Option Valuation with Conditional Heteroskedasticity and Non-Normality

We provide results for the valuation of European style contingent claims for a large class of specifications of the underlying asset returns. Our valuation results obtain in a discrete time, infinite state-space setup using the no-arbitrage principle and an equivalent martingale measure. Our approach allows for general forms of heteroskedasticity in returns, and valuation results for homoskedastic processes can be obtained as a special case. It also allows for conditional non-normal return innovations, which is critically important because heteroskedasticity alone does not suffice to capture the option smirk. We analyze a class of equivalent martingale measures for which the resulting risk-neutral return dynamics are from the same family of distributions as the physical return dynamics. In this case, our framework nests the valuation results obtained by Duan (1995) and Heston and Nandi (2000) by allowing for a time-varying price of risk and non-normal innovations. We provide extensions of these results to more general equivalent martingale measures and to discrete time stochastic volatility models, and we analyze the relation between our results and those obtained for continuous time models. Nous présentons les résultats d’une étude portant sur l’évaluation de créances éventuelles de style européen pour une grande variété de caractéristiques liées au rendement des actifs sous-jacents. Les résultats de notre évaluation proposent en temps discret une formule état-espace infinie, à partir du principe de non-arbitrage et d’une mesure de martingale équivalente. Notre approche permet de tenir compte de formes générales d’hétéroscédasticité dans les rendements et d’obtenir, dans des cas spéciaux, des résultats d’évaluation liés aux processus homoscédastiques. Elle permet aussi de considérer les innovations conditionnellement non normales en matière de rendement, ce qui représente un facteur critique, compte tenu du fait que l’hétéroscédasticité ne permet pas, à elle seule, de saisir pleinement le caractère ironique de l’option. Nous analysons une catégorie de mesures de martingale équivalentes dont la dynamique du rendement risque-neutre obtenu est de la même famille de distribution que la dynamique du rendement physique. Dans ce cas, notre cadre d’étude soutient les résultats d’évaluation obtenus par Duan (1995) et par Heston et Nandi (2000) et tient compte du coût du risque variant dans le temps et des innovations non normales. Nous étendons ces résultats aux mesures de martingale équivalentes plus générales et aux modèles de volatilité stochastique en temps discret et analysons aussi la relation entre nos résultats et ceux obtenus dans le cas des modèles en temps continu.GARCH, risk-neutral valuation, no-arbitrage, non-normal innovations, GARCH (hétéroscédasticité conditionnelle autorégressive généralisée), évaluation du risque neutre, absence d’arbitrage, innovations non normales

Structural The Equity Premium and the Volatility Spread: The Role of Risk-Neutral Skewness

We introduce the Homoscedastic Gamma [HG] model where the distribution of returns is characterized by its mean, variance and an independent skewness parameter under both measures. The model predicts that the spread between historical and risk-neutral volatilities is a function of the risk premium and of skewness. In fact, the equity premium is twice the ratio of the volatility spread to skewness. We measure skewness from option prices and test these predictions. We find that conditioning on skewness increases the predictive power of the volatility spread and that coefficient estimates accord with theory. In short, the data do not reject the model's implications for the equity premium. We also check the model's implications for option pricing and show that the information content of skewness leads to improved in-sample and out-of-sample pricing performances as well as improved hedging performances. Our results imply that expanding around the Gaussian density is restrictive and does not offer sufficient flexibility to match the skewness and kurtosis implicit in option data. Finally, we document the term structure of option-implied volatility, skewness and kurtosis and find that time-dependence in returns has a greater impact on skewness.Financial markets

Bond risk premia and Gaussian term structure models

Cochrane and Piazzesi (2005) show that (i) lagged forward rates improve the predictability of annual bond returns, adding to current forward rates, and that (ii) a Markovian model for monthly forward rates cannot generate the pattern of predictability in annual returns. These results stand as a challenge to modern Markovian dynamic term structure models (DTSMs). We develop the family of conditional mean DTSMs where the yield dynamics depend on current yields and their history. Empirically, we find that (i) current and past yields generate cyclical risk-premium variations, (ii) the model risk premia offer better returns forecasts, and (iii) the model coefficients are close to Cochrane-Piazzesi regressions of long-horizon returns. Yield decompositions differ significantly from what a standard model suggests - the expectation component decreases less in a recession and increases less in the recovery. A small Markovian factor "hidden" in measurement error (Duffee, 2011) explains some of the differences but is not sufficient to match the evidence.Cochrane et Piazzesi (2005) montrent que les taux à terme retardés améliorent la prévisibilité des rendements obligataires en complétant l’apport des taux à terme courants, et qu’un modèle markovien des taux à terme ne peut produire le profil de prévisibilité des rendements annuels. Ces résultats compliquent la tâche des modèles dynamiques de la structure par terme (DTSM) de type markovien. Nous construisons des DTSM nouveaux, où la dynamique de la moyenne conditionnelle dépend des taux courants et passés. De manière empirique, nous constatons que les taux passes contribuent aux variations cycliques de la prime de risque, que le modèle offre de meilleures prévisions des rendements et que ses coefficients avoisinent les résultats des régressions effectuées sur les rendements de long terme, prolongeant l’étude de Cochrane et Piazzesi. Comparativement au modèle standard, la décomposition des taux diffère sensiblement : la composante des anticipations diminue moins durant une récession et s’accroît moins en période de reprise. La présence dans les taux d’un facteur de risqué markovien « caché » dans les erreurs de mesure ne peut rendre compte entièrement du phénomène

Measuring uncertainty in monetary policy using implied volatility and realized volatility

We measure uncertainty surrounding the central bank's future policy rates using implied volatility computed from interest rate option prices and realized volatility computed from intraday prices of interest rate futures. Both volatility measures show that uncertainty decreased following the most important policy actions taken by the Bank of Canada as a response to the financial crisis of 2007 - 08, such as the conditional commitment of 2009 - 10, the unscheduled cut in the target rate coordinated with other major central banks, and the introduction of term purchase and resale agreements. We also find that, on average, uncertainty decreases following the Bank of Canada's policy rate announcements. Furthermore, our measures of policy rate uncertainty improve the estimation of policy rate expectations from overnight index swap (OIS) rates by predicting the risk premium in the OIS market

Affine and generalized affine models : Theory and applications

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Which parametric model for conditional skewness?

This paper addresses an existing gap in the developing literature on conditional skewness. We develop a simple procedure to evaluate parametric conditional skewness models. This procedure is based on regressing the realized skewness measures on model-implied conditional skewness values. We find that an asymmetric GARCH-type specification on shape parameters with a skewed generalized error distribution provides the best in-sample fit for the data, as well as reasonable predictions of the realized skewness measure. Our empirical findings imply significant asymmetry with respect to positive and negative news in both conditional asymmetry and kurtosis processes

Tractable term-structure models and the zero lower bound

We greatly expand the space of tractable term-structure models. We consider one example that combines positive yields with rich volatility and correlation dynamics. Bond prices are expressed in closed form and estimation is straightforward. We find that the early stages of a recession have distinct effects on yield volatility. Upon entering a recession when yields are far from the lower bound, (i) the volatility term structure becomes flatter, (ii) the level and slope of yields are nearly uncorrelated, and (iii) the second principal component of yields plays a larger role. However, these facts are significantly different when yields are close to the lower bound. Entering a recession in such a setting, (i) the volatility term structure instead steepens, (ii) the level and slope factors are strongly correlated, and (iii) the second principal component of yields plays a smaller role. Existing dynamic term-structure models do not capture the changes in the cyclical responses of the volatility term structure near the lower bound

Risk premium, variance premium and the maturity structure of uncertainty

Theoretical risk factors underlying time-variations of risk premium across asset classes are typically unobservable or hard to measure by construction. Important examples include risk factors in Long Run Risk [LRR] structural models (Bansal and Yaron 2004) as well as stochastic volatility or jump intensities in reduced-form affine representations of stock returns (Duffie, Pan, and Singleton 2000). Still, we show that both classes of models predict that the term structure of risk-neutral variance should reveal these risk factors. Empirically, we use model-free measures and construct the ex-ante variance term structure from option prices. This reveals (spans) two risk factors that predict the bond premium and the equity premium, jointly. Moreover, we find that the same risk factors also predict the variance premium. This important contribution is consistent with theory and confirms that a small number of factors underlies common time-variations in the bond premium, the equity premium and the variance premium. Theory predicts that the term structure of higher-order risks can reveal the same factors. This is confirmed in the data. Strikingly, combining the information from the variance, skewness and kurtosis term structure can be summarized by two risk factors and yields similar level of predictability (i.e., R2s). This bodes well for our ability to bridge the gap between the macro-finance literature, which uses very few state variables, and valuations in option markets

Option valuation with observable volatility and jump dynamics

Under very general conditions, the total quadratic variation of a jump-diffusion process can be decomposed into diffusive volatility and squared jump variation. We use this result to develop a new option valuation model in which the underlying asset price exhibits volatility and jump intensity dynamics. The volatility and jump intensity dynamics in the model are directly driven by model-free empirical measures of diffusive volatility and jump variation. Because the empirical measures are observed in discrete intervals, our option valuation model is cast in discrete time, allowing for straightforward filtering and estimation of the model. Our model belongs to the affine class, enabling us to derive the conditional characteristic function so that option values can be computed rapidly without simulation. When estimated on S&P500 index options and returns, the new model performs well compared with standard benchmarks

Resolving the Spanning Puzzle in Macro-Finance Term Structure Models

Previous macro-finance term structure models (MTSMs) imply that macroeconomic state variables are spanned by (i.e., perfectly correlated with) model-implied bond yields. However, this theoretical implication appears inconsistent with regressions showing that much macroeconomic variation is unspanned and that the unspanned variation helps forecast excess bond returns and future macroeconomic fluctuations. We resolve this contradictionor spanning puzzleby reconciling spanned MTSMs with the regression evidence, thus salvaging the previous macro-finance literature. Furthermore, we statistically reject unspanned MTSMs, which are an alternative resolution of the spanning puzzle, and show that their knife-edge restrictions are economically unimportant for determining term premia