Lognormal Process Research Articles

The Pricing of Options on Credit-Sensitive Bonds

We build a three-factor term-structure of interest rates model and use it to price corporate bonds. The first two factors allow the risk-free term structure to shift and tilt. The third factor generates a stochastic credit-risk premium. To implement the model, we apply the Peterson and Stapleton (2002) diffusion approximation methodology. The method approx imates a correlated and lagged-dependent lognormal diffusion processes. We then price options on credit-sensitive bonds. The recombining log-binomial tree methodology allows the rapid computation of bond and option prices for binomial trees with up to forty periods.

The volatility term structure in a lognormal process for the short rate

One-factor Markov models are widely used by practitioners for pricing financial options. Their simplicity facilitates their calibration to the intial conditions and permits fast computer Implementations. Nevertheless, the danger remains that such models behave unrealistically, if the calibration of the volatility is not properly done. Here, we study a lognormal process and investigate how to specify the volatility constraints in such a way that the term structure of volatility at future times, as implied by the short rate process, has a realistic and stable shape. However, the drifting down of the volatility term structure is unavoidable. As a result, there is a tendency to underestimate option prices.

Slip‐size‐dependent renewal processes and Bayesian inferences for uncertainties

This paper is initially concerned with uncertainty of hazard estimates using renewal process models where the parameters are poorly constrained because of the scarcity of large earthquakes occurring along the same fault segment. A Bayesian inference is adopted, taking account of the whole likelihood function of the parameters. Also, new statistical models are considered which make use of knowledge on the slip associated with earthquake ruptures, may help to improve the hazard estimate in a positive way. Including these models, the predictive efficiency is compared by using the Akaike's Bayesian information criterion (ABIC). Three data sets are analyzed for the illustrations. The Brownian Passage Time (BPT) model is selected to fit the first data set, consisting of 10 historical great earthquakes from Nanaki trough in Japan. However, its predictive hazard function shows a large uncertainty (>100 years) of likely occurrence time around 2070. The second data set consists of the last three events of the first data set but associated with record of slip sizes, for which the ABIC selected the extended lognormal renewal process model where the time intervals between successive events are normalized by the corresponding slip sizes of the starting events of the intervals. The estimated predictive hazard function implies that the next event is likely to occur around 2040 ± 10. The last data set, consisting of 4 events with estimated occurrence times and slip sizes from a submarine fault. The ABIC selected the slip‐size‐dependent BPT model for this data. This model indicates that the likelihood of occurrence time of the next event is decreasing from now, and the period of its half decay is more than several hundred years. For the data sets of the last two examples, it was also shown that the slip size records are useful for better prediction of the next event.

Common Correlation Structures for Calibrating the LIBOR Model

In 1997 three papers that introduced very similar lognormal diffusion processes for interest rates appeared virtually simultaneously. These models, now commonly called the 'LIBOR models' are based on either lognormal diffusions of forward rates as in Brace, Gatarek & Musiela (1997) and Miltersen, Sandermann & Sondermann (1997) or lognormal diffusions of swap rates, as in Jamshidian (1997). The consequent research interest in the calibration of the LIBOR models has engendered a growing empirical literature, including many papers by Brigo and Mercurio, and Riccardo Rebonato. The art of model calibration requires a reasonable knowledge of option pricing and a thorough background in statistics - techniques that are quite different to those required to design no-arbitrage pricing models. Researchers will find the book by Brigo and Mercurio (2001) and the forthcoming book by Rebonato (2002) invaluable aids to their understanding. I aim to provide an accessible account of some interesting problems in LIBOR model calibration, but the ideas are complex, so notational complexities are reduced to a minimum. We take fresh look at three important modelling decisions when calibrating the LIBOR model: the parameterization of the correlation matrix for semi-annual forward rates; the use of principal component analysis in the orthogonal transform of the log normal forward rate model; and the iterations for recovering caplet volatilities from 'flat' cap volatilities. The first section provides the briefest of introductions to the lognormal forward rate version of the LIBOR model. Section two considers the calibration of the model to the cap market, where it is shown that the iteration of caplet volatilities from the 'flat' cap volatilities quoted in the market, should be performed by equating a vega weighted sum of caplet volatilities to the cap volatility. Section 3 discusses the more difficult problem of calibration to the swaption market. Two full rank parsimonious parameterizations of the semi-annual correlation matrix are specified, where correlations between annual forward rates are determined by the same parameters. Section 4 reconsiders calibration to the swaption market where the rank of forward rate correlation matrices is reduced by setting all but the three largest eigenvalues to zero. Rebonato (1999a), Rebonato and Joshi (2001) Hull and White (1999, 2000) and Longstaff, Santa-Clara and Schwartz (1999) have all found that this technique is useful for performing the simulations that are necessary for pricing path dependent options. The implication of zeroing eigenvalues is a transformation of the lognormal forward rate model where each forward rate is driven by three orthogonal factors that are derived from a principal component analysis. We show that, in fact, it is the common principal components model of Flury (1988) that should be used. That is, the same eigenvectors should be calibrated to all swaptions of the same tenor, and we advocate the use of market data rather than historical data for the calibration of these common eigenvectors.

Using a lognormal diffusion process to forecast river flows

The logarithm of the flow of a river in Quebec, Canada, is modeled as a Gaussian diffusion process. Using this model, forecasts of the river flow are made for up to 7 days ahead. The forecasts are found to be much more accurate for 1 day ahead than those produced by a deterministic model called PREVIS. The stochastic model also outperforms PREVIS for 2 days ahead forecasts and is comparable to PREVIS for 3 days ahead forecasts. In general, PREVIS is more accurate than the stochastic model proposed from 4 days ahead forecasts. Various possible improvements of the stochastic model are considered, including the incorporation of exogenous variables and the use of linear regression. The best forecasts are actually obtained by taking the average value of the forecasts produced by the stochastic model and by PREVIS. Finally, confidence interval for the forecasted values are also provided.

Economic and Biological Evolution: A Non-Equilibrium Thermodynamic Theory

Economic and biological systems, as open dissipative systems, need to extract low entropy from the environment to compensate for continuous dissipation. This process can be represented by lognormal processes, which in turn can be mapped into a thermodynamic equation. From here, we develop an analytic thermodynamic theory of economics. Since a thermodynamic equation is of first order in temporal dimension, economic and biological systems as thermodynamic systems are intrinsically evolutionary. Most people agree that thermodynamic theory is a sounder foundation to describe living systems than Newtonian mechanics, which is the physical foundation of general equilibrium theory. As it is often the case, an analytical framework that is built on sounder physical foundation delivers more intuitive and simpler results. This theory, for the first time in economic literature, provides an analytic framework that explicitly represents the relation among fixed costs, variable costs, uncertainty of the environment and duration of a project, which is the core concern inmost economic decisions. Since all economic and biological activities represent the extraction and transformation of low entropy from the environment, this analytic thermodynamic theory offers a unified framework to understand the general pattern in economic and biological evolution.

Fixed-Rate Endowment Mortgage and Mortgage Indemnity Valuation

Fixed rate endowment mortgages differ from fixed rate repayment mortgages primarily because, in the event of early termination, the amount owed by the borrower is a function of the evolution of the term structure of interest rates, whereas for a repayment mortgage it is pre-determined. We use a contingent claims framework for valuing the embedded options in certain British fixed-rate endowment mortgages, with a (capped) mortgage indemnity guarantee (MIG). We assume a mean-reverting interest rate and a lognormal house price diffusion process to arrive at a partial differential equation as the basis of the contingent claims and include conventional mortgage terms such as arrangement fees and prepayment penalties. This methodology provides a template for the borrower, lender and insurer to compare mortgage terms, including the fairness of contract rates, fees and any MIG premiums required, when borrowing more than 75% of the house value at origination. Given the conventional terms and the diffusion processes, we assume the parties have common expectations for some diffusion parameters, including expected future rate and price volatilities. Also we assume that borrowers are 'economic actors' regarding default and prepayment options, and that factors such as mobility, divorce, unemployment, and transaction costs are deterministic and can be ignored in this model, along with moral hazard problems between the insurer and the lender. Since the partial differential equation incorporating the general features of these mortgage contracts does not have a closed-form solution, an explicit finite difference method is used for the valuation (and sensitivity) results. We provide graphical representations (and interpretations) of each mortgage component as a function of house prices and interest rate levels. After empirical studies on the input parameters, each mortgage component may be compared with 'market values'. This model may also be useful as a 'mark-to-value' proxy for all parties, as expected parameters change (especially interest rate and house price levels, and expected future volatilities), for purposes of determining 'value added accounting', appropriate reserves, and indeed for setting premiums and business decisions. Finally, we compare endowment and repayment mortgages for different levels of loan-to-value ratios, interest rate and house price volatility.

Fixed-Rate Endowment Mortgage and Mortgage Indemnity Valuation

Fixed rate endowment mortgages differ from fixed rate repayment mortgages primarily because, in the event of early termination, the amount owed by the borrower is a function of the evolution of the term structure of interest rates, whereas for a repayment mortgage it is pre-determined. We use a contingent claims framework for valuing the embedded options in certain British fixed-rate endowment mortgages, with a (capped) mortgage indemnity guarantee (MIG). We assume a mean-reverting interest rate and a lognormal house price diffusion process to arrive at a partial differential equation as the basis of the contingent claims and include conventional mortgage terms such as arrangement fees and prepayment penalties. This methodology provides a template for the borrower, lender and insurer to compare mortgage terms, including the fairness of contract rates, fees and any MIG premiums required, when borrowing more than 75% of the house value at origination. Given the conventional terms and the diffusion processes, we assume the parties have common expectations for some diffusion parameters, including expected future rate and price volatilities. Also we assume that borrowers are 'economic actors' regarding default and prepayment options, and that factors such as mobility, divorce, unemployment, and transaction costs are deterministic and can be ignored in this model, along with moral hazard problems between the insurer and the lender. Since the partial differential equation incorporating the general features of these mortgage contracts does not have a closed-form solution, an explicit finite difference method is used for the valuation (and sensitivity) results. We provide graphical representations (and interpretations) of each mortgage component as a function of house prices and interest rate levels. After empirical studies on the input parameters, each mortgage component may be compared with 'market values'. This model may also be useful as a 'mark-to-value' proxy for all parties, as expected parameters change (especially interest rate and house price levels, and expected future volatilities), for purposes of determining 'value added accounting', appropriate reserves, and indeed for setting premiums and business decisions. Finally, we compare endowment and repayment mortgages for different levels of loan-to-value ratios, interest rate and house price volatility.

Hedging Volatility Risk

Volatility risk has played a major role in several financial debacles (for example,Barings Bank, Long Term Capital Management). This risk could have been managed using options on volatility which were proposed in the past but were never offered for trading mainly due to the lack of a tradable underlying asset.The objective of this paper is to introduce a new volatility instrument, an option on a straddle, which can be used to hedge volatility risk. The design and valuation ofsuch an instrument are the basic ingredients of a successful financial product. Unlike theproposed volatility index option, the underlying of this proposed contract is a traded atthe-money-forward straddle, which should be more appealing to potential participants. In order to value these options, we combine the approaches of compound options and stochastic volatility. We use the lognormal process for the underlying asset, the Orenstein-Uhlenbeck process for volatility, and assume that the two Brownian motions are independent. Our numerical results show that the straddle option price is very sensitive to the changes in volatility which means that the proposed contract is indeed a very powerful instrument to hedge volatility risk.

Inference on some parametric functions in the univeriate lognormal diffusion process with exogenous factors

In this paper we consider, for the univariate lognormal diffusion process with exogenous factors, the inference for some parametric functions that include as particular cases the trend and the covariance function of the process. Concretely, we obtain the UMVU estimators of these functions and the efficiency of them relative to the corresponding ML estimators. Finally, we conclude with an application to a particular case of exogenous factor.

Economic and Biological Evolution: A Real Option Approach

Economic and biological systems, as open dissipative systems, need to extract negative entropy from the environment to compensate for continuous dissipation. This process can be modeled by lognormal processes, which in turn can be represented by the Black-Scholes equation and option theory. Since the Black-Scholes equation and all other thermodynamic equations are of first order in temporal dimension, economic and biological systems as thermodynamic systems are intrinsically evolutionary. With option theory as its analytical foundation, the non-equilibrium thermodynamic theory gives more accurate and simpler descriptions of economic activities than the Newtonian mechanics based general equilibrium theory.

An Analytical Model For Threshold Crossing Rates Of Concentration Fluctuations In Dispersing Plumes

A simple analytical model is developed for the meanupcrossing rate of plume concentration fluctuations assuming that thisprocess can be well approximated by a lognormal process. The resultingexpression requires only the specification of the in-plume fluctuationintensity and in-plume Taylor micro-time scale and, hence, does notexplicitly involve the joint probability density function of theconcentration and its derivative. The analytical model provides agood fit to some field measurements of the mean upcrossing rate ina dispersing plume.

A Libor Market Model with Default Risk

In this paper a new credit risk model for credit derivatives is presented. The model is based upon the Libor market modelling framework for default-free interest rates. We model effective default-free forward rates and effective forward credit spreads as lognormal diffusion processes, and recovery is modelled as a fraction of the par value of the defaulted claim. The newly introduced survival-based pricing measures are a valuable tool in the pricing of defaultable payoffs and allow a straightforward derivation of the no-arbitrage dynamics of forward rates and forward credit spreads. The model can be calibrated to the prices of defaultable coupon bonds, asset swap rates and default swap rates for which closed-form solutions are given. For options on default swaps and caps on credit spreads, approximate solutions of high accuracy exist. This pricing formula for options on default swaps is made exact in a modified modelling framework using an analogy to the swap measure, the default swap measure.

Opções de compra: o ajustamento ao mercado brasileiro de dois modelos de precificação

O presente estudo verificou, para o período de julho de 1994 a junho de 1997, o ajustamento ao mercado brasileiro de dois modelos de precificação de opções de compra: o de Black e Scholes (1973), que reconhece que os retornos dos preços da ação-objeto seguem um processo de difusão log-normal e o proposto por Cox e Ross (1976), que admite um processo de difusão com elasticidade da variância constante. Com a utilização de diferenças percentuais relativas e do teste t, observou-se que ambos os modelos subprecificaram opções fora do preço e no preço, enquanto, para opções dentro do preço, somente as mais próximas do vencimento foram subprecificadas. Verificou-se que o modelo de Black e Scholes (1973) ajustou-se melhor às opções fora do preço e no preço, enquanto o modelo da elasticidade da variância constante ajustou-se melhor às opções dentro do preço. Por fim, os resultados sugerem que, embora o modelo de Black e Scholes (1973) se ajuste ligeiramente melhor ao mercado brasileiro, de uma forma geral, não se pode afirmar que um modelo seja superior ao outro na precificação de opções de compra.

Lognormality of rates and term structure models

A term structure model with lognormal type volatility structure is proposed. The Heath, Jarrow and Morton (HJM) framework, coupled with the theory of stochastic evolution equations in infinite dimensions, is used to show that the resulting instantaneous rates are well defined (they do not explode) and remain positive, contrary to those derived in [2]. They are also bounded from below and above by lognormal processes. The model can be used to price and hedge caps, swaptions and other interest rate and currency derivatives including the Eurodollar futures contract, which requires integrability of one over zero coupon bond. This extends results obtained by Sandmann and Sondermann in [22] and [23] for Markovian lognormal short rates to (non-Markovian) lognormal forward rates. We show also existence of invariant measures for the proposed term structure dynamics

The Nonlinear Gaussian Spectrum of Log-Normal Stochastic Processes and Variables

A procedure is presented in this paper for developing a representation of lognormal stochastic processes via the polynomial chaos expansion. These are processes obtained by applying the exponential operator to a gaussian process. The polynomial chaos expansion results in a representation of a stochastic process in terms of multidimensional polynomials orthogonal with respect to the gaussian measure with the dimension defined through a set of independent normalized gaussian random variables. Such a representation is useful in the context of the spectral stochastic finite element method, as well as for the analytical investigation of the mathematical properties of lognormal processes.

Inference and first-passage-times for the lognormal diffusion process with exogenous factors: application to modelling in economics

Inference and first-passage-times for the lognormal diffusion process with exogenous factors: application to modelling in economics

Revealing a lognormal cascading process in turbulent velocity statistics with wavelet analysis

We use the continuous wavelet transform to extract a cascading process from experimental turbulent velocity signals. We mainly investigate various statistical quantities such as the singularity spe...

A more informative estimation procedure for the parameters of a diffusion process

The estimation procedures for the parameters of a diffusion process with constant coefficients have mainly focused on volatility. Nevertheless, even if the knowledge of the volatility alone suffices to compute the Black and Scholes option prices, other financial application models assume that the price dynamics follows a log-normal process and requires the knowledge of both parameters. On the other hand, while the usual ML estimator of volatility gives satisfactory results, the estimation of drift is much less accurate; moreover, the drift-estimated value highly depends on the phases of the business cycle included in the sample data. This contribution explicitly imposes a risk aversion or risk neutral assumption into the ML estimation procedure and makes a constrained maximization of the sample likelihood function. The aim is twofold: to obtain estimated values which are consistent with a widely accepted assumption and use the risk aversion constraint in order to improve the accuracy of the estimates.

Pricing Lookback and Barrier Options under the CEV Process

This paper examines the pricing of lookback and barrier options when the underlying asset follows the constant elasticity of variance (CEV) process. We construct a trinomial method to approximate the CEV process and use it to price lookback and barrier options. For look-back options, we find that the technique proposed by Babbs for the lognormal case can be modified to value lookbacks when the asset price follows the CEV process. We demonstrate the accuracy of our approach for different parameter values of the CEV process. We find that the prices of barrier and lookback options for the CEV process deviate significantly from those for the lognormal process. For standard options, the corresponding differences between the CEV and Black-Scholes models are relatively small. Our results show that it is much more important to have the correct model specification for options that depend on extrema than for standard options.