•This paper develops a general methodology for modeling and pricing financial derivatives which depend on systems of stochastic diffusion processes.•Weak convergence of the approximation is ...demonstrated, with second order convergence in space.•Numerical experiments demonstrate the accuracy and efficiency of the method for various European and early-exercise options in two and three dimensions
Continuous time Markov Chain (CTMC) approximation techniques have received increasing attention in the option pricing literature, due to their ability to solve complex pricing problems, although existing approaches are mostly limited to one or two dimensions. This paper develops a general methodology for modeling and pricing financial derivatives which depend on systems of stochastic diffusion processes. This is accomplished with a general decorrelation procedure, which reduces the system of correlated diffusions to an uncorrelated system. This enables simple and efficient approximation of the driving processes by univariate CTMC approximations. Weak convergence of the approximation is demonstrated, with second order convergence in space. Numerical experiments demonstrate the accuracy and efficiency of the method for various European and early-exercise options in two and three dimensions.
Deviations from put-call parity contain information about future stock returns. Using the difference in implied volatility between pairs of call and put options to measure these deviations, we find ...that stocks with relatively expensive calls outperform stocks with relatively expensive puts by 50 basis points per week. We find both positive abnormal performance in stocks with relatively expensive calls and negative abnormal performance in stocks with relatively expensive puts, which cannot be explained by short sale constraints. Rebate rates from the stock lending market directly confirm that our findings are not driven by stocks that are hard to borrow. The degree of predictability is larger when option liquidity is high and stock liquidity low, while there is little predictability when the opposite is true. Controlling for size, option prices are more likely to deviate from strict put-call parity when underlying stocks face more information risk. The degree of predictability decreases over the sample period. Our results are consistent with mispricing during the earlier years of the study, with a gradual reduction of the mispricing over time.
Motivated by the theory of demand-based option pricing in imperfect markets, we examine the relation between short-sale constraints and equity option returns, conditional on the level of mispricing ...in the underlying stock. We report a monotonic relation between various measures of short-sale constraints and delta-hedged returns of put options on overpriced stocks. This relation is robust to controls for firm attributes and limits to arbitrage proxies. Our findings suggest that while investors drive up the demand for these put options, dealers command a high premium as compensation for the increased market making risk. We do not find a robust relation for either put options on underpriced stocks or call options.
We find that the demand for stock options that increases exposure to the underlying is positively related to the individual investor sentiments and past market returns, whereas the demand for index ...options is invariant to these factors. These differences in trading patterns are also reflected in the differences in the composition of traders with different types of options-options on stocks are actively traded by individual investors, whereas trades in index options are more often motivated by the hedging demand of sophisticated investors. Consistent with a demand-based view of option pricing, the individual investor sentiments and past market returns are related to time-series variations in the slope of the implied volatility smile of stock options, but they have little impact on the prices of index options. The pricing impact is more pronounced in options with a higher concentration of unsophisticated investors and those with higher delta hedging costs. Our results provide evidence that factors not related to fundamentals also impact security prices.
This paper was accepted by Brad Barber, finance.
Do financial derivatives enhance or impede innovation? We answer this question by examining the relationship between equity options markets and standard measures of firm innovation. We find that ...firms with more options trading activity generate more patents and patent citations per dollar invested in research and development (R&D), after accounting for other confounding factors. These results are confirmed when we use a propensity score matching procedure and an instrumental variable approach to control for the potential endogeneity of options trading. The evidence is consistent with the notion that the enhanced informational efficiency induced by options leads to an improved allocation of corporate resources. We further discuss possible underlying economic mechanisms through which more active options markets boost innovation and show that the effect remains substantial even after controlling for these mechanisms. Considering the average increase in the dollar volume of options traded for our sample firms, we conclude that a 200% move in options volume increases firm innovation by about 31%.
We use a high‐quality microstructure data set of KOSPI 200 index options to examine the patterns of informed options trading around holidays, depending on options market characteristics. The ...information content of options trading increases around holidays, and this holiday effect is pronounced for out‐of‐the‐money calls and at‐the‐money puts. Informed large trades reinforce the holiday effects for out‐of‐the‐money call options. Foreign investors are generally informed, and their out‐of‐the‐money options trades are even more informed after holidays. Although domestic investors are less informed than their foreign competitors, their options trades seem to convey information both before and after holidays.
We present strong evidence that option trading volume contains information about future stock prices. Taking advantage of a unique data set, we construct put-call ratios from option volume initiated ...by buyers to open new positions. Stocks with low put-call ratios outperform stocks with high put-call ratios by more than 40 basis points on the next day and more than 1% over the next week. Partitioning our option ignals into components that are publicly and nonpublicly observable, we find that the economic source of this predictability is nonpublic information possessed by option traders rather than market inefficiency. We also find greater predictability for stocks with higher concentrations of informed traders and from option contracts with greater leverage.
This is a lively textbook providing a solid introduction to financial option valuation for undergraduate students armed with a working knowledge of a first year calculus. Written in a series of short ...chapters, its self-contained treatment gives equal weight to applied mathematics, stochastics and computational algorithms. No prior background in probability, statistics or numerical analysis is required. Detailed derivations of both the basic asset price model and the Black–Scholes equation are provided along with a presentation of appropriate computational techniques including binomial, finite differences and in particular, variance reduction techniques for the Monte Carlo method. Each chapter comes complete with accompanying stand-alone MATLAB code listing to illustrate a key idea. Furthermore, the author has made heavy use of figures and examples, and has included computations based on real stock market data.
This paper proposes a new approach to solve finite-horizon optimal stopping problems for a class of Markov processes that includes one-dimensional diffusions, birth-death processes, and jump ...diffusions and continuous-time Markov chains obtained by time-changing diffusions and birth-and-death processes with Lévy subordinators. When the expectation operator has a purely discrete spectrum in the Hilbert space of square-integrable payoffs, the value function of a discrete optimal stopping problem has an expansion in the eigenfunctions of the expectation operator. The Bellman's dynamic programming for the value function then reduces to an explicit recursion for the expansion coefficients. The value function of the continuous optimal stopping problem is then obtained by extrapolating the value function of the discrete problem to the limit via Richardson extrapolation. To illustrate the method, the paper develops two applications: American-style commodity futures options and Bermudan-style abandonment and capacity expansion options in commodity extraction projects under the subordinate Ornstein-Uhlenbeck model with mean-reverting jumps with the value function given by an expansion in Hermite polynomials.
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