A discrete-time two-factor model for pricing bonds and interest rate derivatives under random volatility

This paper develops a discrete-time two-factor model of interest rates with analytical solutions for bonds and many interest rate derivatives when the volatility of the short rate follows a GARCH process that can be correlated with the level of the short rate itself. Besides bond and bond futures, the model yields analytical solutions for prices of European options on discount bonds (and futures) as well as other interest rate derivatives such as caps, floors, average rate options, yield curve options, etc. The advantage of our discrete-time model over continuous-time stochastic volatility models is that volatility is an observable function of the history of the spot rate and is easily (and exactly) filtered from the discrete observations of a chosen short rate/bond prices. Another advantage of our discrete-time model is that for derivatives like average rate options, the average rate can be exactly computed because, in practice, the payoff at maturity is based on the average of rates that can be observed only at discrete time intervals. ; Calibrating our two-factor model to the treasury yield curve (eight different maturities) for a few randomly chosen intervals in the period 1990–96, we find that the two-factor version does not improve (statistically and economically) upon the nested one-factor model (which is a discrete-time version of the Vasicek 1977 model) in terms of pricing the cross section of spot bonds. This occurs although the one-factor model is rejected in favor of the two-factor model in explaining the time-series properties of the short rate. However, the implied volatilities from the Black model (a one-factor model) for options on discount bonds exhibit a smirk if option prices are generated by our model using the parameter estimates obtained as above. Thus, our results indicate that the effects of random volatility of the short rate are manifested mostly in bond option prices rather than in bond prices.Bonds ; Options (Finance) ; Interest rates ; Derivative securities

A closed-form GARCH option pricing model

This paper develops a closed-form option pricing formula for a spot asset whose variance follows a GARCH process. The model allows for correlation between returns of the spot asset and variance and also admits multiple lags in the dynamics of the GARCH process. The single-factor (one-lag) version of this model contains Heston's (1993) stochastic volatility model as a diffusion limit and therefore unifies the discrete-time GARCH and continuous-time stochastic volatility literature of option pricing. The new model provides the first readily computed option formula for a random volatility model in which current volatility is easily estimated from historical asset prices observed at discrete intervals. Empirical analysis on S&P 500 index options shows the single-factor version of the GARCH model to be a substantial improvement over the Black-Scholes (1973) model. The GARCH model continues to substantially outperform the Black-Scholes model even when the Black-Scholes model is updated every period and uses implied volatilities from option prices, while the parameters of the GARCH model are held constant and volatility is filtered from the history of asset prices. The improvement is due largely to the ability of the GARCH model to describe the correlation of volatility with spot returns. This allows the GARCH model to capture strike-price biases in the Black-Scholes model that give rise to the skew in implied volatilities in the index options market.Econometric models ; Financial markets ; Options (Finance) ; Prices

Preference-free option pricing with path-dependent volatility: A closed-form approach

This paper shows how one can obtain a continuous-time preference-free option pricing model with a path-dependent volatility as the limit of a discrete-time GARCH model. In particular, the continuous-time model is the limit of a discrete-time GARCH model of Heston and Nandi (1997) that allows asymmetry between returns and volatility. For the continuous-time model, one can directly compute closed-form solutions for option prices using the formula of Heston (1993). Toward that purpose, we present the necessary mappings, based on Foster and Nelson (1994), such that one can approximate (arbitrarily closely) the parameters of the continuous-time model on the basis of the parameters of the discrete-time GARCH model. The discrete-time GARCH parameters can be estimated easily just by observing the history of asset prices. ; Unlike most option pricing models that are based on the absence of arbitrage alone, a parameter related to the expected return/risk premium of the asset does appear in the continuous-time option formula. However, given other parameters, option prices are not at all sensitive to the risk premium parameter, which is often imprecisely estimated.Options (Finance)

Derivatives on volatility: some simple solutions based on observables

Proposals to introduce derivatives whose payouts are explicitly linked to the volatility of an underlying asset have been around for some time. In response to these proposals, a few papers have tried to develop valuation formulae for volatility derivatives—derivatives that essentially help investors hedge the unpredictable volatility risk. This paper contributes to this nascent literature by developing closed-form/analytical formulae for prices of options and futures on volatility as well as volatility swaps. The primary contribution of this paper is that, unlike all other models, our model is empirically viable and can be easily implemented. ; More specifically, our model distinguishes itself from other proposed solutions/models in the following respects: (1) Although volatility is stochastic, it is an exact function of the observed path of asset prices. This is crucial in practice because nonobservability of volatility makes it very difficult (in fact, impossible) to arrive at prices and hedge ratios of volatility derivatives in an internally consistent fashion, as it is akin to not knowing the stock price when trying to price an equity derivative. (2) The model does not require an unobserved volatility risk premium, nor is it predicated on the strong assumption of the existence of a continuum of options of all strikes and maturities as in some papers. (3) We show how it is possible to replicate (delta hedge) volatility derivatives by trading only in the underlying asset (on whose volatility the derivative exists) and a risk-free asset. This bypasses the problem of having to trade numerously many options on the underlying asset, a hedging strategy proposed in some other models.Derivative securities ; Hedging (Finance) ; Options (Finance)

Hidden Extra U(1) at the Electroweak/TeV Scale

We propose a simple extension of the Standard Model (SM) by adding an extra U(1) symmetry which is hidden from the SM sector. Such a hidden U(1) has not been considered before, and its existence at the TeV scale can be explored at the LHC. This hidden U(1) does not couple directly to the SM particles, and couples only to new SU(2)_L singlet exotic quarks and singlet Higgs bosons, and is broken at the TeV scale. The dominant signals at the high energy hadron colliders are multi lepton and multi b-jet final states with or without missing energy. We calculate the signal rates as well as the corresponding Standard Model background for these final states. A very distinctive signal is 6 high p_T b-jets in the final state with no missing energy. For a wide range of the exotic quarks masses the signals are observable above the background at the LHC.Comment: 19 pages, 5 figure

A Model for Neutrino and Charged Lepton Masses in Extra Dimensions

We propose a model with one large submm size extra dimension in which the gravity and right-handed (RH) neutrino propagate, but the three Standard Model (SM) families are confined to fat branes of TeV^(-1) size or smaller. The charged leptons and the light neutrinos receive mass from the five dimensional Yukawa couplings with the SM singlet neutrino via electroweak Higgs, while the KK excitations of the SM singlet neutrino gets large TeV scale masses from the five dimensional Yukawa coupling with an electroweak singlet Higgs. The model gives non-hierarchical light neutrino masses, accommodate hierarchical charged lepton masses, and naturally explain why the light neutrino masses are so much smaller compared to the charged lepton masses. Large neutrino mixing is naturally expected in this scenario. The light neutrinos are Dirac particles in this model, hence neutrinoless double beta decay is not allowed. The model has also several interesting collider implications and can be tested at the LHC.Comment: 11 pages, no figure

Comment on "Layering transition in confined molecular thin films: Nucleation and growth"

When fluid is confined between two molecularly smooth surfaces to a few molecular diameters, it shows a large enhancement of its viscosity. From experiments it seems clear that the fluid is squeezed out layer by layer. A simple solution of the Stokes equation for quasi-two-dimensional confined flow, with the assmption of layer-by-layer flow is found. The results presented here correct those in Phys. Rev. B, 50, 5590 (1994), and show that both the kinematic viscosity of the confined fluid and the coefficient of surface drag can be obtained from the time dependence of the area squeezed out. Fitting our solution to the available experimental data gives the value of viscosity which is ~7 orders of magnitude higher than that in the bulk.Comment: 4 pages, 2 figure

A closed-form GARCH option pricing model

This paper develops a closed-form option pricing formula for a spot asset whose variance follows a GARCH process. The model allows for correlation between returns of the spot asset and variance and also admits multiple lags in the dynamics of the GARCH process. The single-factor (one-lag) version of this model contains Heston's (1993) stochastic volatility model as a diffusion limit and therefore unifies the discrete-time GARCH and continuous-time stochastic volatility literature of option pricing. The new model provides the first readily computed option formula for a random volatility model in which current volatility is easily estimated from historical asset prices observed at discrete intervals. Empirical analysis on S&P 500 index options shows the single-factor version of the GARCH model to be a substantial improvement over the Black-Scholes (1973) model. The GARCH model continues to substantially outperform the Black-Scholes model even when the Black-Scholes model is updated every period and uses implied volatilities from option prices, while the parameters of the GARCH model are held constant and volatility is filtered from the history of asset prices. The improvement is due largely to the ability of the GARCH model to describe the correlation of volatility with spot returns. This allows the GARCH model to capture strike-price biases in the Black-Scholes model that give rise to the skew in implied volatilities in the index options market