the outer regions linear in , one imposes the continuity of the second derivative as follows:

 ∂YYWNT(Y=±Y±)=0,

i.e.:

 A(Y+)2+B=0, A(Y−)2+B=0.

From these equations one sees that, at this order, the bandwidth about the Black and Scholes strategy is symmetric, i.e.:

 Y+=Y−=(−BA)1/2 (3.38)

Subtracting the two equations (3.36) and (3.37) to eliminate , and using the above expressions for , one gets:

 23(−B)3/2A−1/2=S.

After some manipulations, and using the expressions for and , the equation (3.31) leads to the following equation:

 L0U10+L2U4=[32fS2√γδν2(y∗z)2]2/3 (3.39)

One can also find an expression for the amplitude of the no-transaction region:

 Y+=Y−=[321f2Sδγν2(y∗z)2]1/3. (3.40)

Equation (3.39) is a Poisson problem of the type of equation (3.18). The solvability condition gives an equation for :

 ⟨L2⟩U4=⟨[32fS2√γδν2(y∗z)2]2/3⟩. (3.41)

Notice also that, adding the two equations (3.36) and (3.37), one gets that . Therefore:

 U14=A12Y4+12BY2+D, (3.42)

which will be useful in the subsection 3.9.

### 3.8 The O(ε5/6) equation

The equation writes as:

 L0U11+L2U5−√2∂U6∂zγδνρf(z)SY−∂U3∂Sγδf(z)2S2Y+∂2U15∂Y2(∂y∗∂z)2ν2=0, (3.43)

This equation can be considered as an ODE for :

 ∂2U15∂Y2=¯AY+¯B,

where we have denoted:

 ¯A = (√2∂U6∂zγδνρf(z)S−∂U3∂Sγδf(z)2S2)/(ν2y∗2z), ¯B = −L0U11+L2U5ν2y∗2z.

Integrating (3.43) with respect to and using the boundary conditions:

 U15Y(Y+)=U15Y(−Y−)=0

which are needed to ensure the continuity of the gradient, one gets:

 L0U11+L2U5=0. (3.44)

The above equation is a Poisson problem for , whose solvability condition reads:

 U5=0. (3.45)

This is a homogeneous Black-Scholes equation for . Given that the final condition is zero, we gets the conclusions:

 U5 ≡ 0, U11 = ~U11(S,t)does not depend on z.

### 3.9 The O(ε) equation

Collecting the terms one gets:

 L2U6+L1U9+L0U12−12f2S2γδ(U3S)2−ν√2fSργδU3SU6z−y∗z(m−z)U14Y (3.46) −γδf2YS2U4S+ν2[−y∗zzU14Y−2y∗zU14Yz+(y∗z)2U16YY−γδ(U6z)2] = 0.

The above equation can be considered as an ODE in for :

 ∂2U16∂Y2=~AY+~B,

where we have defined:

 ~A = γδ