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Abstract:
This paper considers extensions of the Libor market model (Brace et al (1997), Jamshidian (1997), Miltersen et al (1997)) to markets with volatility skews in observable option prices. We expand the family of forward rate processes to include diffusions with non-linear forward rate dependence and discuss efficient techniques for calibration to quoted prices of caps and swaptions. Special emphasis is put on generalized CEV processes for which exact closed-form expressions for cap prices are derived. We also discuss modifications of the CEV process which exhibit appealing growth and boundary characteristics. The proposed models are investigated numerically through Crank-Nicholson finite difference schemes and Monte Carlo simulations.

Abstract:
The standard approach (e.g. Dupire (1994) and Rubinstein (1994)) to fitting stock processes to observed option prices models the underlying stock price as a one-factor diffusion process with state- and time-dependent volatility. While this approach is attractive in the sense that market completeness is maintained, the resulting model is often highly non-stationary, difficult to fit to steep volatility smiles, and generally is not well supported by empirical evidence. In this paper, we attempt to overcome some of these problems by overlaying the diffusion dynamics with a jump-process, effectively assuming that a large part of the observed volatility smiles can be explained by fear of sudden large market movements ("crash-o-phobia"). The first part of this paper derives a forward PIDE (Partial Integro-Differential Equation) satisfied by European call option prices and demonstrates how the resulting equation can be used to fit the model to the observed volatility smile/skew. In the second part of the paper, we discuss efficient methods of applying the calibrated model to the pricing of contingent claims. In particular, we develop an ADI (Alternating Directions Implicit) finite difference method that is shown to be unconditionally stable and, if combined with FFT (Fast Fourier Transform) methods, computationally efficient. The paper also discusses the usage of Monte Carlo methods, and contains several detailed examples from the S&P500 market. We compare pricing results obtained by the jump-diffusion approach with those of pure diffusion, and find significant differences for a range of popular contracts.

Abstract:
This paper presents a number of new theoretical results for replication of barrier options through a static portfolio of European put and call options. Our results are valid for options with completely general knock-out/knock-in sets, and allow for time- and state-dependent volatility as well as discontinuous asset dynamics. We illustrate the theory with numerical examples and discuss the practical implementation.

Abstract:
This paper investigates the effect of interest rate correlation in the pricing of Bermudan swaptions. Investigating both Gaussian Markov models and Libor Market models, we find that Bermudan swaption prices depend only weakly on the number of factors in the underlying interest rate model. Moreover, we find that prices of standard Bermudan swaptions typically decrease slightly in the number of factors, primarily a consequence of effects on the time evolution of volatility induced by calibration of the model dynamics. Our findings are markedly different from those of Longstaff, Schwarz, and Santa-Clara (1999) who conclude that single-factor interest rate models significantly undervalue Bermudan swaptions. We argue that the conclusions of Longstaff, Schwarz, and Santa-Clara are due to non-standard choices of model dynamics and calibration methodology. Our study highlights the importance of using a reasonable set of calibration instruments when applying and comparing interest rate models.

Abstract:
We combine classical ideas of separable volatility structures in the HJM framework with the latest techniques for calibration of stochastic volatility models and create a new class of efficient multi-factor term structure models with stochastic volatility. These models have the flexibility of as the Libor market models but the speed of the short rate models.

Yield curve models, stochastic volatility, Markov property

Abstract:
This paper suggests a new type of model for risky bonds, default swaps, and first-to-default basket swaps in which the instantaneous default intensity is modeled as an explosive process. Survival probabilities, transition densities, explosion time distributions, and Green?s functions are derived in closed form, and we describe how to calibrate the model to market data. For first-to-default basket swaps we show that increasing the explosion probability increases the price effect of correlation between default intensity processes.

Abstract:
The paper surveys eight different derivations that all lead to the celebrated Black and Scholes (1973) formula. Describing these derivations leads us through many of the techniques applied in continuous-time asset pricing. The paper can therefore also be seen as an introduction to continuous-time finance. From pure arbitrage reasoning we have six different derivations: (i) The classical hedge argument that leads to the fundamental partial differential equation for option prices, (ii) the martingale approach where we derive the Black-Scholes formula as a risk-adjusted expectation, (iii) the change of numeraire technique that enables us to solve for the option price without calculating a single integral, (iv) a stop-loss start-gain strategy argument, (v) the European option price also solves a forward partial differential equation where the variables are strike and maturity date whereas current time and spot price are kept fixed, and (vi) convergence of a binomial model. The two last derivations put the Black-Scholes formula in an equilibrium context. The Black-Scholes formula is shown to be consistent with: (vii) the continuous-time capital asset pricing model, and (viii) a single period representative investor economy, where the representative investor has constant relative risk-aversion and is endowed with lognormally distributed terminal wealth.