A Damped Diffusion Framework for Financial Modeling and Closed-form Maximum Likelihood Estimation

Asset price bubbles can arise unintentionally when one uses continuous-time diffusion processes to model financial quantities. We propose a flexible damped diffusion framework that is able to break many types of bubbles and preserve the martingale pricing approach. Damping can be done on either the diffusion or drift function. Oftentimes, certain solutions to the valuation PDE can be ruled out by requiring the solution to be a limit of martingale prices for damped diffusion models. Monte Carlo study shows that with finite time-series length, maximum likelihood estimation often fails to detect the damped diffusion function while fabricates nonlinear drift function.Damped diffusion, asset price bubbles, martingale pricing, maximum likelihood estimation

Analytical Approximations for the Critical Stock Prices of American Options: A Performance Comparison

Many e±cient and accurate analytical methods for pricing American options now exist. However, while they can produce accurate option prices, they often do not give accurate critical stock prices. In this paper, we propose two new analytical approximations for American options based on the quadratic approximation. We compare our methods with existing analytical methods including the quadratic approximations in Barone-Adesi and Whaley (1987) and Barone-Adesi and Elliott (1991), the lower bound approximation in Broadie and Detemple (1996), the tangent approximation in Bunch and Johnson (2000), the Laplace inversion method in Zhu (2006b), and the interpolation method in Li (2008). Both of our methods give much more accurate critical stock prices than all the existing methods above.American option; Analytical approximation; Critical stock price

Asset Pricing - A Brief Review

I first introduce the early-stage and modern classical asset pricing and portfolio theories. These include: the capital asset pricing model (CAPM), the arbitrage pricing theory (APT), the consumption capital asset pricing model (CCAPM), the intertemporal capital asset pricing model (ICAPM), and some other important modern concepts and techniques. Finally, I discuss the most recent development during the last decade and the outlook in the field of asset pricing.Asset Pricing Models

The Impact of Return Nonnormality on Exchange Options

The Margrabe formula is used extensively by theorists and practitioners not only on exchange options, but also on executive compensation schemes, real options, weather and commodity derivatives, etc. However, the crucial assumption of a bivariate normal distribution is not fully satisfied in almost all applications. The impact of nonnormality on exchange options is studied by using a bivariate Gram-Charlier approximation. For near-the-money exchange options, skewness and coskewness induce price corrections which are linear in moneyness, while kurtosis and cokurtosis induce quadratic price corrections. The nonnormality helps to explain the implied correlation smile observed in practice.

A Quasi-analytical Interpolation Method for Pricing American Options under General Multi-dimensional Diffusion Processes

We present a quasi-analytical method for pricing multi-dimensional American options based on interpolating two arbitrage bounds, along the lines of Johnson (1983). Our method allows for the close examination of the interpolation parameter on a rigorous theoretical footing instead of empirical regression. The method can be adapted to general diffusion processes as long as quick and accurate pricing methods exist for the corresponding European and perpetual American options. The American option price is shown to be approximately equal to an interpolation of two European option prices with the interpolation weight proportional to a perpetual American option. In the Black-Scholes model, our method achieves the same e±ciency as Barone-Adesi and Whaley's (1987) quadratic approximation with our method being generally more accurate for out-of-the-money and long-maturity options. When applied to Heston's stochastic volatility model, our method is shown to be extremely e±cient and fairly accurate.American option; Interpolation method; Quasi-analytical approximation; Critical bound- ary; Heston's Stochastic volatility model

Closed-Form Approximations for Spread Option Prices and Greeks

We develop a new closed-form approximation method for pricing spread options. Numerical analysis shows that our method is more accurate than existing analytical approximations. Our method is also extremely fast, with computing time more than two orders of magnitude shorter than one-dimensional numerical integration. We also develop closed-form approximations for the greeks of spread options. In addition, we analyze the price sensitivities of spread options and provide lower and upper bounds for digital spread options. Our method enables the accurate pricing of a bulk volume of spread options with different specifications in real time, which offers traders a potential edge in financial markets. The closed-form approximations of greeks serve as valuable tools in financial applications such as dynamic hedging and value-at-risk calculations.

Reduce computation in profile empirical likelihood method

Since its introduction by Owen in [29, 30], the empirical likelihood method has been extensively investigated and widely used to construct confidence regions and to test hypotheses in the literature. For a large class of statistics that can be obtained via solving estimating equations, the empirical likelihood function can be formulated from these estimating equations as proposed by [35]. If only a small part of parameters is of interest, a profile empirical likelihood method has to be employed to construct confidence regions, which could be computationally costly. In this paper we propose a jackknife empirical likelihood method to overcome this computational burden. This proposed method is easy to implement and works well in practice.profile empirical likelihood; estimating equation; Jackknife

Multi-asset Spread Option Pricing and Hedging

We provide two new closed-form approximation methods for pricing spread options on a basket of risky assets: the extended Kirk approximation and the second-order boundary approximation. Numerical analysis shows that while the latter method is more accurate than the former, both methods are extremely fast and accurate. Approximations for important Greeks are also derived in closed form. Our approximation methods enable the accurate pricing of a bulk volume of spread options on a large number of assets in real time, which offers traders a potential edge in a dynamic market environment.multi-asset spread option, closed-form approximation

An Adaptive Succesive Over-relaxation Method for Computing the Black-Scholes Implied Volatility

A new successive over-relaxation method to compute the Black-Scholes implied volatility is introduced. Properties of the new method are fully analyzed, including global well-definedness, local convergence, as well as global convergence. Quadratic order of convergence is achieved by either a dynamic relaxation or transformation of sequence technique. The method is further enhanced by introducing a rational approximation on initial values. Numerical implementation shows that uniformly in a very large domain, the new method converges to the true implied volatility with very few iterations. Overall, the new method achieves a very good combination of efficiency, accuracy and robustness.

Price Deviations of S&P 500 Index Options from the Black-Scholes Formula Follow a Simple Pattern

It is known that actual option prices deviate from the Black-Scholes formula using the same volatility for different strikes. For the S&P 500 index options, we find that these deviations follow a stable pattern and are described by a simple function of at-the-money-forward total volatility. This im plies that the term structure of at-the-money-forward volatilities is su±cient to determine the entire volatility surface. We also find that the implied risk-neutral density is bimodal. The patterns we find are useful in predicting future implied volatilities.Black Scholes formula; Implied volatility skew; Stable pattern; Risk-neutral density