Option pricing under stochastic volatility models with latent volatility

Abstract

An important challenge regarding the pricing of derivatives is related to the latent nature of volatility. Most studies disregard the uncertain nature of volatility when pricing options; the few authors who account for it typically consider the risk-neutral posterior distribution of the latent volatility. As the latter distribution differs from its physical measure counterpart, this leads to at least two issues: (1) it generates some unwanted path dependence and (2) it oftentimes requires to simultaneously track the physical and risk-neutral distributions of the latent volatility. This article presents pricing approaches purging such a path-dependence issue. This is achieved by modifying conventional pricing approaches (e.g. the Girsanov transform) to formally recognize the uncertainty about the latent volatility during the pricing procedure. The two proposed risk-neutral measures circumventing the aforementioned undesired path-dependence feature are based on the extended Girsanov principle and the Esscher transform. We also show that such pricing approaches are feasible, and we provide numerical implementation schemes.

Keywords:

• Finance
• Option pricing
• Stochastic volatility
• Particle filters
• Path dependence

JEL Classifications:

• G13
• G23

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Note that, very early on, Taylor (1986, 1994) discussed option pricing in the context discrete-time stochastic volatility models.

2 An early paper by Boyle and Ananthanarayanan (1977) investigates the issue of not observing the variance in the context of the Black and Scholes (1973) model. They also provide a pricing approach based on Bayesian statistics that bears similarities with the two-step approach mentioned above.

3 The sequential Monte Carlo approach used in this article is very efficient considering the complex path-dependent problem under study. Indeed, it allows us to focus on relevant paths by means of the resampling step of the particle filter and disregard paths of lesser importance. Note that the proposed method contains more operations than classic Monte Carlo schemes in the context of option pricing (i.e. $\mathcal{O}\left(TP{N}_{\mathrm{S}\mathrm{M}\mathrm{C}}\right)$ for our method versus $\mathcal{O}\left(T{N}_{\mathrm{M}\mathrm{C}}\right)$ for the classic schemes, where T is the number of steps, P is the number of particles, $N_{\mathrm{S}\mathrm{M}\mathrm{C}}$ is the number of paths used in the SMC method, and $N_{\mathrm{M}\mathrm{C}}$ is that of the classic scheme).

4 Conventional Girsanov-type transform approaches refer to methods that first determine option prices assuming the volatilities were observable and then compute a weighted average of such prices with respect to the current conditional latent volatility distribution given observable information. Such approaches are explained in more detail subsequently.

5 In practice, ${\nu }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ depends on the model and the distribution of the innovations. For the normal case used later in this article, it can be set to some very large number (or even to infinity) without impacting any of our results as ${\mathbb{E}}^{\mathbb{P}}\left[{\mathrm{e}}^{\sqrt{{\nu }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{z}_{1,t}^{\mathbb{P}}}\right]<\mathrm{\infty }$ is satisfied for any finite value of ${\nu }_{\mathrm{m}\mathrm{a}\mathrm{x}}$. For other innovation distributions, especially those with fat tails, the value of ${\nu }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ would need to be selected such that the risky asset price is an integrable process under risk-neutral measures proposed subsequently.

6 Note that the jump innovations $z_{3,t}^{\mathbb{P}}$, t = 1,…, T, are very general in our derivation: they do not need to be centred at zero and can be arbitrarily distributed, as long as their moment generating function is finite over some interval that is large enough for the underlying asset price process to be integrable under the risk-neutral measures introduced subsequently. Having a jump innovation that is centred at zero is convenient, nonetheless, because it leads to a better interpretation of the drift function a. In our numerical illustration of Section 5, we use a jump term defined by the product of a Bernouilli random variable and a normal random variable centred at zero.

7 Path-dependent payoffs or derivatives allowing for early exercises are not considered for simplicity, although the ideas put forward in this article could be used in this context.

8 The Esscher transform was first introduced to the context of option pricing by Gerber and Shiu (1994) and further extended by Bühlmann et al. (1996) to the case of the conditional Esscher transform. One notable application of the conditional Esscher transform to GARCH option pricing can be found in Siu et al. (2004).

9 When conditioning on a random variable, the conditional expectation is taken with respect to the σ-field generated by that random variable.

10 Indeed, unless the volatility ${\sigma }_{t-1}$ is ${\mathcal{G}}_{t-1}$-measurable, the $t^{\mathrm{t}\mathrm{h}}$ term in the product of equation (10(10) $\begin{array}{rl}\frac{\mathrm{d}{\tilde{\mathbb{Q}}}^{\mathrm{G}}}{\mathrm{d}\mathbb{P}}& =\prod _{j=1}^T\frac{f_{R_j|{\mathcal{G}}_{j-1}}^{{\tilde{\mathbb{Q}}}^{\mathrm{G}}}\left(R_j\right)}{f_{R_j|{\mathcal{G}}_{j-1}}^{\mathbb{P}}\left(R_j\right)}\\ & =\prod _{j=1}^T\frac{f_{R_j|{\mathcal{G}}_{j-1}}^{\mathbb{P}}(R_j+\log {\tilde{\mu }}_j)}{f_{R_j|{\mathcal{G}}_{j-1}}^{\mathbb{P}}\left(R_j\right)}\\ & =\prod _{j=1}^T\frac{{\int }_0^{\sqrt{v_{max}}}{f}_{R_j|{\sigma }_{j-1}}^{\mathbb{P}}\left(R_j+\log {\tilde{\mu }}_j|\sigma \right)f_{{\sigma }_{j-1}|{\mathcal{G}}_{j-1}}^{\mathbb{P}}\left(\sigma \right)\mathrm{d}\sigma }{{\int }_0^{\sqrt{v_{max}}}{f}_{R_j|{\sigma }_{j-1}}^{\mathbb{P}}\left(R_j|\sigma \right)f_{{\sigma }_{j-1}|{\mathcal{G}}_{j-1}}^{\mathbb{P}}\left(\sigma \right)\mathrm{d}\sigma },\end{array}$(10) ) is a ratio of integrals involving the latent volatility's conditional distribution, which is most likely not of the same form than ${\xi }_t^{{\tilde{\mathbb{Q}}}^{\mathrm{E}}}$.

11 Put-call parity implies that European calls and puts share the same implied volatility for a given maturity and strike price. As our numerical results rely on implied volatility, any of the two types can be used interchangeably in our simulations without impacting the results.

12 A large number of paths is involved in the pricing procedure (i.e. the second part) instead of a single one in the first part, explaining why many more particles are considered in the latter.

13 Although similar filtered volatility distributions are obtained under both the physical and risk-neutral measures for the considered underlying asset returns path, this does not necessarily imply the absence of a volatility risk premium in ${\mathbb{Q}}^{\mathrm{G}}$. Indeed, the frequency at which such paths might be encountered (i.e. its likelihood) might still differ between the two measures; for instance, return paths exhibiting more fluctuations could potentially be encountered more often under the risk-neutral measure.

14 Filtering under the risk-neutral measure is achieved by changing the density $f_{R_s|{\sigma }_{s-1}}^{\mathbb{P}}\left(R_s|{\tilde{\sigma }}_{s-1,p}\right)$ in equation (A3(A3) $\begin{array}{rl}& w_{s,p}\left({\tilde{\sigma }}_{0:s,p}^2\right)\\ & =w_{s-1,p}\left({\sigma }_{0:s-1,p}^2\right)f_{R_s|{\sigma }_{s-1}}^{\mathbb{P}}\left(R_s|{\tilde{\sigma }}_{s-1,p}\right),s=1,\dots ,t.\end{array}$(A3) ) for its risk-neutral counterpart while everything else remains the same.

Additional information

Funding

We thank the Natural Sciences and Engineering Research Council of Canada (Bégin: RGPIN-2018-04377, Godin: RGPIN-2017-06837) for their financial support. Bégin also acknowledges the financial support of Simon Fraser University.