Non-affine Stochastic Volatility Research Articles

Pricing of European Interest Rate Options Under Non-Affine Stochastic Volatility and Stochastic Interest Rates

Pricing of European Interest Rate Options Under Non-Affine Stochastic Volatility and Stochastic Interest Rates

Extremum Options Pricing of Two Assets under a Double Nonaffine Stochastic Volatility Model

In this paper, we consider the pricing problem for the extremum options by constructing a double nonaffine stochastic volatility model. The joint characteristic function of the logarithm of two asset prices is derived by using the Feynman–Kac theorem and one‐order Taylor approximation expansion. The semiclosed analytical pricing formulas of the European extremum options including option on maximum and option on minimum of two underlying assets are derived by using measure change technique and Fourier transform approach. Some numerical examples are provided to analyze the pricing results of extremum options under affine model, nonaffine model, Black–Scholes model, and the influences of some model parameters on the option. Numerical results show that the analytical pricing formulas have higher computational efficiency and accuracy than those of Monte Carlo simulation method. Also, results of sensitivity analysis report that the nonaffine models are more effective than other existing models in capturing the effect of volatility on option pricing.

Analytical approximation of European option prices under a new two-factor non-affine stochastic volatility model

<abstract><p>In this paper, the pricing of European options under a new two-factor non-affine stochastic volatility model is studied. In order to reduce the computational complexity, we use the Taylor expansion and Fourier-cosine method to derive an analytical approximation formula for European option prices. Numerical experiments prove that the proposed formula is fast and efficient for pricing European options compared with Monte Carlo simulations. The sensitivity of the parameters is analyzed to explain the rationality of the model. Finally, we present some preliminary empirical analysis revealing that the pricing performance of our proposed model is superior to that of the single-factor model.</p></abstract>

Valuation of European-style vulnerable options under the non-affine stochastic volatility and double exponential jump

The pricing of European-style vulnerable option when the price process of the underlying asset follows non-affine stochastic volatility and double exponential jump is investigated. An approximate expression for the joint characteristic function of the log-price of underlying asset and the log-value of counterparty asset is derived. An analytical approximate price of European-style vulnerable option is also obtained by means of Fourier-cosine method. Numerical experiments are given to confirm the accuracy and efficiency of the proposed result for pricing the European-style vulnerable option compared with Monte Carlo simulation. Finally, sensitivity analysis is presented to further explain the theoretical results.

An analytical approximation formula for European option prices under a liquidity-adjusted non-affine stochastic volatility model

<abstract><p>In this paper, we investigate the pricing of European options under a liquidity-adjusted non-affine stochastic volatility model. An analytical European option pricing formula is successfully derived with the COS method, based on an approximation for the characteristic function of the underlying log-asset price. Numerical analysis reveals that our results are very efficient and of reasonable accuracy, and we also present some sensitivity analysis to demonstrate the effects of different parameters on option prices.</p></abstract>

Approximate Analytical Pricing of European Options under the Non-affine Stochastic Volatility Model

The non-affine stochastic volatility model has attracted increasing attention in recent years, due to its excellent performance in describing the nonlinear characteristics of asset price path. However, the fact that there is no close-form of option pricing formula under this model, restricts its use badly. To remove this restriction, two approximate analytical option pricing approaches are proposed in this paper, that is, the piecewise first-order Taylor expansion method and the perturbation-based asymptotic expansion method of implied volatility. The first method is used to derive an approximate characteristic function, then based on which the Fourier-Cosine expansion method calculates the European option price. In the second way, implied volatility of the European option price is asymptotically expanded around non-affine term of volatility. Compared with the existing methods in literature, numerical experiments show that the first method has higher accuracy, while the second method is of more practical significance. In addition, the accuracy of the first method is generally higher than that of the second method in most cases, while the second method performs better when the volatility is extremely small.

Non-Affine Stochastic Volatility With Seasonal Trends

We propose a new stochastic volatility model for pricing options on assets that exhibit seasonal trends in volatility. Such assets are prevalent among commodities, with futures on grains and energy being an example. The model is based on the 3/2 stochastic volatility model, but includes a cyclical long-run volatility component. The model yields a closed-form characteristic function which can be used to rapidly calibrate the model to benchmark options. We test the model on market data and show that the model proposed here allows for significantly better empirical fit to option prices on corn and wheat futures than the unmodified 3/2 model.

A Unified Valuation Framework for Variance Swaps under Non-Affine Stochastic Volatility Models

In this article, we investigate the pricing and convergence of general non-affine non-Gaussian GARCH-based variance swap prices. Explicit solutions for fair strike prices under two different sampling schemes are derived using the extended Girsanov principle as our pricing kernel candidate. Following standard assumptions on the time-varying GARCH parameters, we show that these quantities converge to discretely and continuously sampled variance swaps constructed based on the weak diffusion limit of the underlying GARCH model. An empirical study which relies on a joint estimation using both historical returns and VIX data indicates that an asymmetric heavier-tailed distribution is more appropriate for modelling the GARCH innovations. Finally, we provide several numerical exercises to support our theoretical convergence results in which we investigate the effect of the quadratic variation approximation for the realized variance, as well as the impact of discrete versus continuous-time modelling of asset returns.

European Option Pricing With a Fast Fourier Transform Algorithm for Big Data Analysis

Several empirical studies show that, under multiple risks, markets exhibit many new properties, such as volatility smile and cluster fueled by the explosion of transaction data. This paper attempts to capture these newly developed features using the valuation of European options as a vehicle. Statistical analysis performed on the data collected from the currency option market clearly shows the coexistence of mean reversion, jumps, volatility smile, and leptokurtosis and fat tail. We characterize the dynamics of the underlying asset in this kind of environment by establishing a coupled stochastic differential equation model with triple characteristics of mean reversion, nonaffine stochastic volatility, and mixed-exponential jumps. Moreover, we propose a characteristic function method to derive the closed-form pricing formula. We also present a fast Fourier transform (FFT) algorithm-based numerical solution method. Finally, extensive numerical experiments are conducted to validate both the modeling methodology and the numerical algorithm. Results demonstrate that the model behaves well in capturing the properties observed in the market, and the FFT numerical algorithm is both accurate and efficient in addressing large amount of data.

Pricing options under the non-affine stochastic volatility models: An extension of the high-order compact numerical scheme

We consider an improvement of a high-order compact finite difference scheme for option pricing in non-affine stochastic volatility models. Upon applying a proper transformation to equate the different coefficients of second-order non-cross derivatives, a high-order compact finite difference scheme is developed to solve the partial differential equation with nonlinear coefficients that the option values satisfied. Based on the local von Neumann stability analysis, a theoretical stability result is obtained under certain restrictions. Numerical experiments are presented showing the convergence and validity of the expansion methods and the important effects of the non-affine coefficient and volatility of volatility on option values.

Herding and Stochastic Volatility

In this paper we develop a one-factor non-affine stochastic volatility option pricing model where the dynamics of the underlying is endogenously determined from micro-foundations. The interaction and herding of the agents trading the underlying asset induce an amplification of the volatility of the asset over the volatility of the fundamentals. Although the model is non-affine, a closed form option pricing formula can still be derived by using a Gauss-Hermite series expansion methodology. The model is calibrated using S&P 500 index options for the period 1996-2013. When its results are compared to some benchmark models we find that the new non-affine one-factor model outperforms the affine one-factor Heston model and it is competitive, especially out-of-sample, with the affine two-factor double Heston model.

Switching to Non-Affine Stochastic Volatility: A Closed-Form Expansion for the Inverse Gamma Model

This paper introduces the Inverse Gamma (IGa) stochastic volatility model with time-dependent parameters, defined by the volatility dynamics dVt = κt.(θt − Vt).dt λt.Vt.dBt. This non-affine model is much more realistic than classical affine models like the Heston stochastic volatility model, even though both are as parsimonious (only four stochastic parameters). Indeed, it provides more realistic volatility distribution and volatility paths, which translate in practice into more robust calibration and better hedging accuracy, explaining its popularity among practitioners.In order to price vanilla options with IGa volatility, we propose a closed-form volatility-of-volatility expansion. Specifically, the price of a European put option with IGa volatility is approximated by a Black-Scholes price plus a weighted combination of Black-Scholes greeks, where the weights depend only on the four time-dependent parameters of the model. This closed-form pricing method allows for very fast pricing and calibration to market data. The overall quality of the approximation is very good, as shown by several calibration tests on real-world market data where expansion prices are compared favorably with Monte Carlo simulation results.This paper shows that the IGa model is as simple, more realistic, easier to implement and faster to calibrate than classical transform-based affine models. We therefore hope that the present work will foster further research on non-affine models like the Inverse Gamma stochastic volatility model, all the more so as this robust model is of great interest to the industry.

Affine Approximation for Moment Generating Function of Log-Normal Stochastic Volatility Model

While empirical studies have established that the log-normal stochastic volatility (SV) model is superior to its alternatives, the model does not allow for the analytical solutions available for affine models. To circumvent this, we show that the joint moment generating function (MGF) of the log-price and the quadratic variance (QV) under the log-normal SV model can be decomposed into a leading term, which is given by an exponential-affine form, and a residual term, whose estimate depends on the higher order moments of the volatility process. We prove that the second-order leading term is theoretically consistent with the expected values and covariance matrix of the log-price and the quadratic variance. We further extend this approach to the log-normal SV model with jumps. We use Fourier inversion techniques to value vanilla options on the equity and the QV and, by comparison to Monte Carlo simulations, we show that the second-order leading term is precise for the valuation of vanilla options. We generalize the affine decomposition to other non-affine stochastic volatility models with polynomial drift and volatility functions, and with jumps in the volatility process.

Options on realized variance by transform methods: a non-affine stochastic volatility model

In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model by developing efficient transform-based pricing methods. This non-affine model gives prices of options on realized variance that allow upward-sloping implied volatility of variance smiles. Heston's model [Rev. Financial Stud., 1993, 6, 327–343], the benchmark affine stochastic volatility model, leads to downward-sloping volatility of variance smiles—in disagreement with variance markets in practice. Using control variates, we propose a robust method to express the Laplace transform of the variance call function in terms of the Laplace transform of the realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm that gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps.

Volatility dynamics for the S&P 500: Further evidence from non-affine, multi-factor jump diffusions

We apply Markov chain Monte Carlo methods to time series data on S&P 500 index returns, and to its option prices via a term structure of VIX indices, to estimate 18 different affine and non-affine stochastic volatility models with one or two variance factors, and where jumps are allowed in both the price and the instantaneous volatility. The in-sample fit to the VIX term structure shows that the second (stochastic long-term volatility) factor is required to fit the VIX term structure. Out-of-sample tests on the fit to individual option prices, as well as in-sample tests, show that the inclusion of jumps is less important than allowing for non-affine dynamics. The estimation and testing periods together cover more than 21 years of daily data.

Affine Versus Non-Affine Stochastic Volatility and the Impact on Asset Allocation

This paper analyzes the influence of affine versus non-affine stochastic volatility specifications on simulated distributions, option pricing, and asset allocation. We look at models that include stochastic volatility and jumps in the stock price and in its volatility. For the asset allocation problem we consider a CRRA investor who has access to one additional derivative, besides the stock and the money market account, and is restricted to a buy-and-hold strategy. We show that the assumption on the volatility structure has a very strong impact on simulated volatility distributions. Differences among these translate into the simulated stock price distributions, option prices, and the optimal portfolio positions. We are especially interested in the mistake the investor makes by choosing the convenient Heston (1993) specification over a (by assumption correct) non-affine volatility specification. We find that this leads to large utility losses for the investor, which are especially severe in models that do not include jumps. However, the investor can in most cases minimize his losses by including jumps to the affine volatility specification and by choosing the correct moneyness. He therefore can continue to use the convenient and numerically more easily accessible affine specification.