On the RND under Heston’s stochastic volatility model

The stochastic volatility model for option valuation of Heston (1993) is widely accepted nowadays by both, academics and practitioners. It prescribes, under a risk-neutral probability measure $\mathbb{Q}$, say, the dynamics of the spot’s (stock, index) price process $S=\{S_{t},\,t\geq 0\}$, in relation to a corresponding, though unobservable (untradable ) volatility process $V=\{V_{t},\,t\geq 0\}$ via a system of stochastic deferential equations. This system is given by

where $r$ is the risk-free interest rate, $\kappa,\ \theta$ and $\eta$ are some constants (to be discussed below) and where $W_{1}=\{W_{1,t},\,t\geq 0\}$ and $W_{2}=\{W_{2,t},\,t\geq 0\}$ are two Brownian motion processes under $\mathbb{Q}$ with $d(W_{1}W_{2})=\rho dt$.

The quest to incorporate a non-constant volatility in the option valuation model, has risen in the literature (e.g., Wiggins (1987) or Stein and Stein (1991)) ever since the seminal work of Black and Scholes (1973) and of Merton (1973), (abbreviated here as the BSM) in modeling the price of a European call option when the spot’s price was assumed to evolve, with a constant volatility of the spot’s returns, $\sigma$, as a geometric Brownian motion,

Coupled with an ingenious argument of instantaneous portfolio hedging (along with other assumptions such as self-financing, no-cost trading/carry, etc.) and an application of Ito’s Lemma to the underlying PDE, the BSM model provides an exact solution for the price of an European call option $C(\cdot)$. Specifically, given the current spot price $S_{\tau}=S$ and the risk-free interest rate $r$, the price of the corresponding call option with price-strike $K$ and duration $T$,

where $t=T-\tau$ is the remaining time to expiry. Here, using the conventional notation, $\Phi(\cdot)$ and $\phi(\cdot)$ denote the standard Normal $cdf$ and $pdf$, respectively, and

In similarity to the form of the BSM solution in (3), Heston (1993) obtained that the solution to the system of PDE resulting from the stochastic volatility model, (1), is given by

where $P_{j}$ $j=1,2$, are two related (under a risk-neutral probability measure $\mathbb{Q}$) conditional probabilities that the option will expire in-the-money, conditional on the given current stock price $S_{\tau}=S$ and the current volatility, $V_{\tau}=V_{0}$. However, unlike the explicit BSM solution in (3) which is given in terms of the normal (or log-normal) distribution, Heston (1993) provided (semi) closed-form solutions to these two probabilities, $P_{1}$ and $P_{2}$ in terms of their characteristic functions (for more details, see the Appendix). Hence, $C_{S}(K)$ in (5) is readily computable, via complex integration, for any choice of the parameters $\vartheta=(\kappa,\theta,\eta,\rho)$ in (1), all in addition to $S,\ V_{0}$ and $r$. These parameters have particular meaning in context of the SV model (1): $r$ is the prevailing risk-free interest rate; $\rho$ is the correlation between the two Brownian motions comprising it; $\theta$ is the long-run average, $\kappa$ is the mean-reversion speed and $\eta^{2}$ is the variance of the volatility $V$ (see also Section 4 below). It should be noted that different choices of $\vartheta$ will lead to different values $C_{S}(K)$ in (5) and hence, the value $\vartheta=(\kappa,\theta,\eta,\rho)$ must be appropriately ‘calibrated’ first for $C_{S}(K)$ to actually match the option market data.

The role of the risk-neutral probability measure $\mathbb{Q}$ in option valuation in general and in determining the specific solution in (5) (or in (3)) in particular, cannot be overstated (in the ‘risk-neutral’ world). As was established by Cox and Ross (1976), the risk-neutral equilibrium requires that for $T>\tau$ (with $t=T-\tau$),

and that (in the case of a European call option), $C_{S_{\tau}}(K)$ must alo satisfy

where, for any $x\in{\mathbb{R}}$, $x^{+}:=max(x,0)$. Here $q(\cdot)$ is the risk-neutral density (RND) under $\mathbb{Q}$, reflective of the conditional distribution of the spot price $S_{T}$ at time $T$, given the spot price, $S_{\tau}$ at time $\tau<T$. The risk-neutral probability $\mathbb{Q}$ links together the option evaluation and the distribution of spot price $S_{T}$ and its stochastic dynamics governing the model (as in (1), say). As was mentioned earlier, in the case of the BSM in (2) the RND is unique and is given by the log-normal distribution. However, since the Heston (1993) model involves the dynamics of two stochastic processes, one of which (the volatility, V) is untradable and hence not directly observable, there are innumerable many choices of RNDs, $q(\cdot)$, that would satisfy (6)-(7) and hence, the general solutions of $P_{1}$ an $P_{2}$ in (5) by means of characteristic functions (per each choice of $\vartheta=(\kappa,\theta,\eta,\rho)$).

Needless to say, there is an extensive body of literature dealing with the ‘extraction’, ‘recovery’, ‘estimation’ or ‘approximation’, in parametric or non-parametric frameworks, of the RND, $q(\cdot)$ from the available (market) option prices; see Jackwerth (2004), Figlewski (2010), Grith and Krätschmer (2012) and Figlewski (2018), for comprehensive reviews of the subject. With the parametric approach in particular, one strives to estimate by various means (maximum likelihood, method of moments, least squares, etc.) the parameters of some assumed distribution so as to approximate available option data or implied volatilities (c.f. Jackwerth and Rubinstein (1996)). This type of assumed multi-parameter distributions includes some mixtures of log-normal distribution (Mizrach (2010), Grith and Krätschmer (2012)), generalized gamma (Grith and Krätschmer (2012)), generalized extreme value (Figlewski (2010)), the gamma and the Weibull distributions (Savickas (2005)), among others. While empirical considerations have often led to suggesting these parametric distributions as possible RNDs, the motivation to these considerations did not include direct link to the governing pricing model and it dynamics, as was the case in the BSM model, linking directly the log-normal distribution and the price dynamics of model (2).

In this paper, we present in our Theorem 2 a more direct approach (and hence, a link) to the RND quest in the case of Heston’s (1993) SV model (1). By expanding the last term in (7), with $S_{\tau}=S$, we obtain,

Clearly, by comparing (5) to (8), it follows that $P_{2}=\mathbb{Q}(S_{T}>K)\equiv 1-Q(K)$ (the risk-neutral probability of the option expiring in the money), whereas by, (6) and (5),

We note that since by (6), we have,

the probability $P_{1}$ is also being interpreted (see for example xxxx) as the probability of the option expiring in the money, but under the so-called physical probability measure that is being dominated by $\mathbb{Q}$. However here, in the case of of the Heston’s (1993) model, we consider a different interpretation of this term $P_{1}$, which enables us to characterize a class of RND candidates that satisfy (5).

It is a standard notation to denote by $\Delta(K)$ the so-call delta function (or hedging fraction) in the option valuation, as defined by

In the Appendix, we show that for Heston’s call option price $C_{S}(K)$ as given in (5), one has (see also Bakshi, Cao and Chen (1997)),

Hence, under model (1), Heston’s solution for the option price in (5) can be written in an equivalent form as:

We point out in passing that this presentation (12) also trivially applies to the BSM option price in (3) since in that case, $\Phi(d_{1})\equiv\Delta(K)$ and $\Phi(d_{2})$ is just the probability that option will expire in the money (calculated under the log-normal distribution).

In Section 2, we identify the class of distributions (and hence of RNDs) that admit the presentation in (12) of Heston’s (1993) option price as as is given in (5). Specifically, we show that any risk-neutral probability distribution $Q$ that satisfies (6)-(7) with a scale parameter $\mu=S\cdot e^{rt}$ would admit the presentation in (12) and hence would satisfy Hestons’ (1993) option pricing model in (5) . In fact, we also show in the Appendix that the RNDs that may be calculated directly from Heston’s characteristic functions (corresponding to $P_{1}$ and $P_{2}$) are members of this class of distributions as well. In Theorem 2 below we establish the direct link, through Heston’s (1993) solution in (5) (or (12)) between this class of RNDs and the assumed stochastic volatility model in (1) governing the spot price dynamics.

In Section 3 we provide some specific examples of well known distributions that satisfy (12). These include the Gamma, Inverse Gaussian, Log-Normal, the Weibull and the Inverse Weibull distributions (all with a particular parametrization) as possible RND solutions for option valuation under Heston’s SV model. The extent agreement between each of these five particular distributions as a possible RND for the Heston’s model, and the actual Heston’s RND (calculated from $P_{2}$ in the Appendix) and the simulated distribution of the spot prices obtained under (a discretized version of) model (1) is illustrated numerically in Section 4. We demonstrated the applicability and suitability of these explicit RNDs using already published Index data with a calibrated Heston model (S&P500, Bakshi, Cao and Chen (1997), and ODAX, Mrázek and Pospíšil (2017)), as well as current option market data (AMD). In the Appendix, we present the expressions for for Heston’s characteristic functions and discuss some of the immediate properties leading to our main results as stated in Theorem 2.

In this section we identify the class of distributions, and therefore of possible RNDs that admit the presentation in (12) for the price of a European call option. Specifically, we show that any RND candidate that satisfies (6)-(7) with a scale parameter $\mu=S\cdot e^{rt}$ would admit the presentation in (12) and hence in light of the result in (11) (see Claim 7 in the Appendix), would equivalently satisfy Hestons’ (1993) option pricing model in (5).

To that end and to simplify the presentation, we consider a continuous positive random variable $X$ with mean $\mu>0$ (with respect to some underlying probability measure $\mathbb{Q}$). We denote by $Q_{\mu}(\cdot)$ and $q_{\mu}(\cdot)$ the $cdf$ and $pdf$ of $X$, respectively, to emphasize their dependency on $\mu$, as a parameter. Similarly, for a given $\mu>0$, we write $E_{\mu}(\cdot)$ for the expectation of $X$ (or functions thereof) under $Q_{\mu}$ so that,

In similarity to (7), we define, for each $s\geq 0$,

Clearly, $c_{\mu}(0)=\mu$. Note that $c_{\mu}(\cdot)$ is merely the undiscounted version of $C_{S}(\cdot)$ in (7), so that with $\mu=S\cdot e^{rt}$ as in (6), we have, $c_{\mu}(K)\equiv e^{rt}\cdot C_{S}(K)$.

It is straightforward to see that, as in (8),

or equivalently,

Hence, it follows immediately from (14) that for each $s>0$,

As we proceed to explore more of the basic properties of the function $c_{\mu}(\cdot)$, we add the simple assumption that $\mu$ is a scale-parameter of the underlying distribution of $X$.

In Lemma 1 below we establish the linear homogeneity of $c_{\mu}(s)$ under Assumption A and provide in Lemma 2 the implied re-scaling property of this function and the consequential specific derived form of $c_{\mu}(s)$ as presented in Theorem 1 below. For the linear homogeneity property of the European options, in general, see Theorems 6 & 9 of Merton (1973) or Theorem 2.3 in Jiang (2005).

It should be clear from the above results that this function, $c_{\mu}(s)$, can be re-scaled or ”standardized” so that $c_{\mu}(s)/\mu$ is independent of $\mu$. In particular, if $s=b\,\mu$ for some $b>0$, then again by (16), $c_{\mu}(b\,\mu)=\mu\,c_{1}(b)$.

Next, as in (10), we define the ’Delta’-function corresponding to the function $c_{\mu}(s)$ in (13) or (14), as $\Delta_{\mu}(s):={{\partial}c_{\mu}(s)/{\partial\mu}}$. In the next theorem we show that under Assumption A, $\Delta_{\mu}(\cdot)$ may be expressed in terms of the truncated mean of $X$ and the consequential representation of $c_{\mu}(\cdot)$.

An immediate conclusion of Theorem 1 is that if $Q_{\mu}$ is a member of the scale-family of distribution ${\cal Q}$ (by Assumption A) then the functions $c_{\mu}(s)$ can easily be evaluated by calculating first the values of $c_{1}(\cdot)$ for the ratio $b=s/\mu$. Specifically,

The results of the Theorem, either as given in (19) or in (21), can be used directly for the risk-neutral valuation of European call option with a strike $K$ and a current spot price $S_{\tau}\equiv S$, providing the expression for $C_{S}(K)$ as is given in (12). That is, if $Q\in{\cal Q}$, then with $\mu=S\,e^{rt}$ applied to (21), we have

We summarize these findings in the following Corollary.

It should be clear from (22) that since the probability distribution $Q_{\mu}$ is assumed here to be a member of a scale family ${\cal Q}$, its values depend on $K$ and $S$ only through the ratio $K/S$. In the Appendix, we assert in Proposition 1 that any risk neutral probability distribution $Q_{\mu}$ that satisfies the solution (5) for Heston’s option pricing model, must also be a member of a scale family of distributions, with a scale parameter $\mu=S\,e^{rt}$ (or $\mu=S\,e^{(r-q)t}$, in the case of a dividend yielding spot). This assertion follows directly from the specific form of Heston’s RND established in the Appendix which is given in terms of characteristic function corresponding to the term $P_{2}$ (see (31) and the subsequent comments there). Hence combined, the statements of Corollary 1 and Proposition 1, can be summarized in the following theorem.

In view of Theorem 2, the quest for finding appropriate RND for Heston’ SV model for a particular parametrization of $\vartheta=(\kappa,\theta,\eta,\rho)$ must be focused only on those members of a scale-family of distributions with a scale parameter $\mu=S\,e^{rt}$. Accordingly, we provide in this section, five particular examples of well-known distributions, that satisfy the conditions of Assumption A and hence admit, per Corollary 1, the presentation (12) for the Heston’s option pricing model in (5). These well-known distributions, namely, the Log-Normal, the Gamma, the Inverse Gaussian, the Weibull and the Inverse Weibull distributions, are re-parametrized under Assumption A to a standardized, one-parameter version having mean $1$ and a second moment that depends on a single parameter $\nu>0$ (in fact, we take $\nu\equiv\sigma\sqrt{t}$, for some $\sigma>0$). Due to their relative simplicity (involving only one parameter), we view these distributions as inexpensive RNDs, easy to obtain, to calculate and calibrate as compared to the alternatives approaches available in the literature. We note that while the gamma and the Weibull distribution were considered by Savickas (2002, 2005) for deriving ‘alternative’ option pricing formulas, the motivation for the parametrization there was made without regard to the spot price dynamics (but rather for fitting kurtosis and skewness) and therefore are different.

With these standardized distributions in hand and the corresponding explicit expressions for $Q_{1}(\cdot)$ as obtained under Assumption A, we utilize (21) (with $\mu\equiv 1$) to first calculate in each case the expression for

which is then used to obtain, with $\mu>0$, the expression for the undiscounted option price as,

Finally, as in (22), the corresponding expression for the call option price is obtained as $C_{S}(K)=e^{-rt}c_{\mu}(K)$ (with $\mu=Se^{rt},\ s=K$ and $\nu=\sigma\sqrt{t}$). We point out again, that each of these five distributions would satisfy as RND, Heston’s (1993) general solution for the valuation of a European call option as is given in (5). We begin with the log-normal distribution which results with the classical Back-Scholes option pricing model (as given in (3)-(4)) .