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2019 | OriginalPaper | Buchkapitel

European Option Pricing with Stochastic Volatility Models Under Parameter Uncertainty

verfasst von : Samuel N. Cohen, Martin Tegnér

Erschienen in: Frontiers in Stochastic Analysis–BSDEs, SPDEs and their Applications

Verlag: Springer International Publishing

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Abstract

We consider stochastic volatility models under parameter uncertainty and investigate how model derived prices of European options are affected. We let the pricing parameters evolve dynamically in time within a specified region, and formalise the problem as a control problem where the control acts on the parameters to maximise/minimise the option value. Through a dual representation with backward stochastic differential equations, we obtain explicit equations for Heston’s model and investigate several numerical solutions thereof. In an empirical study, we apply our results to market data from the S&P 500 index where the model is estimated to historical asset prices. We find that the conservative model-prices cover 98% of the considered market-prices for a set of European call options.

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Vorheriges Kapitel An Example of Martingale Representation in Progressive Enlargement by an Accessible Random Time

Nächstes Kapitel Construction of an Aggregate Consistent Utility, Without Pareto Optimality. Application to Long-Term Yield Curve Modeling

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A supporting case for this assumption is the fact pointed out for instance by [53]: while volatilities may be estimated within reasonable confidence with a few years of data, drift estimation requires data from much longer time periods.

Also know as a CIR process from its use as a model for short-term interest rates by [24]. The square root process goes back to [29].

Heston motivates this choice from the model of [18] under the assumption that the equilibrium consumption process also follows a square-root process; the risk premium is then proportional to variance. Aggregate risk preferences aside, a consequence is that the pricing equation (3) conveniently allows for Heston’s pricing formula.

We use $\bullet $ to denote the stochastic integral of d-dimensional processes: $H\bullet M =\sum _{i=1}^d \int _0^. H^i_tdM^i_t$ for H, M taking values in $\mathbb {R}^d$.

Here we could be a bit more finical on notation, for instance with $(r_t,\tilde{\kappa }_t,\tilde{\theta }_t)\in \tilde{U}$ representing the pricing uncertainty deduced from calibration. For brevity, we refrain from such a notional distinction.

This draws on the interpretation that $\{Q^u:u\in \mathcal {U}\}$ is the set of equivalent martingale measures of an incomplete market model, such that the most conservative risk-neutral price of an option equals the super-replication cost of a short position in the same: with $\Pi _t(G)=\inf _{\phi }\{\tilde{V}_t(\phi ):V_T(\phi )\ge G,\,\text {a.s.} \}$ being the discounted portfolio value of the (cheapest) admissible strategy $\phi $ that super-replicates G, then $\Pi _t(G)={{\,\mathrm{ess\,sup}\,}}_{u\in \mathcal {U}}\mathbb {E}_u[\tilde{G}|\mathcal {F}_t]$ and the supremum is attained. See for instance [23], Sect. 10.2.

As we assume the driver f to be sufficiently integrable for the J(u)-BSDE to admit a unique solution (i.e. it is a stochastic Lipschitz driver) the integrability carries over to H such that the Y-BSDE admits a unique solution as well.

The function for the martingale representation Z is obtained explicitly by applying Itô’s lemma to $D_t=D(t,S_t,V_t)$ and using the semilinear pricing PDE (14), which gives

$$\begin{aligned} dD(t,S_t,V_t) = -H(S_t,V_t,D_t,Z_t)dt + \partial _xD(t,S_t,V_t) \sigma (S_t,V_t)d\tilde{W}_t \end{aligned}$$

where $\partial _xf\equiv (\partial _sf,\partial _vf)$ and $\sigma (s,v)$ should be understood as the diffusion matrix of (2). Hence, by uniqueness of the BSDE solution, $z(t,s,v) \equiv \partial _xD(t,s,v) \sigma (s,v)$ is the deterministic generating function for Z.

The Laplace transform of the integrated variance $\mathbb {E}[\exp (-\beta \int _0^TV_tdt)]$ goes back to [24] and is well defined for $-\beta \le \kappa ^2/(2\sigma ^2)$, see also [20].

[21] gives an expression for the joint transform of the log-price and integrated variance of a 3-over-2 process. Applying Itô’s formula to $1/V_t$ we find that the inverse-CIR $(\kappa ,\theta ,\sigma )$ process is a 3-over-2 process with parameters $(\hat{\kappa }\equiv \kappa \theta -\sigma ^2,\hat{\theta }\equiv \kappa /(\kappa \theta -\sigma ^2),\hat{\sigma }\equiv -\sigma )$. Using their transform, provided $\hat{\kappa }>-\hat{\sigma }^2/2$,

$$\begin{aligned} \mathbb {E}\left[ e^{-\lambda \int _0^T \frac{1}{V_t}dt} \right] = \frac{\Gamma (\gamma -\alpha )}{\Gamma (\gamma )}\left( \frac{2}{\hat{\sigma }^2y(0,1/V_0)} \right) ^\alpha M\left( \alpha ,\gamma ,-\frac{2}{\hat{\sigma }^2 y(0,1/V_0)} \right) \end{aligned}$$

where

$$\begin{aligned}&y(t,x) \equiv x(e^{\hat{\kappa }\hat{\theta } (T-t)}-1)/(\hat{\kappa }\hat{\theta })= x(e^{\kappa (T-t)}-1)/\kappa \\&\alpha \equiv -(1/2+\hat{\kappa }/\sigma ^2) + \sqrt{(1/2+\hat{\kappa }/\sigma ^2)^2 + 2\lambda /\sigma ^2} \\&\gamma \equiv 2(\alpha +1+\hat{\kappa }/\sigma ^2) = 1+2\sqrt{(1/2+\hat{\kappa }/\sigma ^2)^2 + 2\lambda /\sigma ^2} \end{aligned}$$

and M is the confluent hypergeometric function. From this, we see that

$$\begin{aligned} \lambda \ge -\left( \frac{2\hat{\kappa }+\sigma ^2}{2\sqrt{2}\sigma } \right) ^2=-\left( \frac{2{\kappa }\theta -\sigma ^2}{2\sqrt{2}\sigma } \right) ^2 \end{aligned}$$

is a sufficient condition for the transform to being well defined.

The Realised Library version 0.2 by Heber, Gerd, Lunde, Shephard and Sheppard (2009), http://realized.oxford-man.ox.ac.uk.

Note that (25) may generate negative outcomes of ${V}^{\pi }_{t+\delta }$ and is thus not suitable for simulation in its standard form. Alternative schemes are discussed in Appendix A.3. Here we use (25) for an approximative Gaussian likelihood—the Euler contrast (26)—which is well defined.

http://realized.oxford-man.ox.ac.uk and https://wrds-web.wharton.upenn.edu/wrds/.

Alternatively, we may employ the (approximative) likelihood with exact conditional moments. For daily observations, the numerical optimisation does not converge while for weekly data, this yields very similar parameter estimates and standard errors as with approximative moments.

The quadratic covariation of logarithmic data gives $\frac{1}{t}[\log S ,\frac{1}{\sigma }\log V ]_t = \frac{1}{t}\int _0^t\sqrt{V_s}\frac{1}{\sqrt{V_s}}d[\rho W^1+\sqrt{1-\rho ^2}W^2,W^1]_s = \rho $ and we use a realized covariation estimate thereof.

The time-stepping of the scheme fails whenever $V^{\pi }_{t_i}<0$ due to the computation of $\sqrt{V^{\pi }_{t_i}}$ and the truncation step is simply to replace with $\sqrt{(V^{\pi }_{t_i})^+}$. Although this prevents the scheme to fail, note that negative values may still be generated and in particular when the time-step $\Delta _i$ is large.

Bookkeeping the sign of the generated variance values yields positive outcomes $99.3\%$ of the time when using $n=1{,}000$ time steps and $96.7\%$ for $n=25$.

As the jumps are generated by a Poisson random measure, S will have jumps given by

$$ \Delta S_t=\int _{z\in \mathbb {R}^d} h(z,S_{t-},V_t)\tilde{\mu }(dz,\{t\}) = h(z_t,S_{t-},V_t)\mathbf {1}_{\{\Delta S_t\ne 0 \}} $$

where $z_t\in \mathbb {R}^d$ is a (unique) point in the set where $\mu (\{z_t\})=1$.

We are using the R package “earth” by [49].

The calculation of the pricing-formula for the call relies on numerical integration and we need $N=100{,}000$ such evaluations for each of $n=25$ time-step which makes the scheme very computer intensive. For this reason, we calculate a subset of 500 call prices and use a polynomial regression to predict the remaining call prices. As the pricing formula is a “nice” function of S and V, this approximation only has a limited impact.

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Titel: European Option Pricing with Stochastic Volatility Models Under Parameter Uncertainty
verfasst von: Samuel N. Cohen
Martin Tegnér
Verlag: Springer International Publishing
Buch: Frontiers in Stochastic Analysis–BSDEs, SPDEs and their Applications
Print ISBN: 978-3-030-22284-0

Electronic ISBN: 978-3-030-22285-7

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-22285-7_5

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