11.1 Asset Price Dynamics

The dynamics of asset prices are reflected by uncertain movements of their values over time. Cuthbertson (1996, ch. 5) and Wilmott et al. (1997, ch. 2) state that the efficient market hypothesis (EMH) is one possible reason for the random behavior of the asset price. In spite of its different forms, the EMH basically states that past history is fully reflected in present prices and markets respond immediately to any new information about the asset. These two assumptions imply that changes in the asset price are a Markov process.

In this context, modelling the asset price is concerned with modelling the arrival of new information, which affects the price. Depending on the appearance of the so called "normal" and "rare" events, there are two basic blocks in modelling the continuous-time asset price. Neftci (2000, ch. 8) states that the main difference between the "normal" and the "rare" behavior concerns the size of the events and their probability to occur. As the interval of observation ($h$) gets smaller, the size of "normal" events also gets smaller
and it becomes unimportant as $h \rightarrow 0$. Even in a short time interval, there is always a non-zero probability that some non-noticeable news will arrive. In contrast to a "normal" event, when a "rare" event (or shock) occurs, the value of the random variable can change significantly over a short period of time. An example of a "rare" event is a market crash, such as the one that occurred in 1987. In other words, as $h \rightarrow 0$, the probability of a "rare" event moves to zero, but its size may not shrink.

If markets are dominated by "normal" events, then a Brownian motion process can be used. This is a continuous-time stochastic process, where extremes occur only infrequently according to the probabilities in the tails of the normal distribution. The continuous-time diffusion (Brownian motion-based) process is written in form of the following stochastic differential equation for the asset return:

$$\frac{dS_t}{S_t} = {\mu}\, dt + {\sigma}\, d W_t$$ (11.1)

with

$$S_t = \textrm{the current price of the underlying asset,}$$

$$\mu = \textrm{the constant trend or drift,}$$

$$\sigma = \textrm{the constant annualized volatility,}$$

$$W_t = \textrm{the standard Wiener process.}$$

These asset returns follow the so called geometric Brownian motion. Expression (11.1) is called a differential equation, because the asset price $S_t$ is only defined implicitly by describing its changes through time. Since this process has a continuous-time sample path, it does not allow for discontinuities or jumps in its values when "rare" events occur. In this case, the Poisson jump process can be useful. In particular, the time series of the asset price can be modelled as the sum of the continuous-time diffusion process and Poisson jump processes. XploRe uses the Merton jump-diffusion model to simulate and estimate simultaneously the "normal" and "rare" events in the asset price. The stochastic differential equation for $S_t$ is:

$$\frac{dS_t}{S_t}=\mu dt + \sigma dW_t + d\sum_{j=1}^{N_t}(Y_j-1)$$ (11.2)

with the following addition of variables to (11.1):

$$Y_j-1 = \textrm{a log-normal distributed random variable representing the }$$

$$\textrm{jump size,}$$

$$N_t = \textrm{jumps (shocks) in the interval (0,t) governed by a Poisson }$$

$$\textrm{ process with parameter $\lambda t$,}$$

$$\ d = \textrm{constant.}$$

Jump-diffusion models undoubtedly capture a real phenomenon that is missing from the Black-Scholes models. Yet, they are rarely used in practice. There are three main reasons for this, difficulty in parameter estimation, solution and impossibility of perfect hedging (Wilmott; 1999, ch. 26).

11.1.1 Software Application

XploRe offers the following three quantlets to simulate and estimate the price of the underling asset according to (11.1) and (11.2):

stocksim ()

stocksim (S,sigma,tau,rate,shocks,jumps,jumpvola)

simulates a random process for the stock price, specifying either interactively or directly the input parameters.

stockest (data)

estimates the parameters of the Brownian motion process or the Merton jump-diffusion process for the given dataset.

stockestsim (data)

estimates the parameters of the Brownian motion process or the Merton jump-diffusion process for the given data and uses simulation to compare both models with the real dataset.

The quantlet stocksim provides three means to simulate the asset price as a random process by using i) the geometric Brownian motion (11.1), ii) a compounded Poisson jump process with the lognormal distribution of the jump height and iii) a mixture of both, namely the Merton jump-diffusion model (11.2).

If a direct approach is preferable, then the following input parameters have to be defined: $\tt {S}$ - the starting value of the underlying asset, $\tt {sigma}$ - the volatility of the asset return for the continuous-time diffusion process, $\tt {tau}$ - time (days) to expiration, $\tt {rate}$ - shocks per days, $\tt {jumps}$ - the expected number of jumps, which corresponds to $\lambda$ in the Poisson process and $\tt {jumpvola}$ - the volatility of the jump height. The following example simulates the asset price by directly giving the values of the input parameters.

library("finance")
 S=200          ; starting value of the underlying asset
 sigma=10       ; annualized volatility in percentage
 tau=200        ; days to expiration
 rate=5         ; increasing rate of return
 shocks=2       ; number of shocks per day
 jumps=20       ; the expected number of jumps
 jumpvola=0.5   ; volatility for the height of jump
 stocksim(S,sigma,tau,rate,shocks,jumps,jumpvola)

XLGfindex1.xpl

The quantlet returns a display as output (Figure 11.1), where all three processes are plotted together.

$\parbox{10cm}{\caption{\small Simulatio... ...rocess\newline \textcolor{blue}{-----} mixed process (Merton jump-diffusion)}}$

In the example above, the stock price is simulated at 400 points. The number of discretization points $\tt {n}$ is calculated as a product of days to expirations $\tt {tau}$ and $\tt {jumps}$ (shocks) per day. The jump consists of two random variables: one for the shock arrival process, which is Poisson distributed with parameter $\lambda$ and the other one for the jump size, which is log-normal distributed with volatility $\tt {jumpvola}$. A jump is detected if $\left(\frac{\lambda}{n}\ge y\right)$, where $y$ is a random variable with a standard uniform distribution, i.e. $y\sim U[0,1]$. For the mixed process, the geometric Brownian motion is simulated with an overlaying Poisson process.

For a given dataset, the quantlet stockest estimates the parameters of a random process, when the asset returns are assumed to follow either a geometric Brownian motion, as in (11.1), or a mix of the geometric Brownian motion and of a compounded Poisson Jump Process, namely the Merton jump-diffusion process given in (11.2). The following illustrates stockest on the data for Motorola stock prices.

library("finance")
 data=read("motorola")   ; read the data
 data=data[,2]           ; select the stock prices in the
                         ; second column
 stockest(data)

XLGfindex2.xpl

The output window contains the following information:

Contents of _tmp.mue [1,]   7.0066

Contents of _tmp.sigma [1,]   44.191

Contents of _tmp.lambda [1,]        4

Contents of _tmp.mue2 [1,]   3.2302

Contents of _tmp.sigma2 [1,]   38.819

Contents of _tmp.jump [1,]     10.9

When no jumps are assumed, the estimated parameters for $\mu$ and $\sigma$ are 7.0066 and 44.191, respectively. The intensity of the Poisson process is $\lambda=4$. When the jump-diffusion process is assumed, then the estimated values for $\mu$, $\sigma$ and the jump volatility are 3.2302, 38.819 and 10.9, respectively.

The quantlet stockestsim is a combination of both stocksim and stockest . First, the parameters for the given dataset are estimated for both, the Brownian motion based and the compounded Poisson jump process. Then both models are compared with the real dataset by means of a simulation. stockestsim is illustrated in the following example. The results are graphically displayed in Figure (11.2).

library("finance")
 data=read("motorola")    ; read the data
 data=data[,2]            ; select the stock prices in the
                          ; second column
 stockestsim(data)

XLGfindex3.xpl

$\parbox{10cm}{\caption{\small Estimatio... ...ss (Merton jump-diffusion)\newline \textcolor{blue}{-----} the original data}}$

11.1.2 Asset Price as a Stochastic Process

11.1.2.1 Geometric Brownian Motion

The stochastic process for asset price movement, when only "normal" events are considered, is assumed to be continuous in time, so that analytical tools, such as stochastic calculus can be employed. In general, the asset price is assumed to follow a Markov process, which means that only the present value of the asset price is relevant for the future price movements. This feature is consistent with the weak form of market efficiency, which assumes that the present value of the asset price already involves all information contained in past prices and history is irrelevant.

A special case of a Markov process is the Brownian motion. The Brownian motion with drift is defined as a stochastic process $\{X_t;t\ge 0\}$ with the following properties:

(i)

for $t>0$ and $s>0$, every increment $X_{t+s}-X_s$ is normally distributed with mean $\mu t$ and variance $\sigma^2t$, where $\mu$ and $\sigma$ are fixed parameters,

(ii)

for every $t_1<t_2<...<t_n$, the increments $X_{t_2}-X_{t_1},...,X_{t_n}-X_{t_{n-1}}$ are independent random variables with distribution given in (i),

(iii)

$X_0=0$ and the sample paths of $X_t$ are continuous.

Note that $X_{t+s}-X_{s}$ is independent of history, i.e. knowing $X_{\tau}$, $\tau<s$ has no effect on the probability distribution of $X_{t+s}-X_s$. This is exactly the Markovian property for the Brownian motion. For the particular case, when $\mu=0$ and $\sigma^2=1$, the Brownian motion is called standard Brownian motion (or standard Wiener process) denoted as $W_t$.

Fluctuations in asset price can be explained by using the following stochastic differential equation, also known as Ito's process:

$$dS_t = m(t,S_t) \, dt + \sigma(t,S_t) \, dW_t,$$ (11.3)

where

$$S_t = \textrm{the current underlying asset price,}$$

$$m(t,S_t) = \textrm{the trend or drift,}$$

$$\sigma(t,S_t) = \textrm{the annualized instantaneous volatility of the underlying asset,}$$

$$W_t = \textrm{the standard Wiener process.}$$

Different assumptions about the form of volatility give rise to different solutions for $S_t$ to this stochastic differential equation. Classical models, such as the Black-Scholes option price model (section 11.2), assume that $\sigma(t,S_t)=\sigma S_t$ and $\mu(t,S_t)=\mu S_t$. In this case, the asset return ($dS_t/S_t$) follows a so called geometric Brownian motion, written as:

$$\frac{dS_t}{S_t} = {\mu}\, dt + {\sigma}\, d W_t$$ (11.4)

The first term $\mu dt$ represents the deterministic return within a short interval $dt$, where $\mu$ is the average growth rate of the asset price. The second term $\sigma dW_t$, where $\sigma$ is assumed to be constant, takes into account the random changes of the asset price to external effects, such as unexpected news.

Brownian motion and normal distribution have been widely used in the Black-Scholes option pricing framework (section 11.2). However, two puzzles have emerged from several empirical investigations and are subject to current research. These are the leptokurtic distribution of asset returns and evidence on volatility smile (section 11.5).

A distribution is leptokurtic, if it has a higher peak and heavier tails than those of the normal distribution. One possible reason for fat tails can be the discontinuous path of the asset returns. To reflect this discontinuity, jump-diffusion processes have been widely used to model financial time series. The jump-diffusion model gives higher values than the Black-Scholes model for deep out-of-the-money and in-the-money options, especially when the time to maturity is short. The main reason for this is that a jump-diffusion process results in a distribution with fatter tails than the normal distributed returns (log-normal asset price). For low values of gamma, the jump-diffusion model gives similar results as the Black-Scholes model.

11.1.2.2 The Jump-Diffusion Model

Merton (1976) was one of the first, who applied Poisson jumps to normal Brownian motion process, in order to approximate the movement of stock prices subject to occasional discontinuous breaks. In this section the concept behind the jump-diffusion model is described following Merton (1976), and the later works of Jiang (1998) and Neftci (2000, ch. 8 and ch. 7.2).

In the jump-diffusion model, changes in the asset price are a mixture of normal events that occur in continuous fashion and of rare events, which are modelled as jumps that occur sporadically. The continuous component of the change in the asset price is a Wiener process. The jump component is a Poisson-driven process. It is assumed that the arrivals of rare events are independently, identically distributed (i.i.d.). The probability of a jump that occurs during a time interval of length $h$ is written as

$$\ (t,t+h)\} = \lambda h,$$

$$\ (t,t+h)\} = 1-\lambda h,$$

where $\lambda$ is the intensity of the jump process, i.e. the number of jumps per unit of time, which do not depend on the information set available at time $t$. It can be calculated by dividing the corresponding probability $\lambda h$ by $h$.

The jump process can be modelled using a Poisson counting process, which has the following properties:

1.

as $h \rightarrow 0$, at most one event can occur with probability very close to 1,

2.

the information up to time t does not help to predict the occurrence (non-occurrence) of the event in the next instant $h$,

3.

the events occur at a constant rate $\lambda$.

Note that the time interval between two successive jumps is exponentially distributed with parameter $\lambda$. Given that the rare event occurs in the time interval $(t,t+h)$ causing a jump, the asset price $S_{t+h}$ at time $t+h$ will be the random variable $S_{t+h}=S(t)Y$. The discontinuous change in the asset price will be $S_{t+h}-S(t)=S(t)(Y-1)$. The random variable $Y-1$, also called the jump size, gives the percentage change in the stock price, if the Poisson event occurs. In general, $\tilde{Y}$ may be a random variable. It is assumed that the successive jump sizes ( $\tilde{Y}-1$) are i.i.d.

One aspect of the jump should once again be highlighted. The process has two sources of randomness. The occurrence of a jump is a random event. But once the jump occurs, the size of the jump is also random. Moreover, it is assumed that these two sources of randomness are independent of each other. Under this structure, the general parametric jump-diffusion process, as a mixture of both, continuous diffusion path and discontinuous jump path, can be written in general form as (Jiang; 1998):

$$\frac{dS_t}{S_t}=\{\alpha_t(\beta)-\lambda \mu_0\}dt + \sigma_t(\beta)d W_t + dQ_t(\lambda)$$ (11.5)

where

$$S_t = \textrm{asset price at time t}$$

$$\alpha_t = \textrm{the instantaneous expected return}$$

$$\mu_0 = \textrm{expectation of the relative jump size, i.e. } \mu_0=E[Y_t-1],$$

$$\lambda = \textrm{the intensity parameter of the Poisson distribution,}$$

$$\sigma_{t} = \textrm{the instantaneous volatility of the asset's return }$$

$$\textrm{conditional on the Poisson jump event not to occur,}$$

$$W_t = \textrm{a standard Wiener process,}$$

$$Q_t(\lambda) = \textrm{a Poisson process with parameter } \lambda,$$

$$(\beta,\mu_0,\lambda)\in \Theta = \textrm{the parameter space, which parametrizes the}$$

$$\textrm{coefficient functions, the jump sizes, as well as the}$$

$$\textrm{intensity of the Poisson process.}$$

Both, the Wiener process $W_t$ and the Poisson process $dQ_t(\lambda)$ are infinitely divisible in time and appropriated scaled; $dQ_t(\lambda)$ and $dW_t$ are statistically independent. By definition $\mu_0$ is the mean jump size. Thus, the expected change in $S_t$ from the jump component $dQ_t$ over the time interval $dt$ is $\lambda\mu_0dt$. Therefore, if $\alpha_t$ denotes the total expected return (rate of change) on $S_t$, $\lambda\mu_0dt$ needs to be subtracted from the drift term of $S_t$:

$$E\left(\frac{dS_t}{S_t}\right) = E\{(\alpha_t-\lambda \mu_0)dt\} + E(\sigma_t dW_t) + E(dQ_t)$$ (11.6)

$$\ = (\alpha_t-\lambda \mu_0)dt + 0 + \lambda \ \mu_0dt=\alpha_tdt$$

Note, that a sample path $S_t$ for a process described by equation (11.5) will be continuous most of the time, but has finite jumps of different signs and sizes at discrete points in time. Assuming $\alpha_t$ and $\sigma_t$ are constant, so that the continuous component of $S(t)$ is lognormally distributed, and conditional upon there being $N_t$ jumps in the time interval ($0,t$), the asset price at time $t$ can be written as

$$S_t=S_0\; exp{\left(\alpha_t-\frac{1}{2}\sigma_t^2-\lambda\mu_0\right)t+\sigma W_t}\tilde{Y}(N_t),$$ (11.7)

where $W_t\sim N(0,t)$ and $\left\{\tilde{Y}_j\right\}_{j=1}^n$ is a set of i.i.d. jumps, such that:

$$\left\{ \begin{array}{rc} \ \tilde{Y}(N_t)=0 & \quad \text{for} \... ...d\limits_{j=1}^{N_t}Y_j & \quad \text{for} \quad N_t\geq 1 \ \end{array}\right.$$

Special cases of this model include those by Press (1967) with $\sigma_t(\cdot)=0$ and $\mu_t(\cdot)=0$, Merton (1976) with $\sigma_t(\cdot)=\sigma$, $\mu_t(\cdot)=\mu$ and lognormal jumps, Lo (1998) with $ln(Y_t)=f(s_t)$, i.e. the jump size is determined by the process itself, and Ornstein-Uhlenbeck process with $\mu_t(\cdot)=\mu s_t$ and exponentially decaying jumps (see Gourieroux (2001, pp. 249-253) and Küchler et al. (1997, pp. 27-28) for the explanation of the Ornstein-Uhlenbeck process).

In XploRe the geometric Brownian motion with i.i.d. lognormal jumps is used to simulate and estimate the asset price in a mixed process. This method was proposed by Merton (1976) and is now known as the Merton's jump-diffusion Model. It assumes that in equation (11.6) $\mu_t(\cdot)=\mu$, $\sigma_t(\cdot)=\sigma$, and $lnY_t\sim \textrm{i.i.d. }N(\mu_0,\nu^2)$, where all $\mu,\mu_0,\sigma,\nu$ are constant. Then the stochastic differential equation for $S_t$ is written as:

$$\frac{dS_t}{S_t}=\mu dt + \sigma dW_t + d\sum_{j=1}^{N_t}(Y_j-1)$$ (11.8)

where $d$ is constant. The term $\sum_{j=1}^{N_t}(Y_j-1)$ is also known as a compounded Poisson process (Küchler et al.; 1997, pp. 9). Equation (11.8) can be equivalently written as:

$$S_t=S_0\;exp\left(\mu dt + \sigma dW_t\right)\prod \limits_{j=1}^{N_t}Y_j$$ (11.9)

with $\left(\prod \limits_{j=1}^{N_t}Y_j\right)=0$ if $N_t=0$ and $\left(\prod \limits_{j=1}^{N_t}Y_j\right)\not=0$ if $N_t\ge1$, where $Y_j$ are i.i.d. and $N_t$ is a Poisson distributed process with parameter $\lambda t$.