CHAPTER 1
The Heston Model for European Options

Abstract

Here, we present the European call price under the Heston model. We first present the model and then illustrate that the call price in the Heston model can be expressed as the sum of two terms that each contains an in-the-money probability but obtained under a separate measure, a result demonstrated by Bakshi and Madan (2000). We then show how to incorporate a continuous dividend yield and how to compute the price of a European put, and demonstrate that the numerical integration can be speed up by consolidating the two numerical integrals into a single integral. Finally, we derive the Black-Scholes model as a special case of the Heston model.

CIR process, European call, characteristic function, dividend yield, put-call parity, Black-Scholes

In this chapter, we present the European call price under the Heston model. We first present the model and then illustrate that the call price in the Heston model can be expressed as the sum of two terms that each contains an in-the-money probability, but obtained under a separate measure, a result demonstrated by Bakshi and Madan (2000). We then show how to incorporate a continuous dividend yield and how to compute the price of a European put, and demonstrate that the numerical integration can be speeded up by consolidating the two numerical integrals into a single integral. Finally, we derive the Black-Scholes model as a special case of the Heston model.

MODEL DYNAMICS

The Heston model assumes that the underlying stock price, St, follows a Black-Scholes-type stochastic process, but with a stochastic variance, vt, that follows a Cox, Ingersoll, and Ross (1985) process. Hence, the Heston model is represented by the bivariate system of stochastic differential equations (SDEs),

(1.1)

where . We will sometimes drop the time index and write S = St, v = vt, W1 = W1, t and W2 = W2, t for notational convenience. The parameters of the model are

• µ the drift of the process for the stock;
• ? > 0 the mean reversion speed for the variance;
• ? > 0 the mean reversion level for the variance;
• s > 0 the volatility of the variance;
• v0 > 0 the initial (time zero) level of the variance;
• ? ? [ - 1, 1] the correlation between the two Brownian motion W1 and W2; and
• ? the volatility risk parameter (discussed below).

We will see in Chapter 2 that these parameters affect the distribution of the terminal stock price in a manner that is intuitive. Some authors refer to v0 as an unobserved initial state variable rather than a parameter. Because volatility cannot be observed, only estimated, and because v0 represents this state variable at time zero, this characterization is sensible. For the purposes of estimation, however, many authors treat v0 as a parameter like any other. Parameter estimation is covered in Chapter 6.

The stock price and variance follow the processes in Equation (1.1) under the historical measure , also called the physical measure. For pricing purposes, however, we need the processes for (St, vt) under the risk-neutral measure . The risk-neutral process is

(1.2)

where . The risk-neutral parameters of the variance process are where ?* = ? + ? and ?* = ??/(? + ?).

Note that when ? = 0, we have ?* = ? and ?* = ? so that the parameters under the physical and risk-neutral measures are the same. Throughout this book, we set ? = 0, but this is not always needed. Indeed, ? is embedded in the risk-neutral parameters ?* and ?*. Hence, when we estimate the risk-neutral parameters to price options, we do not need to estimate ?. Estimation of ? is the subject of its own research, such as that by Bollerslev et al. (2011). For notational simplicity, throughout this book we will drop the asterisk on the parameters and the tilde on the Brownian motion when it is obvious that we are dealing with the risk-neutral measure.

For details on how the risk-neutral process is constructed, see Rouah (2013).

THE HESTON EUROPEAN CALL PRICE

In this section, we show that the call price in the Heston model can be expressed in a manner that resembles the call price in the Black-Scholes model, which we present in Equation (1.16). Authors sometimes refer to this characterization of the call price as "Black-Scholes-like" or "à la Black-Scholes." The time-t price of a European call on a non-dividend-paying stock with spot price St, when the strike is K and the time to maturity is t = T - t, is the discounted expected value of the payoff under the risk-neutral measure

(1.3)

The last line of (1.3) is the "Black-Scholes-like" call price formula, with P1 replacing F(d1), and P2 replacing F(d2) in the Black-Scholes call price (1.16).

By introducing two measures, and , the European call price of Equation (1.3) can be written

(1.4)

The measure uses the bond Bt as the numeraire, while the measure uses the stock price St. Bakshi and Madan (2000) present a derivation of the call price expressed as (1.4), but under a general setup. As shown in their paper, the change of measure that leads to (1.4) is valid for a wide range of models, including the Black-Scholes and Heston models. We will see later in this chapter that when ST follows the lognormal distribution specified in the Black-Scholes model, then and . Hence, the characteristic function approach to pricing options, pioneered by Heston (1993), applies to the Black-Scholes model also.

In the Heston model, it can be shown that P1 and P2 in (1.3) can be written

(1.5)

where is the imaginary unit, Pr1 = , Pr2 = , and fj(?; x, v) are the characteristic functions for the logarithm of the terminal stock price xT = ln ST. Heston (1993) postulates that these characteristic functions are of the log linear form,

(1.6)

where Cj and Dj are constant coefficients and t = T - t is the time to maturity.

The coefficients can be shown to be

(1.7)

and

(1.8)

where a = ??,

and with , , b1 = ? + ? - ?s, and b2 = ? + ?. See Rouah (2013) for a complete derivation of the coefficients (1.7) and (1.8).

We use two functions to implement the Heston call price in VBA using the trapezoidal rule, HestonProb and HestonPriceTrapz. The first function calculates the characteristic functions and returns the real part of the integrand. The functions allow to price calls or puts, and allow for a dividend yield, as explained in the following section. The following code snippet is taken from the Excel file Ch1Trapz.xls. To conserve space, parts of the functions have been omitted.

The second function calculates the price of a European call C(K), or European put P(K), by put-call parity in Equation (1.13). The function calls the HestonProb function at every point of the integration grid and uses the trapezoidal rule for integration when all the integration points have been calculated.

Pricing European calls and puts is straightforward. For example, the price of a six-month European put with...