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

Authors: Zhaoqiang Yang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

A new framework for pricing the American fractional lookback option is developed in the case where the stock price follows a mixed 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.


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## Body

## 1. 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 [1–9]. 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 [10–12]. 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 [14–18] also considered the problem of pricing options under a jump-diffusion 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 log-return 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 heavy-tails, in both autocorrelations and cross-correlations, and volatility clustering [2–4]. Since fractional Brownian motion has two substantial features such as self-similarity and long-range 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 [19–21]).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 [22–28], 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 Section2, 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.


## 2. 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 parametersα, β, and H is a linear combination of fractional Brownian motion (fBm) with Hurst parameter H and Brownian motion, defined on the probability space (Ω,F,P) for any t∈R+ by(1)MtH=αBt+βBtH,where Bt is a Brownian motion, BtH is an independent fBM with H∈(0,1), and α, β are two real constants such that (α,β)≠(0,0).
Now we list some properties in [1] by the following proposition.Proposition 2.
The mfBmMtH satisfies the following properties:(i)
MtH is a centered Gaussian process and not a Markovian one for all H∈(0,1)∖1/2;(ii)
M0H=0  P-almost surely;(iii)
the covariation function ofMtH(α,β) and MsH(a,b) for any t,s∈R+ is given by (2)CovMtH,MsH=α2t∧s+β22t2H+s2H-t-s2H,where∧ denotes the minimum of two numbers;(iv)
the increments ofMtH(α,β) are stationary and mixed-self-similar for any h>0(3)MhtHα,β≜MtHαh1/2,βhH,where≜ means “to have the same law”;(v)
the increments ofMtH are positively correlated if 1/2<H<1, uncorrelated if H=1/2, and negatively correlated if 0<H<1/2;(vi)
the increments ofMtH are long-range dependent if, and only if, H>1/2;(vii)
for allt∈R+, we have (4)EMtHα,βn=0,n=2l+12l!2ll!α2t+β2t2Hl,n=2l.Proof.
These properties are easily obtained based on [1, 2].Now, let(Ω,F,P) be a probability field such that Bt is a Brownian motion with respect to P and BtH 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 every0<t<T and σ∈C one has (5)E~teσBT+BTH=eσBT+BTH+1/2T-tσ2+1/2σ2T2H-t2H,where E~t denotes the quasi-conditional expectation with respect to the risk-neutral measure.Lemma 4.
Letf be a function such that E~t[f(BT,BTH)]<∞. Then for every 0<t≤T and σ∈C, (6)E~tfσBT+σBTH=∫R12πσ2T-t+T2H-t2Hexp⁡-x-σBt-σBtH22σ2T-t+T2H-t2H·fxdx.Lemma 5.
LetA∈B(R), f(x)=1A(x). Then(7)E~t1AσBT+σBTH=∫R12πσ2T-t+T2H-t2Hexp⁡-x-σBt-σBtH22σ2T-t+T2H-t2H1Axdx.Let σ1,σ2∈R. Consider the process (8)Zt∗=σ1Bt∗+σ2BtH∗=σ1Bt+σ12t+σ2BtH+σ22Bt2H,0≤t≤T,where σ1 and σ2 are two real constants such that (σ1,σ2)≠(0,0). From Girsanov’s theorem, there exists a measure P∗, such that Zt∗ is a new mfBm. We denote by Et∗[·] the quasi-conditional expectation with respect to P∗ as follows:(9)Et∗eσ1BT+σ2BTH∗=eσ1BT+BTH+1/2σ12T-t+1/2σ22T2H-t2H.Lemma 6.
Letf be a function such that E~t[f(σ1BT+σ2BTH)]<∞. Let Xt=exp(-σ1Bt-σ12/2t-σ2BtH-σ22/2t2H). Then for every t≤T, we have(10)E~t∗fσ1BT+σ2BTH=1XtE~tfσ1BT+σ2BTHXT.Lemma 7.
The price at everyt∈[0,T] of a bounded FTH-measurable claim F∈L2 is given by (11)Ft=e-rT-tE~tF,where r represents the constant riskless interest rate.The proof of Lemmas3–7 can be seen in [2, 3, 5].


### 2.1. Mixed Jump-Diffusion Fractional Brownian Motion Pricing Model

Suppose an option whose valueVt=V(St,t) depends only on St, t, and option pricing formula while the dynamics of stock log-price St satisfies(12)dSt=μStdt+σ1StdBtH+σ2StdBt,where μ is the instantaneous expected return, σ1 and σ2 are the instantaneous volatility, Bt is a standard Brownian motion, and BtH is a fractional Brownian motion. Assume that Bt and BtH are independent.Lemma 8.
LetVt=V(St,t) be a binary differential function; if stochastic process St is suitable for stochastic differential equation (12), then(13)dVt=∂V∂t+μSt∂V∂S+Hσ12St2t2H-1∂2V∂S2+12σ22St2∂2V∂S2dt+σ1St∂V∂SdBtH+σ2St∂V∂SdBt.Proof.
According to Taylor expansion we have(14)dVt=∂V∂tdt+∂V∂SdSt+12∂2V∂S2dSt2+odtdSt.Notice that CH(t,s)=1/2{|t|2H+|s|2H-|t-s|2H}; then E[BtH]2=t2H, E[Bt]2=t; since we can approximate that (dBtH)2=2Ht2H-1dt, (dBt)2=dt, (dt)2=0, dtdBt=0, we get(15)dSt2=μStdt+σ1StdBtH+σ2StdBt2=μ2St2dt2+σ12St2dBtH2+σ22St2dBt2+2μσ1St2dBtHdt+μσ2St2dBtdt+σ1σ2St2dBtHdBt=2Hσ12St2t2H-1dt+σ22St2dt+odt.Combining (14) with (15) we get(16)dVt=∂V∂tdt+∂V∂SμStdt+σ1StdBtH+σ2StdBt+12∂2V∂S22Hσ12St2t2H-1dt+σ22St2dt=∂V∂t+μSt∂V∂S+Hσ12St2t2H-1∂2V∂S2+12σ22St2∂2V∂S2dt+σ1St∂V∂SdBtH+σ2St∂V∂SdBt.
Consider a continuous-time financial market in[0,T]. It can be described by a filtered complete probability space {Ω,F,Ft,P}. {Ft}0≤t≤T≡F is a natural σ-filtration generated by a standard Brownian motion Bt, fractional Brownian motion BtH, and a Poisson process Pt. Here Pt is an (Ft,P)-Poisson-jump process with intensity λ, independent of Bt and BtH. jt is jump percent at time t and i.i.d.; jt satisfies Merton assumptions ln(1+jt)~N[ln(1+μj)-1/2σj2,σj2], where μj is the unconditional expectation of jt, and σj2 is the variance of ln(1+jt). Notice that the unconditional expectation μj is deterministic and can be calculated; jt is a bounded function of t, so we assume that σ3=ln(1+μj) is a constant. By Lemma 8 we have Lemma 9 as follows.Lemma 9.
LetVt=V(St,t) be a binary differential function; if stochastic process St is suitable for stochastic differential equation(17)dSt=μStdt+σ1StdBtH+σ2StdBt+σ3StdPt,then(18)dVt=∂V∂t+μSt∂V∂S+Hσ12St2t2H-1+12σ22St2+12λσ32St2∂2V∂S2dt+σ1St∂V∂SdBtH+σ2St∂V∂SdBt+σ3St∂V∂SdPt+λEVS1+jt,t-VS,tdt,where σ3St∂V/∂SdPt is the change volume of Poisson-jump process within dt for ∂V/∂S, λE[V(S(1+jt),t)-V(S,t)]dt is the change volume of Poisson-jump process within dt, and E is the expectation operator of V.Proof.
LetdPt admit two-point distribution as follows:(19)dPt=Pjt=ω1=p,jumpscannotoccur,Pjt=ω2=1-p,jumpscanoccur.
Moreover, during the time interval[t,t+dt], we can write the probability that jumps can not occur as Prob(ω1)=1-λdt and the probability that jumps can occur as Prob(ω2)=λdt. In the case jt>0, St+>St describes stock price St has upward jump at time t. In the case jt>0, while St+<St describes stock price St has downward jump at time t. Hence jt>-1 can ensure the stock price is positive, such that St+=St(1+jt)>0.
LetΠt=Vt-∇tSt be a riskless portfolio and ∇t is stock shares at time t. In a complete financial market, there are no risk-free arbitrage opportunities. Then(20)EdΠt=rΠtdt.
No matter the jumps occur or not, by Merton assumptionsln(1+jt)~N[ln(1+μj)-1/2σj2,σj2] and model (17), the variance σj2 exists surely. Either P(X=x)>0 for all x>0 or P(X=x)=0 for all x>0. In the first case, every positive point can be hit continuously in X and this phenomenon is called creep [18]. In the second case, only jumps can occur (almost surely). In time interval [t,t+dt], the pricing satisfies the following hypothesis:
(i) If jumps can not occur, for the eventsω1, by Vt=V(St,t) second-order differentiable, hence we can use Itô formula; then(21)dΠtω1=dVt-∇tdSt=∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2St,tdt+∂V∂S-∇St,tdSt.
(ii) If jumps can occur, for the eventsω2, we have (22)dΠtω2=VSt+-VSt,t-∇tSt+-St=V1+jtSt,t-VSt,t-∇tjtSt. Then(23)EdΠt=rΠtdt=1-λdtdΠtω1+λdtdΠtω2=1-λdt∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2St,tdt+∂V∂S-∇St,tdSt+λdtV1+jtSt,t-VSt,t-∇tjtSt.
Let∇t=∂V/∂S(St,t) take expectations for jt on both sides of above equation, and cancel items of dt2; then we have a parabolic partial differential equation as follows:(24)∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+r-λσ3S∂V∂S+λEVS1+jt,t-VS,t-r+λV=0,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.
Hence we can use Taylor expansion and imitate proof of Lemma8, and by(25)dPt,dPt=dP,Pt=dPt2=λdt,and then by approximate that (dBtH)2=2Ht2H-1dt, (dBt)2=dt, (dt)2=0, dtdBt=0, we obtain(26)dSt2=μStdt+σ1StdBtH+σ2StdBt+σ3StdPt2=μ2St2dt2+σ12St2dBtH2+σ22St2dBt2+σ32St2dPt2+2μStσ1StdBtHdt+μStσ2StdBtdt+μStσ3StdtdPt+σ1Stσ2StdBtHdBt+σ1Stσ3StdBtHdPt+σ2Stσ3StdBtdPt=2Hσ12St2t2H-1dt+σ22St2dt+λσ32St2dt+odtdStdPt.Combine (25) with (26); then(27)dVt=∂V∂tdt+∂V∂SμStdt+σ1StdBtH+σ2StdBt+σ3StdPt+12∂2V∂S2·2Hσ12St2t2H-1dt+σ22St2dt+λσ32St2dt=∂V∂t+μSt∂V∂S+Hσ12St2t2H-1+12σ22St2+12λσ32St2∂2V∂S2dt+σ1St∂V∂SdBtH+σ2St∂V∂SdBt+σ3St∂V∂SdPt+λEVS1+jt,t-VS,tdt.


### 2.2. Wick-Itô-Skorohod Integral and Mixed Jump-Diffusion Fractional Stochastic Differential Equation

Definition 10 (see [10]).
LetY: R→(δ)∗ be a given function such that Yt⋄BtH is dt-integrable in (δ)∗. The Wick-Itô-Skorohod integral of Yt with respect to BtH is given by(28)∫RYtdBtH≔∫RYt⋄BtHdt,where ⋄ is the Wick product and BtH is the fractional Brownian motion.Lemma 11 (fractional Girsanov formula I, see [10]).
Letψ∈Lp(PH), for some p>1 and let γ∈Lϕ2(R)∩C(R)⊂δ′(R). Let γ~ be defined by γ~=∫R‍ϕ(t,s)ds. Then the map ω→ψ(ω+γ~) belongs to Lρ(PH), for all ρ<p and (29)∫δ′Rψω+γ~dPHω=∫δ′Rψω·exp⋄⁡ω,γdPHω,where equip δ(R) with the inner product (30)ω,γH≔∫R∫Rωsγtψs,tdsdt,ω,γ∈δR.Lemma  11∗(fractional Girsanov formula II, see [10]).LetT>0, and letγbe a continuous function with suppγ⊂[0,T]. LetKbe a function with suppK⊂[0,T]and such that(31)K,fH=γ,fL2R,∀f∈δR,suppf⊂0,T,that is, (32)∫RKsϕs,tds=γt,0≤t≤T.On theσ-algebraFTHgenerated by{BsH:0≤t≤T}, define a probability measurePH,γby(33)dPH,γdPH=exp⋄⁡-ω,K,and thenB^H(t)=BtH+∫0t‍γsds,0≤t≤T,is a fBm underPH,γ.Lemma 12 (Wick products on different white noise spaces, see [10]).
LetP=PH, Q=PH,γ, and B^H(t)=BtH+∫0t‍γsds. Let the Wick products corresponding to P and Q be denoted by ⋄P and ⋄Q, respectively. Then(34)F⋄PG=F⋄QG,for all F,G∈(δ)H∗.Suppose the price of the risky assetSt and interest rate rt satisfy the following equation:(35)dSt=rtStdt,S0=1.By Definition10–Lemma 12 we can consider the mixed jump-diffusion fractional stochastic differential equation(36)dSt=St⋄r-q-λσ3dt+σ1dBtH+σ2dBt+σ3dPt,St=S,where r is the instantaneous expected return, q≥0 is the continuous dividend rate, σ1 and σ2 are denoted as in Lemma 5, and σ3 is the unconditional expectation of jt. Assume that Bt, BtH, and Pt are independent.LetJ be the path-dependent variable. For looking back put option on the maturity T, the stock price St satisfies St≤Jt=max0≤t≤TSt; for looking back call option on the maturity T, the stock price St satisfies St≥Jt=min0≤t≤TSt. Thus, the looking back put option value V(St,Jt,t) is the function of S, J, and t. Then we construct a riskless portfolio Π=V-∇⋄S, select the appropriate variable ∇, and make the investment portfolio Π in the interval (t,t+dt) risk-free. Then dΠ=r(V-∇⋄S)dt; according to Itô’s formula we have(37)dΠ=dV-∇⋄dS-∇q⋄Sdt=∂V∂tdt+∂V∂JdJ+12λσ32S2∂2V∂S2dt+∂V∂SdS+Hσ12S2t2H-1∂2V∂S2dt+12σ22S2∂2V∂S2dt+σ3∂V∂SdPt-∇⋄dS-∇q⋄Sdt=∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2-∇q⋄Sdt+∂V∂JdJ+σ3∂V∂SdPt+∂V∂S-∇⋄dS.Notice that Jt is nondifferentiable about t, and define Jn(t)=1/t∫0t‍(Sτ)ndτ1/n, such that approximate amount Jn(t) is differentiable about t and satisfies(38)nJnn-1tdJndt=Stn-Jnntt.Hence St is continuous function with t, for n→∞, then limn→∞Jn(t)=max0≤t≤TSt=Jt, and we get(39)dΠ=∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2-∇q⋄Sdt+∂V∂JdJ+σ3∂V∂SdPt+∂V∂S-∇⋄dS=∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+∂V∂JndJndt-∇q⋄Sdt+σ3∂V∂SdPt+∂V∂S-∇⋄dS.By ∇=∂V/∂S, then(40)∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+S/Jnn-1S-Jnnt∂V∂Jn+r-q-λσ3S∂V∂S+λEVS1+jt,t-VS,t-r+λV=0,where S≤Jn and E is the expectation operator of V.For fixed(J,t), when n→∞, then (S/Jn)n-1S-Jn/nt→0, since we obtain the general equations of lookback option under the environment of mixed jump-diffusion fraction Brownian motion as follows:(41)∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+r-q-λσ3S∂V∂S+λEVS1+jt,t-VS,t-r+λV=0,VS,J,t=J-S,where 0≤S≤J<∞, 0≤t≤T, and the terminal condition is given by(42)∂V∂JS=J=0.


## 2.1. Mixed Jump-Diffusion Fractional Brownian Motion Pricing Model

Suppose an option whose valueVt=V(St,t) depends only on St, t, and option pricing formula while the dynamics of stock log-price St satisfies(12)dSt=μStdt+σ1StdBtH+σ2StdBt,where μ is the instantaneous expected return, σ1 and σ2 are the instantaneous volatility, Bt is a standard Brownian motion, and BtH is a fractional Brownian motion. Assume that Bt and BtH are independent.Lemma 8.
LetVt=V(St,t) be a binary differential function; if stochastic process St is suitable for stochastic differential equation (12), then(13)dVt=∂V∂t+μSt∂V∂S+Hσ12St2t2H-1∂2V∂S2+12σ22St2∂2V∂S2dt+σ1St∂V∂SdBtH+σ2St∂V∂SdBt.Proof.
According to Taylor expansion we have(14)dVt=∂V∂tdt+∂V∂SdSt+12∂2V∂S2dSt2+odtdSt.Notice that CH(t,s)=1/2{|t|2H+|s|2H-|t-s|2H}; then E[BtH]2=t2H, E[Bt]2=t; since we can approximate that (dBtH)2=2Ht2H-1dt, (dBt)2=dt, (dt)2=0, dtdBt=0, we get(15)dSt2=μStdt+σ1StdBtH+σ2StdBt2=μ2St2dt2+σ12St2dBtH2+σ22St2dBt2+2μσ1St2dBtHdt+μσ2St2dBtdt+σ1σ2St2dBtHdBt=2Hσ12St2t2H-1dt+σ22St2dt+odt.Combining (14) with (15) we get(16)dVt=∂V∂tdt+∂V∂SμStdt+σ1StdBtH+σ2StdBt+12∂2V∂S22Hσ12St2t2H-1dt+σ22St2dt=∂V∂t+μSt∂V∂S+Hσ12St2t2H-1∂2V∂S2+12σ22St2∂2V∂S2dt+σ1St∂V∂SdBtH+σ2St∂V∂SdBt.
Consider a continuous-time financial market in[0,T]. It can be described by a filtered complete probability space {Ω,F,Ft,P}. {Ft}0≤t≤T≡F is a natural σ-filtration generated by a standard Brownian motion Bt, fractional Brownian motion BtH, and a Poisson process Pt. Here Pt is an (Ft,P)-Poisson-jump process with intensity λ, independent of Bt and BtH. jt is jump percent at time t and i.i.d.; jt satisfies Merton assumptions ln(1+jt)~N[ln(1+μj)-1/2σj2,σj2], where μj is the unconditional expectation of jt, and σj2 is the variance of ln(1+jt). Notice that the unconditional expectation μj is deterministic and can be calculated; jt is a bounded function of t, so we assume that σ3=ln(1+μj) is a constant. By Lemma 8 we have Lemma 9 as follows.Lemma 9.
LetVt=V(St,t) be a binary differential function; if stochastic process St is suitable for stochastic differential equation(17)dSt=μStdt+σ1StdBtH+σ2StdBt+σ3StdPt,then(18)dVt=∂V∂t+μSt∂V∂S+Hσ12St2t2H-1+12σ22St2+12λσ32St2∂2V∂S2dt+σ1St∂V∂SdBtH+σ2St∂V∂SdBt+σ3St∂V∂SdPt+λEVS1+jt,t-VS,tdt,where σ3St∂V/∂SdPt is the change volume of Poisson-jump process within dt for ∂V/∂S, λE[V(S(1+jt),t)-V(S,t)]dt is the change volume of Poisson-jump process within dt, and E is the expectation operator of V.Proof.
LetdPt admit two-point distribution as follows:(19)dPt=Pjt=ω1=p,jumpscannotoccur,Pjt=ω2=1-p,jumpscanoccur.
Moreover, during the time interval[t,t+dt], we can write the probability that jumps can not occur as Prob(ω1)=1-λdt and the probability that jumps can occur as Prob(ω2)=λdt. In the case jt>0, St+>St describes stock price St has upward jump at time t. In the case jt>0, while St+<St describes stock price St has downward jump at time t. Hence jt>-1 can ensure the stock price is positive, such that St+=St(1+jt)>0.
LetΠt=Vt-∇tSt be a riskless portfolio and ∇t is stock shares at time t. In a complete financial market, there are no risk-free arbitrage opportunities. Then(20)EdΠt=rΠtdt.
No matter the jumps occur or not, by Merton assumptionsln(1+jt)~N[ln(1+μj)-1/2σj2,σj2] and model (17), the variance σj2 exists surely. Either P(X=x)>0 for all x>0 or P(X=x)=0 for all x>0. In the first case, every positive point can be hit continuously in X and this phenomenon is called creep [18]. In the second case, only jumps can occur (almost surely). In time interval [t,t+dt], the pricing satisfies the following hypothesis:
(i) If jumps can not occur, for the eventsω1, by Vt=V(St,t) second-order differentiable, hence we can use Itô formula; then(21)dΠtω1=dVt-∇tdSt=∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2St,tdt+∂V∂S-∇St,tdSt.
(ii) If jumps can occur, for the eventsω2, we have (22)dΠtω2=VSt+-VSt,t-∇tSt+-St=V1+jtSt,t-VSt,t-∇tjtSt. Then(23)EdΠt=rΠtdt=1-λdtdΠtω1+λdtdΠtω2=1-λdt∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2St,tdt+∂V∂S-∇St,tdSt+λdtV1+jtSt,t-VSt,t-∇tjtSt.
Let∇t=∂V/∂S(St,t) take expectations for jt on both sides of above equation, and cancel items of dt2; then we have a parabolic partial differential equation as follows:(24)∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+r-λσ3S∂V∂S+λEVS1+jt,t-VS,t-r+λV=0,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.
Hence we can use Taylor expansion and imitate proof of Lemma8, and by(25)dPt,dPt=dP,Pt=dPt2=λdt,and then by approximate that (dBtH)2=2Ht2H-1dt, (dBt)2=dt, (dt)2=0, dtdBt=0, we obtain(26)dSt2=μStdt+σ1StdBtH+σ2StdBt+σ3StdPt2=μ2St2dt2+σ12St2dBtH2+σ22St2dBt2+σ32St2dPt2+2μStσ1StdBtHdt+μStσ2StdBtdt+μStσ3StdtdPt+σ1Stσ2StdBtHdBt+σ1Stσ3StdBtHdPt+σ2Stσ3StdBtdPt=2Hσ12St2t2H-1dt+σ22St2dt+λσ32St2dt+odtdStdPt.Combine (25) with (26); then(27)dVt=∂V∂tdt+∂V∂SμStdt+σ1StdBtH+σ2StdBt+σ3StdPt+12∂2V∂S2·2Hσ12St2t2H-1dt+σ22St2dt+λσ32St2dt=∂V∂t+μSt∂V∂S+Hσ12St2t2H-1+12σ22St2+12λσ32St2∂2V∂S2dt+σ1St∂V∂SdBtH+σ2St∂V∂SdBt+σ3St∂V∂SdPt+λEVS1+jt,t-VS,tdt.


## 2.2. Wick-Itô-Skorohod Integral and Mixed Jump-Diffusion Fractional Stochastic Differential Equation

Definition 10 (see [10]).
LetY: R→(δ)∗ be a given function such that Yt⋄BtH is dt-integrable in (δ)∗. The Wick-Itô-Skorohod integral of Yt with respect to BtH is given by(28)∫RYtdBtH≔∫RYt⋄BtHdt,where ⋄ is the Wick product and BtH is the fractional Brownian motion.Lemma 11 (fractional Girsanov formula I, see [10]).
Letψ∈Lp(PH), for some p>1 and let γ∈Lϕ2(R)∩C(R)⊂δ′(R). Let γ~ be defined by γ~=∫R‍ϕ(t,s)ds. Then the map ω→ψ(ω+γ~) belongs to Lρ(PH), for all ρ<p and (29)∫δ′Rψω+γ~dPHω=∫δ′Rψω·exp⋄⁡ω,γdPHω,where equip δ(R) with the inner product (30)ω,γH≔∫R∫Rωsγtψs,tdsdt,ω,γ∈δR.Lemma  11∗(fractional Girsanov formula II, see [10]).LetT>0, and letγbe a continuous function with suppγ⊂[0,T]. LetKbe a function with suppK⊂[0,T]and such that(31)K,fH=γ,fL2R,∀f∈δR,suppf⊂0,T,that is, (32)∫RKsϕs,tds=γt,0≤t≤T.On theσ-algebraFTHgenerated by{BsH:0≤t≤T}, define a probability measurePH,γby(33)dPH,γdPH=exp⋄⁡-ω,K,and thenB^H(t)=BtH+∫0t‍γsds,0≤t≤T,is a fBm underPH,γ.Lemma 12 (Wick products on different white noise spaces, see [10]).
LetP=PH, Q=PH,γ, and B^H(t)=BtH+∫0t‍γsds. Let the Wick products corresponding to P and Q be denoted by ⋄P and ⋄Q, respectively. Then(34)F⋄PG=F⋄QG,for all F,G∈(δ)H∗.Suppose the price of the risky assetSt and interest rate rt satisfy the following equation:(35)dSt=rtStdt,S0=1.By Definition10–Lemma 12 we can consider the mixed jump-diffusion fractional stochastic differential equation(36)dSt=St⋄r-q-λσ3dt+σ1dBtH+σ2dBt+σ3dPt,St=S,where r is the instantaneous expected return, q≥0 is the continuous dividend rate, σ1 and σ2 are denoted as in Lemma 5, and σ3 is the unconditional expectation of jt. Assume that Bt, BtH, and Pt are independent.LetJ be the path-dependent variable. For looking back put option on the maturity T, the stock price St satisfies St≤Jt=max0≤t≤TSt; for looking back call option on the maturity T, the stock price St satisfies St≥Jt=min0≤t≤TSt. Thus, the looking back put option value V(St,Jt,t) is the function of S, J, and t. Then we construct a riskless portfolio Π=V-∇⋄S, select the appropriate variable ∇, and make the investment portfolio Π in the interval (t,t+dt) risk-free. Then dΠ=r(V-∇⋄S)dt; according to Itô’s formula we have(37)dΠ=dV-∇⋄dS-∇q⋄Sdt=∂V∂tdt+∂V∂JdJ+12λσ32S2∂2V∂S2dt+∂V∂SdS+Hσ12S2t2H-1∂2V∂S2dt+12σ22S2∂2V∂S2dt+σ3∂V∂SdPt-∇⋄dS-∇q⋄Sdt=∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2-∇q⋄Sdt+∂V∂JdJ+σ3∂V∂SdPt+∂V∂S-∇⋄dS.Notice that Jt is nondifferentiable about t, and define Jn(t)=1/t∫0t‍(Sτ)ndτ1/n, such that approximate amount Jn(t) is differentiable about t and satisfies(38)nJnn-1tdJndt=Stn-Jnntt.Hence St is continuous function with t, for n→∞, then limn→∞Jn(t)=max0≤t≤TSt=Jt, and we get(39)dΠ=∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2-∇q⋄Sdt+∂V∂JdJ+σ3∂V∂SdPt+∂V∂S-∇⋄dS=∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+∂V∂JndJndt-∇q⋄Sdt+σ3∂V∂SdPt+∂V∂S-∇⋄dS.By ∇=∂V/∂S, then(40)∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+S/Jnn-1S-Jnnt∂V∂Jn+r-q-λσ3S∂V∂S+λEVS1+jt,t-VS,t-r+λV=0,where S≤Jn and E is the expectation operator of V.For fixed(J,t), when n→∞, then (S/Jn)n-1S-Jn/nt→0, since we obtain the general equations of lookback option under the environment of mixed jump-diffusion fraction Brownian motion as follows:(41)∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+r-q-λσ3S∂V∂S+λEVS1+jt,t-VS,t-r+λV=0,VS,J,t=J-S,where 0≤S≤J<∞, 0≤t≤T, and the terminal condition is given by(42)∂V∂JS=J=0.


## 3. Main Results

In this section we present the American lookback options factorization formula and the optimal exercise boundary.


### 3.1. Fundamental Solution Derivation of the General Equations

DefineG(S,t;ξ,T) as the fundamental solution of the following solution problem: (43)∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+r-q-λσ3S∂V∂S+λEVS1+jt,t-VS,t-r+λV=0,(44)VS,T=δS-ξ,where 0<S<∞, 0<ξ<∞, 0<t<T, and δ(x) is Dirac function.Lemma 13 (see [29], Theorem 4.3).
The price of a derivative on the stock price with a bounded payofff(St) is given by D(t,St), where D(t,S) is the solution of the PDE:(45)∂D∂t+Hσ2S2t2H-1∂2D∂S2+rS∂D∂S-rD=0,DS,T=fS.Lemma 14.
The solution of stochastic partial differential equations (43) and (44) is(46)GS,t;ξ,T=∑n=0∞λnT-tnexp⁡-λT-tn!En·exp⁡-rT-tξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t·exp⁡-12σ12T2H-t2H+σ22T-t+λσ32T-t·ln⁡S∏i=1n1+jtiξ+r-q-λσ3T-t-σ12T2H-t2H-12σ22+λσ32T-t2,where En denotes the expectation operator over the distribution of ∏i=1n‍(1+jti).Proof.
Letx=lnS∏i=1n‍(1+jti)/ξ∈R, W=V·exp{-r(T-t)}; then solution problem of (43) and (44) is equivalent to the following solution problem:(47)∂W∂t-Hσ12t2H-1+12σ22+12λσ32∂2W∂x2-r-q-λσ3-Hσ12t2H-1+12σ22+12λσ32·∂W∂x-r+λW=0.Wx,T=exp⁡-rT-tξδx,Let W=V·exp{α(T,t)+β(T,t)x}, where(48)αT,t=-rT-t-r-q-λσ3T-t-σ12T2H-t2H-1/2σ22+λσ32T-t22σ12T2H-t2H+σ22T-t+λσ32T-t,βT,t=∑n=0∞Pjt=nEnξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t=∑n=0∞λnT-tnexp⁡-λT-tn!-1Enξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t,here Poisson distribution with intensity λ(T-t). Then by (47) we get(49)∂V∂t-Hσ12t2H-1+12σ22+12λσ32∂2V∂x2-r-q-λσ3-Hσ12t2H-1+12σ22+12λσ32∂V∂x-r+λV=0,(50)Vx,0=exp⁡-βxξδx=1ξδx.The solution of stochastic parabolic partial differential equations (49) and (50) can be written as(51)Vx,ξ;t,T=1ξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t·exp⁡-x22σ12T2H-t2H+σ22T-t+λσ32T-t.Combining (51) with (48) we have(52)Wx,ξ;t,T=∑n=0∞λnT-tnexp⁡-λT-tn!-1Enξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t·exp⁡-rT-t-12σ12T2H-t2H+σ22T-t+λσ32T-tx+r-q-λσ3·T-t-σ12T2H-t2H-12σ22+λσ32T-t2.Substitute original variables (S,t) in (52); we obtain(53)GS,t;ξ,T=∑n=0∞λnT-tnexp⁡-λT-tn!En·exp⁡-rT-tξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t·exp⁡-12σ12T2H-t2H+σ22T-t+λσ32T-t·ln⁡S∏i=1n1+jtiξ+r-q-λσ3T-t-σ12T2H-t2H-12σ22+λσ32T-t2.Theorem 15.
The fundamental solutionG(S,t;ξ,T) is considered to be the function of ξ and η, and G(S,t;ξ,T) satisfies the adjoint equation of (43) and (44). If we note(54)GS,t;ξ,η=vξ,η,then v(ξ,η) satisfies(55)-∂v∂t-Hσ12t2H-1+12σ22+12λσ32S2∂2v∂S2-r-q-λσ3S∂v∂S-λEvS1+jt,t-vS,t-r+λv=0,(56)vξ,η=δξ-S,where 0<S<∞, 0<ξ<∞, t<η.Corollary 16.
By Lemmas13 and 14 and Theorem 15, if one notes G∗(ξ,η;S,t) is the fundamental solution of (55) and (56), then (57)GS,t;ξ,η=G∗ξ,η;S,t.Proof of Theorem15.
For anyϵ>0, consider integral as follows:(58)0=∫0∞∫t+ϵη-ϵG∗x,y;S,t·Gx,y;ξ,η-Gx,y;ξ,η·G∗x,y;S,tdxdy=∫0∞dx∫t+ϵη-ϵ∂G∗G∂y+Hσ12t2H-1+12σ22+12λσ32∂∂xx2G∗∂G∂x+λEvS1+jt,t-vS,t-Hσ12t2H-1+12σ22+12λσ32∂∂xG∂∂xx2G∗+r-q-λσ3∂∂xxGG∗-λEvS1+jt,t-vS,tdy.When x→0, we have(59)x2G∗∂G∂x⟶0,G∂∂xx2G∗⟶0,xGG∗⟶0,since(60)∫0∞G∗x,η-ϵ;S,t·Gx,η-ϵ;ξ,ηdx=∫0∞G∗x,t+ϵ;S,t·Gx,t+ϵ;ξ,ηdx.Let ϵ→0; by conditions (44) and (56) we get(61)∫0∞G∗x,η;S,t·δx-ξdx=∫0∞δx-S·Gx,t;η,ξdx.Then(62)GS,t;ξ,η=G∗ξ,η;S,t.


### 3.2. 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 T are given, respectively, by (ST-amT)+ and (bMT-ST)+, 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 T>0, let C=C(t,S,m) be the value of the American fractional lookback call option at time t∈[0,T]. 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(t,S,m) is a solution of an optimal stopping problem(63)Ct,S,m=supTt∈t,T⁡Eexp⁡-rTt-tSTt-amTt+∣St=S,mt=m,where Tt is a stopping time of the filtration F and the conditional expectation is calculated under the risk-neutral probability measure P. The random variable Tt∗∈[t,T] 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 S and nonincreasing in t, m, and a. Solving the optimal stopping problem (63) is equivalent to finding the points (t,St,mt) for which early exercise before maturity is optimal. Let(64)D=t,S,m∈0,T×m,∞×R+be the whole domain, and let E and C denote the exercise region and continuation region, respectively. In terms of the value function C(t,S,m), the exercise region E is defined by (65)E=t,S,m∣Ct,S,m=S-am+,for which the optimal stopping time Tt∗ satisfies(66)Tt∗=inf⁡η∈t,T∣η,Sη,mη∈E.The continuation region C is the complement of S in D, such as(67)C=t,S,m∣Ct,S,m>S-am+.The boundary that separates S from C is referred to as the early exercise boundary, which is defined by(68)S¯t,m=sup⁡S≥m∣t,S,m∈C,t∈0,T.At the early exercise boundary [S¯(t,m)]t∈[0,T], the American fractional lookback call option would be optimally exercised. In terms of S(t,m), the continuation region C can be represented as(69)C=t,S,m;m≤S<S¯t,m.LetV(S,J,t) be the lookback option price at time t with stock price S and path-dependent variable J. 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: (70)0=∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+r-q-λσ3S∂V∂S-r+λV+λEVS1+jt,min⁡J,S1+jt,t,for  lookback  call  option,λEVS1+jt,max⁡J,S1+jt,t,for  lookback  put  option,VS,J,t=J-S,where 0≤t≤T, and satisfies an order continuous differentiable on domain(71)Σ=S,J:0<J≤S<∞,for  lookback  call  option,S,J:0<S≤J<∞,for  lookback  put  option,and the terminal condition is given by (72)∂V∂JS=J=0.We define the differential operator Lt,S by(73)Lt,S=∂∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2∂S2+r-q-λσ3S∂∂S-r+λ.Then the free boundary problem can be written in a linear complementary form as(74)Lt,SCt,S,m·Ct,S,m-S-am+=0,Lt,SCt,S,m≤0,Ct,S,m-S-am+≥0together with auxiliary conditions(75)CT,S,m-S-am+=0,limS↓m⁡∂C∂m=0.For the free boundary[S¯(t,m)]t∈[0,T], this problem is equivalent to solving the Black-Scholes-Merton partial differential equations (76)Lt,SCt,S,m=0,m≤S<S¯t,m,together with the boundary conditions (77)limS↑S¯⁡Ct,S,m=S¯t,m-am,limS↑S¯⁡∂C∂S=1,limS↓m⁡∂C∂m=0.and the terminal condition (78)CT,S,m=S-am+.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(t,S,m) be the value of the American fractional lookback call option at time t∈[0,T]. The value P(t,S,M) is a solution of an optimal stopping problem (79)Pt,S,M=supTt∈t,T⁡Eexp⁡-rTt-tbMTt-STt+∣St=S,Mt=M,and P(t,S,M) satisfies the same PDE as (76); then(80)Lt,SPt,S,M=0,S_t,M<S≤M,where [S_(t,M)]t∈[0,T] is the early exercise boundary for put. The boundary conditions for put are(81)limS↓S_⁡Pt,S,M=bM-S_t,M,limS↓S_⁡∂P∂S=-1,limS↑M⁡∂P∂M=0.and the terminal condition is given by(82)PT,S,M=bM-S+.


### 3.3. 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(t,S,M), we can obtain the early exercise representation V(t,S,M) under our mixed jump-diffusion fractional Brownian motion environment, which is shown in the following theorem.Theorem 17.
LetV(t,S,M) be American fractional lookback put option; then(83)Vt,S,M=VEt,S,M+et,S,M,where VE(t,S,M) is price of the hedging portfolio of the equivalent European option, and(84)VEt,S,M=∑n=0∞λnT-tnexp⁡-λT-tn!En·bMexp⁡-rT-tN-d^2-S∏i=1n1+jtiexp⁡-q+λσ3T-tN-d^1,where En denotes the expectation operator over the distribution of ∏i=1n(1+jti), M is contractual strike price, b is positive constants, N(·) is the cumulative normal distribution function, and(85)d^1=ln⁡S∏i=1n1+jtiξ+r-q-λσ3T-t+σ12T2H-t2H+12σ22+λσ32·T-t2σ12T2H-t2H+σ22T-t+λσ32T-t-1,d^2=d^1-2σ12T2H-t2H+σ22T-t+λσ32T-t,e(t,S,M) is early exercise premium, and its explicit integral can be represented by(86)et,S,M=∫tTdη∫0SηbMr-q+λσ3ξ·GS,t;ξ,ηdξ.Proof.
SupposeV(t,S,M) satisfies an order continuous differentiable on domain Σ: Σ={0≤S≤M<∞,0≤t<T}, and V(t,S,M) is piecewise continuous on domain Σ, and the second-order derivative exists; then(87)-LVt,S,M=0,t,S,M∈Σ1,bMr-q+λσ3S,t,S,M∈Σ2,where L is the Black-Scholes operator.
Multiplying both sides of (87) to G∗(ξ,η;S,t) and quadrature on domain {0≤ξ≤M<∞,t+ϵ≤η≤T}, for Σ2={0≤ξ≤M≤Sη,0≤η<T} and Sη is continuous monotonous of the asset price, then(88)∫t+ϵTdη∫0SηbMr-q+λσ3ξ·G∗ξ,η;S,tdξ=-∫t+ϵTdη∫0∞G∗ξ,η;S,tLVdξ=-∫t+ϵTdη∫0∞G∗ξ,η;S,tLVη,ξ,M-Vη,ξ,ML∗G∗ξ,η;S,tdξ=-∫t+ϵTdη∫0∞∂G∗V∂η+Hσ12t2H-1+12σ22+12λσ32∂∂ξξ2G∗∂V∂ξ+λEVt,S1+jt,M-Vt,S,M-Hσ12t2H-1+12σ22+12λσ32∂∂ξ·V∂∂ξξ2G∗+r-q-λσ3∂∂ξξVG∗-λEVt,S1+jt,M-Vt,S,Mdξ,where L∗ is the adjoint Black-Scholes operator, and when ξ→0, we have(89)ξ2G∗∂V∂ξ⟶0,V∂∂ξξ2G∗⟶0,ξVG∗⟶0,and hence(90)∫0∞G∗ξ,t+ϵ;S,tVt+ϵ,ξ,Mdξ=∫0∞G∗ξ,T;S,tVT,ξ,Mdξ+∫t+ϵTdη∫0SηbMr-q+λσ3ξG∗ξ,η;S,tdξ.Let ϵ→0; by (56) and Corollary 16, we obtain(91)Vt,S,M=∫0∞GS,t;ξ,TbM-ξ+dξ+∫tTdη∫0SηbMr-q+λσ3ξGS,t;ξ,ηdξ=VEt,S,M+et,S,M.Then the proof is finished.Remark 18.
If the optimal exercise boundary is given byS=St, 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 byS=St, 0≤t<T, and then S=St satisfy the following Volterra integral equation(92)St=bM+Sexp⁡-q+λσ3T-t·N--ln⁡S∏i=1n1+jtibM+r-q-λσ3T-t+σ12T2H-t2H+12σ22+λσ32T-t·2σ12T2H-t2H+σ22T-t+λσ32T-t-1-bMexp⁡-rT-t∑n=0∞λnT-tnexp⁡-λT-tn!En·N--ln⁡S∏i=1n1+jtibM+r-q-λσ3T-t-σ12T2H-t2H-12σ22+λσ32T-t·2σ12T2H-t2H+σ22T-t+λσ32T-t-1-bMr∫tTexp⁡-rη-t∑n=0∞λnη-tnexp⁡-λη-tn!En·1-Nln⁡S∏i=1n1+jtiSη+r-q-λσ3η-t-σ12η2H-t2H-12σ22+λσ32η-t·2σ12η2H-t2H+σ22η-t+λσ32η-t-1dη+q+λσ3S∫tTexp⁡-q+λσ3η-t∑n=0∞λnη-tnexp⁡-λη-tn!En·1-NlnS∏i=1n1+jtiSη+r-q-λσ3η-t+σ12η2H-t2H+12σ22+λσ32η-t·2σ12η2H-t2H+σ22η-t+λσ32η-t-1dη,where N(·) is the cumulative normal distribution function and Sη≥M>0, 0<η≤T.Proof.
By (86) and (50), e(S,t) can be represented by(93)et,S,M=∫tTdη∫0SηbMr-q+λσ3ξ·∑n=0∞λnT-tnexp⁡-λT-tn!En·exp⁡-rT-tξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t·exp⁡-12σ12T2H-t2H+σ22T-t+λσ32T-t·ln⁡S∏i=1n1+jtiξ+r-q-λσ3T-t-σ12T2H-t2H-12σ22+λσ32T-t2dξ.
Let(94)x∗=12σ12η2H-t2H+σ22η-t+λσ32η-t·ln⁡S∏i=1n1+jtiξ+r-q-λσ3η-t-σ12η2H-t2H-12σ22+λσ32η-t,dx∗=-1ξ2σ12T2H-t2H+σ22T-t+λσ32T-tdξ,and then(95)et,S,M=bMr2π∫tTexp⁡-rη-t∑n=0∞λnη-tnexp⁡-λη-tn!Endη·∫ln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t-σ12η2H-t2H-1/2σ22+λσ32η-t/2σ12η2H-t2H+σ22η-t+λσ32η-t∞exp⁡-x2dx-q+λσ3S2π∫tTexp⁡-q+λσ3η-t·∑n=0∞λnη-tnexp⁡-λη-tn!Endη·∫ln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t-σ12η2H-t2H-1/2σ22+λσ32η-t/2σ12η2H-t2H+σ22η-t+λσ32η-t∞exp⁡-x22+2σ12η2H-t2H+σ22η-t+λσ32η-txdx=bMr∫tTexp⁡-rη-t∑n=0∞λnη-tnexp⁡-λη-tn!En·1-Nln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t-σ12η2H-t2H-σ22+λσ32η-t/22σ12η2H-t2H+σ22η-t+λσ32η-tdη-q+λσ3S∫tTexp⁡-q+λσ3η-t∑n=0∞λnη-tnexp⁡-λη-tn!Endη·∫ln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t-σ12η2H-t2H-1/2σ22+λσj2η-t/2σ12η2H-t2H+σ22η-t+λσ32η-t∞exp⁡-12·x+2σ12η2H-t2H+σ22η-t+λσ32η-t2+r-q-λσ3η-t+σ12η2H-t2H+σ22+λσ32η-t2dx=bMr∫tTexp⁡-rη-t∑n=0∞λnη-tnexp⁡-λη-tn!En·1-Nln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t-σ12η2H-t2H-σ22+λσ32η-t/22σ12η2H-t2H+σ22η-t+λσ32η-tdη-q+λσ3S∫tTexp⁡-q+λσ3η-t∑n=0∞λnη-tnexp⁡-λη-tn!En·1-Nln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t+σ12η2H-t2H+σ22+λσ32η-t/22σ12η2H-t2H+σ22η-t+λσ32η-tdη.Substituting (95) into (83), we have(96)Vt,St,M=bM-St,and then we obtain (97)bM-St=VEt,St,M+et,St,M,such that (98)St=bM+Sexp⁡-q+λσ3T-t·N--ln⁡S∏i=1n1+jtibM+r-q-λσ3T-t+σ12T2H-t2H+12σ22+λσ32T-t·2σ12T2H-t2H+σ22T-t+λσ32T-t-1-bMexp⁡-rT-t∑n=0∞λnT-tnexp⁡-λT-tn!En·N--ln⁡S∏i=1n1+jtibM+r-q-λσ3T-t-σ12T2H-t2H-12σ22+λσ32T-t·2σ12T2H-t2H+σ22T-t+λσ32T-t-1-bMr∫tTexp⁡-rη-t∑n=0∞λnT-tnexp⁡-λT-tn!En·1-Nln⁡S∏i=1n1+jtiSη+r-q-λσ3η-t-σ12η2H-t2H-12σ22+λσ32η-t·2σ12η2H-t2H+σ22η-t+λσ32η-t-1dη+q+λσ3S∫tTexp⁡-q+λσ3η-t∑n=0∞λnT-tnexp⁡-λT-tn!En·1-Nln⁡S∏i=1n1+jtiSη+r-q-λσ3η-t+σ12η2H-t2H+12σ22+λσ32η-t·2σ12η2H-t2H+σ22η-t+λσ32η-t-1dη.
In practice, it is very difficult to calculate nonlinear integral in (92); we only show that this assumption holds in Volterra integral equation for the optimal exercise boundary; by the relation (92) and Theorem 17, asymptotic expression of optimal exercise boundary S=St is obtained near the maturity t=T. Comparing with Theorem 17 and Corollary 20, the optimal exercise boundary and Volterra integral equation of American fractional lookback call option are obvious as follows.Corollary 21.
LetV∗(t,S,m) be American fractional lookback call option; then(99)V∗t,S,m=VE∗t,S,m+e∗t,S,m,where VE∗(t,S,m) is price of the hedging portfolio of the equivalent European option, and(100)VE∗t,S,m=∑n=0∞λnT-tnexp⁡-λT-tn!En·S∏i=1n1+jtiexp⁡-q+λσ3T-tN-d^1-amexp⁡-rT-tN-d^2,where En denotes the expectation operator over the distribution of ∏i=1n(1+jti), am is contractual strike price, N(·) is the cumulative normal distribution function, and(101)d^1=ln⁡S∏i=1n1+jtiξ+r-q-λσ3T-t+σ12T2H-t2H+12σ22+λσ32T-t·2σ12T2H-t2H+σ22T-t+λσ32T-t-1,d^2=d^1-2σ12T2H-t2H+σ22T-t+λσ32T-t,e∗(t,S,m) is early exercise premium, and its explicit integral can be represented by (102)e∗t,S,m=∫tTdη∫0Sηq+λσ3ξ-amr·GS,t;ξ,ηdξ.


## 3.1. Fundamental Solution Derivation of the General Equations

DefineG(S,t;ξ,T) as the fundamental solution of the following solution problem: (43)∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+r-q-λσ3S∂V∂S+λEVS1+jt,t-VS,t-r+λV=0,(44)VS,T=δS-ξ,where 0<S<∞, 0<ξ<∞, 0<t<T, and δ(x) is Dirac function.Lemma 13 (see [29], Theorem 4.3).
The price of a derivative on the stock price with a bounded payofff(St) is given by D(t,St), where D(t,S) is the solution of the PDE:(45)∂D∂t+Hσ2S2t2H-1∂2D∂S2+rS∂D∂S-rD=0,DS,T=fS.Lemma 14.
The solution of stochastic partial differential equations (43) and (44) is(46)GS,t;ξ,T=∑n=0∞λnT-tnexp⁡-λT-tn!En·exp⁡-rT-tξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t·exp⁡-12σ12T2H-t2H+σ22T-t+λσ32T-t·ln⁡S∏i=1n1+jtiξ+r-q-λσ3T-t-σ12T2H-t2H-12σ22+λσ32T-t2,where En denotes the expectation operator over the distribution of ∏i=1n‍(1+jti).Proof.
Letx=lnS∏i=1n‍(1+jti)/ξ∈R, W=V·exp{-r(T-t)}; then solution problem of (43) and (44) is equivalent to the following solution problem:(47)∂W∂t-Hσ12t2H-1+12σ22+12λσ32∂2W∂x2-r-q-λσ3-Hσ12t2H-1+12σ22+12λσ32·∂W∂x-r+λW=0.Wx,T=exp⁡-rT-tξδx,Let W=V·exp{α(T,t)+β(T,t)x}, where(48)αT,t=-rT-t-r-q-λσ3T-t-σ12T2H-t2H-1/2σ22+λσ32T-t22σ12T2H-t2H+σ22T-t+λσ32T-t,βT,t=∑n=0∞Pjt=nEnξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t=∑n=0∞λnT-tnexp⁡-λT-tn!-1Enξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t,here Poisson distribution with intensity λ(T-t). Then by (47) we get(49)∂V∂t-Hσ12t2H-1+12σ22+12λσ32∂2V∂x2-r-q-λσ3-Hσ12t2H-1+12σ22+12λσ32∂V∂x-r+λV=0,(50)Vx,0=exp⁡-βxξδx=1ξδx.The solution of stochastic parabolic partial differential equations (49) and (50) can be written as(51)Vx,ξ;t,T=1ξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t·exp⁡-x22σ12T2H-t2H+σ22T-t+λσ32T-t.Combining (51) with (48) we have(52)Wx,ξ;t,T=∑n=0∞λnT-tnexp⁡-λT-tn!-1Enξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t·exp⁡-rT-t-12σ12T2H-t2H+σ22T-t+λσ32T-tx+r-q-λσ3·T-t-σ12T2H-t2H-12σ22+λσ32T-t2.Substitute original variables (S,t) in (52); we obtain(53)GS,t;ξ,T=∑n=0∞λnT-tnexp⁡-λT-tn!En·exp⁡-rT-tξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t·exp⁡-12σ12T2H-t2H+σ22T-t+λσ32T-t·ln⁡S∏i=1n1+jtiξ+r-q-λσ3T-t-σ12T2H-t2H-12σ22+λσ32T-t2.Theorem 15.
The fundamental solutionG(S,t;ξ,T) is considered to be the function of ξ and η, and G(S,t;ξ,T) satisfies the adjoint equation of (43) and (44). If we note(54)GS,t;ξ,η=vξ,η,then v(ξ,η) satisfies(55)-∂v∂t-Hσ12t2H-1+12σ22+12λσ32S2∂2v∂S2-r-q-λσ3S∂v∂S-λEvS1+jt,t-vS,t-r+λv=0,(56)vξ,η=δξ-S,where 0<S<∞, 0<ξ<∞, t<η.Corollary 16.
By Lemmas13 and 14 and Theorem 15, if one notes G∗(ξ,η;S,t) is the fundamental solution of (55) and (56), then (57)GS,t;ξ,η=G∗ξ,η;S,t.Proof of Theorem15.
For anyϵ>0, consider integral as follows:(58)0=∫0∞∫t+ϵη-ϵG∗x,y;S,t·Gx,y;ξ,η-Gx,y;ξ,η·G∗x,y;S,tdxdy=∫0∞dx∫t+ϵη-ϵ∂G∗G∂y+Hσ12t2H-1+12σ22+12λσ32∂∂xx2G∗∂G∂x+λEvS1+jt,t-vS,t-Hσ12t2H-1+12σ22+12λσ32∂∂xG∂∂xx2G∗+r-q-λσ3∂∂xxGG∗-λEvS1+jt,t-vS,tdy.When x→0, we have(59)x2G∗∂G∂x⟶0,G∂∂xx2G∗⟶0,xGG∗⟶0,since(60)∫0∞G∗x,η-ϵ;S,t·Gx,η-ϵ;ξ,ηdx=∫0∞G∗x,t+ϵ;S,t·Gx,t+ϵ;ξ,ηdx.Let ϵ→0; by conditions (44) and (56) we get(61)∫0∞G∗x,η;S,t·δx-ξdx=∫0∞δx-S·Gx,t;η,ξdx.Then(62)GS,t;ξ,η=G∗ξ,η;S,t.


## 3.2. 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 T are given, respectively, by (ST-amT)+ and (bMT-ST)+, 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 T>0, let C=C(t,S,m) be the value of the American fractional lookback call option at time t∈[0,T]. 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(t,S,m) is a solution of an optimal stopping problem(63)Ct,S,m=supTt∈t,T⁡Eexp⁡-rTt-tSTt-amTt+∣St=S,mt=m,where Tt is a stopping time of the filtration F and the conditional expectation is calculated under the risk-neutral probability measure P. The random variable Tt∗∈[t,T] 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 S and nonincreasing in t, m, and a. Solving the optimal stopping problem (63) is equivalent to finding the points (t,St,mt) for which early exercise before maturity is optimal. Let(64)D=t,S,m∈0,T×m,∞×R+be the whole domain, and let E and C denote the exercise region and continuation region, respectively. In terms of the value function C(t,S,m), the exercise region E is defined by (65)E=t,S,m∣Ct,S,m=S-am+,for which the optimal stopping time Tt∗ satisfies(66)Tt∗=inf⁡η∈t,T∣η,Sη,mη∈E.The continuation region C is the complement of S in D, such as(67)C=t,S,m∣Ct,S,m>S-am+.The boundary that separates S from C is referred to as the early exercise boundary, which is defined by(68)S¯t,m=sup⁡S≥m∣t,S,m∈C,t∈0,T.At the early exercise boundary [S¯(t,m)]t∈[0,T], the American fractional lookback call option would be optimally exercised. In terms of S(t,m), the continuation region C can be represented as(69)C=t,S,m;m≤S<S¯t,m.LetV(S,J,t) be the lookback option price at time t with stock price S and path-dependent variable J. 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: (70)0=∂V∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2V∂S2+r-q-λσ3S∂V∂S-r+λV+λEVS1+jt,min⁡J,S1+jt,t,for  lookback  call  option,λEVS1+jt,max⁡J,S1+jt,t,for  lookback  put  option,VS,J,t=J-S,where 0≤t≤T, and satisfies an order continuous differentiable on domain(71)Σ=S,J:0<J≤S<∞,for  lookback  call  option,S,J:0<S≤J<∞,for  lookback  put  option,and the terminal condition is given by (72)∂V∂JS=J=0.We define the differential operator Lt,S by(73)Lt,S=∂∂t+Hσ12t2H-1+12σ22+12λσ32S2∂2∂S2+r-q-λσ3S∂∂S-r+λ.Then the free boundary problem can be written in a linear complementary form as(74)Lt,SCt,S,m·Ct,S,m-S-am+=0,Lt,SCt,S,m≤0,Ct,S,m-S-am+≥0together with auxiliary conditions(75)CT,S,m-S-am+=0,limS↓m⁡∂C∂m=0.For the free boundary[S¯(t,m)]t∈[0,T], this problem is equivalent to solving the Black-Scholes-Merton partial differential equations (76)Lt,SCt,S,m=0,m≤S<S¯t,m,together with the boundary conditions (77)limS↑S¯⁡Ct,S,m=S¯t,m-am,limS↑S¯⁡∂C∂S=1,limS↓m⁡∂C∂m=0.and the terminal condition (78)CT,S,m=S-am+.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(t,S,m) be the value of the American fractional lookback call option at time t∈[0,T]. The value P(t,S,M) is a solution of an optimal stopping problem (79)Pt,S,M=supTt∈t,T⁡Eexp⁡-rTt-tbMTt-STt+∣St=S,Mt=M,and P(t,S,M) satisfies the same PDE as (76); then(80)Lt,SPt,S,M=0,S_t,M<S≤M,where [S_(t,M)]t∈[0,T] is the early exercise boundary for put. The boundary conditions for put are(81)limS↓S_⁡Pt,S,M=bM-S_t,M,limS↓S_⁡∂P∂S=-1,limS↑M⁡∂P∂M=0.and the terminal condition is given by(82)PT,S,M=bM-S+.


## 3.3. 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(t,S,M), we can obtain the early exercise representation V(t,S,M) under our mixed jump-diffusion fractional Brownian motion environment, which is shown in the following theorem.Theorem 17.
LetV(t,S,M) be American fractional lookback put option; then(83)Vt,S,M=VEt,S,M+et,S,M,where VE(t,S,M) is price of the hedging portfolio of the equivalent European option, and(84)VEt,S,M=∑n=0∞λnT-tnexp⁡-λT-tn!En·bMexp⁡-rT-tN-d^2-S∏i=1n1+jtiexp⁡-q+λσ3T-tN-d^1,where En denotes the expectation operator over the distribution of ∏i=1n(1+jti), M is contractual strike price, b is positive constants, N(·) is the cumulative normal distribution function, and(85)d^1=ln⁡S∏i=1n1+jtiξ+r-q-λσ3T-t+σ12T2H-t2H+12σ22+λσ32·T-t2σ12T2H-t2H+σ22T-t+λσ32T-t-1,d^2=d^1-2σ12T2H-t2H+σ22T-t+λσ32T-t,e(t,S,M) is early exercise premium, and its explicit integral can be represented by(86)et,S,M=∫tTdη∫0SηbMr-q+λσ3ξ·GS,t;ξ,ηdξ.Proof.
SupposeV(t,S,M) satisfies an order continuous differentiable on domain Σ: Σ={0≤S≤M<∞,0≤t<T}, and V(t,S,M) is piecewise continuous on domain Σ, and the second-order derivative exists; then(87)-LVt,S,M=0,t,S,M∈Σ1,bMr-q+λσ3S,t,S,M∈Σ2,where L is the Black-Scholes operator.
Multiplying both sides of (87) to G∗(ξ,η;S,t) and quadrature on domain {0≤ξ≤M<∞,t+ϵ≤η≤T}, for Σ2={0≤ξ≤M≤Sη,0≤η<T} and Sη is continuous monotonous of the asset price, then(88)∫t+ϵTdη∫0SηbMr-q+λσ3ξ·G∗ξ,η;S,tdξ=-∫t+ϵTdη∫0∞G∗ξ,η;S,tLVdξ=-∫t+ϵTdη∫0∞G∗ξ,η;S,tLVη,ξ,M-Vη,ξ,ML∗G∗ξ,η;S,tdξ=-∫t+ϵTdη∫0∞∂G∗V∂η+Hσ12t2H-1+12σ22+12λσ32∂∂ξξ2G∗∂V∂ξ+λEVt,S1+jt,M-Vt,S,M-Hσ12t2H-1+12σ22+12λσ32∂∂ξ·V∂∂ξξ2G∗+r-q-λσ3∂∂ξξVG∗-λEVt,S1+jt,M-Vt,S,Mdξ,where L∗ is the adjoint Black-Scholes operator, and when ξ→0, we have(89)ξ2G∗∂V∂ξ⟶0,V∂∂ξξ2G∗⟶0,ξVG∗⟶0,and hence(90)∫0∞G∗ξ,t+ϵ;S,tVt+ϵ,ξ,Mdξ=∫0∞G∗ξ,T;S,tVT,ξ,Mdξ+∫t+ϵTdη∫0SηbMr-q+λσ3ξG∗ξ,η;S,tdξ.Let ϵ→0; by (56) and Corollary 16, we obtain(91)Vt,S,M=∫0∞GS,t;ξ,TbM-ξ+dξ+∫tTdη∫0SηbMr-q+λσ3ξGS,t;ξ,ηdξ=VEt,S,M+et,S,M.Then the proof is finished.Remark 18.
If the optimal exercise boundary is given byS=St, 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 byS=St, 0≤t<T, and then S=St satisfy the following Volterra integral equation(92)St=bM+Sexp⁡-q+λσ3T-t·N--ln⁡S∏i=1n1+jtibM+r-q-λσ3T-t+σ12T2H-t2H+12σ22+λσ32T-t·2σ12T2H-t2H+σ22T-t+λσ32T-t-1-bMexp⁡-rT-t∑n=0∞λnT-tnexp⁡-λT-tn!En·N--ln⁡S∏i=1n1+jtibM+r-q-λσ3T-t-σ12T2H-t2H-12σ22+λσ32T-t·2σ12T2H-t2H+σ22T-t+λσ32T-t-1-bMr∫tTexp⁡-rη-t∑n=0∞λnη-tnexp⁡-λη-tn!En·1-Nln⁡S∏i=1n1+jtiSη+r-q-λσ3η-t-σ12η2H-t2H-12σ22+λσ32η-t·2σ12η2H-t2H+σ22η-t+λσ32η-t-1dη+q+λσ3S∫tTexp⁡-q+λσ3η-t∑n=0∞λnη-tnexp⁡-λη-tn!En·1-NlnS∏i=1n1+jtiSη+r-q-λσ3η-t+σ12η2H-t2H+12σ22+λσ32η-t·2σ12η2H-t2H+σ22η-t+λσ32η-t-1dη,where N(·) is the cumulative normal distribution function and Sη≥M>0, 0<η≤T.Proof.
By (86) and (50), e(S,t) can be represented by(93)et,S,M=∫tTdη∫0SηbMr-q+λσ3ξ·∑n=0∞λnT-tnexp⁡-λT-tn!En·exp⁡-rT-tξ2π2σ12T2H-t2H+σ22T-t+λσ32T-t·exp⁡-12σ12T2H-t2H+σ22T-t+λσ32T-t·ln⁡S∏i=1n1+jtiξ+r-q-λσ3T-t-σ12T2H-t2H-12σ22+λσ32T-t2dξ.
Let(94)x∗=12σ12η2H-t2H+σ22η-t+λσ32η-t·ln⁡S∏i=1n1+jtiξ+r-q-λσ3η-t-σ12η2H-t2H-12σ22+λσ32η-t,dx∗=-1ξ2σ12T2H-t2H+σ22T-t+λσ32T-tdξ,and then(95)et,S,M=bMr2π∫tTexp⁡-rη-t∑n=0∞λnη-tnexp⁡-λη-tn!Endη·∫ln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t-σ12η2H-t2H-1/2σ22+λσ32η-t/2σ12η2H-t2H+σ22η-t+λσ32η-t∞exp⁡-x2dx-q+λσ3S2π∫tTexp⁡-q+λσ3η-t·∑n=0∞λnη-tnexp⁡-λη-tn!Endη·∫ln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t-σ12η2H-t2H-1/2σ22+λσ32η-t/2σ12η2H-t2H+σ22η-t+λσ32η-t∞exp⁡-x22+2σ12η2H-t2H+σ22η-t+λσ32η-txdx=bMr∫tTexp⁡-rη-t∑n=0∞λnη-tnexp⁡-λη-tn!En·1-Nln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t-σ12η2H-t2H-σ22+λσ32η-t/22σ12η2H-t2H+σ22η-t+λσ32η-tdη-q+λσ3S∫tTexp⁡-q+λσ3η-t∑n=0∞λnη-tnexp⁡-λη-tn!Endη·∫ln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t-σ12η2H-t2H-1/2σ22+λσj2η-t/2σ12η2H-t2H+σ22η-t+λσ32η-t∞exp⁡-12·x+2σ12η2H-t2H+σ22η-t+λσ32η-t2+r-q-λσ3η-t+σ12η2H-t2H+σ22+λσ32η-t2dx=bMr∫tTexp⁡-rη-t∑n=0∞λnη-tnexp⁡-λη-tn!En·1-Nln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t-σ12η2H-t2H-σ22+λσ32η-t/22σ12η2H-t2H+σ22η-t+λσ32η-tdη-q+λσ3S∫tTexp⁡-q+λσ3η-t∑n=0∞λnη-tnexp⁡-λη-tn!En·1-Nln⁡S∏i=1n1+jti/Sη+r-q-λσ3η-t+σ12η2H-t2H+σ22+λσ32η-t/22σ12η2H-t2H+σ22η-t+λσ32η-tdη.Substituting (95) into (83), we have(96)Vt,St,M=bM-St,and then we obtain (97)bM-St=VEt,St,M+et,St,M,such that (98)St=bM+Sexp⁡-q+λσ3T-t·N--ln⁡S∏i=1n1+jtibM+r-q-λσ3T-t+σ12T2H-t2H+12σ22+λσ32T-t·2σ12T2H-t2H+σ22T-t+λσ32T-t-1-bMexp⁡-rT-t∑n=0∞λnT-tnexp⁡-λT-tn!En·N--ln⁡S∏i=1n1+jtibM+r-q-λσ3T-t-σ12T2H-t2H-12σ22+λσ32T-t·2σ12T2H-t2H+σ22T-t+λσ32T-t-1-bMr∫tTexp⁡-rη-t∑n=0∞λnT-tnexp⁡-λT-tn!En·1-Nln⁡S∏i=1n1+jtiSη+r-q-λσ3η-t-σ12η2H-t2H-12σ22+λσ32η-t·2σ12η2H-t2H+σ22η-t+λσ32η-t-1dη+q+λσ3S∫tTexp⁡-q+λσ3η-t∑n=0∞λnT-tnexp⁡-λT-tn!En·1-Nln⁡S∏i=1n1+jtiSη+r-q-λσ3η-t+σ12η2H-t2H+12σ22+λσ32η-t·2σ12η2H-t2H+σ22η-t+λσ32η-t-1dη.
In practice, it is very difficult to calculate nonlinear integral in (92); we only show that this assumption holds in Volterra integral equation for the optimal exercise boundary; by the relation (92) and Theorem 17, asymptotic expression of optimal exercise boundary S=St is obtained near the maturity t=T. Comparing with Theorem 17 and Corollary 20, the optimal exercise boundary and Volterra integral equation of American fractional lookback call option are obvious as follows.Corollary 21.
LetV∗(t,S,m) be American fractional lookback call option; then(99)V∗t,S,m=VE∗t,S,m+e∗t,S,m,where VE∗(t,S,m) is price of the hedging portfolio of the equivalent European option, and(100)VE∗t,S,m=∑n=0∞λnT-tnexp⁡-λT-tn!En·S∏i=1n1+jtiexp⁡-q+λσ3T-tN-d^1-amexp⁡-rT-tN-d^2,where En denotes the expectation operator over the distribution of ∏i=1n(1+jti), am is contractual strike price, N(·) is the cumulative normal distribution function, and(101)d^1=ln⁡S∏i=1n1+jtiξ+r-q-λσ3T-t+σ12T2H-t2H+12σ22+λσ32T-t·2σ12T2H-t2H+σ22T-t+λσ32T-t-1,d^2=d^1-2σ12T2H-t2H+σ22T-t+λσ32T-t,e∗(t,S,m) is early exercise premium, and its explicit integral can be represented by (102)e∗t,S,m=∫tTdη∫0Sηq+λσ3ξ-amr·GS,t;ξ,ηdξ.


## 4. Simulation Studies

In the whole domainD={(t,S,M)∈[0,T]×[M,∞)×R+} of the American fractional floating strike lookback put option model, the American put is alive when S<M and becomes dead when S>M. The boundary which divides the continuation region (option remains alive) and the stopping region (option becomes dead) is time dependent; that is, S is a function of t. Similar to the usual argument for American floating strike lookback put options the critical exercise price should be denoted as [30](103)S_t,M=Stt,S,MM.As deduced from (92), at a given time t, the optimal exercise boundary St(t,S,M) increases linearly with M. Since both S_(t,M) and M are increasing function of time t, St(t,S,M) increases as time is approaching expiration.Similarly, the critical exercise price for the American fractional floating strike lookback call option should be denoted as(104)S¯t,m=Stt,S,mm,where S¯(t,m) is a monotonically decreasing function of time t. The plots of S_(t,M) and S¯(t,m) against time t with varying interest rate are shown in Figure 1, where m=M=100, T=1, r=0.05, q=0.02,0.04,0.06, σ1=0.2, σ2=0.3, σ3=0.0071, λ=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].Figure 1
Plot ofS_(t,M) and S¯(t,m) against time t for American fractional lookback options with varying q=0.02,0.04,0.06.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(105)limt→T-⁡Stt,S,M=1amin⁡rq+λσ3,1limt→T-⁡Mt,limt→T-⁡Stt,S,m=1bmax⁡rq+λσ3,1limt→T-⁡mt.Hence the strike prices of the American fractional lookback put and call options are set to be M0T and m0T, 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(St)t≥0. If r-q-λσ3<0, then St decreases to mT as t→T in an average sense, so that (St-amt)+→0 and (bMt-St)+ grows as t→T 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 r-q-λσ3>0, then the reverse would be true. For put options with r-q-λσ3<0 and call options with r-q-λσ3>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 r-q-λσ3<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 r-q-λσ3=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 S, and the option values are relatively insensitive to the asset price S when the option is out-of-the-money. Notice that the call (put) values are almost increasing (decreasing) in S when the option is in-the-money but not when S=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.Figure 2
Values of American and European fractional lookback options(t=0, T=1, m=80, M=120, r-q-λσ3=0.05, a=1.0,1.1,1.2,1.3, b=0.7,0.8,0.9,1.0).


## 5. 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 r≥q+λσ3(r≤q+λσ3), 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.



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*Source: 5904125-2017-07-17.xml*