Javier F. Navas

On the Robustness of Least - Squares Monte Carlo (LSM) for Pricing American Derivatives

This paper analyses the robustness of Least-Squares Monte Carlo, a technique proposed by Longstaff and Schwartz (2001) for pricing American options. This method is based on least-squares regressions in which the explanatory variables are certain polynomial functions. We analyze the impact of different basis functions on option prices. Numerical results for American put options show that this approach is quite robust to the choice of basis functions. For more complex derivatives, this choice can slightly affect option prices.

Consistent Versus Non-Consistent Term Structure Models: Some Evidence from the Spanish Market

This article prices cap and swaptions in the Spanish market using the Vasicek, Cox, Ingersoll, and Ros and Hull and White (HW) models. Derivatives prices obtained with the Vasicek and CIR models estimated from time series data are very similar, but they differ substantially from the values given by the HW model fited to the term structureof interest rate swap yields (especially for at-the-money and out-of-the-money options). When the former models are estimated cross-sectionally, they produce option proces similar to those of the HW model. In samples of caps and swaptions, the Vasicek model estimated cross-sectionally, they produce option prices similar to those of the HW model. In samples of caps and swaptions, the Vasicek model estimated cross-sectionally outperforms the HW model. The Vasicek and CIR models estimated from time series produce very large pricing errors.

Correct Calculation of Volatility in a Jump-Diffusion Model

The jump-diffusion model as an extension of the Black-Scholes pure logarithmic diffusion process was first introduced by Merton and others in the 1970s. The underlying asset follows a regular diffusion, but occasionally experiences a large discrete jump of random size, whose arrival is governed by a Poisson process. To estimate the volatility parameters for a jump-diffusion process, it is important to take into account the impact of both random jump arrival and also the uncertainty over the size of a jump if it should occur. In this article, Navas points out that the influence of jump size uncertainty on stock volatility was left out by a number of the early, and some not-so-early, investigators. The effect on theoretical option values is not huge, but also not negligible, as the results presented here show.

The Stochastic String Model as a Unifying Theory of the Term Structure of Interest Rates

We present the stochastic string model of Santa-Clara and Sornette (2001), as reformulated by Bueno-Guerrero et al. (2014), as a unifying theory of the continuous-time modeling of the term structure of interest rates. We provide several new results, such as: a) an orthogonality condition for the volatilities in the Heath, Jarrow, and Morton (1992) (HJM) model, b) the interpretation of multi-factor HJM models as approximations to a full infinite-dimensional model, c) a result of consistency based on Hilbert spaces, and d) a theorem for option valuation.

Stochastic String Models with Continuous Semimartingales

This paper reformulates the stochastic string model of Santa-Clara and Sornette (2001) using stochastic calculus with continuous semimartingales. We present some new results, such as: a) the dynamics of the short-term interest rate, b) the PDE that must be satisfied by the bond price, and c) an analytic expression for the price of a European bond call option. Additionally, we clarify some important features of the stochastic string model and show its relevance to price derivatives and the equivalence with a infinite dimensional HJM model to price European options.

Pricing levered warrants with dilution using observable variables

Abstract We propose a valuation framework for pricing European call warrants on the issuer’s own stock that allows for debt in the issuer firm. In contrast to other works that also price warrants with dilution issued by levered firms, ours uses only observable variables. Thus, we extend the models of Crouhy and Galai [J. Bank. Finance, 1994, 18, 861–880] and Ukhov [J. Financ. Res., 2004, 27(3), 329–339]. We provide numerical examples to study some implementation issues and to compare the model with existing ones.

Valuation of Caps and Swaptions under a Stochastic String Model

We develop a Gaussian stochastic string model that provides closed-form expressions for the prices of caps and swaptions that, under certain conditions, reduce to Black (1976) formulas. We also propose a stochastic string LIBOR market model that generalizes the models of Brace et al. (1997) and Longstaff et al. (2001a) and allows us to obtain cap prices using the Black (1976) formula. This model can be approximated by one that is compatible with the previous Gaussian models. Our main results are as follows: we show that one of the assumptions of Longstaff et al. (2001a) is incompatible with our model and, then, we obtain a possible explanation for some problems related to the relative valuation of caps and swaptions. We attain the observational equivalence of Kerkhof and Pelsser (2002) and we prove that, under the stochastic string approach, the models proposed by Brace et al. (1997) and Longstaff et al. (2001a) are more parsimonious than previously stated in the original papers. Finally, we derive a a close-form expression for the price of a portfolio option that is a multi-factor extension of the result obtained by Jamshidian (1989) and an analytical formula for swaption prices.

Land valuation using a real option approach

This paper uses real option theory to asses the value of agricultural land that can be seeded with crops. We consider one- and two-factor models for the evolution of crop prices through time and derive a partial differential equation (PDE) for the land value. We model the potential selling decision of the land owner as a put option and incorporate it as a boundary condition in the PDE for the land price. We solve this equation numerically and show that theoretical prices are close to market land prices and that the value of the put option accounts for, at least, 25% of the total land value. Valoraci ón de un terreno agrícola mediante opciones realesResumenEn este trabajo utilizamos la teoría de opciones reales para valorar un terreno agrícola donde pueden cultivarse diferentes productos. Consideramos modelos unifactoriales y bifactoriales para la evolución del precio de estos productos y obtenemos una ecuación diferencial en derivadas parciales (EDP) para valorar el terreno. Modelizamos la posible decisión de venta del dueño del terreno como una opción put y la incorporamos en la EDP del valor de la tierra. Resolvemos numéricamente dicha ecuación y mostramos que los precios obtenidos son similares a los precios de mercado del terreno y que el valor de la opción de venta representa, al menos, un 25% del valor total del terreno.

Bond Market Completeness Under Stochastic Strings with Distribution-Valued Strategies

We study bond market completeness under infinite-dimensional models and show that, with stochastic string models, the market is complete if we consider strategies as generalized functions. We also obtain completeness for infinite-dimensional HJM models within the stochastic string framework. This result is not at odds with the incompleteness obtained in Barski et al. (2011). For a wide class of options, we obtain a new result, referred to as T-forward hedging, and we show that T-forward and delta hedging are equivalent in the Gauss-Markov case. As an application, we obtain a closed-form expression for the price of some compound options. Finally, we prove that in the stochastic string HJM case the martingale measure is unique, whereas in the general stochastic string case uniqueness is equivalent to a condition on the form of specific market risk premia.

Sensitivity Analysis and Hedging in Stochastic String Models

We analyze certain results on the stochastic string modeling of the term structure of interest rates and we apply them to study the sensitivities and the hedging of options with payoff functions homogeneous of degree one. Under the same framework, we use an exact multi-factor extension of Jamshidian (1989) to find the sensitivities for swaptions and we prove that it cannot be applied to captions. We present a new approximate result for pricing options on coupon bonds based on the Fenton-Wilkinson method and we show that it generalizes the fast coupon bond option pricing proposed in Munk (1999). This result can be easily applied to the approximate valuation of swaptions and captions.