A new closed-form solution as an extension of the Black–Scholes formula allowing smile curve plotting

In this paper, as a generalization of the Black–Scholes (BS) model, we elaborate a new closed-form solution for a uni-dimensional European option pricing model called the J-model. This closed-form solution is based on a new stochastic process, called the J-process, which is an extension of the Wiener process satisfying the martingale property. The J-process is based on a new statistical law called the J-law, which is an extension of the normal law. The J-law relies on four parameters in its general form. It has interesting asymmetry and tail properties, allowing it to fit the reality of financial markets with good accuracy, which is not the case for the normal law. Despite the use of one state variable, we find results similar to those of Heston dealing with the bi-dimensional stochastic volatility problem for pricing European calls. Inverting the BS formula, we plot the smile curve related to this closed-form solution. The J-model can also serve to determine the implied volatility by inverting the J-formula and can be used to price other kinds of options such as American options.     

Pricing equity options everywhere

Finite difference methods are a popular technique for pricing American options. Since their introduction to finance by Brennan and Schwartz their use has spread from vanilla calls and puts on one stock to path-dependent and exotic options on multiple assets. Despite the breadth of the problems they have been applied to, and the increased sophistication of some of the newer techniques, most approaches to pricing equity options have not adequately addressed the issues of unbounded computational domains and divergent diffusion coefficients. In this article it is shown that these two problems are related and can be overcome using multiple grids. This new technique allows options to be priced for all values of the underlying, and is illustrated using standard put options and the call on the maximum of two stocks. For the latter contract, I also derive a characterization of the asymptotic continuation region in terms of a one-dimensional option pricing problem, and give analytic formulae for the perpetual case.     

The term structure of S&P 100 model-free volatilities

We develop an improved method to obtain the model-free volatility more accurately despite the limitations of a finite number of options and large strike price intervals. Our method computes the model-free volatility from European-style S&P 100 index options over a horizon of up to 450 days, the first time that this has been attempted, as far as we are aware. With the estimated daily term structure over the long horizon, we find that (i) changes in model-free volatilities are asymmetrically more positively impacted by a decrease in the index level than negatively impacted by an increase in the index level; (ii) the negative relationship between the daily change in model-free volatility and the daily change in index level is stronger in the near term than in the far term; and (iii) the slope of the term structure is positively associated with the index level, having a tendency to display a negative slope during bear markets and a positive slope during bull markets. These significant results have important implications for pricing and hedging index derivatives and portfolios.     

Evaluating stochastic discount factors from term structure models

This paper examines the feasibility of applying the stochastic discount factor methodology to fixed-income data using modern term structure models. Using this approach the researcher can examine returns on bond portfolios whose exact composition is unknown, as is often the case. This paper proposes an observable proxy for the SDF from continuous-time models and documents via Monte Carlo methods the properties of the GMM estimator based on using this proxy.     

Efficient estimation of drift parameters in stochastic volatility models

We study the parametric problem of estimating the drift coefficient in a stochastic volatility model , where Y is a log price process and V the volatility process. Assuming that one can recover the volatility, precisely enough, from the observation of the price process, we construct an efficient estimator for the drift parameter of the diffusion V. As an application we present the efficient estimation based on the discrete sampling with δ _n →0 and n δ _n →∞. We show that our setup is general enough to cover the case of ‘microstructure noise’ for the price process as well.      

Detecting volatility persistence in GARCH models in the presence of the leverage effect

Most asset prices are subject to significant volatility. The arrival of new information is viewed as the main source of volatility. As new information is continually released, financial asset prices exhibit volatility persistence, which affects financial risk analysis and risk management strategies. This paper proposes a nonlinear regime-switching threshold generalized autoregressive conditional heteroskedasticity model which can be used to analyse financial data. The empirical results based on quasi-maximum likelihood estimation presented in this paper suggest that the proposed model is capable of extracting information about the sources of volatility persistence in the presence of the leverage effect.     

Analyzing interest rate risk: Stochastic volatility in the term structure of government bond yields

We propose a Nelson–Siegel type interest rate term structure model where the underlying yield factors follow autoregressive processes with stochastic volatility. The factor volatilities parsimoniously capture risk inherent to the term structure and are associated with the time-varying uncertainty of the yield curve’s level, slope and curvature. Estimating the model based on US government bond yields applying Markov chain Monte Carlo techniques we find that the factor volatilities follow highly persistent processes. We show that yield factors and factor volatilities are closely related to macroeconomic state variables as well as the conditional variances thereof.     

Options on realized variance and convex orders

Realized variance option and options on quadratic variation normalized to unit expectation are analysed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds the risk-neutral densities are said to be increasing in the convex order. For Lévy processes, such prices decrease with maturity. A time series analysis of squared log returns on the S&P 500 index also reveals such a decrease. If options are priced to a slightly increasing level of acceptability, then the resulting risk-neutral densities can be increasing in the convex order. Calibrated stochastic volatility models yield possibilities in both directions. Finally, we consider modeling strategies guaranteeing an increase in convex order for the normalized quadratic variation. These strategies model instantaneous variance as a normalized exponential of a Lévy process. Simulation studies suggest that other transformations may also deliver an increase in the convex order.     

Forecasting volatility in GARCH models with additive outliers

We model the volatility of a single risky asset using a multifactor (matrix) Wishart affine process, recently introduced in finance by Gourieroux and Sufana. As in standard Duffie and Kan affine models the pricing problem can be solved through the Fast Fourier Transform of Carr and Madan. A numerical illustration shows that this specification provides a separate fit of the long-term and short-term implied volatility surface and, differently from previous diffusive stochastic volatility models, it is possible to identify a specific factor accounting for the stochastic leverage effect, a well-known stylized fact of the FX option markets analysed by Carr and Wu.     

Bayesian range-based estimation of stochastic volatility models

Alizadeh, Brandt, and Diebold [2002. Journal of Finance 57, 1047–1091] propose estimating stochastic volatility models by quasi-maximum likelihood using data on the daily range of the log asset price process. We suggest a related Bayesian procedure that delivers exact likelihood based inferences. Our approach also incorporates data on the daily return and accommodates a nonzero drift. We illustrate through a Monte Carlo experiment that quasi-maximum likelihood using range data alone is remarkably close to exact likelihood based inferences using both range and return data.     

Regression-based Monte Carlo methods for stochastic control models: variable annuities with lifelong guarantees

We present regression-based Monte Carlo simulation algorithm for solving the stochastic control models associated with pricing and hedging of the guaranteed lifelong withdrawal benefit (GLWB) in variable annuities, where the dynamics of the underlying fund value is assumed to evolve according to the stochastic volatility model. The GLWB offers a lifelong withdrawal benefit, even when the policy account value becomes zero, while the policyholder remains alive. Upon death, the remaining account value will be paid to the beneficiary as a death benefit. The bang-bang control strategy analysed under the assumption of maximization of the policyholder’s expected cash flow reduces the strategy space of optimal withdrawal policies to three choices: zero withdrawal, withdrawal at the contractual amount or complete surrender. The impact on the GLWB value under various withdrawal behaviours of the policyholder is examined. We also analyse the pricing properties of GLWB subject to different model parameter values and structural features.     

Pricing via recursive quantization in stochastic volatility models

We provide the first recursive quantization-based approach for pricing options in the presence of stochastic volatility. This method can be applied to any model for which an Euler scheme is available for the underlying price process and it allows one to price vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein [Rev. Financ. Stud. 1991, (4), 727–752] model and the SABR model introduced in Hagan et al. [Wilmott Mag., 2002, 84–108]. A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation.     

Quadratic hedging in affine stochastic volatility models

We determine the variance-optimal hedge for a subset of affine processes including a number of popular stochastic volatility models. This framework does not require the asset to be a martingale. We obtain semiexplicit formulas for the optimal hedging strategy and the minimal hedging error by applying general structural results and Laplace transform techniques. The approach is illustrated numerically for a Lévy-driven stochastic volatility model with jumps as in Carr et al. (Math Finance 13:345–382, 2003).