4,179 research outputs found
Crash of ’87 - Was it Expected? Aggregate Market Fears and Long Range Dependence
We develop a dynamic framework to identify aggregate market fears ahead of a major market crash through the skewness premium of European options. Our methodology is based on measuring the distribution of a skewness premium through a q-Gaussian density and a maximum entropy principle. Our findings indicate that the October 19th, 1987 crash was predictable from the study of the skewness premium of deepest out-of-the-money options about two months prior to the crashNon-additive Entropy, Shannon Entropy, Tsallis Entropy, q-Gaussian Distribution, Skewness Premium
A Family of Maximum Entropy Densities Matching Call Option Prices
We investigate the position of the Buchen-Kelly density in a family of
entropy maximising densities which all match European call option prices for a
given maturity observed in the market. Using the Legendre transform which links
the entropy function and the cumulant generating function, we show that it is
both the unique continuous density in this family and the one with the greatest
entropy. We present a fast root-finding algorithm that can be used to calculate
the Buchen-Kelly density, and give upper boundaries for three different
discrepancies that can be used as convergence criteria. Given the call prices,
arbitrage-free digital prices at the same strikes can only move within upper
and lower boundaries given by left and right call spreads. As the number of
call prices increases, these bounds become tighter, and we give two examples
where the densities converge to the Buchen-Kelly density in the sense of
relative entropy when we use centered call spreads as proxies for digital
prices. As pointed out by Breeden and Litzenberger, in the limit a continuous
set of call prices completely determines the density.Comment: 22 pages, 6 figure
Maximum Entropy Distributions Inferred from Option Portfolios on an Asset
We obtain the maximum entropy distribution for an asset from call and digital
option prices. A rigorous mathematical proof of its existence and exponential
form is given, which can also be applied to legitimise a formal derivation by
Buchen and Kelly. We give a simple and robust algorithm for our method and
compare our results to theirs. We present numerical results which show that our
approach implies very realistic volatility surfaces even when calibrating only
to at-the-money options. Finally, we apply our approach to options on the S&P
500 index.Comment: 23 pages, 5 figures, to appear in Finance and Stochastic
Growth Optimal Investment and Pricing of Derivatives
We introduce a criterion how to price derivatives in incomplete markets,
based on the theory of growth optimal strategy in repeated multiplicative
games. We present reasons why these growth-optimal strategies should be
particularly relevant to the problem of pricing derivatives. We compare our
result with other alternative pricing procedures in the literature, and discuss
the limits of validity of the lognormal approximation. We also generalize the
pricing method to a market with correlated stocks. The expected estimation
error of the optimal investment fraction is derived in a closed form, and its
validity is checked with a small-scale empirical test.Comment: 21 pages, 5 figure
Efficient option pricing with transaction costs
A fast numerical algorithm is developed to price European options with proportional transaction costs using the utility-maximization framework of Davis (1997). This approach allows option prices to be computed by solving the investor’s basic portfolio selection problem without insertion of the option payoff into the terminal value function. The properties of the value function can then be used to drastically reduce the number of operations needed to locate the boundaries of the no-transaction region, which leads to very efficient option valuation. The optimization problem is solved numerically for the case of exponential utility, and comparisons with approximately replicating strategies reveal tight bounds for option prices even as transaction costs become large. The computational technique involves a discrete-time Markov chain approximation to a continuous-time singular stochastic optimal control problem. A general definition of an option hedging strategy in this framework is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed
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