Pricing Perpetual American Put Option in the Mixed Fractional Brownian M

Mathematical Computation June 2015, Volume 4, Issue 2, PP.41-45

Pricing Perpetual American Put Option in the Mixed Fractional Brownian Motion Feng Xu Business School, Suzhou Vocational University, 215104, China Email: xfoooym@163.com

Abstract Under the assumption of the underlying asset is driven by the mixed fractional Brownian motion, we obtain the mixed fractional Black-Scholes partial differential equation by fractional Itô formula, and the pricing formula of perpetual American put option by this partial differential equation theory.

Keywords: Mixed Fractional Brownian Motion; Perpetual American Put Option; Mixed Fractional Black-Scholes Model; Option Pricing

1 INTRODUCTION The Black-Scholes model has be the most popular method for option pricing and its generalized version has provided mathematically beautiful and powerful results on option pricing. However, behavioral finance as well as empirical studies shows that there exists long-range dependence in stock returns and verifies that long-range dependence is one of the genuine features of financial markets. Behavioral finance also suggests the return distributions of stocks are leptokurtic and have longer and fatter tails than normal distribution and there exists long-range dependence in stock returns. These features have some differences with the standard Brownian motion, while are in accordance with the fractional Brownian motion. The fractional Black-Scholes model is a generalization of the Black-Scholes model, which is based on replacing the standard Brownian motion by a fractional Brownian motion in the Black-Scholes model.

However, the fractional Brownian motion is neither a Markov process nor a semi-martingale (except in the Brownian motion case), and we cannot use the usual stochastic calculus to analyze it. Although some research have testified that the purely fractional Black-Scholes market becomes arbitrage-free with Wick-self-financing strategies [1, 2], Björk and Hult [3] showed recently, that the use of fractional Brownian motion in finance does not make much economic sense, while Wick integration leads to no-arbitrage, the definition of the corresponding self-financing trading strategies is quite restrictive and, for example, in the setup of [4], the simple buy-and-hold strategy is not self-financing. Thus, the fractional market based on Wick integrals is considered a beautiful mathematical construct but with limited applicability in finance. From the above analyses, it is quite evident that the classical Itô theory could not apply to fractional Brownian motion and defining a suitable stochastic integral with respect to fractional Brownian motion is difficult. Actually, the main problem of applying fractional Brownian motion in finance is that fBm is not a semi-martingale. To get around this problem and to take into account the long memory property, it is reasonable to use the mixed fractional Brownian motion (mfBm hereafter) to capture fluctuations of the financial asset [5, 6].

Pricing and hedging American option is an important yet difficult problem in the finance research. As we know, the perpetual American put is the simplest interesting American option. It is interesting because the optimal exercise policy is not obvious, and it is simple as we will know, because this policy can be determined explicitly.

The remainder of this paper is organized as follows. In section 2, we briefly state the formal definitions and properties related to mfBm that will be used in the forthcoming sections. In section 3, we use a fractional Itô formula to obtain the mixed fractional Black-Scholes partial differential equation. In section 4, a closed-form solution for the - 41 www.ivypub.org/mc

perpetual American put options with mfBm is obtained.

2 PREPARATION KNOWLEDGE Before describing the model, we first review some background concerning mfBm. For a more detailed treatment, we refer to Cheridito [7] and Zili [8].

Definition2.1 A mixed fractional Brownian motion of parameters α , β and H is a linear combination of Brownian motion and fractional Brownian motion, defined on the probability space ( Ω, F , Ρ ) for any t ∈ R+ by: H M= α Bt + β BtH , t

Where Bt is a Brownian motion, BtH is an independent fBm with Hurst parameter H ∈ ( 0,1) , α and β are two real constants such that (α , β ) ≠ ( 0, 0 ) .

Now we list some properties by the following proposition.

Proposition 2.1 The mfBm M tH satisfies the following proposition. i) M tH is a centered Gaussian process and not a Markovian one; ii) M 0H = 0 P-almost surely; iii) The covariance function of M tH (α , β ) and M sH (α , β ) for any t , s ∈ R+ is given by

(

)

Cov M tH , M sH= α 2 ( t ∧ s ) +

β2 2

(t

2H

+ s2H − t − s

2H

),

where ∧ denotes the minimum of two numbers; iv) The increments of M tH (α , β ) are stationary and mixed-self-similar for any h > 0 M htH

(α , β ) 

M tH

 1 α h 2 , β hH  

 ,  

where  means “to have the same law”; v) The increments of M tH are positively correlated if if 0 < H <

1 1 < H < 1 , uncorrelated if H = and negatively correlated 2 2

1 ; 2

vi) The increments of M tH are long-range dependent if, and only if H >

1 .

2

3 THE MIXED FRACTIONAL BLACK-SCHOLES EQUATION FOR PERPETUAL AMERICAN PUT OPTION Since mfBm is a well-developed mathematical model of strongly correlated stochastic processes, it is the most efficient tool for capturing the long memory behavior of the financial asset. In what follows, we state some basic assumptions that will be used in this paper.

3.1 Assumptions i) These are no transaction costs or taxed and all securities are perfectly divisible; ii) Dividends are not paid during the time of the outstanding options; iii) A money market account dM t = rM t dt , M 0 = 1, 0 ≤ t ≤ T ,

where r is a constant risk-free rate.

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iv) The dynamics of the stock price process takes the following form

(

)

dSt = µ St dt + St σ dBtH + e dBt ,

where µ is the mean rate of return, σ and e are constants. Bt is a Brownian motion, and

BtH

(1) is an independent fBm

1  with Hurst parameter H ∈  ,1 . 2 

The solution of this stochastic differential equation (1) is 1 1   (2) St S0 exp  µ t + σ BtH + e Bt − σ 2t 2 H − e 2t  = 2 2   Let V = V ( t , St ) be the value of a perpetual American put option on the above underlying stock at time t with exercise

price K . A portfolio θ = (θ1 ,θ 2 ) is given by the number of units of the money market account M t , and risky

asset St respectively. The value process V ( St ) of a trading strategy θ is

V= ( St ) θ1M t + θ 2 St .

(3)

Then dV= ( St ) θ1dM t + θ 2 dSt

(

(

)) + e dB ) .

=θ1rM t dt + θ 2 µ St dt + St σ dBtH + e dBt

(

= (θ1rM t + θ 2 µ St ) dt + θ 2 St σ dBtH On the other hand, using Equation (2) and fractional Itô formula, we have

t

(4)

 1  ∂V  ∂V 1   ∂ 2V ∂V   dV ( St ) =  µ − H σ 2t 2 H −1 − e 2  St σ dBtH + e dBt +  H σ 2t 2 H −1 + e 2   St2 2 + St  dt  dt + St 2  ∂St  ∂St 2   ∂St ∂St     ∂V  1  ∂ 2V  ∂V (5) = +  H σ 2t 2 H −1 + e 2  St2 2  dt + St σ dBtH + e dBt .  µ St ∂St  2  ∂St  ∂St  Comparing Equation (4) and (5), we obtain

(

)

(

)

1  ∂V  2 2 H −1 + e 2  St2 2  Hσ t 2  ∂St  ∂V . ,θ 2 θ1 = = rM ( t ) ∂St 2

According to Equation (3), then the following equation is derived.

θ1 =

V ( St ) − St M (t )

∂V ∂St

.

Then the mixed fractional Black-Scholes equation for perpetual American option is giving by the following equation.

1  ∂ 2V ∂V  2 2 H −1 + ε 2  St2 2 + rSt − rV = 0.  Hσ t 2  ∂St ∂St  1 In particular, if= H = , e 0 , from Equation (6) we have 2 ∂V 1 2 2 ∂ 2V + rSt − rV = σ St 0, 2 ∂St 2 ∂St

(6)

which is the Black-Scholes equation for perpetual American option. We may also rewrite equation (6) in the form which resembles the Black-Scholes equation ∂V 1 2 2 ∂ 2V σ St + rSt − rV = 0.

2 ∂St 2 ∂St - 43 www.ivypub.org/mc

Where the modified Volatility is given by 2 H σ 2t 2 H −1 + ε 2 .

= σ

(7)

Moreover, noted that the boundary conditions, we have σ 2 2 ∂ 2V ∂V 0, + rSt − rV =  St 2 ∂St ∂St  2 ) K − L, V ( L= V ( ∞ ) =0.  

(8)

4 THE PRICING FORMULA FOR PERPETUAL AMERICAN PUT OPTION IN THE MIXED FRACTIONAL BROWNIAN MOTION MODEL Now, we will prove Theorem4.1 The function V ( St ) is given by K − S , 2r  V ( St ) =  L σ   ( K − L ) .  S 

if

L≥S ≥0

if

S≥L

2

(9)

Proof According to the ODE Theory, we firstly need to find two linear independent particular solutions. Let V = xα , and we get

σ 2



σ 2 

0, α − r = 2   2r it has two solutions: α = 1, and α = − 2 , so the basic solution can be written as σ 2

α 2 +  r −

V ( x= ) Ax + Bx

−

(10)

2r

σ 2 .

(11)

By the boundary condition in (8), we know 2r

= A 0,= B Lσ

2

( K − L ).

Then (11) becomes 2r

 L σ = V ( S )   ( K − L). S * Theorem4.2 The optimal exercise boundary L and the value of perpetual American put option V ( S ) are given by 2

2r

2r σ 2  2r σ , L K V S = = ( ) 2   K σ 2 + 2r σ + 2r  σ 2 + 2r  Proof Firstly, the optimal exercise boundary L* satisfy 2

*

(

2r

)

2r σ 2 + 2 r − σ 2 S σ 2 .

 L σ = V S, L = max V ( S , L ) max   ( K − L ).

0≤ L ≤ K 0≤ L ≤ K  S  It is easy to know that L* is the value of L that maximizes this quantity. We thus define *

2

f ( L= )

2r

( K − L ) Lσ

And seek the value of L that maximizes this function over L ≥ 0 .

Because - 44 www.ivypub.org/mc

2

2r

f ′( L) = − Lσ + 2

2r

2r ( K − L ) Lσ σ 2

2

−1

,

Setting this equal to zero, we have L* =

2r

σ + 2r 2

K.

Furthermore,

( )

2r

σ 2

 2r σ f L = 2   K σ + 2r  σ 2 + 2r  *

Because

2r

σ 2

>0

2

σ 2 + 2 r σ 2

.

, we have f ( 0 ) = 0 . Therefore, when L = L* , V has a maximal value, that is

σ 2

2r

 2r σ V (S ) = 2   K σ + 2r  σ 2 + 2r  2

σ 2 + 2 r 2r − σ 2 S σ 2 .

5 CONCLUSIONS Without using the arbitrage argument, we obtain a perpetual American put option pricing formula for the mixed fractional Brownian motion. It has been shown that Hurst exponent H and e play an important role in option 1 pricing. In particular, when e = 0 and H = , the results is exactly the same as Refs[9].

2

ACKNOWLEDGMENT This work is supported by Innovation Project of Suzhou Vocational University of China. (No. 2013SZDYY05)

REFERENCES [1]

Cheridito P. “Arbitrage in fractional Brownian motion models.” Finance Stoch. 7(2003) 533-553

[2]

Christian B, Robert J E. “Arbitrage in discrete version of the wick-fractional Black-Scholes market.” Math. Oper. Res. 29(2004)

[3]

Björk T, Hult H. “A note on Wick products and the fractional Black–Scholes model.” Finance Stoch. 9 (2005) 197-209

[4]

Elliott R J, Van der Hoek J. “A general fractional white noise theory and applications to finance.” Math. Finance 13 (2003) 301-

935-945

330 [5]

EI-Nouty C. “The fractional mixed fractional Brownian motion.” Statist. Probab. Lett. 65 (2003) 111-120

[6]

Mishura Y. “Stochastic Calculus for Fractional Brownian Motions and Related Processes.” Springer, 2008

[7]

Cheridito P. “Mixed fractional Brownian motion.” Bernoulli 7 (2001) 913-934

[8]

Zili M. “On the mixed fractional Brownian motion.” J. Appl. Math. Stoch. Anal. 2006 (2006) 1-9

[9]

Merton Robert C. “Continuous-Time Finance.” Wiley-Blackwell, 1992

AUTHOR Feng Xu (1980- ) Male; Nationality: Han; Degree: Master Degree; Research Area: Financial mathematics.

Work Exeprence: 2008.09-present, Lecturer, worked in the Business School, Suzhou Vocational University.

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