Modern Stochastics: Theory and Applications

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 $H>1/2$ 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 $H\in (1/2,1)$. The asymptotic behaviour of a modified fractional geometric Ornstein-Uhlenbeck process variance as $H\to 1/2$ and $H\to 1$ 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.

Let $(\varOmega ,\mathcal{F},\{\mathcal{F}\}_{t\ge 0},P)$ be the probability space which satisfies standard assumptions. Let $W=\{W_{t},\mathcal{F}_{t},t\ge 0\}$ be the Wiener process defined on this probability space. The Ornstein-Uhlenbeck process is defined as the solution of the following stochastic differential equation where $a>0$ and $\gamma >0$ are constants. This stochastic differential equation has the following solution: The process $\tilde{X}$ is Gaussian and Markov.

In what follows for technical simplicity we consider $\tilde{X}_{0}=1$. The Ornstein-Uhlenbeck process has the following numerical characteristics: $E\tilde{X}_{t}={e}^{-at}\to 0$, $t\to \infty $ and $\operatorname{Var}\tilde{X}_{t}={\gamma }^{2}\frac{1-{e}^{-2at}}{2a}\to \frac{{\gamma }^{2}}{2a}$, $t\to \infty $. 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: Let us consider a modified geometric Ornstein-Uhlenbeck process Z with additional parameter for the variance: where $X_{t}={e}^{-\mu t}+\gamma {e}^{-\mu t}{\int _{0}^{t}}{e}^{as}dW_{s}$ is the modified Ornstein-Uhlenbeck process, $a>0$, $\mu \in R$. Mathematical expectation and variance for $X_{T}$ are the following:

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.

We investigate the behaviour of the European call option price (2) as a function of the mean m and the variance ${\sigma }^{2}$.

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.

Let the financial market be considered on $[0,T]$.

The existence of the martingale measure ${P}^{\ast }$ 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 $B(t)={e}^{rt}$ and the risk price process governed by a modified geometric Ornstein-Uhlenbeck process X. Note that X satisfies the following linear stochastic differential equation: In order of technical simplification further on we consider the following discounted price process: where This discounted price process can be represented as follows: We look for the likelihood ratio $\frac{d{P}^{\ast }}{dP}|_{t}$ of the form where $\beta _{t}$, $t\in [0,T]$, is $\mathcal{F}_{t}$-adapted process and the discounted price process ${Z_{t}^{\ast }}$ is the martingale with respect to the measure ${P}^{\ast }$. Consider the product of the price process and the likelihood ratio: Since this process must be a martingale with respect to the objective measure P, it must have the form: Comparing integrands in (5) and (6), we obtain the following equations for the processes α and β: From these equations we obtain that For all that the process $\frac{d{P}^{\ast }}{dP}|_{t}$ must be a martingale. According to Theorem 6.1 [5] the process $\varphi _{t}(c)=\exp \{{\int _{0}^{t}}c_{s}\hspace{0.1667em}dW_{s}-\frac{1}{2}{\int _{0}^{t}}{c_{s}^{2}}\hspace{0.1667em}ds\}$ is a martingale on $[0,T]$ with $E\varphi _{t}(c)=1$, $t\in [0,T]$, when c is a Gaussian process such that In our case $\beta _{t}$, $t\in [0,T]$, is indeed Gaussian and $\mathcal{F}_{t}$-adapted. We also know that Then So the martingale conditions hold.

Since equations (5) and (6) explicitly define the process β, then the martingale measure $\frac{d{P}^{\ast }}{dP}|_{T}$ on $[0,T]$ is also defined uniquely.

Thus the following result holds:

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 ${Z}^{\ast }$, 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 ${K}^{\ast }:=K{e}^{-rT}$.

Using the results of the previous section we obtain the following representation for the fair price $C_{f}$ of the European call option, issued on the bond with discounted price, governed by the process ${Z}^{\ast }$: where ${E}^{\ast }$ is the mean with respect to the risk-neutral measure, ${Y_{t}^{\ast }}:=\gamma {\int _{0}^{t}}{e}^{(a-\mu )s}\hspace{0.1667em}dW_{s}-\frac{{\gamma }^{2}}{2}{\int _{0}^{t}}{e}^{2(a-\mu )s}\hspace{0.1667em}ds$. The process ${Y}^{\ast }$ is Gaussian, the mean and the variance of ${Y_{T}^{\ast }}$ are the following: where We apply Lemma 2.1 and obtain the following fair price of the option: Using the same lemma we obtain the following objective price of the option, issued on the specified bond: where $m_{{o}^{\ast }}$ and ${\sigma _{{o}^{\ast }}^{2}}$ are the mean and the variance respectively of the process ${X_{T}^{\ast }}$:

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 $r<\frac{{e}^{-\mu T}}{T}$. Compare the mean and the variance for the case of the fair price with the corresponding values for the objective price case.

For the mean of the corresponding processes we have the inequality $m_{f}<m_{{o}^{\ast }}$. Investigate the mutual location of the variances.

First we consider the case $a\ne \mu $. Then It is obvious that when $\mu >0$, and the inverse inequality holds when $\mu <0$.

Now consider the case $a=\mu $. Then ${\sigma _{f}^{2}}={\gamma }^{2}T>{\gamma }^{2}\frac{1-{e}^{-2aT}}{2a}={\sigma _{{o}^{\ast }}^{2}}$.

(2) Let $r>\frac{{e}^{-\mu T}}{T}$, $a>0$, $\mu >0$. Then $m_{f}>m_{{o}^{\ast }}$ and ${\sigma _{f}^{2}}>{\sigma _{{o}^{\ast }}^{2}}$. Thus we have proved the following result:

The standard fractional Brownian motion with Hurst index $H\in (0,1)$ is a Gaussian process ${B}^{H}=\{{B_{t}^{H}},t\in R_{+}\}$ on the $(\varOmega ,\mathcal{F},P)$ with the following properties:

Let us introduce a fractional Ornstein-Uhlenbeck process where ${B_{s}^{H}}$ is a fractional Brownian motion with Hurst index $H>1/2$. The existence of the integral ${\int _{0}^{t}}{e}^{as}d{B_{s}^{H}}$ follows from [8]. The process ${X}^{H}$ satisfies the following stochastic differential equation: The mean and the variance of the Ornstein-Uhlenbeck process ${X}^{H}$ at the moment T equal and correspondingly. Consider a European call option issued on a bond, governed by a geometric fractional Ornstein-Uhlenbeck process

We calculate the objective price of this option using formula $E{[{Y_{T}^{H}}-K]}^{+}$. 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: where the mean m is determined by the formula (8), and the variance ${\sigma }^{2}$ is determined by the formula (9).

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.

Therefore the variance of the geometric fractional Ornstein-Uhlenbeck process when $H\to \frac{1}{2}$ converges to the variance of the geometric Ornstein-Uhlenbeck process with a Wiener process.

Now we investigate the monotonicity of variance as a function of the Hurst index. Consider The multiplier $2{e}^{aT}$ does not depend on H, so it is omitted. Use integration by parts: We obtain Using integration by parts again we get Now we consider the term Using the following transformation $\frac{z}{T}=x$; $dz=Tdx$; $z\in [0;T]$; $x\in [0;1]$, we obtain Use the change of variables $aT=p$: Let us calculate the derivative Up to some constant multiplier ${R^{\prime }}(H)$ has the same sign as $\frac{{R^{\prime }}(H)}{R(H)}$. 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:

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 ${\sigma }^{2}$, for the price as a function of Hurst index there are also three cases:

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 ${\sigma }^{2}$ (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.