Nengjiu Ju

Bio: Nengjiu Ju is an academic researcher from University of Maryland, College Park. The author has contributed to research in topics: Binomial options pricing model & Exponential function. The author has an hindex of 1, co-authored 1 publications receiving 207 citations.

Pricing by American Option by Approximating its Early Exercise Boundary as a Multipiece Exponential Function

Abstract: This article proposes to price an American option by approximating its early exercise boundary as a multipiece exponential function. Closed form formulas are obtained in terms of the bases and exponents of the multipiece exponential function. It is demonstrated that a three-point extrapolation scheme has the accuracy of an 800-time-step binomial tree, but is about 130 times faster. An intuitive argument is given to indicate why this seemingly crude approximation works so well. Our method is very simple and easy to implement. Comparisons with other leading competing methods are also included. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

Option Pricing Under a Double Exponential Jump Diffusion Model

Abstract: This paper aims to extend the analytical tractability of the Black--Scholes model to alternative models with arbitrary jump size distributions. More precisely, we propose a jump diffusion model for asset prices whose jump sizes have a mixed-exponential distribution, which is a weighted average of exponential distributions but with possibly negative weights. The new model extends existing models, such as hyperexponential and double-exponential jump diffusion models, because the mixed-exponential distribution can approximate any distribution as closely as possible, including the normal distribution and various heavy-tailed distributions. The mixed-exponential jump diffusion model can lead to analytical solutions for Laplace transforms of prices and sensitivity parameters for path-dependent options such as lookback and barrier options. The Laplace transforms can be inverted via the Euler inversion algorithm. Numerical experiments indicate that the formulae are easy to implement and accurate. The analytical solutions are made possible mainly because we solve a high-order integro-differential equation explicitly. A calibration example for SPY options shows that the model can provide a reasonable fit even for options with very short maturity, such as one day. This paper was accepted by Michael Fu, stochastic models and simulation.

Randomization and the American Put

Abstract: While American calls on non-dividend paying stocks may be valued as European, there is no completely explicit exact solution for the values of American puts. We introduce a novel technique called randomization to value American puts and calls on dividend-paying stocks. This technique yields a new semi-explicit approximation for American option values in the Black Scholes model. Numerical results indicate that the approximation is both accurate and computationally efficient.

Financial valuation of guaranteed minimum withdrawal benefits

Abstract: Financial valuation OF GMWBs: We develop a variety of methods for assessing the cost and value of a very popular ‘rider’ available to North American investors on variable annuity (VA) policies called a Guaranteed Minimum Withdrawal Benefit (GMWB). The GMWB promises to return the entire initial investment, albeit spread over an extended period of time, regardless of subsequent market performance. First, we take a static approach that assumes individuals behave passively and holds the product to maturity. We show how the product can be decomposed into a Quanto Asian Put plus a generic term-certain annuity. At the other extreme of consumer behavior, the dynamic approach leads to an optimal stopping problem akin to pricing an American put option, albeit complicated by the non-traditional payment structure. Our main result is that the No Arbitrage hedging cost of a GMWB ranges from 73 to 160 basis points of assets. In contrast, most products in the market only charge 30–45 basis points. Although we suggest a number of behavioral reasons for the apparent under-pricing of this feature in a typically overpriced VA market, we conclude by arguing that current pricing is not sustainable and that GMWB fees will eventually have to increase or product design will have to change in order to avoid blatant arbitrage opportunities.

An exact and explicit solution for the valuation of American put options

Abstract: In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, i...

Continuous-Time Methods in Finance: A Review and an Assessment

Abstract: I survey and assess the development of continuous-time methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuous-time models. Capital market frictions and bargaining issues are being increasingly incorporated in continuous-time theory. THE ROOTS OF MODERN CONTINUOUS-TIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuous-time modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting. Merton ~1973b! also showed how such a framework can be used to develop equilibrium asset pricing implications, thereby significantly extending the asset pricing theory to richer dynamic settings and expanding the scope of applications of continuous-time methods to study problems in financial economics. 1 Within a span of about 30 years from the publication of Merton’s inf luential papers, continuous-time methods have become an integral part of financial economics. Indeed, in certain core areas in finance ~such as, e.g., asset pricing, derivatives valuation, term structure theory, and portfolio selection! continuoustime methods have proved to be the most attractive way to conduct research and gain economic intuition. The continuous-time approach in these areas has produced models with a rich variety of testable implications. The econometric theory for testing continuous-time models has made rapid strides in the last decade and has thus kept pace with the impressive progress on the theoretical front. One hopes that the actual empirical investigations and estimation using the new procedures will follow suit soon.