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On the Monotone Stability Approach to BSDEs with Jumps: Extensions, Concrete Criteria and Examples Read chapter Read first chapter

2019 | OriginalPaper | Chapter

European Option Pricing with Stochastic Volatility Models Under Parameter Uncertainty

Authors : Samuel N. Cohen, Martin Tegnér

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

Publisher: 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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previous chapter An Example of Martingale Representation in Progressive Enlargement by an Accessible Random Time
next chapter Construction of an Aggregate Consistent Utility, Without Pareto Optimality. Application to Long-Term Yield Curve Modeling
Appendix
Available only for authorised users
Footnotes
1
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.
 
2
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].
 
3
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.
 
4
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 \(\mathbb {R}^d\).
 
5
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.
 
6
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.
 
7
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.
 
8
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.
 
9
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].
 
10
[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.
 
11
The Realised Library version 0.2 by Heber, Gerd, Lunde, Shephard and Sheppard (2009), http://​realized.​oxford-man.​ox.​ac.​uk.
 
12
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.
 
14
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.
 
15
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.
 
16
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.
 
17
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 \(n=25\).
 
18
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 \(\mu (\{z_t\})=1\).
 
19
We are using the R package “earth” by [49].
 
20
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.
 
Literature
1.
Aït-Sahalia, Y., Kimmel, R.: Maximum likelihood estimation of stochastic volatility models. J. Financ. Econ. 83(2), 413–452 (2007)CrossRef
2.
Alanko, S., Avellaneda, M.: Reducing variance in the numerical solution of BSDEs. Comptes Rendus Mathematique 351(3), 135–138 (2013)MathSciNetMATHCrossRef
3.
Alfonsi, A.: Affine Diffusions and Related Processes: Simulation. Springer, Theory and Applications (2016)MATH
4.
Andersen, L.B., Jckel, P., Kahl, C.: Simulation of Square-Root Processes. Wiley (2010)
5.
Andersen, T., Benzoni, L.: Realized volatility. In: Handbook of Financial Time Series 555–575 (2009)
6.
Avellaneda, M., Levy, A., Parás, A.: Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Financ. 2(2), 73–88 (1995)CrossRef
7.
Avellaneda, M., Paras, A.: Managing the volatility risk of portfolios of derivative securities: the lagrangian uncertain volatility model. Appl. Math. Financ. 3(1), 21–52 (1996)MATHCrossRef
8.
Bannör, K.F., Scherer, M.: Model risk and uncertainty—illustrated with examples from mathematical finance. In: Risk—A Multidisciplinary Introduction, pp. 279–306. Springer (2014)
9.
Barczy, M., Pap, G.: Asymptotic properties of maximum-likelihood estimators for Heston models based on continuous time observations. Statistics 50(2), 389–417 (2016)MathSciNetMATH
10.
Bates, D.S.: Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options. Rev. Financ. Stud. 9(1), 69–107 (1996)CrossRef
11.
12.
Beneš, V.: Existence of optimal strategies based on specified information, for a class of stochastic decision problems. SIAM J. Control 8, 179–188 (1970)MathSciNetMATHCrossRef
13.
Bibby, B.M., Sørensen, M.: Martingale estimation functions for discretely observed diffusion processes. Bernoulli 17–39 (1995)
14.
Black, F.: Studies of stock price volatility changes. In: Proceedings of the 1976 Meetings of the American Statistical Association, pp. 171–181 (1976)
15.
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 637–654, (1973)
16.
Bouchard, B., Elie, R.: Discrete-time approximation of decoupled forward-backward SDE with jumps. Stochast. Process. Appl. 118(1), 53–75 (2008)MathSciNetMATHCrossRef
17.
Bouchard, B., Touzi, N.: Discrete-time approximation and monte-carlo simulation of backward stochastic differential equations. Stochast. Process. Appl. 111(2), 175–206 (2004)MathSciNetMATHCrossRef
18.
Breeden, D.T.: An intertemporal asset pricing model with stochastic consumption and investment opportunities. J. Financ. Econ. 7(3), 265–296 (1979)MATHCrossRef
19.
Bunnin, F.O., Guo, Y., Ren, Y.: Option pricing under model and parameter uncertainty using predictive densities. Stat. Comput. 12(1), 37–44 (2002)MathSciNetMATHCrossRef
20.
Carr, P., Geman, H., Madan, D.B., Yor, M.: Stochastic volatility for lévy processes. Math. Financ. 13(3), 345–382 (2003)MATHCrossRef
21.
Carr, P., Sun, J.: A new approach for option pricing under stochastic volatility. Rev. Deriv. Res. 10(2), 87–150 (2007)MATHCrossRef
22.
Cohen, S.N., Elliott, R.J.: Stochastic Calculus and Applications. Springer (2015)
23.
Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC (2004)
24.
Cox, J.C., Ingersoll Jr., J.E., Ross, S.A.: A theory of the term structure of interest rates. Econ.: J. Econ. Soc. 385–407 (1985)
25.
26.
Derman, E.: Model risk. Risk Mag. 9(5), 34–37 (1996)
27.
28.
El Karoui, N., Quenez, M.-C.: Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33(1), 29–66 (1995)MathSciNetMATHCrossRef
29.
Feller, W.: Two singular diffusion problems. Ann. Math. 173–182 (1951)
30.
Filippov, A.: On certain questions in the theory of optimal control. Vestnik Moskov. Univ. Ser. Mat. Meh. Astronom. 2, 25–42 (1959). English trans. J. Soc. Indust. Appl. Math. Ser. A. Control 1, 76–84 (1962)
31.
Gatheral, J.: The Volatility Surface: A Practitioner’s Guide, vol. 357. Wiley (2011)
32.
33.
Gobet, E., Lemor, J.-P., Warin, X.: A regression-based monte carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. (2005)
34.
Gobet, E., Turkedjiev, P.: Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Math. Comput. 85(299), 1359–1391 (2016)MathSciNetMATHCrossRef
35.
Godambe, V., Heyde, C.: Quasi-likelihood and optimal estimation. Int. Stat. Rev./Revue Internationale de Statistique 231–244 (1987)
36.
Gupta, A., Reisinger, C., Whitley, A.: Model uncertainty and its impact on derivative pricing. In: Rethinking Risk Measurement and Reporting, p. 122 (2010)
37.
Hastie, T., Tibshirani, R., Friedman, J., Franklin, J.: The elements of statistical learning: data mining, inference and prediction. Math. Intell. 27(2), 83–85 (2005)
38.
Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)MathSciNetMATHCrossRef
39.
Heston, S.L.: A simple new formula for options with stochastic volatility (1997)
40.
Hull, J., White, A.: The pricing of options on assets with stochastic volatilities. J. Financ. 42(2), 281–300 (1987)MATHCrossRef
41.
Jacquier, E., Jarrow, R.: Bayesian analysis of contingent claim model error. J. Econ. 94(1), 145–180 (2000)MATHCrossRef
42.
43.
Keynes, J.M.: A treatise on probability (1921)
44.
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations, vol. 23. Springer (1992)
45.
Knight, F.H.: Risk, Uncertainty and Profit. Houghton Mifflin, Boston (1921)
46.
Lehmann, E.L., Romano, J.P.: Testing Statistical Hypotheses. Springer Science & Business Media (2006)
47.
Lyons, T.J.: Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Financ. 2(2), 117–133 (1995)CrossRef
48.
McShane, E., Warfield Jr., R.: On Filippov’s implicit functions lemma. Proc. Am. Math. Soc. 18, 41–47 (1967)MathSciNetMATH
49.
Milborrow. Derived from mda:mars by T. Hastie and R. Tibshirani, S.: Earth: Multivariate Adaptive Regression Splines. R package (2011)
50.
Peng, S.: A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochast. Stochast. Rep. 38, 119–134 (1992)MathSciNetMATHCrossRef
51.
Prakasa-Rao, B.: Asymptotic theory for non-linear least squares estimator for diffusion processes. Statis.: J. Theor. Appl. Statis. 14(2), 195–209 (1983)MathSciNetMATHCrossRef
52.
Quenez, M.-C.: Stochastic Control and BSDEs. Addison Wesley Longman, Harlow (1997)MATH
53.
54.
Stein, E.M., Stein, J.C.: Stock price distributions with stochastic volatility: an analytic approach. Rev. Financ. Stud. 4(4), 727–752 (1991)MATHCrossRef
55.
Stein, J.: Overreactions in the options market. J. Financ. 44(4), 1011–1023 (1989)CrossRef
56.
57.
Wong, B., Heyde, C.: On changes of measure in stochastic volatility models. Int. J. Stochast. Anal. (2006)
Metadata
Title
European Option Pricing with Stochastic Volatility Models Under Parameter Uncertainty
Authors
Samuel N. Cohen
Martin Tegnér
Copyright Year
2019
Book
Frontiers in Stochastic Analysis–BSDEs, SPDEs and their Applications

Print ISBN: 978-3-030-22284-0

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

Copyright Year: 2019

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