This paper investigates the predictive power of implied variances extracted from the dollar/yen option prices. Implied variances are estimated from transaction prices of currency options traded on PHLX using the option pricing model of Garman and Kohlhagen (1983). In contrast to recent findings on stock and stock index options, the out-of-sample tests indicate that the implied variance is an upward biased estimator of future variance; and that the variance forecasts from GARCH and historical models do not contain significant incremental information in predicting future variance. Trading strategies are also developed to exploit the observed overstatement of variance in the dollar/yen option market. Traders that can execute the delta-neutral trading strategies at the observed market transaction prices could lock in a significant profits during the period examined. However, for investors that facing higher transaction costs, the magnitude of the profits is generally not large enough to al...

of the Dissertation Analytical Approximate Solution for the Prices of American Exotic Options. by Tatiana S. Taksar in Applied Mathematics and Statistics State University of New York at Stony Brook 1997 Dissertation Advisor: Professor Qiang Zhang In this thesis we will present analytical approximate solutions to the values of American barrier, lookback and Asian options. Both "out" barrier and "in" barrier options have been considered. At an "out" ("in") barrier, an option becomes worthless (starts to exist). A path-dependent option is an option iii whose payoff at exercise or maturity depends on the past history of the price of the underlying asset. Lookback options are one of the most common types of path-dependent options. They are path-dependent options whose payoff depends on the maximum or the minimum realized price of the underlying asset over the life of the option. Asian options are path-dependent options whose payoff depends on the average stock price observed during the li...

The paper investigates a contingent claim replicating problem for a diffusion market model. It is proposed an approach which does not call to solve the backward parabolic equation, unlike for the classical method, and this approach is applied for a case when the volatility coefficient is random and depends on time. The replicating strategy is decomposed to series of explicit strategies which does not use current estimations of volatility. A mean variance convergence of the series is proved. Keywords: diffusion market model, option pricing, time varying random volatility, approximate series 1 Introduction The paper investigates optimal investment problem and hedging problem for a market which consists of a risks free bond and a risky stock. It is assumed that the dynamics of the stock is described by a diffusion random process. The dynamics of the bond is deterministic and exponentially increasing with a given risk free rate. The most well-known strategies for this market model were ob...

In this short note we show how virtual arbitrage opportunities can be modelled and included in the standard derivative pricing without changing the general framework. Whatever people say about the drawbacks of the Black-Scholes (BS) approach [1] to derivative pricing, it is a standard method and almost any pricing and hedging software in financial institutions is based on it. Practitioners have got used to BSlike partial differential equations, martingales and other related mathematical animals. Both analytical and numerical methods are well developed and it is hardly surprising that practitioners are rather reluctant to "buy" complicated new theories. That is why it is interesting to see how some limitations of BS analysis can be overcome in the same mathematical framework without disturbing the foundations. One way to improve BS is to use a more realistic price process instead of the geometrical random walk. The most popular alternatives are ARCH-GARCH models where the volatility of...

We consider the hedging of derivative securities when the price movement of the underlying asset can exhibit jumps of random size. Working in a single factor Markovian setting, we derive a new spanning relation between a given option and a continuum of shorter term options written on the same asset. In the portfolio of shorter term options, the portfolio weights do not vary with changes in stock price or time. We then implement this static relation using a finite set of shorter term options based on a quadrature rule and use Monte Carlo simulation to determine the hedging error thereby introduced. We compare this hedging error to that of a delta hedging strategy based on daily rebalancing in the underlying futures. The simulation results indicate that the two types of hedging strategies exhibit comparable performance in the classic Black-Scholes environment, but that our static hedge strongly outperforms delta-hedging when the underlying asset price is governed by Merton (1976)'s jump-di#usion model. Further simulation exercises indicate that these results are robust to model misspecification, so long as one performs ad hoc adjustments based on the observed implied volatility.

Unconditional alpha estimates are biased when conditional beta covaries with market risk premia (“market-timing”) or volatility (“volatility-timing”). We demonstrate an additional bias (“overconditioning”) that can occur any time an empiricist uses a risk proxy not in the investor information set — for example when asset payoffs are nonlinear and the conditional loading is proxied by contemporaneous realized beta. Calibrating to U.S. equity returns, volatility-timing and overconditioning plausibly impact alphas much more than market-timing, which has been the focus of prior literature. A variety of instrumental variables estimators using realized betas can substantially correct market- and volatility-timing biases, while eliminating overconditioning. Empirically, appropriate instrumentation reduces momentum alphas by 20-40 % relative to unconditional, whereas overconditioned alphas overstate performance by up to 2.5 times. Volatility-timing inflates unconditionally estimated momentum alpha because the formationperiod market return (i) positively predicts holding-period beta (Grundy and Martin, 2001) and (ii) negatively predicts holding-period market volatility (French, Schwert, and Stambaugh,

Statistics plays a leading role in finance. The explosive development of increasingly complex markets makes it more and more difficult for practitioners to correctly value financial asset. Statistical analysis has become a powerful tool for a better market valuation, taking a leading role in the development of new financial products that try to hedge the increasing amount of risks that an investor has to take. Statistics knowledge demand is steadily increasing in Hedge Funds, Investment Banking and Financial Institutions in general, where statistics students could developed a professional career. Finance can be seen as a way to motivate students on the applications of almost any statistical tool we would like to teach them, since we could always find an example where these techniques are put into practice.

ABSTRACT This paper describes and analyses the history of the fundamental equation of modern financial economics: the Black-Scholes (or Black-Scholes-Merton) option pricing equation. In that history, several themes of potentially general importance are revealed. First, the key mathematical work was not rule-following but bricolage, creative tinkering. Second, it was, however, bricolage guided by the goal of finding a solution to the problem of option pricing analogous to existing exemplary solutions, notably the Capital Asset Pricing Model, which had successfully been applied to stock prices. Third, the central strands of work on option pricing, although all recognizably ‘orthodox ’ economics, were not unitary. There was significant theoretical disagreement amongst the pioneers of option pricing theory; this disagreement, paradoxically, turns out to be a strength of the theory. Fourth, option pricing theory has been performative. Rather than simply describing a preexisting empirical state of affairs, it altered the world, in general in a way that made itself more true.

The predictability of an asset's returns will affect the prices of options on that asset, even though predictability is typically induced by the drift, which does not enter the option pricing formula. For discretely-sampled data, predictability is linked to the parameters that do enter the option pricing formula. We construct an adjustment for predictability to the Black-Scholes formula and show that this adjustment can be important even for small levels of predictability, especially for longer maturity options. We propose several continuous-time linear diffusion processes that can capture broader forms of predictability, and provide numerical examples that illustrate their importance for pricing options. THEREIS NOW A substantial body of evidence that documents the predictability of financial asset returns. ' Despite the lack of consensus as to the sources of such predictability-some attribute it to time-varying expected returns, perhaps due to changes in business conditions, while others argue that predictability is a symptom of inefficient markets or irrational investors-there is a growing consensus that predictability is a genuine feature of many financial asset returns. In this article, we investigate the impact of asset return predictability on the prices of an asset's options. A comparison between the polar cases of perfect predictability (certainty) and perfect unpredictability (the random walk) suggests that predictability must have an effect on option prices, although what that effect might be is far from obvious. However, in the

Although it is widely believed that option prices provide the best possible forecasts of the future variance of the assets which underlie them, a large body of empirical evidence concludes that option prices consistently yield biased forecasts of future variance. The prevailing interpretation of these findings is that option investors may be forming unbiased forecasts of the future variance of underlying assets but that these unbiased forecasts fail to get impounded into option prices because of either (1) the difficulty of carrying out the necessary arbitrage strategies that would force the prices to their proper levels, or (2) the availability to market makers of lucrative alternative strategies in which they simply profit from the large bid-ask spreads in the options markets. This interpretation has significant consequences for nearly the entire range of option pricing research, since it implies that non-continuous trading, bid-ask spreads, and other market imperfections substantially influence option prices. This implication is important, both because incorporating these types of market imperfections into option pricing models is much more difficult than, for example, altering the dynamics of the underlying asset and also because it suggests that researchers cannot learn about option investor expectations by filtering option