STOCHASTIC VOLATILITY

Pricing Interest Rate Derivatives Under Stochastic Volatility: Diffussion Model

Pricing Interest Rate Derivatives Under Stochastic Volatility: Diffussion Model

Introduction and Background

The interest rate dynamics in the short term receives considerable attention in the financial engineering literature as the main factor in modeling the temporal structure of interest rates. However, there is no unanimous consensus on the best way of modeling the interest rate short-term and, in particular, on treatment with- Decree of volatility, a key, at least to form a theoretical point of view, in the determination of risk premiums (Ball and Torous, 1999, 2339-2359). In order to provide evidence for that, in this paper we analyze the interest rate derivatives under stochastic volatility models in continuous time with a single factor, besides presenting the advantages of its simplicity, its uses to estimate and to provide, under certain conditions, closed solutions for the valuation of zero coupon bonds and derivatives, by virtue of being the most frugal in the number of factors, will allow us to highlight the influence of different treatment that contribute to the volatility.

In the above context, the different models developed in the literature can be classified in response to specific treatment given to interest rate volatility in the short term into three groups: (1) nested models, which specify the interest rate volatility in the short term according to their level and, (2) GARCH models, where volatility is a function of its own impact of interest rate innovations in the short term, and (3) the mixed model combining the effect level and GARCH effects, models initially introduced by Kearns (1992). The international empirical evidence has shown that the models tend to overestimate the volatility effect itself; mainly for not picking up the serial correlation in the conditional variance.

Since the work of Black-Scholes-Merton in 1973, finance has changed radically with the emergence of a new discipline of financial engineering. On the one hand, markets (exchange rates, interest rates, commodity prices, etc. have become more volatile, creating an increasing demand for derivatives (options, futures, swaps, derivatives hybrid and exotic credit derivatives) to control, build, speculate and manage risk (Singleton and Umantsev, 2002, 427-446).

On the other hand, technological advances have enabled financial institutions and other stakeholders to create, and put a price on the market for products and services not only to protect against these risks but also to generate revenue from these risks. The design, analysis and development of these complex financial products and services, or financial engineering, require, in addition to a thorough knowledge of financial theories advanced mastery of mathematics, statistics and sophisticated numerical calculations (Chacko and Das, 2002, 195-241).

It is widely accepted in the literature that interest rate volatility is stochastic. The volatility of interest rates implicit in the price of options on swaps has risen sharply between 2001 and early 2004. This thrust was much stronger in the rate of the euro, especially in the short segment and options on swaps horizon of 6 months or ...