The Model

This paper considers the pricing model of options under illiquidity and the following several sections are the core of this paper. In this section, following Frey (2000) and Frey and Patie (2001), we assume that there are two traded assets: bond and stock in the market where bond is a risk-free asset (i.e. cash account) and stock is a risky asset which follows a stochastic process. Simultaneously, we consider the bond as a numeraire (i.e., sometimes called discount factor) and assume that bond market is perfect liquidity that there is no liquidity problem exist. Now we focus on liquidity problem in the stock market.

The BS model assumes that the underlying stock have perfect liquidity, meaning that investors can buy or sell a large amount of stock without affecting the stock price in market so that there is no feedback effect in the market. However, we take the market liquidity variable into account in the model due to the liquidity problem is an existent fact in the stock market. In this study, we do not assume the parameter of liquidity following a certain stochastic process, meaning that the liquidity is deterministic and it is not stochastic.¹⁵

The following sections will introduce the basic assumptions and asset dynamics firstly. Secondly, the Frey model will be conducted and then we introduce the tracking error of the model. After that we explore the numerical method applications in the model. Finally, we present the smooth version of the model which proposed by Frey and Patie (2001).

15 In Esser and Moench (2003), the liquidity follows a certain stochastic process. Their framework generalizes the constant liquidity model of Frey (2000) and they impose a stochastic factor into the liquidity. Hence, the stochastic liquidity model of Esser and Moench (2003) is more sophisticated than the Frey model; it’s becoming very complex in the modeling of option pricing and in constructing the hedging strategies when the market liquidity is considered as a random source. In this paper, we do not deal with this kind of complicated circumstance in the parameter of liquidity which follows a certain stochastic process.

2.1 Basic assumptions and asset dynamics

We now introduce the basic model setup proposed by Frey and Patie (2001). The risky asset (i.e. the stock) follows the stochastic process without drift term

( )

t t t t t t

dS

=

σ S dW

₋ +

ρλ S

₋

S d

₋

α

⁺, (1) where α is the number of stock shares held by large investor, i.e. the trading strategy of the large trader. The variable

α

⁺ denotes the right-continuous process, and

ρ

is a non-negative constant liquidity parameter. A large value of the parameter

ρ

means that the market becomes more illiquid. Moreover, we state that the parameter

ρ

is equal to zero as the market reduces to the BS world with perfect liquid. Recall that the drift term plays a role in stock dynamics in the assumption of the BS model.

After the change of measure, however, the drift term is removed from the BS-PDE which is dominated by risk-free rate in risk neutral measure.

Frey (2000) and Frey and Patie (2001) discuss the influence of the trading strategy on the asset process with a smooth stock trading strategyα and suppose that the large trader utilize the strategy of the form

α φ

= ( ,

t S

_t). Thus, the asset dynamic becomes a new dynamics and then we can obtain the new effective asset dynamics by Ito formula¹⁶ with the following form

( , ) ( , )

Derivation of the new asset dynamics

We suppose that the large trader utilize the strategy of the form

α φ

= ( ,

t S

_t) for a

16 See Shreve (2004) chapter 4.

function α and it is satisfying a mathematical assumption with two variables which are once continuously differentiable in time and twice continuously differentiable in stock¹⁷. The trading strategy of large trader expanded by Ito formula and thus we can get the form

Firstly, we have already known the stock prices are controlled by the following stochastic process

t t t ( )t t t

dS

=

σ S dW

+

ρλ S S d α

.

Secondly, we substitute the Equation (5) into the second term of the RHS of the Equation (1) and thus we obtain the Equation (6).

2 2 Therefore, generates the following explicit form for asset dynamics

[ ] [ ]

In this section, we provide a simple proof of the new effective asset dynamics. In next section, we interpret how the Frey model is controlled by the Equation (12) and clarify all of basic assumptions in the model.

2.2 The Frey model (nonlinear parabolic PDE)

The Frey model has two significant characteristics different from Black-Scholes PDE.

17 The stock is often designated as the space in the FDM application

First, the risk-free rate does not play a role in Frey model. Second, the Frey model argues that the volatility is not a constant volatility. In the Frey model, the volatility term is dominated by three main parameters

ρ

, λ and

u in the Frey model.

_SS However, we can utilize the three main parameters to capture the volatility behavior in real markets. The parameterλ can be utilized to describe the asymmetry of liquidity¹⁸. Generally, markets tend to be more liquid in the bull market than in the bear market. Thus, Frey and Patie (2001) denote the parameter in the following form

{

0 0

}

2

0 1 ( ) 2 ( )

( ) 1 (

S S S

)

a I

_{S S}

a I

_{S S}

λ

= + − _≤ + _> , (9) where the parameter

a is usually larger than

₁

a in empirical study. The parameter

₂ λ plays an important role in this model. The asymmetry of liquidity can be explained byλ with financial sense. The third critical factor is

u . We are familiar with the

_SS Greeks in options pricing and hedging. In the Frey model, the parameter

u

_SS represents the value of gamma and it is also a crucial factor in the model.

First,

u plays a role in Equation (3) & (4) and thus it would affect the asset

_SS dynamic. If we want to simulate the sample path of the stock price afterward, the parameter

u must be calculated by FDM before the simulation. Second, it goes

_SS without saying that

u has a great influence on the size of hedging error and the

_SS large trader’s trading strategy. In section 2.3, we will demonstrate the relationship between the parameter

u and tracking error with mathematical equation.

_SS

There are two major trading strategy which include positive feedback trading and contrarian feedback trading. When

u

_SS > the large trader adopting the positive 0 feedback trading strategy. On the other hand, the larger trader employs the contrarian feedback trading strategy as

u

_SS < . Moreover, 0

u is also vital when we discrete

_SS

18 Kamara and Miller (1995) show that the relationship between moneyness and liquidity is asymmetric. Etling and Miller, Jr. (2000) also state that although the maximum value of liquidity is near the money, liquidity does not decrease symmetrically as strike price move away from at the money (ATM).

the Frey model in numerical computation and the detail of this part will be presented in section 2.4.

The Frey Model (nonlinear parabolic PDE)

We denote that there is a solution uof the Frey nonlinear PDE model. Specifically, the Frey model and terminal condition is given by

( )

∂ is the large traders’ strategy and it must satisfies a critical assumption with

ρλ

( )

S Su

_SS( , ) 1

t S

< . Obviously, we observe the Frey model setting risk-free rate equal to zero and illustrate the PDE formula under risk neutral measure without drift term (i.e. risk-free rate equal to zero). However, we can improve the Frey model by taking risk-free rate into account, making the pricing model more general than Frey’s and we present the form of the Frey model with risk-free rate. For tractability, we still assume that the parameter, risk-free rate, equal to zero the same as Frey’s model setting for “parsimonious principle” in numerical analysis.

2.3 Tracking error (hedging error)

Firstly, we realize the large traders following the trading strategy with

α

_t =

u t S

_S( , ) and the volatility of asset price is

( )

employ the Ito formula to uso that we can obtain

1 ^{2 2}

where we denote ( ,

u T S

_T)is the payoff of derivative at maturity day. Hence, ( _T) ( , _T)

h S

=

u T S

and the payoff of derivative at maturity day can be represented in this form Assuming we have already known the Frey nonlinear PDE model. If the Frey model holds, we can eliminate the last term in the right hand side (abbreviated RHS) of the Equation (14). Now we denote the tracking error

e

_T^M =

h S

( _T)−

V

_T^M . The tracking error measures the difference between the terminal payoff of the European option (i.e. (

h S

_T)) and the replication of derivative (i.e.

V

_T^M) which duplicated by bond and stocks with the self-financing trading strategy. Using the tracking error can easily track and judge the performance of hedging strategy. In fact, we regard (

h S

_T) and

M

V

T as the total cost and the total revenue respectively in economic sense.

We conclude that a positive value of

e displays the large trader who loss the

_T^M money in the hedging strategy, meaning the payoff of the replication of derivative can not completely cover the payment of European style option at maturity date. Thus, the large traders suffer loss from under-hedging at maturity day.

According to self-financing trading strategy, we can obtain the option payoff at terminal time by the following representation

0 0

( _T( , )) ^T _{t t}( , )

h S ρ α =V +

∫

αdS ρ α , (15) where ( ,

α

_t =

u

_S^BS

t S

_t) and

V

₀ =

u

^BS(0,

S

₀). Next we demonstrate that the tracking error 0

e

_T^M = under BS world and as self-financing trading strategy holds.

(

^{0 0}

)

Frey (2000) demonstrates that the Black-Scholes hedging is costly under imperfectly liquid market and the tracking error is always absolutely positive-value in BS world. We can display the tracking error in the following form¹⁹

( )

tracking error is positive. When

u

_SS^BS < , we can get the same result in this integral 0 form. In a word, the value of the tracking error is always positive if the large trader uses the Black-Scholes option pricing model in illiquid market.

2.4 Numerical method applications in the Frey model

In this section, we are interested in how to discretize the Frey model which is a complex nonlinear PDE. Obviously, the coefficient of the Frey model is an unknown number which includes the solution that we want to solve it. In Frey and Patie (2001), they use the Newton method to solve the whole nonlinear system. However, using the Newton method might quite sophisticate and spending much more computational time in programming procedure. As a result, we provide an alternative approach that transfers the nonlinear problem into the linear system and this approach can reduce the computational costs. We will demonstrate the detail of methodology in the

19 Frey (2000) shows the basic concept of tracking error. Theoretically, the tracking error can be treated as a “cumulative dividend stream with instantaneous dividend.” By the way, we should notice that Frey (2000) does not consider the parameterλin the model and in the tracking error.

following.

Firstly, we use the explicit method for the calculation of the coefficient. After we solve the coefficient in the first step, the nonlinear PDE becomes the linear PDE and therefore the coefficient of the model is known at this moment. Secondly, we use the implicit method²⁰ to solve every linear system at each time step. Recall the Frey model without the risk-free rate (zero drift)

( )

However, we impose the risk free rate term in the model which improves the model more general than the Frey model.

( )

Using the finite difference methods, the Frey model can be represented in this form

1

( )

1 1 1 1 1 time i−1. Therefore, all the gird of call option value can be obtained by FDM easily.

Figure 1 displays the basic concept of the implicit method.

20 See Appendix, the implicit method is unconditional stable. Thus, we purpose to solve the Frey model via the implicit method.

[Insert Figure 1 here]

2.5 The smooth version of nonlinear PDE

We find a serious problem in this term (1−

ρλ

( )

S Su

_SS)as we check the numerical data and find that this term violates the basic assumption “

ρλ

( )

S Su

_SS( , ) 1

t S

< ” in programming process. If we do not address this numerical problem, the nonlinear PDE will display a non-smooth solution in option intrinsic value and generating bad numerical solution. Therefore, using some skill in nonlinear PDE, we revise the violation of the basic assumption by the following form:

2 version of nonlinear PDE. Frey and Patie (2001) do not explain how the parameter would be selected and the detail of methodology does not appear in their paper. If we do not impose artificial conditions in the PDE model, the denominator of second coefficient²² in Equation (21) could be greater than one. Frey and Patie (2001) provide this approach to settle the non-smooth numerical solution problem in option pricing. General speaking, the more smooth PDE we have, the more precise solution we get.

The second coefficient of PDE in Equation (21) is controlled by some of the factors and the numerical boundary value are governed by

α

₀ and

α

₁. We can treat

α

0and

α

₁as the artificial condition or the barrier. We state that

α

₁is the maximum value of

ρλ

( )

S Su

_SS and use the parameter

α

₁to control (1−

ρλ

( )

S Su

_SS) and thus

21 Two artificial conditions are imposed into the PDE model.

22 The term is min(

α ρλ

₁, ( )

S Su

_SS).

this term will not be negative anymore. Hence, we obtain smooth solution in the Frey model. Secondly, the parameter

α

₀control the volatility term in nonlinear PDE. The term

σ α

² ₀²can be explained by the minimum value of the volatility under illiquidity market condition. The smooth version of PDE proposed by Frey and Patie (2001) sounds great but this approach might affect the stability of the numerical solution.

Since the disadvantage of this revised PDE is too artificial that designedly limit the numerical value in certain boundary to avoid the violation of the basic assumption.

We test the call option value, delta (i.e., first derivative) and gamma (i.e., second derivative) and finding the value of gamma will explode as

ρ

>0.4 . This phenomenon tells us that the smoothed version of PDE is absolutely not a unique approach as we calculate the second derivative. The second derivative represents the variation of the hedging strategy for large trader. Consequently, we should carefully use the smoothed version of PDE when we need to build up a hedging strategy in the certain period.

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