TUG‐OF‐WAR,MARKET MANIPULATION,AND OPTION PRICING

We develop an option pricing model based on a tug‐of‐war game. This two‐player zero‐sum stochastic differential game is formulated in the context of a multidimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the nonlinear and completely degenerate infinity Laplace operator.     

THE RANGE OF TRADED OPTION PRICES

Suppose we are given a set of prices of European call options over a finite range of strike prices and exercise times, written on a financial asset with deterministic dividends which is traded in a frictionless market with no interest rate volatility. We ask: when is there an arbitrage opportunity? We give conditions for the prices to be consistent with an arbitrage-free model (in which case the model can be realized on a finite probability space). We also give conditions for there to exist an arbitrage opportunity which can be locked in at time zero. There is also a third boundary case in which prices are recognizably misspecified, but the ability to take advantage of an arbitrage opportunity depends upon knowledge of the null sets of the model.     

THE MOMENT FORMULA FOR IMPLIED VOLATILITY AT EXTREME STRIKES

Consider options on a nonnegative underlying random variable with arbitrary distribution. In the absence of arbitrage, we show that at any maturity T , the large-strike tail of the Black-Scholes implied volatility skew is bounded by the square root of  2| x |/ T   , where x is log-moneyness. The smallest coefficient that can replace the 2 depends only on the number of finite moments in the underlying distribution. We prove the moment formula , which expresses explicitly this model-independent relationship. We prove also the reciprocal moment formula for the small-strike tail, and we exhibit the symmetry between the formulas. The moment formula, which evaluates readily in many cases of practical interest, has applications to skew extrapolation and model calibration.     

Statistical manipulability of social choice functions

We consider the proportion of profiles at which an anonymous neutral social choice function cannot be manipulated to any coalition's advantage. We fix the number of alternatives and consider the limit as the number of individuals goes to infinity, and assume that the limit of the fraction of all profiles in the domain is 1. We show that there is a continuous function on “profile vectors’ which gives the maximum probability of strategy-proofness, and that this is strictly between 0 and 1 for three alternatives. We also analyze a number of common social choice functions along these lines.     

Selling digital music: business models for public goods

This paper considers the market for digital music. We claim that the combination of the MP3 format and peer-to-peer networks has made music non-excludable and this feature is essential for the understanding of the economics of the music market. We study optimal business models for selling non-excludable goods and show that despite promising theoretical results, adding just a slight uncertainty about the number of customers has significant negative implications for profitability. Indeed, as the average number of customers tends to infinity the average payment per customer converges to zero. Therefore, the music industry should concentrate on alternative ways of creating profit such as selling access to listeners, concerts, merchandise, ringtones etc.     

Asymptotics of the price oscillations of a European call option in a tree model

It is well known that the price of a European vanilla option computed in a binomial tree model converges toward the Black-Scholes price when the time step tends to zero. Moreover, it has been observed that this convergence is of order 1/ n in usual models and that it is oscillatory. In this paper, we compute this oscillatory behavior using asymptotics of Laplace integrals, giving explicitly the first terms of the asymptotics. This allows us to show that there is no asymptotic expansion in the usual sense, but that the rate of convergence is indeed of order 1/ n in the case of usual binomial models since the second term (in     ) vanishes. The next term is of type   C ₂( n )/ n   , with   C ₂( n )  some explicit bounded function of n that has no limit when n tends to infinity.     

EXACT SOLUTION OF A MARTINGALE STOCHASTIC VOLATILITY OPTION PROBLEM AND ITS EMPIRICAL EVALUATION

Exact explicit solution of the log-normal stochastic volatility (SV) option model has remained an open problem for two decades. In this paper, I consider the case where the risk-neutral measure induces a martingale volatility process, and derive an exact explicit solution to this unsolved problem which is also free from any inverse transforms. A representation of the asset price shows that its distribution depends on that of two random variables, the terminal SV as well as the time average of future stochastic variances. Probabilistic methods, using the author's previous results on stochastic time changes, and a Laplace–Girsanov Transform technique are applied to produce exact explicit probability distributions and option price formula. The formulae reveal interesting interplay of forces between the two random variables through the correlation coefficient. When the correlation is set to zero, the first random variable is eliminated and the option formula gives the exact formula for the limit of the Taylor series in Hull and White's (1987) approximation. The SV futures option model, comparative statics, price comparisons, the Greeks and practical and empirical implementation and evaluation results are also presented. A PC application was developed to fit the SV models to current market prices, and calculate other option prices, and their Greeks and implied volatilities (IVs) based on the results of this paper. This paper also provides a solution to the option implied volatility problem, as the empirical studies show that, the SV model can reproduce market prices, better than Black–Scholes and Black-76 by up to 2918%, and its IV curve can reproduce that of market prices very closely, by up to within its 0.37%.     

The log-moment formula for implied volatility

We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that in the absence of arbitrage, if the underlying stock price at time T admits finite log-moments $\mathbb{E}[|\log S_T{|}^q]$ for some positive q, the arbitrage-free growth in the left wing of the implied volatility smile for T is less constrained than Lee's bound. The result is rationalized by a market trading discretely monitored variance swaps wherein the payoff is a function of squared log-returns, and requires no assumption for the underlying price to admit any negative moment. In this respect, the result can be derived from a model-independent setup. As a byproduct, we relax the moment assumptions on the stock price to provide a new proof of the notorious Gatheral–Fukasawa formula expressing variance swaps in terms of the implied volatility.     

GREED,LEVERAGE, AND POTENTIAL LOSSES: A PROSPECT THEORY PERSPECTIVE

This paper quantifies the notion of greed, and explores its connection with leverage and potential losses, in the context of a continuous‐time behavioral portfolio choice model under (cumulative) prospect theory. We argue that the reference point can serve as the critical parameter in defining greed. An asymptotic analysis on optimal trading behaviors when the pricing kernel is lognormal and the S‐shaped utility function is a two‐piece CRRA shows that both the level of leverage and the magnitude of potential losses will grow unbounded if the greed grows uncontrolled. However, the probability of ending with gains does not diminish to zero even as the greed approaches infinity. This explains why a sufficiently greedy behavioral agent, despite the risk of catastrophic losses, is still willing to gamble on potential gains because they have a positive probability of occurrence whereas the corresponding rewards are huge. As a result, an effective way to contain human greed, from a regulatory point of view, is to impose a priori bounds on leverage and/or potential losses.     

Measuring the effect of the zero lower bound on yields and exchange rates in the U.K. and Germany

The zero lower bound on nominal interest rates began to constrain many central banks' setting of short-term interest rates in late 2008 or early 2009. According to standard macroeconomic models, this should have greatly reduced the effectiveness of monetary policy and increased the efficacy of fiscal policy. However, these models also imply that asset prices and private-sector decisions depend on the entire path of expected future short-term interest rates, not just the current level of the monetary policy rate. Thus, interest rates with a year or more to maturity are arguably more relevant for asset prices and the economy, and it is unclear to what extent those yields have been affected by the zero lower bound. In this paper, we apply the methods of Swanson and Williams (2013) to medium- and longer-term yields and exchange rates in the U.K. and Germany. In particular, we compare the sensitivity of these rates to macroeconomic news during periods when short-term interest rates were very low to that during normal times. We find that: 1) USD/GBP and USD/EUR exchange rates have been essentially unaffected by the zero lower bound, 2) yields on German bunds were essentially unconstrained by the zero bound until late 2012, and 3) yields on U.K. gilts were substantially constrained by the zero lower bound in 2009 and 2012, but were surprisingly responsive to news in 2010–11. We compare these findings to the U.S. and discuss their broader implications.     

TERM STRUCTURES OF IMPLIED VOLATILITIES: ABSENCE OF ARBITRAGE AND EXISTENCE RESULTS

This paper studies modeling and existence issues for market models of stochastic implied volatility in a continuous-time framework with one stock, one bank account, and a family of European options for all maturities with a fixed payoff function h . We first characterize absence of arbitrage in terms of drift conditions for the forward implied volatilities corresponding to a general convex h . For the resulting infinite system of SDEs for the stock and all the forward implied volatilities, we then study the question of solvability and provide sufficient conditions for existence and uniqueness of a solution. We do this for two examples of h , namely, calls with a fixed strike and a fixed power of the terminal stock price, and we give explicit examples of volatility coefficients satisfying the required assumptions.     

LEVERAGED ETF IMPLIED VOLATILITIES FROM ETF DYNAMICS

The growth of the exchange‐traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). We study the relationship between the ETF and LETF implied volatility surfaces when the underlying ETF is modeled by a general class of local‐stochastic volatility models. A closed‐form approximation for prices is derived for European‐style options whose payoffs depend on the terminal value of the ETF and/or LETF. Rigorous error bounds for this pricing approximation are established. A closed‐form approximation for implied volatilities is also derived. We also discuss a scaling procedure for comparing implied volatilities across leverage ratios. The implied volatility expansions and scalings are tested in three settings: Heston, limited constant elasticity of variance (CEV), and limited SABR; the last two are regularized versions of the well‐known CEV and SABR models.     

ASYMPTOTICS OF IMPLIED VOLATILITY IN LOCAL VOLATILITY MODELS

Using an expansion of the transition density function of a one‐dimensional time inhomogeneous diffusion, we obtain the first‐ and second‐order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the first‐ and second‐order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate.     

How do US options traders “smirk” on China? Evidence from FXI options

In this paper, we study the implied volatility smirk (IVS) of options written on the FXI, the Financial Times Stock Exchange/Xinhua China 50 Index exchange-traded fund (ETF). Using the methodology of Zhang and Xiang (2008, Quant Financ, 8, pp. 263–284), we document the empirical characteristics of the level, slope, and curvature of IVS of the FXI options. We find that, on average, IVS becomes steeper and more convex as time to maturity increases. The level and curvature are usually positive, and the slope is negative. We provide evidence that the information in the quantified IV factors has some predictive power for the future monthly FXI ETF returns.     

The Effect of Government Spending on the Debt‐to‐GDP Ratio: Some Keynesian Arithmetic

According to the standard approach to the issue of public debt sustainability, fiscal austerity is the route that many countries must currently follow to reduce their debt‐to‐GDP ratios back to sustainable paths. We challenge this conventional wisdom and argue that, below full employment, an increase in government spending may paradoxically reduce the debt‐to‐GDP ratio. This claim is particularly relevant today because with Central Bank interest rates near zero there is no alternative to fiscal policy, and the only argument against increasing government expenditure as a way to fight unemployment is its supposed negative effect on the state of public finances.     

Show me the money: Option moneyness concentration and future stock returns

Informed traders often use options that are not in-the-money due to higher potential gains for a smaller upfront cost. Thus, trading activity by option moneyness should be a gauge of informed option trading. We construct a dollar volume-weighted average moneyness measure to capture option trading activity at different moneyness levels. Stock returns increase with this measure, suggesting more trading activity in options with higher leverage predicts future stock returns. Our results hold cross-sectionally and at the portfolio level yielding a Fama–French five-factor α of 12% per year for all stocks and 33% per year for high implied volatility stocks.     

REGULAR VARIATION AND SMILE ASYMPTOTICS

We consider risk-neutral returns and show how their tail asymptotics translate directly to asymptotics of the implied volatility smile, thereby sharpening Roger Lee's celebrated moment formula. The theory of regular variation provides the ideal mathematical framework to formulate and prove such results. The practical value of our formulae comes from the vast literature on tail asymptotics and our conditions are often seen to be true by simple inspection of known results.     

偏馈法距离校零误差校正

针对偏馈法距离校零产生的误差,提出一种几何校正方法。与传统的零值分离方法不同,该方法基于距离测量原理,用几何法分解对塔校零和偏馈校零的信号路径,并计算对塔校零的距离零值,然后采用参数传递的方式代入到偏馈校零的表达式中导出数学模型,使两种校零方法的距离零值完全等效,从机理上消除了偏馈法的零值误差。同时,考虑近场时延误差和漏射误差,可确定校零天线的最佳安装位置。试验结果表明,校正后的偏馈校零与对塔校零相比,距离零值的误差小于1 m。由于计算模型简单易用,该几何校正方法对当前无塔标校的应用具有一定参考价值。     

Firm-level political risk and implied cost of equity capital

I find a strong positive association between firms' implied cost of equity capital and firm-level political risk. This effect is above and beyond the firm-level cost of equity implications of economywide political risk. Firm-level political risk contributes to elevating stock illiquidity, increases dispersion of analyst forecasts and dampens analyst coverage and these attributes, in turn, have positive cost of equity capital implications. Overall, the findings of this study suggest firm-level political risk has a non-trivial effect on increasing equity market illiquidity, increasing dispersion of earnings forecasts and decreasing analyst coverage thus increasing financing costs.     

An approximation formula for normal implied volatility under general local stochastic volatility models

We approximate normal implied volatilities by means of an asymptotic expansion method. The contribution of this paper is twofold: to our knowledge, this paper is the first to provide a unified approximation method for the normal implied volatility under general local stochastic volatility models. Second, we applied our framework to polynomial local stochastic volatility models with various degrees and could replicate the swaptions market data accurately. In addition we examined the accuracy of the results by comparison with the Monte‐Carlo simulations.