General closed-form basket option pricing bounds

This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known. Moreover, the basket value is not required to be positive. We test our new price approximations on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms; hence, they do not suffer from the curse of dimensionality and can be applied also to high-dimensional problems where most existing methods fail. In particular, we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. We also show how to improve Monte Carlo accuracy by using one of our bounds as a control variate.     

Non-parametric method for European option bounds

There is much research whose efforts have been devoted to discovering the distributional defects in the Black–Scholes model, which are known to cause severe biases. However, with a free specification for the distribution, one can only find upper and lower bounds for option prices. In this paper, we derive a new non-parametric lower bound and provide an alternative interpretation of Ritchken’s (J Finance 40:1219–1233, 1985) upper bound to the price of the European option. In a series of numerical examples, our new lower bound is substantially tighter than previous lower bounds. This is prevalent especially for out of the money options where the previous lower bounds perform badly. Moreover, we present how our bounds can be derived from histograms which are completely non-parametric in an empirical study. We discover violations in our lower bound and show that those violations present arbitrage profits. In particular, our empirical results show that out of the money calls are substantially overpriced (violate the lower bound).     

It takes a model to beat a model: Volatility bounds

Models like the CAPM and Fama–French three-factor models are commonly used as benchmarks for calculating cost of capital and evaluating portfolio performance, despite the empirical evidence to reject them. For many practical purposes, “it takes a model to beat a model.” In this paper we derive restrictions on models that could “beat” a bench-mark model but might still be misspecified. In these “takes-a-model-to-beat-a-model” (TMBM) bounds, model A beats model B if model A's quadratic form of pricing errors is smaller. The bounds generalize the Hansen–Jagannathan bound and distance measure. We use the TMBM bounds to evaluate various linear factor models and consumption-based models. The failure of the power utility model is much less extreme when it is compared with the CAPM and Fama–French model. For reasonable utility curvature, the Ferson–Constantinides model and Epstein–Zin model perform best among the consumption-based models, beating the model of Campbell and Cochrane, in which model the value of the persistence parameter that matches the time-series properties of aggregate stock market returns seems too low for cross-sectional asset pricing.     

Do Bonds Span Volatility Risk in the U.S. Treasury Market? A Specification Test for Affine Term Structure Models

We propose using model‐free yield quadratic variation measures computed from intraday data as a tool for specification testing and selection of dynamic term structure models. We find that the yield curve fails to span realized yield volatility in the U.S. Treasury market, as the systematic volatility factors are largely unrelated to the cross‐section of yields. We conclude that a broad class of affine diffusive, quadratic Gaussian, and affine jump‐diffusive models cannot accommodate the observed yield volatility dynamics. Hence, the Treasury market per se is incomplete, as yield volatility risk cannot be hedged solely through Treasury securities.     

Gaussian models for Euro high grade government yields

This paper tests affine, quadratic and Black-type Gaussian models on Euro area triple A Government bond yields for maturities up to 30 years. Quadratic Gaussian models beat affine Gaussian models both in-sample and out-of-sample. A Black-type model best fits the shortest maturities and the extremely low yields since 2013, but worst fits the longest maturities. Even for quadratic models we can infer the latent factors from some yields observed without errors, which makes quasi-maximum likelihood (QML) estimation feasible. New specifications of quadratic models fit the longest maturities better than does the ‘classic’ specification of Ahn et al. [2002. ‘Quadratic Term Structure Models: Theory and Evidence.’ The Review of Financial Studies 15 (1): 243–288], but the opposite is true for the shortest maturities. These new specifications are more suitable to QML estimation. Overall quadratic models seem preferable to affine Gaussian models, because of superior empirical performance, and to Black-type models, because of superior tractability. This paper also proposes the vertical method of lines (MOL) to solve numerically partial differential equations (PDEs) for pricing bonds under multiple non-independent stochastic factors. ‘Splitting’ the PDE drastically reduces computations. Vertical MOL can be considerably faster and more accurate than finite difference methods.     

Polynomial processes and their applications to mathematical finance

We introduce a class of Markov processes, called m-polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Lévy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Lévy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.     

Conditioning Information and Variance Bounds on Pricing Kernels

Gallant, Hansen, and Tauchen (1990) show how to use conditioninginformation optimally to construct a sharper unconditional variancebound (the GHT bound) on pricing kernels. The literature predominantlyresorts to a simple but suboptimal procedure that scales returnswith predictive instruments and computes standard bounds usingthe original and scaled returns. This article provides a formalbridge between the two approaches. We propose an optimally scaledbound that coincides with the GHT bound when the first and secondconditional moments are known. When these moments are misspecified,our optimally scaled bound yields a valid lower bound for thestandard deviation of pricing kernels, whereas the GHT bounddoes not. We illustrate the behavior of the bounds using a numberof linear and nonlinear models for consumption growth and bondand stock returns. We also illustrate how the optimally scaledbound can be used as a diagnostic for the specification of thefirst two conditional moments of asset returns.     

Can interest rate volatility be extracted from the cross section of bond yields?

Most affine models of the term structure with stochastic volatility predict that the variance of the short rate should play a ‘dual role’ in that it should also equal a linear combination of yields. However, we find that estimation of a standard affine three-factor model results in a variance state variable that, while instrumental in explaining the shape of the yield curve, is essentially unrelated to GARCH estimates of the quadratic variation of the spot rate process or to implied variances from options. We then investigate four-factor affine models. Of the models tested, only the model that exhibits ‘unspanned stochastic volatility’ (USV) generates both realistic short rate volatility estimates and a good cross-sectional fit. Our findings suggest that short rate volatility cannot be extracted from the cross-section of bond prices. In particular, short rate volatility and convexity are only weakly correlated.     

Bounds on the autocorrelation of admissible stochastic discount factors

We show how to use asset market data to restrict the admissible region for the first-order autocorrelation of the stochastic discount factor (SDF). We interpret this statistic as a measure of a model’s economic time variation across two periods. Estimating bounds for nominal and real SDFs at monthly and quarterly frequencies, we find that the admissible autocorrelations are significantly negative, but greater than −0.02, implying that the bounds impose a strong restriction on candidate SDFs. We illustrate the relevancy of these findings by showing that some widely used consumption-based models are misspecified with respect to the autocorrelation bound. Finally, we examine the implications of our results for the admissibility of linear factor models and the appropriateness of empirical pricing factors.     

Identification of Maximal Affine Term Structure Models

Building on Duffie and Kan (1996) , we propose a new representation of affine models in which the state vector comprises infinitesimal maturity yields and their quadratic covariations. Because these variables possess unambiguous economic interpretations, they generate a representation that is globally identifiable. Further, this representation has more identifiable parameters than the “maximal” model of Dai and Singleton (2000) . We implement this new representation for select three‐factor models and find that model‐independent estimates for the state vector can be estimated directly from yield curve data, which present advantages for the estimation and interpretation of multifactor models.     

Quadratic term structure models in discrete time

This paper extends the results on quadratic term structure models in continuous time to the discrete time setting. The continuous time setting can be seen as a special case of the discrete time one. Discrete time quadratic models have advantages over their continuous time counterparts as well as over discrete time affine models. Recursive closed form solutions for zero coupon bonds are provided even in the presence of multiple correlated underlying factors, time-dependent parameters, regime changes and “jumps” in the underlying factors. In particular regime changes and “jumps” cannot so easily be accommodated in continuous time quadratic models. Pricing bond options requires simple integration and model estimation does not require a restrictive choice of the market price of risk.     

Estimating affine multifactor term structure models using closed-form likelihood expansions

We develop and implement a technique for closed-form maximum likelihood estimation (MLE) of multifactor affine yield models. We derive closed-form approximations to likelihoods for nine Dai and Singleton (2000) affine models. Simulations show our technique very accurately approximates true (but infeasible) MLE. Using US Treasury data, we estimate nine affine yield models with different market price of risk specifications. MLE allows non-nested model comparison using likelihood ratio tests; the preferred model depends on the market price of risk. Estimation with simulated and real data suggests our technique is much closer to true MLE than Euler and quasi-maximum likelihood (QML) methods.     

Market price of risk specifications for affine models: Theory and evidence

We extend the standard specification of the market price of risk for affine yield models, and apply it to U.S. Treasury data. Our specification often provides better fit, sometimes with very high statistical significance. The improved fit comes from the time-series rather than cross-sectional features of the yield curve. We derive conditions under which our specification does not admit arbitrage opportunities. The extension has extremely strong statistical significance for affine yield models with multiple square-root type variables. Although we focus on affine yield models, our specification can be used with other asset pricing models as well.     

Option pricing for GARCH-type models with generalized hyperbolic innovations

In this paper, we provide a new dynamic asset pricing model for plain vanilla options and we discuss its ability to produce minimum mispricing errors on equity option books. Given the historical measure, the dynamics of assets being modeled by Garch-type models with generalized hyperbolic innovations and the pricing kernel is an exponential affine function of the state variables, we show that the risk-neutral distribution is unique and again implies a generalized hyperbolic dynamics with changed parameters. We provide an empirical test for our pricing methodology on two data sets of options, respectively written on the French CAC 40 and the American SP 500. Then, using our theoretical result associated with Monte Carlo simulations, we compare this approach with natural competitors in order to test its efficiency. More generally, our empirical investigations analyse the ability of specific parametric innovations to reproduce market prices in the context of an exponential affine specification of the stochastic discount factor.     

Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities

In quantitative risk management, it is important and challenging to find sharp bounds for the distribution of the sum of dependent risks with given marginal distributions, but an unspecified dependence structure. These bounds are directly related to the problem of obtaining the worst Value-at-Risk of the total risk. Using the idea of complete mixability, we provide a new lower bound for any given marginal distributions and give a necessary and sufficient condition for the sharpness of this new bound. For the sum of dependent risks with an identical distribution, which has either a monotone density or a tail-monotone density, the explicit values of the worst Value-at-Risk and bounds on the distribution of the total risk are obtained. Some examples are given to illustrate the new results.     

“Extended Black” term structure models

This paper examines “Extended Black” term structure models (EBTSM), which are multi-factor extensions of the one-factor Black model (Black, F., 1995. Interest rates as options. Journal of Finance 50, 1371-1376). EBTSM are not affected by the admissibility restrictions that plague canonical affine models. EBTSM encompass quadratic models, but unlike in quadratic models bond yields are sufficient statistics to infer the latent factors driving the short interest rate. EBTSM are amenable to econometric estimation despite the need to solve bond pricing equations through finite difference numerical methods. Estimation through the Iterated Extended Kalman filter reveals that a two-factor EBTSM fit well the observed cross section and time series of Japanese Government bond yields. A three-factor EBTSM is also proposed.     

Minimum option prices under decreasing absolute risk aversion

We establish bounds on option prices in an economy where the representative investor has an unknown utility function that is constrained to belong to the family of nonincreasing absolute risk averse functions. For any distribution of terminal consumption, we identify a procedure that establishes the lower bound of option prices. We prove that the lower bound derives from a particular negative exponential utility function. We also identify lower bounds of option prices in a decreasing relative risk averse economy. For this case, we find that the lower bound is determined by a power utility function. Similar to other recent findings, for the latter case, we confirm that under lognormality of consumption, the Black Scholes price is a lower bound. The main advantage of our bounding methodology is that it can be applied to any arbitrary marginal distribution for consumption. This revised version was published online in June 2006 with corrections to the Cover Date.     

Tight bounds on American option prices

In contrast to the constant exercise boundary assumed by Broadie and Detemple (1996) [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies 9, 1211–1250], we use an exponential function to approximate the early exercise boundary. Then, we obtain lower bounds for American option prices and the optimal exercise boundary which improve the bounds of Broadie and Detemple (1996). With the tight lower bound for the optimal exercise boundary, we further derive a tight upper bound for the American option price using the early exercise premium integral of Kim (1990) [Kim, I.J., 1990. The analytic valuation of American options. Review of Financial Studies 3, 547–572]. The numerical results show that our lower and upper bounds are very tight and can improve the pricing errors of the lower bound and upper bound of Broadie and Detemple (1996) by 83.0% and 87.5%, respectively. The tightness of our upper bounds is comparable to some best accurate/efficient methods in the literature for pricing American options. Moreover, the results also indicate that the hedge ratios (deltas and gammas) of our bounds are close to the accurate values of American options.     

When do jumps matter for portfolio optimization?

We consider the continuous-time portfolio optimization problem of an investor with constant relative risk aversion who maximizes expected utility of terminal wealth. The risky asset follows a jump-diffusion model with a diffusion state variable. We propose an approximation method that replaces the jumps by a diffusion and solve the resulting problem analytically. Furthermore, we provide explicit bounds on the true optimal strategy and the relative wealth equivalent loss that do not rely on quantities known only in the true model. We apply our method to a calibrated affine model. Our findings are threefold: Jumps matter more, i.e. our approximation is less accurate, if (i) the expected jump size or (ii) the jump intensity is large. Fixing the average impact of jumps, we find that (iii) rare, but severe jumps matter more than frequent, but small jumps.     

Efficient discretization of stochastic integrals

Sharp asymptotic lower bounds on the expected quadratic variation of the discretization error in stochastic integration are given when the integrator admits a predictable quadratic variation and the integrand is a continuous semimartingale with nondegenerate local martingale part. The theory relies on inequalities for the kurtosis and skewness of a general random variable which are themselves seemingly new. Asymptotically efficient schemes which attain the lower bounds are constructed explicitly. The result is directly applicable to a practical hedging problem in mathematical finance; for hedging a payoff which is replicated by a continuous-time trading strategy, it gives an asymptotically optimal way to choose discrete rebalancing dates and portfolios with respect to transaction costs. The asymptotically efficient strategies in fact reflect the structure of the transaction costs. In particular, a specific biased rebalancing scheme is shown to be superior to unbiased schemes if the transaction costs follow a convex model. The problem is discussed also in terms of exponential utility maximization.