Computation of Greeks Using Binomial Tree

This chapter presents the binomial tree approach to the option pricing problem. We first illustrate the basic ideas of option pricing by considering the one-period binomial tree model and then extend to a multi-period binomial tree model. We then show that, by taking limits in an appropriate way, the binomial expression for the option price converges to the Black–Scholes option price and pricing equation. Alternatively, the continuous time model can be discretised in a way that yields the same expressions as obtained by the binomial tree approach.

Many important factors affect the spread of childhood infectious disease. To better understand the fundamental drivers of infectious disease spread, several researchers have estimated seasonal transmission coefficients in discrete-time models. In this paper, we build upon this previous work and also develop a framework for efficient estimation using continuous differential equation models. We introduce nonlinear programming formulations to efficiently estimate model parameters and seasonal transmission profiles from existing case count data for the childhood disease, measles. We compare results from discrete time and continuous time models and address several shortcomings of the discrete-time method, including removing the need for the data reporting interval to match the time between successive cases in the chain of transmission or serial interval of the disease. Using a simultaneous approach for optimization of differential equation systems, we demonstrate that seasonal transmission parameters can be effectively estimated using continuous time models instead of discrete-time models.

The recombining binomial tree approach, which has been initiated by Cox et al. (J Financ Econ 7: 229–263, 1979) and extended to arbitrary diffusion models by Nelson and Ramaswamy (Rev Financ Stud 3(3): 393–430, 1990) and Hull and White (J Financ Quant Anal 25: 87–100, 1990a), is applied to the simultaneous evaluation of price and Greeks for the amortized fixed and variable rate mortgage prepayment option. We consider the simplified binomial tree approximation to arbitrary diffusion processes by Costabile and Massabo (J Deriv 17(3): 65–85, 2010) and analyze its numerical applicability to the mortgage valuation problem for some Vasicek and CIR-like interest rate models. For fixed rates and binomial trees with about thousand steps, we obtain very good results. For the Vasicek model, we also compare the closed-form analytical approximation of the callable fixed rate mortgage price by Xie (IAENG Int J Appl Math 39(1): 9, 2009) with its binomial tree counterpart. With respect to the binomial tree values one observes a systematic underestimation (overestimation) of the callable mortgage price (prepayment option price) analytical approximation. This numerical discrepancy increases at longer maturities and becomes impractical for a valuable estimation of the prepayment option price.

Computation of Greeks is an important task in financial risk management. In the last decade, there have been many studies on the computational methods for a variety of financial options. The Malliavin calculus approach has become one of the main approaches to derive Greeks for these options, for example. In this article, we show a new relationship between Greeks (vega, rho, and theta) for Asian options. This result is obtained using elementary mathematics.

Limitations record on a variable which have continuous variable changes make the 'time' attached to the data is discrete. That underlies development of continuous time model. Compared to discrete time model, the continuous time model has advantages that can make prediction at different lag. In addition, it still can perform continuous time model parameter estimation, although the objects observations do not always have the same time interval (irregular time). Thus, the purpose of this study is to examine the consistency of continuous time model coefficient parameter with structural equation model (SEM) approach in irregular time data. The method used is the Exact Discrete Model (EDM) with SEM approach. EDM is a model that connects the parameter to value which is underlying parameter of the continuous time model with non-linear relationships. The result analysis of the research indicated that the coefficients of continuous time model parameter are still consistent until 20% of irregular time with 5% consistency limits.

In this chapter we review continuous time series modeling and estimation by extended structural equation models (SEM) for single subjects and N > 1. First-order as well as higher-order models will be dealt with. Both will be handled by the general state space approach which reformulates higher-order models as first-order models. In addition to the basic model, the extensions of exogenous variables and traits (random intercepts) will be introduced. The connection between continuous time and discrete time for estimating the model by SEM will be made by the exact discrete model (EDM). It is by the EDM that the exact estimation procedure in this chapter differentiates from many approximate procedures found in the literature. The proposed analysis procedure will be applied to the well-known Wolfer sunspot data, an N = 1 time series that has been analyzed by several continuous time analysts in the past. The analysis will be carried out by ctsem, an R-package for continuous time modeling that interfaces to OpenMx, and the results will be compared to those reported in the previous studies.

American Option Pricing with Discrete and Continuous Time Models: An Empirical Comparison

On Sampling and Reconstruction of Continuous Time Stochastic Systems

The Euro/Dollar exchange rate: Chaotic or non-chaotic? A continuous time model with heterogeneous beliefs

In this paper, we address the problem of the Greeks' evaluation for European and American options, when the model is defined by a general stochastic differential equation. We represent the Greeks as expectations, in order to allow their computations using Monte Carlo simulations. We avoid the use of Malliavin calculus techniques since in general, it leads to random variables whose simulations are costly in terms of computational time. We take advantage of the Markovian structure to derive simple formulas in a great generality. Moreover, they appear to be efficient in practice.

This paper presents and demonstrates a general approach to solving spatial dynamic models in continuous space and continuous time that characterize the behaviour of intertemporally and interspatially optimizing agents and estimating from discrete data the parameters of such models. The approach involves the use of a projection method to solve the models and a quasi-Newton algorithm to update quasi-FIML parameter estimates.

This paper presents some empirical evidence for a time series model of aggregate consumer' expenditure which is formulated in continuous time. It is argued that such models, expressed in terms of stochastic differential equations, are more realistic in representing the dynamic evolution of many economic time series, particularly at the aggregate level. However, such models impose complicated restrictions on the discretely observed data, which must be taken account of in estimation. The appropriate methodology is applied to a version of the consumption function for UK data, and is found to provide a good description of the data. The estimated continuous time model is compared with more popular discrete models, and is found to provide superior dynamic post-sample predictions over 16 quarters. The formulation of econometric models in continuous time may provide an alternative and potentially successful approach to the modelling of time series in many aspects of economics.

Continuous-time (CT) models are a flexible approach for modeling longitudinal data of psychological constructs. When using CT models, a researcher can assume one underlying continuous function for the phenomenon of interest. In principle, these models overcome some limitations of discrete-time (DT) models and allow researchers to compare findings across measures collected using different time intervals, such as daily, weekly, or monthly intervals. Theoretically, the parameters for equivalent models can be rescaled into a common time interval that allows for comparisons across individuals and studies, irrespective of the time interval used for sampling. In this study, we carry out a Monte Carlo simulation to examine the capability of CT autoregressive (CT-AR) models to recover the true dynamics of a process when the sampling interval is different from the time scale of the true generating process. We use two generating time intervals (daily or weekly) with varying strengths of the AR parameter and assess its recovery when sampled at different intervals (daily, weekly, or monthly). Our findings indicate that sampling at a faster time interval than the generating dynamics can mostly recover the generating AR effects. Sampling at a slower time interval requires stronger generating AR effects for satisfactory recovery, otherwise the estimation results show high bias and poor coverage. Based on our findings, we recommend researchers use sampling intervals guided by theory about the variable under study, and whenever possible, sample as frequently as possible. (PsycInfo Database Record (c) 2025 APA, all rights reserved).

New distribution theory for the estimation of structural break point in mean

For most continuous time models formulated in finance, there is no closed form for the likelihood function and estimation of the parameters on the basis of discrete data will be based on an approximation rather than an exact discretization. For example, the Euler method introduces discretization bias because it ignores the internal dynamics that can be excessively erratic. We view the approximation as a difference equation and note that the solution of the continuous time model does not satisfy this difference equation. The effectiveness of the approximation will depend on the rate at which the underlying process is sampled. We investigate how much it matters: can we get significantly different estimates of the same structural parameter when we use say hourly data as compared with using monthly data under given discretization? If yes, then that discretization when applied to a data set in hand, as is done in practice, cannot be said to give robust results. We compare numerically the application of methods by Yu and Phillips (2001), Shoji and Ozaki (1998) and Ait-Sahalia (2002) in the maximum likelihood estimation of the unrestricted interest rate model proposed by Chan et al. (1992). We find that reducing the sampling rate yield large biases in the estimation of the parameters. The AitSahalia method is shown to offer a good approximation and has the advantage of reducing some of the temporal aggregation bias.