European call option issued on a bond governed by a modified geometric Ornstein-Uhlenbeck process, is investigated. Objective price of such option as a function of the mean and the variance of a geometric Ornstein-Uhlenbeck process is studied. It is proved that the “Ornstein-Uhlenbeck” market is arbitrage-free and complete. We obtain risk-neutral measure and calculate the fair price of a call option. We consider also the bond price, governed by a modified fractional geometric Ornstein-Uhlenbeck process with Hurst index H∈(1/2,1). Limit behaviour of the variance of the process as H→1/2 and H→1 is studied, the monotonicity of the variance and the objective price of the option as a function of Hurst index is established.
Modern financial world requires increasingly more accurate and convenient models for simulation of the dynamics of real financial markets. Classical stochastic models for interest rates dynamics are Vasicek, Cox-Ingersoll-Ross and Hull–White models [1–4, 6, 7, 9, 10]. These models turn out to be convenient for interest rates dynamics modelling as well as for analysis of dynamics of financial instruments, depending on interest rates.
Actually any model gives only a certain approximation of the real markets dynamics. In particular, interest rates of real financial markets typically have jumping dynamics at some moments of time. Also, a serious shortcoming of Vasicek and Hull-White models is the possibility of interest rate under the model to become negative.
To avoid negative values, we consider the geometric Vasicek model (Ornstein-Uhlenbeck model). For instance, we consider a modified geometric Ornstein-Uhlenbeck process and modified geometric fractional Ornstein-Uhlenbeck process with the fractional Brownian motion with Hurst index H>1/21/2$]]> instead of the Wiener process. This process has a long memory property and flexibility, necessary for simulation of specific features of financial markets.
The paper is devoted to the investigation of European call option, issued on a bond, governed by a geometric or a fractional geometric Ornstein-Uhlenbeck process. The paper is organized as follows. In Section 2 we obtain the objective price of a call option, issued on a bond, governed by a modified geometric Ornstein-Uhlenbeck process. Its behaviour as a function of mean and variance of the modified Ornstein-Uhlenbeck process is studied in Section 3. Section 4 is devoted to the arbitrage-free property and completeness of the market generated by a modified Ornstein-Uhlenbeck process. The call option fair price is obtained in Section 5. In Section 6 we get the objective price of the call option, governed by a modified fractional geometric Ornstein-Uhlenbeck process with Hurst index H∈(1/2,1). The asymptotic behaviour of a modified fractional geometric Ornstein-Uhlenbeck process variance as H→1/2 and H→1 is investigated in Section 7. Finally, the monotonicity property of the variance and the objective price of the option as the Hurst index function is presented in Section 8.
The objective price of a European call option issued on a bond governed by a modified geometric Ornstein-Uhlenbeck process
Let (Ω,F,{F}t≥0,P) be the probability space which satisfies standard assumptions. Let W={Wt,Ft,t≥0} be the Wiener process defined on this probability space. The Ornstein-Uhlenbeck process is defined as the solution of the following stochastic differential equation
dX˜t=−aX˜tdt+γdWt,X˜|t=0=X˜0,
where a>00$]]> and γ>00$]]> are constants. This stochastic differential equation has the following solution:
X˜t=X˜0e−at+γe−at∫0teasdWs.
The process X˜ is Gaussian and Markov.
In what follows for technical simplicity we consider X˜0=1. The Ornstein-Uhlenbeck process has the following numerical characteristics: EX˜t=e−at→0, t→∞ and VarX˜t=γ21−e−2at2a→γ22a, t→∞. That is mean and variance are asymptotically stable, so this process is convenient for simulation of interest rates or stock values, but it can take negative values. Therefore for interest rate dynamics simulation a geometric Ornstein-Uhlenbeck process appears to be more acceptable:
Z˜t=exp{X˜t}=exp{e−at+γe−at∫0teasdWs}.
Let us consider a modified geometric Ornstein-Uhlenbeck process Z with additional parameter for the variance:
Zt=exp{Xt},
where Xt=e−μt+γe−μt∫0teasdWs is the modified Ornstein-Uhlenbeck process, a>00$]]>, μ∈R. Mathematical expectation and variance for XT are the following:
mo:=EXT=e−μT,σo2:=VarXT=γ2e−2μT∫0Te2asds=γ2e−2μT12a(e2aT−1).
Consider the model of financial market where a bond price is governed by a geometric Ornstein-Uhlenbeck process (1). Let us calculate the objective price of the European call option issued on this bond. In what follows in this section we assume that all values are discounted.
First we prove a simple auxiliary result.
Let the bond price be governed by the stochastic processeY, whereY={Yt,t∈[0,T]}is a Gaussian process. Then the price C of the issued on this bond European call option with the strike price K and maturity date T equalsC(m,σ2)=em+12σ2Φ(m+σ2−lnKσ)−KΦ(m−lnKσ),wherem=EYT,σ2=VarYT.
We express the option price in terms of density of the distribution:
C(m,σ2)=E[eYT−K]+=∫lnK∞(ex−K)1σ2πexp{−(x−m)22σ2}dx:=I1+I2.
Now we calculate either of the integrals:
I1=∫lnK∞ex1σ2πexp{−(x−m)22σ2}dx=exp{m+σ22}(1−Φ(lnK−m−σ2σ))=exp{m+σ22}Φ(m+σ2−lnKσ),
where Φ is the distribution function of the standard normal distribution. Similarly
I2=−K∫lnK∞1σ2πexp{−(x−m)22σ2}dx=−K∫lnK−mσ∞12πe−y22dy=−K(1−Φ(lnK−mσ))=−KΦ(m−lnKσ).
Thus the equality (2) holds. □
Using the result of Lemma2.1we obtain the objective priceCoof the European call option issued on the bond governed by the modified geometric Ornstein-Uhlenbeck process Z:Co=emo+12σo2Φ(mo+σo2−lnKσo)−KΦ(mo−lnKσo).
The behaviour of the option price as a function of mean and variance
We investigate the behaviour of the European call option price (2) as a function of the mean m and the variance σ2.
The option price (2) is increasing in m and inσ2.
We calculate the derivatives with respect to s:=σ2 and m. The derivative in s is equal
C(m,s)s′=12exp{m+s2}Φ(m+s−lnKs)+s−m+lnK22πssexp{m+s2}exp{−12(m+s−lnKs)2}+K(m−lnK)22πssexp{−12(m−lnKs)2}=12exp{m+s2}Φ(m+s−lnKs)+K22πsexp{−12(m−lnKs)2}.
The derivative in m is equal
C(m,s)m′=exp{m+s2}Φ(m+s−lnKs)+12πsexp{m+s2}exp{−12(m+s−lnKs)2}−K2πsexp{−12(m−lnKs)2}=exp{m+s2}Φ(m+s−lnKs).
From the equalities (3) and (4) it follows, that both derivatives are positive, and therefore the option price increases in m and in σ2. □
Arbitrage-free property and completeness of the financial market generated by a modified Ornstein-Uhlenbeck process
We investigate the arbitrage-free property and completeness of the financial market generated by a modified geometric Ornstein-Uhlenbeck process. Let us recall the necessary definitions.
The probability measure P∗∼P is called the martingale measure if the discounted price process is P∗-martingale.
Let the financial market be considered on [0,T].
A financial market is complete, if every FT-measurable integrated contingent claim is achievable, that is for such claim the generating portfolio exists.
The existence of the martingale measure P∗ is equivalent to the arbitrage-free property of the market, its uniqueness is equivalent to the completeness of the market.
Let us prove the arbitrage-free property and completeness of the financial market. On the market under consideration we have the risk-free interest rate B(t)=ert and the risk price process governed by a modified geometric Ornstein-Uhlenbeck process X. Note that X satisfies the following linear stochastic differential equation:
dXt=−μXtdt+γe(a−μ)tdWt,X0=1.
In order of technical simplification further on we consider the following discounted price process:
Zt∗=Ztexp{−rt−γ22∫0te2(a−μ)sds}=exp{Xt∗},
where
Xt∗:=e−μt−rt−γ22∫0te2(a−μ)sds+γe−μt∫0teasdWs.
This discounted price process can be represented as follows:
Zt∗=exp{Xt−rt−γ22∫0te2(a−μ)sds}=exp{1−μ∫0tXsds+γ∫0te(a−μ)sdWs−rt−γ22∫0te2(a−μ)sds}=exp{γ∫0te(a−μ)sdWs−γ22∫0te2(a−μ)sds−∫0t(μXs+r)ds+1}.
We look for the likelihood ratio dP∗dP|t of the form
dP∗dP|t=exp{∫0tβsdWs−12∫0tβs2ds},
where βt, t∈[0,T], is Ft-adapted process and the discounted price process Zt∗ is the martingale with respect to the measure P∗. Consider the product of the price process and the likelihood ratio:
Zt∗dP∗dP|t=exp{∫0t[γe(a−μ)s+βs]dWs−12∫0t[γ2e2(a−μ)s+2(μXs+r)+βs2]ds+1}.
Since this process must be a martingale with respect to the objective measure P, it must have the form:
exp{∫0tαsdWs−12∫0tαs2ds}.
Comparing integrands in (5) and (6), we obtain the following equations for the processes α and β:
αt=γe(a−μ)t+βt,αt2=γ2e2(a−μ)t+βt2+2(μXt+r).
From these equations we obtain that
βt=μγe(μ−a)tXt+rγe(μ−a)t.
For all that the process dP∗dP|t must be a martingale. According to Theorem 6.1 [5] the process φt(c)=exp{∫0tcsdWs−12∫0tcs2ds} is a martingale on [0,T] with Eφt(c)=1, t∈[0,T], when c is a Gaussian process such that
supt≤TE|ct|<∞andsupt≤TVarct2<∞.
In our case βt, t∈[0,T], is indeed Gaussian and Ft-adapted. We also know that
EXt=e−μtandVarXt=γ2e−2μt2a(e2at−1).
Then
Eβt=μγe−at+rγe(μ−a)tandVarβt=12aμ2(1−e−2at).
So the martingale conditions hold.
Since equations (5) and (6) explicitly define the process β, then the martingale measure dP∗dP|T on [0,T] is also defined uniquely.
Thus the following result holds:
The financial market generated by a modified geometric Ornstein-Uhlenbeck process is arbitrage-free and complete.
The comparison of objective and fair price of the European call option
Our further goal is to calculate the fair price of the European call option issued on the bond with the discounted price governed by the price process Z∗, and to compare it with the objective price of the option issued on the specified bond.
Let K be the option strike price, the option maturity time is T, r is the annual compounded risk-free bank interest rate. The discounted strike price of the option is equal K∗:=Ke−rT.
Using the results of the previous section we obtain the following representation for the fair price Cf of the European call option, issued on the bond with discounted price, governed by the process Z∗:
Cf=E∗[ZT∗−K∗]+=E[eYT∗−K∗]+,
where E∗ is the mean with respect to the risk-neutral measure, Yt∗:=γ∫0te(a−μ)sdWs−γ22∫0te2(a−μ)sds. The process Y∗ is Gaussian, the mean and the variance of YT∗ are the following:
mf=EYT∗=−12γ2∫0Te2(a−μ)sds=−12γ2bT2,σf2:=VarYT∗=γ2∫0Te2(a−μ)sds=γ2bT2,
where
bt2=∫0te2(a−μ)sds=12(a−μ)[e2(a−μ)t−1],a≠μ,t,a=μ.
We apply Lemma 2.1 and obtain the following fair price of the option:
Cf=emf+12σf2Φ(mf+σf2−lnK∗σf)−K∗Φ(mf−lnK∗σf).
Using the same lemma we obtain the following objective price of the option, issued on the specified bond:
Co∗=emo∗+12σo∗2Φ(mo∗+σo∗2−lnK∗σo∗)−K∗Φ(mo∗−lnK∗σo∗),
where mo∗ and σo∗2 are the mean and the variance respectively of the process XT∗:
mo∗:=E[XT∗]=e−μT−rT−γ22∫0Te2(a−μ)sds=e−μT−rT−12γ2bT2,σo∗2:=VarXT∗=γ2e−2μT∫0Te2asds=γ2e−2μT12a(e2aT−1).
Using Lemma 3.1 we obtain that the option price is an increasing function of the mean and the variance. Therefore comparing the mean and the variance of the corresponding processes, we can compare the objective and the fair option price.
(1) Suppose that r<e−μTT. Compare the mean and the variance
mf=−12γ2bT2andσf2=γ2bT2
for the case of the fair price with the corresponding values
mo∗=e−μT−rT−12γ2bT2andσo∗2=γ2e−2μT12a(e2aT−1)
for the objective price case.
For the mean of the corresponding processes we have the inequality mf<mo∗. Investigate the mutual location of the variances.
First we consider the case a≠μ. Then
σf2=γ212(a−μ)[e2(a−μ)T−1],σo∗2=γ2e−2μT12a[e2aT−1].
It is obvious that
σf2=γ2∫0Te2(a−μ)sds>γ2e−2μT∫0Te2asds=σo∗2,{\gamma }^{2}{e}^{-2\mu T}{\int _{0}^{T}}{e}^{2as}\hspace{0.1667em}ds={\sigma _{{o}^{\ast }}^{2}},\]]]>
when μ>00$]]>, and the inverse inequality holds when μ<0.
Now consider the case a=μ. Then σf2=γ2T>γ21−e−2aT2a=σo∗2{\gamma }^{2}\frac{1-{e}^{-2aT}}{2a}={\sigma _{{o}^{\ast }}^{2}}$]]>.
(2) Let r>e−μTT\frac{{e}^{-\mu T}}{T}$]]>, a>00$]]>, μ>00$]]>. Then mf>mo∗m_{{o}^{\ast }}$]]> and σf2>σo∗2{\sigma _{{o}^{\ast }}^{2}}$]]>. Thus we have proved the following result:
(1) Let the interest rater<e−μTT,a>00$]]>,μ<0. Then the fair price is less than the objective priceCo∗>CfC_{f}$]]>.
The price of a bond governed by a fractional geometric Ornstein-Uhlenbeck process
The standard fractional Brownian motion with Hurst index H∈(0,1) is a Gaussian process BH={BtH,t∈R+} on the (Ω,F,P) with the following properties:
EBtH=0, t∈R+,
EBtHBsH=12(t2H+s2H−|t−s|2H), s,t∈R+.
Let us introduce a fractional Ornstein-Uhlenbeck process
XtH=e−μt+γe−μt∫0teasdBsH,
where BsH is a fractional Brownian motion with Hurst index H>1/21/2$]]>. The existence of the integral ∫0teasdBsH follows from [8]. The process XH satisfies the following stochastic differential equation:
dXtH=−μXtHdt+γe(a−μ)tdBtH.
The mean and the variance of the Ornstein-Uhlenbeck process XH at the moment T equal
m=e−μT
and
σ2=H(2H−1)γ2e−2μT∫0T∫0Teas+au|s−u|2H−2duds,
correspondingly. Consider a European call option issued on a bond, governed by a geometric fractional Ornstein-Uhlenbeck process
YtH=exp{XtH}.
We calculate the objective price of this option using formula E[YTH−K]+. The random variable we calculate the mean for has the same structure as in the case of a modified geometric Ornstein-Uhlenbeck process. This random variable consists of the non-random and Gaussian component. Therefore, according to the formula (2), the price of such option is equal:
E[YTH−K]+=exp{m+σ22}Φ(m+σ2−lnKσ)−KΦ(m−lnKσ),
where the mean m is determined by the formula (8), and the variance σ2 is determined by the formula (9).
The behaviour of a fractional geometric Ornstein-Uhlenbeck process variance as a function of Hurst index
At first, we investigate the behaviour of a fractional geometric Ornstein-Uhlenbeck process variance as a function of Hurst index when H approaches its bounds 1/2 and 1. In this order, rewrite the variance in the following way.
The fractional geometric Ornstein-Uhlenbeck process variance can be represented as follows:σ2=H(2H−1)γ2e−2μt1a(e2at∫0te−azz2H−2dz−∫0teazz2H−2dz).
We consider the integral ∫0t∫0teas+au|s−u|2H−2duds. Replace the double integral with the repeated integral:
∫0t∫0teas+au|s−u|2H−2duds=2∫0teas∫0seau(s−u)2H−2duds.
Then it is easy to obtain the following representation for the variance:
σ2=H(2H−1)γ2e−2μt1a(e2at∫0te−azz2H−2dz−∫0teazz2H−2dz).
□
limH→12σ2=γ2e−2μt2a(e2at−1);
limH→1σ2=γ2e−2μta2(1−eat)2.
We consider two integrals from the previous lemma, multiplied by (2H−1), and use the formula of integration by parts for the first and second integrals respectively:
u=e−az;du=−ae−azdz;dv=(2H−1)z2H−2;v=z2H−1,u=eaz;du=aeazdz;dv=(2H−1)z2H−2;v=z2H−1.
We obtain
(2H−1)∫0te−azz2H−2dz=e−att2H−1+a∫0te−azz2H−1dz,(2H−1)∫0teazz2H−2dz=eatt2H−1−a∫0teazz2H−1dz.
When H→12, it is obvious that t2H−1→1, z2H−1→1. So
limH→12(2H−1)∫0te−azz2H−2dz=e−att2H−1+a∫0te−azz2H−1dz=1,limH→12(2H−1)∫0teazz2H−2dz=eatt2H−1−a∫0teazz2H−1dz=1.
Now we investigate the asymptotic behaviour of the variance when H→12.
limH→12σ2=γ2e−2μt2a(e2at−1).
When H→1 we have
limH→1σ2=γ2e−2μta(e2at∫0te−azdz−∫0teazdz),limH→1σ2=γ2e−2μta2(1−eat)2.
□
Therefore the variance of the geometric fractional Ornstein-Uhlenbeck process when H→12 converges to the variance of the geometric Ornstein-Uhlenbeck process with a Wiener process.
Variance and the objective price of the option monotonicity as a function of the Hurst index
Now we investigate the monotonicity of variance as a function of the Hurst index. Consider
H(2H−1)(e2aT∫0Te−azz2H−2dz−∫0Teazz2H−2dz)=2eaTH(2H−1)∫0Tea(T−z)−ea(z−T)2z2H−2dz=2eaTH(2H−1)∫0Tsinh(a(T−z))z2H−2dz.
The multiplier 2eaT does not depend on H, so it is omitted. Use integration by parts:
u=sinh(a(T−z));du=−a·cosh(a(T−z));dv=(2H−1)z2H−2dz;v=z2H−1.
We obtain
H(2H−1)∫0Tsinh(a(T−z))z2H−2dz=Ha∫0Tz2H−1cosh(a(T−z))dz.
Using integration by parts again
u=cosh(a(T−z));du=−a·sinh(a(T−z));dv=Hz2H−1dz;v=12z2H,
we get
Ha∫0Tz2H−1cosh(a(T−z))dz=a2(T2H+a∫0Tz2Hsinh(a(T−z))dz).
Now we consider the term
R(H)=T2H+a∫0Tz2Hsinh(a(T−z))dz.
Using the following transformation zT=x; dz=Tdx; z∈[0;T]; x∈[0;1], we obtain
R(H)=T2H+aT2H+1∫01x2Hsinh(aT(1−x))dx.
Use the change of variables aT=p:
R(H)=T2H(1+p∫01x2Hsinh(p(1−x))dx).
Let us calculate the derivative
R′(H)=2T2HlnT(1+p∫01x2Hsinh(p(1−x))dx)+2T2Hp∫01x2Hsinh(p(1−x))lnxdx.
Up to some constant multiplier
R′(H)R(H)=lnT+p∫01x2Hsinh(p(1−x))lnxdx1+p∫01x2Hsinh(p(1−x))dx.R′(H) has the same sign as R′(H)R(H). Therefore we investigate the sign of the right part. The numerator is negative, and due to the negative logarithm it increases in H. The denominator is positive and is decreasing in H. Thus the fraction is increasing in H.
We obtain three possible cases:
If T and p are such that under H=12lnT+p∫01xsinh(p(1−x))lnxdx1+p∫01xsinh(p(1−x))dx>0,0,\]]]>
then the variance increases in H. This situation is presented on the graph
If T and p are such that
lnT+p∫01xsinh(p(1−x))lnxdx1+p∫01xsinh(p(1−x))dx<0,
and at the same time
lnT+p∫01x2sinh(p(1−x))lnxdx1+p∫01x2sinh(p(1−x))dx>0,0,\]]]>
then the variance first decreases to the minimum and then increases.
If
lnT+p∫01x2sinh(p(1−x))lnxdx1+p∫01x2sinh(p(1−x))dx<0,
then the variance decreases.
In turn we can conclude on the monotonicity of the objective price of the option issued on the bond driven by the fractional geometric Ornstein-Uhlenbeck process, as a function of the Hurst index. Since the objective price increases in σ2, for the price as a function of Hurst index there are also three cases:
The objective price increases in H. This case corresponds to the first case of the variance monotonicity, i.e. for T and p such that under H=12 inequality (10) holds true.
The variance first decreases to the minimum and then increases. This case corresponds to the second case of the variance monotonicity, i.e. when for T and p we have (11) and (12).
The price decreases in H. This case corresponds to the third case of the variance monotonicity, i.e. when (13) holds true.
Conclusions
We calculate the objective price of the European call option issued on a bond governed by a modified geometric Ornstein-Uhlenbeck process. The behaviour of the objective option price as a function of m and σ2 (the mean and the variance of the corresponding modified Ornstein-Uhlenbeck process) is investigated. We show the arbitrage-free property and completeness of the financial market generated by the modified Ornstein-Uhlenbeck process. The risk-neutral measure and the fair price for the specified option are obtained. We compare the fair and objective price of the indicated option. Then we consider the model of the bond governed by a modified fractional Ornstein-Uhlenbeck process. The objective price of the option issued on such bond is calculated. The asymptotic behaviour and the monotonicity of the variance of a modified fractional geometric Ornstein-Uhlenbeck process as a function of the Hurst index are investigated. In particular, the cases of the monotonicity of variance are obtained.
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