Optimal Exercise Boundary of American Fractional Lookback Option in a Mixed Jump-Diffusion Fractional Brownian Motion Environment

A new framework for pricing theAmerican fractional lookback option is developed in the casewhere the stock price follows amixed jump-diffusion fraction Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions.The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.


Introduction
The mixed fractional Brownian motion (mfBm) is a family of Gaussian processes, which is a linear combination of Brownian motion and fractional Brownian motion.The mfBm models have been extensively studied in the literature [1][2][3][4][5][6][7][8][9].Cheridito [2] derived an European call pricing option on an asset driven by a linear combination of a Brownian motion and an independent fractional Brownian motion.Based on this theory, Sun and Yan [9] proposed a mixed fractional Brownian motion version of a credit risk pricing Merton model and assumed that the value of the firm obeys a geometric mixed fractional Brownian motion; the result shows that the mixed fractional model to simulate credit risk pricing is a reasonable one.The case even for the fractional Brownian motion with arbitrary Hurst parameter and Wick products of the fractional Black-Scholes model have been proposed as an improvement of the classical Black-Scholes model [10][11][12].As is known the Black-Scholes model is inadequate to describe the asset returns and the behavior of the option markets.Chang et al. [13] proposed a jump-diffusion process with Poisson-jump to match the abnormal fluctuation of stock price.Several authors [14][15][16][17][18] also considered the problem of pricing options under a jumpdiffusion environment in a larger setting.Actually, various empirical studies on the statistical properties of log-returns show that the log-returns are not necessarily independent and also not Gaussian.One way to a more realistic modelling is to change the geometric Brownian motion to a geometric fractional Brownian motion: the dependence of the logreturn increments can now be modelled with the Hurst parameter of the fractional Brownian motion.It can be said that the properties of financial return series are nonnormal, nonindependent, and nonlinear, self-similar, with heavytails, in both autocorrelations and cross-correlations, and volatility clustering [2][3][4].Since fractional Brownian motion has two substantial features such as self-similarity and longrange dependence, using it is more applicable to capture behavior from financial asset [3].Also, the fractional Brownian motion is neither a Markov process nor a semimartingale, and thus we cannot apply the common stochastic calculus to analyze it.To get around this problem and to take into account the long memory property, it is reasonable to use the mixed fractional Brownian motion with jumps model to capture fluctuations of the financial asset (see [19][20][21]).
Fortunately, Hu and Øksendal [12] employed the Wick product rather than the pathwise product to define a fractional stochastic integral whose mean is indeed zero.This property was very convenient for both theoretical developments and practical applications.Further, in [10], it was stated that if one uses the Wick-Itô-Skorohod integral, then they can obtain an arbitrage-free model; while Wick integration leads to no-arbitrage, the definition of the corresponding self-financing trading strategies is quite restrictive.Therefore, the fractional market based on Wick integrals is considered which is a beautiful mathematical construction but with restricted applicability in finance.
Further, to capture jumps or discontinuous, fluctuations problem or take into account long memory property, we present here a new method to solve option pricing problem for American lookback option in a mixed jump-diffusion fractional Brownian motion environment.It is different from the mixed fractional Brownian motion ones which are based on a linear combination of a Brownian motion and an independent fractional Brownian motion.We establish mixed jump-diffusion fraction Brownian motion.The mixed jump-diffusion fraction Brownian motion is a linear combination of Brownian motion, fraction Brownian motion, and Poisson process.By using Itô formula and fractional Wick-Itô-Skorohod integral a new market pricing model is built, and the fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions.Then the explicit integral representation of early exercise premium and the American fractional lookback options factorization formula are also given.Based on some researches of lookback options pricing and early exercise premium in the literature [22][23][24][25][26][27][28], to achieve quick and accurate pricing for practical purposes, this paper adopts the critical exercise price to value American fractional lookback options, and numerical simulation illustrates some notable features of American fractional lookback options.
The rest of this paper is organized as follows.In Section 2, we present some basic lemmas and preliminary results of mixed jump-diffusion fractional pricing model and Wick-Itô-Skorohod integral which will be used throughout this paper.In Section 3, the fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions and the optimal exercise boundary is derived; then the American fractional lookback options factorization formula is obtained.In Section 4, some simulation results and notable features are provided.The paper is ended with conclusive remarks in the last section.

Mixed Jump-Diffusion Fractional Brownian Motion Pricing Model and Wick-Itô-Skorohod Integral
In this section we construct mixed Poisson-jump-diffusion processes as a suitable alternative to fractional Brownian motion.
Definition 1 (see [1,3,5]).A mixed fractional Brownian motion of parameters , , and  is a linear combination of fractional Brownian motion (fBm) with Hurst parameter  and Brownian motion, defined on the probability space (Ω, F, P) for any  ∈ R + by where   is a Brownian motion,    is an independent fBM with  ∈ (0, 1), and ,  are two real constants such that (, ) ̸ = (0, 0).Now we list some properties in [1] by the following proposition.

Proposition 2. The mfBm 𝑀 𝐻
satisfies the following properties: is a centered Gaussian process and not a Markovian one for all  ∈ (0, 1) \ 1/2; (ii)   0 = 0 P-almost surely; (iii) the covariation function of    (, ) and    (, ) for any ,  ∈ R + is given by where ∧ denotes the minimum of two numbers; (iv) the increments of    (, ) are stationary and mixedself-similar for any ℎ > 0 where ≜ means "to have the same law"; (v) the increments of    are positively correlated if 1/2 <  < 1, uncorrelated if  = 1/2, and negatively correlated if 0 <  < 1/2; (vi) the increments of    are long-range dependent if, and only if,  > 1/2; (vii) for all  ∈ R + , we have Proof.These properties are easily obtained based on [1,2].Now, let (Ω, F, P) be a probability field such that   is a Brownian motion with respect to P and    is an independent fractional Brownian motion with respect to P. Some results presented that is needed for the following Lemma [2,3,5].Lemma 3.For every 0 <  <  and  ∈ C one has where Ẽ denotes the quasi-conditional expectation with respect to the risk-neutral measure.
Lemma 4. Let  be a function such that Ẽ [(  ,    )] < ∞.Then for every 0 <  ≤  and  ∈ C, Let  1 ,  2 ∈ R. Consider the process where  1 and  2 are two real constants such that ( 1 ,  2 ) ̸ = (0, 0).From Girsanov's theorem, there exists a measure P * , such that Z *  is a new mfBm.We denote by  *  [⋅] the quasiconditional expectation with respect to P * as follows: Lemma 6.Let  be a function such that Ẽ [( ).Then for every  ≤ , we have Lemma 7. The price at every  ∈ [0, ] of a bounded   measurable claim  ∈  2 is given by where  represents the constant riskless interest rate.

Mixed Jump-Diffusion
Proof.According to Taylor expansion we have = ; since we can approximate that (d   ) 2 = 2 2−1 d, (d  ) 2 = d, (d) 2 = 0, dd  = 0, we get Mathematical Problems in Engineering Combining (14) with (15) we get where   is the unconditional expectation of   , and  2  is the variance of ln(1 +   ).Notice that the unconditional expectation   is deterministic and can be calculated;   is a bounded function of , so we assume that  3 = ln(1 +   ) is a constant.By Lemma 8 we have Lemma 9 as follows.
Lemma 9. Let   = (  , ) be a binary differential function; if stochastic process   is suitable for stochastic differential equation then where  3   (/)d  is the change volume of Poisson-jump process within d for /, E[((1 +   ), ) − (, )]d is the change volume of Poisson-jump process within d, and E is the expectation operator of .
Proof.Let d  admit two-point distribution as follows: Moreover, during the time interval [, +d], we can write the probability that jumps can not occur as Prob( 1 ) = 1−d and the probability that jumps can occur as Prob( 2 ) = d.
In the case   > 0,  + >   describes stock price   has upward jump at time .In the case   > 0, while  + <   describes stock price   has downward jump at time .Hence   > −1 can ensure the stock price is positive, such that  + =   (1 +   ) > 0.
Let Π  =   − ∇    be a riskless portfolio and ∇  is stock shares at time .In a complete financial market, there are no risk-free arbitrage opportunities.Then No matter the jumps occur or not, by Merton assumptions ln(1 ] and model ( 17), the variance  2  exists surely.Either ( = ) > 0 for all  > 0 or ( = ) = 0 for all  > 0. In the first case, every positive point can be hit continuously in  and this phenomenon is called creep [18].In the second case, only jumps can occur (almost surely).In time interval [,  + d], the pricing satisfies the following hypothesis: (i) If jumps can not occur, for the events  1 , by   = (  , ) second-order differentiable, hence we can use Itô formula; then (ii) If jumps can occur, for the events  2 , we have Then Let ∇  = (/)| (  ,) take expectations for   on both sides of above equation, and cancel items of d 2 ; then we have a parabolic partial differential equation as follows: and then under the environment of mixed jump-diffusion fractional Brownian motion, lookback option pricing model can be expressed as a parabolic integral equation which contains expectation.
Definition 10 (see [10]).Let : R → () * be a given function such that   ⬦    is d-integrable in () * .The Wick-Itô-Skorohod integral of   with respect to    is given by where ⬦ is the Wick product and    is the fractional Brownian motion.

Mathematical Problems in Engineering
Suppose the price of the risky asset   and interest rate   satisfy the following equation: By Definition 10-Lemma 12 we can consider the mixed jump-diffusion fractional stochastic differential equation where  is the instantaneous expected return,  ≥ 0 is the continuous dividend rate,  1 and  2 are denoted as in Lemma 5, and  3 is the unconditional expectation of   .Assume that   ,    , and   are independent.Let  be the path-dependent variable.For looking back put option on the maturity , the stock price   satisfies   ≤   = max 0≤≤   ; for looking back call option on the maturity , the stock price   satisfies   ≥   = min 0≤≤   .Thus, the looking back put option value (  ,   , ) is the function of , , and .Then we construct a riskless portfolio Π =  − ∇ ⬦ , select the appropriate variable ∇, and make the investment portfolio Π in the interval (,  + d) risk-free.Then dΠ = ( − ∇ ⬦ )d; according to Itô's formula we have Notice that   is nondifferentiable about , and define By ∇ = /, then where  ≤   and E is the expectation operator of .

Main Results
In this section we present the American lookback options factorization formula and the optimal exercise boundary.
Lemma 13 (see [29], Theorem 4.3).The price of a derivative on the stock price with a bounded payoff (  ) is given by (,   ), where (, ) is the solution of the PDE: Lemma 14.The solution of stochastic partial differential equations ( 43) and ( 44) is where E  denotes the expectation operator over the distribution of ∏  =1 (1 +    ).
Corollary 16.By Lemmas 13 and 14 and Theorem 15, if one notes G * (, ; , ) is the fundamental solution of ( 55) and ( 56), then G (, ; , ) = G * (, ; , ) . (57) Proof of Theorem 15.For any  > 0, consider integral as follows: When  → 0, we have where   is a stopping time of the filtration F and the conditional expectation is calculated under the risk-neutral probability measure P. The random variable  *  ∈ [, ] is called an optimal stopping time if it gives the supremum value of the right-hand side of (63).It is clear from (63) that C is nondecreasing in  and nonincreasing in , m, and a. Solving the optimal stopping problem (63) is equivalent to finding the points (,   , m  ) for which early exercise before maturity is optimal.Let be the whole domain, and let E and C denote the exercise region and continuation region, respectively.In terms of the value function C(, , m), the exercise region E is defined by for which the optimal stopping time  *  satisfies The continuation region C is the complement of S in D, such as The boundary that separates S from C is referred to as the early exercise boundary, which is defined by At the early exercise boundary [(, m)] ∈[0,] , the American fractional lookback call option would be optimally exercised.
In terms of (, m), the continuation region C can be represented as Let (, , ) be the lookback option price at time  with stock price  and path-dependent variable .Using argument similar to Section 2.2, it can be shown that the American fractional lookback option price solves the following stochastic partial differential equations: where 0 ≤  ≤ , and satisfies an order continuous differentiable on domain and the terminal condition is given by We define the differential operator L , by Then the free boundary problem can be written in a linear complementary form as together with auxiliary conditions For the free boundary [(, m)] ∈[0,] , this problem is equivalent to solving the Black-Scholes-Merton partial differential equations together with the boundary conditions and the terminal condition In the same way as in the call case, by (63), (70), (71), (72), (76), (77), and (78), we can formulate the put case: Let P = P(, , m) be the value of the American fractional lookback call option at time  ∈ [0, ].The value P(, , M) is a solution of an optimal stopping problem and P(, , M) satisfies the same PDE as (76); then (82)

Optimal Exercise Boundary.
It is well known that the value of an American fractional option can be represented as the sum of the value of the corresponding European option and the early exercise premium.For American fractional lookback options, Lai and Lim [28] proved that the value has such a decomposition and that the premium has an integral representation.Applying the same solution method as in Theorem 15 to the PDE (82) for P(, , M), we can obtain the early exercise representation (, , M) under our mixed jump-diffusion fractional Brownian motion environment, which is shown in the following theorem.
Theorem 17.Let (, , M) be American fractional lookback put option; then where   (, , M) is price of the hedging portfolio of the equivalent European option, and where E  denotes the expectation operator over the distribution of ∏  =1 (1 +    ), M is contractual strike price, b is positive constants, N(⋅) is the cumulative normal distribution function, and where L * is the adjoint Black-Scholes operator, and when  → 0, we have Then the proof is finished.
Remark 18.If the optimal exercise boundary is given by  =   , then pricing of American fractional lookback options can be represented as (83)-(86).
Remark 19.This results show that American fractional lookback options are equal to the option of a hedging portfolio: a risk premium associated with the European values plus the early exercise premium.Using the similar techniques, the optimal exercise boundary can be given by the following corollary.

Corollary 20. The optimal exercise boundary of looking back
American fractional put options is defined by  =   , 0 ≤  < , and then  =   satisfy the following Volterra integral equation ) q = 0.06 q = q = 0.04 q = 0.04 q = 0.02 q = 0.02 Similarly, the critical exercise price for the American fractional floating strike lookback call option should be denoted as where (, m) is a monotonically decreasing function of time .The plots of (, M) and (, m) against time  with varying interest rate are shown in Figure 1, where m = M = 100,  = 1,  = 0.05,  = 0.02, 0.04, 0.06,  1 = 0.2,  2 = 0.3,  3 = 0.0071,  = 2.68, a = 1.2, and b = 0.8.Since lower value of interest rate leads to the loss of the time value of the floating strike price to be smaller when the American fractional floating strike lookback call option is exercised prematurely, then the critical exercise price decreases when the interest rate assumes lower value.The similar phenomenon also appeared in [30].
The limiting behaviors at times close to maturity of the critical exercise prices for the American fractional floating strike lookback put and call options are, respectively, given by lim Hence the strike prices of the American fractional lookback put and call options are set to be M  0 and m  0 , respectively.The usual argument and proof of analysis of limiting behaviors can be applied in a similar manner like literature [28,30].
With respect to the early exercise premium of the American fractional lookback option, we intuitively suppose that it strongly depends on the sign of the drift coefficient of the price process (  ) ≥0 .If  −  −  3 < 0, then   decreases to m  as  →  in an average sense, so that (  − am  ) + → 0 and (bM  −   ) + grows as  →  in the same sense.This implies that early exercise before maturity is optimal for call but not for put, at least in the average sense.On the contrary, if  −  −  3 > 0, then the reverse would be true.For put options with  −  −  3 < 0 and call options with  −  −  3 > 0, the early exercise premiums should be negligible, and hence there should be almost no difference between the values of American and European options.To confirm this expectation, we compare the values of American and European options with  −  −  3 < 0 in Figure 2 for call and put case, where American (European) values are drawn in blue (red) lines.Further numerical experiments confirmed that the same expectation is realized for the critical case  − − 3 = 0, too.Also, we observed from these figures that the call (put) values are monotonically decreasing (increasing) in Figure 2, the option values are convex functions of , and the option values are relatively insensitive to the asset price  when the option is out-of-the-money.Notice that the call (put) values are almost increasing (decreasing) in  when the option is in-the-money but not when  = M and b = 1.From other numerical experiments, we can conclude that the adjustable constants a ≥ 2 and b ≤ 0.5 are practically insignificant, because the option values are negligible for those cases.

Conclusions
In this paper, we compound the Brownian motion, fraction Brownian motion, and Poisson process by fractional Wick-Itô-Skorohod integral.In our new pricing model, the fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions.By solving the free boundary problem and the Black-Scholes-Merton partial differential equations, we characterized asymptotic behaviors of the early exercise boundaries at a time close to expiration and at infinite time to expiration.Since the fundamental solution of stochastic partial differential equations plays an important role in numerical inversion for the put and call case, then we characterized asymptotic behaviors of the early exercise boundaries at a time close to expiration and at infinite time to expiration.The numerical simulation reveals some notable features of the fractional lookback options: (1) For call (put) options with  ≥  +  3 ( ≤  +  3 ), there is little difference in value between American and European options.This readily suggests that investors should buy American options, because the holder of American options has a benefit of clearing his investment position at any time, even if he unexpectedly encounters the opportunity of terminating the contract.The early exercise boundary is of practical value for making decisions by the option holders.(2) The fractional lookback scheme with the adjustable constants a ≥ 1 and b ≤ 1 is extremely effective for reducing option premiums, especially for low volatile cases.With a = 1.2 and b = 0.8, we can achieve larger reductions of the option premiums from the standard cases with a = b = 1.The fractional lookback options must be more attractive to investors than the standard floating strike options.
Fractional Brownian Motion Pricing Model.Suppose an option whose value   = (  , ) depends only on   , , and option pricing formula while the dynamics of stock log-price   satisfiesd  =   d +  1   d   +  2   d  , (12)where  is the instantaneous expected return,  1 and  2 are the instantaneous volatility,   is a standard Brownian motion, and    is a fractional Brownian motion.Assume that   and    are independent.
Consider a continuous-time financial market in [0, ].It can be described by a filtered complete probability space {Ω, F, F  , P}. {F  } 0≤≤ ≡ F is a natural -filtration generated by a standard Brownian motion   , fractional Brownian motion    , and a Poisson process   .Here   is an (F  , P)-Poisson-jump process with intensity , independent of   and .  is jump percent at time  and i.i.d.;   satisfies Merton assumptions ln(1 Hence   is continuous function with , for  → ∞, then lim →∞   () = max 0≤≤   =   , and we get 1/ , such that approximate amount   () is differentiable about  and satisfies [23]Early Exercise Boundary of American Lookback Option.Conze and Viswanathan[23]introduced a fractional or partial lookback option, where the strike is fixed at some fraction over (for a call) or below (for a put) the extreme value.Specifically, the payoffs for European lookback call and put with fractional floating strikes and maturity date  are given, respectively, by (  − am  ) + and (bM  −   ) + , where a and b are positive constants, allowing flexible adjustment of option premiums.To reduce option premiums, we assume that a ≥ 1 and 0 < b ≤ 1.Given a finite time horizon  > 0, let C = C(, , m) be the value of the American fractional lookback call option at time  ∈ [0, ].Note that the values of American and European call options are equal if the underlying asset pays no dividends.In the absence of arbitrage opportunities, the value C(, , m) is a solution of an optimal stopping problem C (, , m)