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Abstract:
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.

Spread options, exercise boundary, closed-form approximation

Abstract:
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 proximations 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. The availability of a closed-form formula for spread options also helps us understand and design real and financial contracts with embedded spread-option-like features.

Spread option, Closed-form approximation, Greeks

Abstract:
The Black-Scholes formula is often used in the backward direction to invert the implied volatility, usually with some solver method. Solver methods, being aesthetically unappealing, are also slower than closed-form approximations. However, closed-form approximations in previous works lack accuracy, often providing option pricing errors well exceeding the bid-ask spreads. We develop a new closed-form method based on the rational approximation. By exploiting the homogeneity in the Black-Scholes formula, we are able to show explicitly our domain of approximation and investigate thoroughly the accuracy of our method. The rational approximation is much faster than typical solver methods and very accurate for both at-the-money and away-from-the-money options. Its accuracy can be further improved by one or two steps of Newton-Raphson iterations.

Implied volatility, Black-Scholes formula, rational approximation

Abstract:
The last three decades have witnessed a whole array of option pricing models. We compare the predictive performances of a selection of models by carrying out a horse race on S&P 500 index options along the lines of Jackwerth and Rubinstein (2001). The models we consider include: Black-Scholes, trader rules, Heston's stochastic volatility model, Merton's jump diffusion models with and without stochastic volatility, and more recent Levy type models. Trader rules still dominate mathematically more sophisticated models, and the performance of the trader rules is further improved by incorporating the stable index skew pattern documented in Li and Pearson (2005). Furthermore, after incorporating the stable index skew pattern, the Black-Scholes model beats all mathematically more sophisticated models in almost all cases. Mathematically more sophisticated models vary in their overall performance and their relative accuracy in forecasting future volatility levels and future volatility skew shapes.

option pricing models, performance comparison, implied skew

Abstract:
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 options, boundary approximation, Kirk approximation

Abstract:
There are a number of circumstances in finance where it is useful to estimate diffusion processes conditional on some event. In this paper, we develop the theoretical and numerical tools necessary to perform conditional estimation of diffusion processes within a generalized method of moments framework. We illustrate our method by estimating a univariate diffusion process for a standard time-series of interest rate data conditioned to remain between lower and upper boundaries. A test statistic fails to reject by a wide margin the linearity of the conditionally estimated drift coefficient.

Abstract:
A new successive over-relaxation method to compute the Black-Scholes implied volatility is introduced. Properties of the new method are fully analyzed, including the well-definedness, and local and global convergence patterns. Quadratic order of convergence is achieved by either a transformation of sequence technique or dynamic relaxation. The method is further enhanced by introducing a rational approximation on initial values. Numerical implementation shows that uniformly in a very large approximation 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.

Successive over-relaxation, Black-Scholes formula, Implied volatility, Convergence acceleration, Rational approximation

Abstract:
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 bivariate normal distribution is not fully satisfied in almost all applications. We study the impact of nonnormality on exchange options 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.

multivariate Gram-Charlier approximation, nonnormality, exchange option, Margrabe formula

Abstract:
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 bivariate normal distribution is not fully satisfied in almost all applications. We study the impact of nonnormality on exchange options 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.

multivariate Gram-Charlier approximation, nonnormality, exchange option, Margrabe formula

Abstract:
Many efficient 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

Abstract:
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. An alternative method based on A\"{i}t-Sahalia's specification test on parametric models is proposed.

Damped diffusion, asset price bubbles, martingale pricing, maximum likelihood estimation, parametric specification test

Abstract:
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 efficiency 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 efficient and fairly accurate.

American option, Interpolation method, Quasi-analytical approximation, Critical boundary, Heston's Stochastic volatility model

Abstract:
The Black-Scholes formula is often used in the backward direction to invert the implied volatility, usually with some solver method. Solver methods, being aesthetically unappealing, are also slower than closed-form approximations. However, closed-form approximations in previous works lack accuracy, often providing option pricing errors well exceeding the bid-ask spreads. We develop a new closed-form method based on the rational approximation. The rational approximation is much faster than typical solver methods and very accurate for both at-the-money and away-from-the-money options. Its accuracy can be further improved by one or two steps of Newton-Raphson iterations.

Implied volatility, Black-Scholes formula, Rational functions

Abstract:
It is well known that the actual prices of options deviate from values computed using the Black-Scholes formula or the binomial model with the same volatility for different strikes. For the S&P 500 index options, we find that these deviations from the Black-Scholes formula follow a simple pattern. Loosely, the slope and curvature of the differences between option prices and Black-Scholes values are described by a simple function of at-the-money-forward total volatility. Similarly, the slope and curvature of the volatility skew are described by a simple function of at-the-money-forward total volatility. This implies that the term structure of at-the-money-forward volatilities is sufficient to determine the entire volatility surface. Finally, we find that the implied risk-neutral probability density is bimodal. This finding has interesting implications for models of stochastic volatility.

Implied volatility, volatility skew, index options