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Abstract:
We construct a statistical model for term-structure of implied volatilities of currency options based on daily historical data for 13 currency pairs in a 19-month period. We examine the joint evolution of 1 month, 2 month, 3 month, 6 month and 1 year 50-delta options in all the currency pairs. We show that there exist three uncorrelated state variables (principal components) which account for the parallel movement, slope oscillation, and curvature of the term structure and which explain, on average, the movements of the term-structure of volatility to more than 95% in all cases. We test and construct an exponential ARCH, or E-ARCH, model for each state variable. One of the applications of this model is to produce confidence bands for the term- structure of volatility.

Abstract:
We present a framework for calibrating a pricing model to a prescribed set of option prices quoted in the market. Our algorithm yields an arbitrage-free diffusion process that minimizes the relative entropy distance to a prior diffusion. We solve a constrained (minimax) optimal control problem using a finite-difference scheme for a Bellman parabolic equation combined with a gradient-based optimization routine. The number of unknowns in the optimization step is equal to the number of option prices that need to be matched, and is independent of the mesh-size used for the scheme. This results in an efficient, non- parametric calibration method that can match an arbitrary number of option prices to any desired degree of accuracy. The algorithm can be used to interpolate, both in strike and expiration date, between implied volatilities of traded options and to price exotics. The stability and qualitative properties of the computed volatility surface are discussed, including the effect of the Bayesian prior on the shape of the surface and on the implied volatility smile/skew. The method is illustrated by calibrating to market prices of Dollar-Deutschemark over-the-counter options and computing interpolated implied-volatility curves.

Abstract:
We study model-driven statistical arbitrage strategies in U.S. equities. Trading signals are generated in two ways: using Principal Component Analysis and using sector ETFs. In both cases, we consider the residuals, or idiosyncratic components of stock returns, and model them as a mean-reverting process, which leads naturally to "contrarian'' trading signals.

The main contribution of the paper is the back-testing and comparison of market-neutral PCA- and ETF- based strategies over the broad universe of U.S. equities. Back-testing shows that, after accounting for transaction costs, PCA-based strategies have an average annual Sharpe ratio of 1.44 over the period 1997 to 2007, with a much stronger performances prior to 2003: during 2003-2007, the average Sharpe ratio of PCA-based strategies was only 0.9. On the other hand, strategies based on ETFs achieved a Sharpe ratio of 1.1 from 1997 to 2007, but experience a similar degradation of performance after 2002. We introduce a method to take into account daily trading volume information in the signals (using "trading time'' as opposed to calendar time), and observe significant improvements in performance in the case of ETF-based signals. ETF strategies which use volume information achieve a Sharpe ratio of 1.51 from 2003 to 2007.

The paper also relates the performance of mean-reversion statistical arbitrage strategies with the stock market cycle. In particular, we study in some detail the performance of the strategies during the liquidity crisis of the summer of 2007. We obtain results which are consistent with Khandani and Lo (2007) and validate their "unwinding'' theory for the quant fund drawndown of August 2007.

Abstract:
We propose a formula for calculating the implied volatility of index options based on the volatility skews of the options on the underlying stocks and on a given correlation matrix for the basket. The derivation uses the steepest-descent approximation for the multivariate probability distribution function of forward prices. A simple financial justification is provided. We apply the formula to compute the implied volatilities of liquidly-traded options on exchange-traded funds (ETF) across different strikes. Our theoretical results were found to be in good agreement with contemporaneous quotes on the Chicago Board of Options Exchange (CBOE) and the American Stock Exchange (AMEX).

Abstract:
We use Principal Component Analysis (PCA) to study the Brady Bond Debt of the four primary Latin American sovereign issuers: Argentina, Brazil, Mexico, and Venezuela. Our dataset covers a period of 5 years starting in July 1994 and consists of daily sovereign ("stripped") yield levels for the par and discount debt securities of each country. We examine the behavior of the characteristic roots and eigenvectors of the empirical covariance matrices computed sequentially over different periods.

We show that, by and large, there exist two statistically significant components, or factors, which explain up to 90% of the realized variance. The eigenvector with largest eigenvalue corresponds to the variance attributable to "regional" ("Latin") risk. The second component strongly suggests the existence of a volatility risk factor associated to Venezuelan debt in relation to the rest of the region. A time-dependent factor analysis reveals that the importance of the variance explained by the factor changes over time and that this variation can be interpreted, to some extent, in terms of market events. In particular, we investigate the relation between the evolution of the PCA factors with the market dislocations that occurred during the observation period, including the so-called Tequila effect, Asian flu, Ruble devaluation, and Real devaluation.

Abstract:
We propose a model to describe stock pinning on option expiration dates. We argue that if the open interest in a particular contract is unusually large, Delta-hedging in aggregate by floor market-makers can impact the stock price and drive it to the strike price of the option. We derive a stochastic differential equation for the stock price which has a singular drift that accounts for the price-impact of Delta-hedging. According to this model, the stock price has a finite probability of pinning at a strike. We calculate analytically and numerically this probability in terms of the volatility of the stock, the time-to-maturity, the open interest for the option under consideration and a "price-elasticity" constant that models price impact.

Stock clustering, options, open interest, volume, pinning

Abstract:
We propose a model to describe stock pinning on option expiration dates. We argue that if the open interest in a particular contract is unusually large, Delta-hedging in aggregate by floor market-makers can impact the stock price and drive it to the strike price of the option. We derive a stochastic differential equation for the stock price which has a singular drift that accounts for the price-impact of Delta-hedging. According to this model, the stock price has a finite probability of pinning at a strike. We calculate analytically and numerically this probability in terms of the volatility of the stock, the time-to-maturity, the open interest for the option under consideration and a "price-elasticity" constant that models price impact.

Stock clustering, options, open interest, volume, pinning

Abstract:
It is well-known that leveraged exchange-traded funds (LETFs) don't reproduce the corresponding multiple of index returns over extended (quarterly or annual) investment horizons. In 2008, most leveraged ETFs underperformed the corresponding static strategies. In this paper, we study this phenomenon in detail. We give an exact formula linking the return of a leveraged fund with the corresponding multiple of the return of the unleveraged fund and its realized variance. This formula is tested empirically over quarterly horizons for 56 leveraged funds (44 double-leveraged, 12 triple-leveraged) using daily prices since January 2008 or since inception, according to the fund considered. The results indicate excellent agreement between the formula and the empirical data. The study also shows that leveraged funds can be used to replicate the returns of the underlying index, provided we use a dynamic rebalancing strategy. Empirically, we find that rebalancing frequencies required to achieve this goal are moderate, on the order of one week between rebalancings. Nevertheless, this need for dynamic rebalancing leads to the conclusion that leveraged ETFs as currently designed may be unsuitable for buy-and-hold investors.

ETFs, Leveraged ETFs, Realized variance

Abstract:
We study the price-evolution of stocks that are subject to restrictions on short-selling, generically referred to as hard-to-borrow. Such stocks are either subject to regulatory short-selling restrictions or have insufficient float available for lending. Traders with short positions risk being bought-in, in the sense that their positions may be closed out by the clearing firm at market prices. The model we present consists of a coupled system of stochastic differential equations describing the stock price and the buy-in rate, an additional factor absent in standard models. The conclusion of the model is that short-sale restrictions result in increased prices and volatilities. Our model prices options as if the stock paid a continuous dividend, reflecting a modified form of Put-Call parity. Another consequence is that stocks that do not pay a dividend may have calls subject to early exercise. Both features are in agreement with empirical observations on hard-to-borrow stocks.

stock-loan, hard-to-borrow stocks

Abstract:
We introduce a new class of strategies for hedging derivative securities taking into account transaction costs, assuming lognormal continuous-time prices for the underlying asset. We do not assume that the payoff is convex as in Leland (J of Finance, 1985), or that the transaction costs are small compared to the price changes between portfolio adjustments, as in Hoggard, Whalley and Wilmott (Adv. in Futures and Options Res., 1993). The Leland number, A, which is proportional to the ratio of the round-trip tansaction cost over the typical price movement during the period between transactions, is a measure of the importance of transaction costs versus hedging risk. If A is greater than or equal to one, standard delta-hedging methods fail unless the payoff of the derivative security is a convex function of the price of the underlying asset. In contrast, our new strategies can be used effectively in the presence of large transaction costs to control simultaneously hedge-slippage as well as hedging costs. These strategies are associated with the solution an "obstacle problem" for a Black-Scholes diffusion equation with Leland's "augmented" volatility, a parameter which depends on the volatility of the underlying asset as well as on A. The new strategies are such that the frequency for rebalancing the portfolio is variable. There are periods in which rehedging takes place often to control gamma-risk and other periods, which can be relatively long, when no transactions are needed. Moreover, instead of replicating exactly the final payoff, the strategies can yield a positive cash flow at expiration, according to the price history of the underlying security. The solution to the "obstacle problem" is often simple to calculate. There exist closed-form solutions for various securities of practical interest, such as digital options.

Abstract:
We construct a sequence of trinomial trees in which an asset's price becomes lognormally distributed with given drift mu and a volatility between given sigma_min and sigma_max as the time between trades approaches zero. In this simple model of an incomplete market, we show that, as the time between trading approaches zero, the bid or ask prices of a derivative security are given by the solution of a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. In this equation, the input volatility is "dynamically" selected from the two values sigma_min, sigma_max, according to the sign of the second-order price derivative of the value function. This approach gives a new way of pricing derivative securities in markets with uncertain volatilities. It can be shown that any stochastic volatility process that stays between sigma_min and sigma_max, will give rise to a price between the bid and ask prices.

Abstract:
We consider a financial market where the volatility of the interest rate is not known exactly, but rather it is assumed to lie within two a-priori known bounds. These bounds represent the extreme values of the volatility implied by traded options. In this market, the interest rate process which allows no arbitrage and fits exactly the initial term structure of the forward interest rates, is not determined uniquely: for each volatility path in a band between the minimal and maximal volatility, there exists a different interest rate process.The asking and the bidding prices in our model are functions of the time, the interest rate, and the accumulated volatility, and they satisfy a new non-linear partial differential pricing equation. In this equation, the volatility used for pricing a claim is chosen dynamically: it is either the minimal or the maximal volatility depending on the claim's curvature with respect to both the interest rate and the accumulated volatility. We compare our model to the standard Ho-Lee model. We illustrate the effectiveness of our pricing scheme with numerical calculations for a calendar spread.

Abstract:
We present an algorithm for hedging option portfolios and custom-tailored derivative securities which uses options to manage volatility risk. The algorithm uses a volatility band to model heteroskedasticity and a non-linear partial differential equation to evaluate worst-case volatility scenarios for any given forward liability structure. This equation gives sub-additive portfolio prices and hence provides a natural ordering of preferences in terms of hedging with options. The second element of the algorithm consists of a portfolio optimization taking into account the prices of options available in the market. Several examples are discussed, including possible applications to market-making in equity and foreign-exchange derivatives.