The Enron Bankruptcy: When did the options market in Enron lose it’s smirk?

The Enron Corporation went from a $65 billion dollar market capitalization to bankruptcy in just 16 months. Using statistical techniques for extracting the implied probability distributions built into option prices, I examine the market’s expectation of Enron’s risk of collapse. I find that the options market remained far too optimistic about the stock until just weeks before their bankruptcy filing. I thank Oded Palmon and an anonymous referee for helpful comments. JEL Classification G13 · G14     

The implied volatility smirk

This paper provides an industry standard on how to quantify the shape of the implied volatility smirk in the equity index options market. Our local expansion method uses a second-order polynomial to describe the implied volatility–moneyness function and relates the coefficients of the polynomial to the properties of the implied risk-neutral distribution of the equity index return. We present a formal, two-way representation of the link between the level, slope and curvature of the implied volatility smirk and the risk-neutral standard deviation, skewness and excess kurtosis. We then propose a new semi-analytical method to calibrate option-pricing models based on the quantified implied volatility smirk, and investigate the applicability of two option-pricing models.     

Hedging derivatives with model error

The current derivatives pricing technology enables users to hedge derivatives with the underlying asset or any other traded derivative. In theory, there is no reason to prefer one hedging instrument to another. However, given model errors, this is not true. Imposing some simple assumptions on the structure of model errors, this paper shows that to maximize hedging accuracy, there is an ordering to the hedging instruments utilized. Holding constant market illiquidities, one should always hedge first with ‘like’ derivatives, next with derivatives one layer down the hierarchy of derivatives, and lastly using the underlying.     

Hedging labor income risk

We use a detailed panel data set of Swedish households to investigate the relation between their labor income risk and financial investment decisions. In particular, we relate changes in wage volatility to changes in the portfolio holdings for households that switched industries between 1999 and 2002. We find that households do adjust their portfolio holdings when switching jobs, which is consistent with the idea that households hedge their human capital risk in the stock market. The results are statistically and economically significant. A household going from an industry with low wage volatility to one with high volatility ceteris paribus decreases its portfolio share of risky assets by up to 35%, or $15,575.     

Explaining asset pricing puzzles associated with the 1987 market crash

The 1987 market crash was associated with a dramatic and permanent steepening of the implied volatility curve for equity index options, despite minimal changes in aggregate consumption. We explain these events within a general equilibrium framework in which expected endowment growth and economic uncertainty are subject to rare jumps. The arrival of a jump triggers the updating of agents' beliefs about the likelihood of future jumps, which produces a market crash and a permanent shift in option prices. Consumption and dividends remain smooth, and the model is consistent with salient features of individual stock options, equity returns, and interest rates.     

基于支持向量机的套期保值技术研究

本文在结构风险最小化的准则下,从提高样本外套期保值效率的视角,建立了基于支持向量机的套期保值新模型,并利用我国沪深300股票指数和沪深300股指期货仿真交易的历史数据进行了实证检验,并与基于最小二乘回归的套期保值模型进行了对比分析。实证结果表明本文提出的新套期保值技术能够有效提高样本外的保值效果,且该方法具有良好的鲁棒性,从而具有较好的理论和应用价值。     

New evidence on the implied-realized volatility relation

We consider the relation between the volatility implied in an option's price and the subsequently realized volatility. Earlier studies on stock index options have found biases and inefficiencies in implied volatility as a forecast of future volatility. More recently, Christensen and Prabhala find that implied volatility in at-the-money one-month OEX call options on the S&P 100 index in fact is an unbiased and efficient forecast of ex-post realized index volatility after the 1987 stock market crash. In this paper, the robustness of the unbiasedness and efficiency result is extended to a more recent period covering April 1993 to February 1997. As a new contribution, implied volatility is constructed as a trade weighted average of implied volatilities from both in-the-money and out-of-the-money options and both puts and calls. We run a horse race between implied call, implied put, and historical return volatility. Several robustness checks, including a new simultaneous equation approach, underscore our conclusion, that implied volatility is an efficient forecast of realized return volatility.     

Implied volatilities,stochastic interest rates,and currency futures options valuation: an empirical investigation

Different models of pricing currency call and put options on futures are empirically tested. Option prices are determined using different models and compared to actual market prices. Option prices are determined using historical as well as implied volatility. The different models tested include both constant and stochastic interest rate models. To determine if the model prices are different from the market prices, regression analysis and paired t-tests are performed. To see which model misprices the least, root mean square errors are determined. It is found that better results are obtained when implied volatility is used. Stochastic interest rate models perform better than constant interest rate models.     

Implied volatility skews and stock return skewness and kurtosis implied by stock option prices

The Black-Scholes* option pricing model is commonly applied to value a wide range of option contracts. However, the model often inconsistently prices deep in-the-money and deep out-of-the-money options. Options professionals refer to this well-known phenomenon as a volatility ‘skew’ or ‘smile’. In this paper, we examine an extension of the Black-Scholes model developed by Corrado and Su that suggests skewness and kurtosis in the option-implied distributions of stock returns as the source of volatility skews. Adapting their methodology, we estimate option-implied coefficients of skewness and kurtosis for four actively traded stock options. We find significantly nonnormal skewness and kurtosis in the option-implied distributions of stock returns.     

No-dynamic-arbitrage and market impact

Starting from a no-dynamic-arbitrage principle that imposes that trading costs should be non-negative on average and a simple model for the evolution of market prices, we demonstrate a relationship between the shape of the market impact function describing the average response of the market price to traded quantity and the function that describes the decay of market impact. In particular, we show that the widely assumed exponential decay of market impact is compatible only with linear market impact. We derive various inequalities relating the typical shape of the observed market impact function to the decay of market impact, noting that, empirically, these inequalities are typically close to being equalities.     

General equilibrium pricing of options with habit formation and event risks

This paper proposes a general equilibrium model that explains the pricing of the S&P 500 index options. The central ingredients are a peso component in the consumption growth rate and the time-varying risk aversion induced by habit formation which amplifies consumption shocks. The amplifying effect generates the excess volatility and a large jump-risk premium which combine to produce a pronounced volatility smirk for index options. The time-varying volatility and jump-risk premiums explain the observed state-dependent smirk patterns. Besides volatility smirks, the model has a variety of other implications which are broadly consistent with the aggregate stock and option market data.     

对冲基金、对冲策略以及市场中性

本文梳理了对冲基金的概念、特征及对冲策略,指出市场中性是这些对冲策略普遍存在的内在一致性要求。在此基础上,本文进一步讨论了市场中性策略的收益来源,分析了市场中性策略的做空优势,指出对冲策略拓宽了传统组合边界。     

The smirk in the S&P500 futures options prices: a linearized factor analysis

In the S&P500 futures options, we identify three factors, corresponding to movements in the underlying, parallel movements, and tilting of the cross section of implied volatilities (the “smirk factor”). We relate these factors non-linearly to movements in the option prices. They seem to be diffusive in nature, have significant associated risk premia, and can account for an overwhelming part of the option price movements. We interpret the options smirk, which is the notion that out-of-the-money (OTM) puts seem expensive relative to OTM calls, in terms of the prices of these risk factors. Going short OTM puts and long OTM calls, corresponding to the third factor, makes a profit on average, but this corresponds to its risk premium, and does not represent a market inefficiency. Our smirk factor is useful for hedging option portfolios, but seems unrelated to movements in the underlying, and does not fit into the framework of the jump-diffusion models.      

论国内市场寻找最佳避险比率的过程

国内即将推出指数期货，投资者可以利用期货来规避现货投资的风险，但如何寻找最佳的避险比率呢？本文通过欧美日港台等市场的比较研究，认为国内可能需要采用二阶段估计法寻找最佳的避险比率，而且，对于国内股票市场来说，β值在某些时候出现较大的跳动，因此，应该采用相关修正模型调整历史β值。本文的结论为即将推出指数期货的国内市场提供了重要的实践指导建议。     

公司套期保值的决定因素——基于有色金属行业的经验研究

利用公司披露的数据,发现了样本公司套期保值决策与财务困境成本和流动性需求的高度相关性,同时也发现了与理论预期相反的证据,对于我国有色金属上市公司来说,协调投融资的工具并不是套期保值,而是其它;管理者持股激励机制有可能并未起到应有的作用。     

DCC-GARCH、M-Copula-GARCH和Copula-SV在商品期货对冲中的应用研究

中国商品期货市场已经成为全球期货市场不可或缺的重要组成部分。文章比较研究了DCC-GARCH、M-Copula-GARCH和Copula-SV三种模型对我国最重要的期货合约——铜和棉花的对冲比率的影响。结果表明:Copula-SV是最优的对冲模型,文章还发现:二月期的铜期货合约和三月期的棉花期货合约对冲现货的效率最高。     

基于VaR模型的最优套期保值比例探析——以美元远期套保为例

VaR模型被认为是兼顾投机套利与套期保值两大动机的衍生工具决策模型。国内文献对此讨论较少。基于其理论推导,可以归纳地认为VaR模型具有兼容性、一般性、"期权"特征和可操作性。以美元远期套保为例,研究发现,基于VaR模型的最优套保比较之传统最优套保比、最小方差最优套保比更有优势,能够解释中国企业运用衍生工具失败的原因。但据此推论,基于VaR模型确定最优套保比在实践中仍然存在一些问题,如非正态分布、现货头寸不确定、损益报告困难等。     

期货最优套期保值比率研究评述

套期保值、价格发现、资产配置是期货市场的三大基本职能。而在这三种职能中,最为重要的是套期保值,它是期货市场得以生存和发展的关键动因。在国内外套期保值研究中,最优套期保值比率的估计是套期保值研究最为核心的问题。而运用期货套期保值理论进行实践更是随着套期保值比率估计模型的不断优化、完善而向前发展的,本文通过对国内外套期保值研究相关文献进行分类,整理,综述,梳理出最优套期保值比率估计的研究思路和相关实证技术路线,概括出国内外最优套保比率的研究框架,以此来向国内相关研究学者指出未来的进一步研究方向,同时对国内期货交易主体进行套期保值操作提供了估计模型的选择建议。     

Hedging efficiently under correlation

We show that when a derivative portfolio has different correlated underlyings, hedging using classical greeks (first-order derivatives) is not the best possible choice. We first show how to adjust greeks to take correlation into account and reduce P&L volatility. Then we embed correlation-adjusted greeks in a global hedging strategy that reduces cost of hedging without increasing P&L volatility, by optimization of hedge re-adjustments. The strategy is justified in terms of a balance between transaction costs and risk-aversion, but, unlike more complex proposals from previous literature, it is completely defined by observable parameters, geometrically intuitive, and easy to implement for an arbitrary number of risk factors. We test our findings on a CVA hedging example. We first consider daily re-hedging: in this test, correlation-adjusted greeks allow the reduction of P&L volatility by more than 30% compared to standard deltas. Then we apply our general strategy to a context where a CVA portfolio is exposed to both credit and interest rate risk. The strategy keeps P&L volatility in line with daily standard delta-hedging, but with massive cost-saving: only six rebalances of the illiquid credit hedge are performed, over a period of six months.     

The term structure of S&P 100 model-free volatilities

We develop an improved method to obtain the model-free volatility more accurately despite the limitations of a finite number of options and large strike price intervals. Our method computes the model-free volatility from European-style S&P 100 index options over a horizon of up to 450 days, the first time that this has been attempted, as far as we are aware. With the estimated daily term structure over the long horizon, we find that (i) changes in model-free volatilities are asymmetrically more positively impacted by a decrease in the index level than negatively impacted by an increase in the index level; (ii) the negative relationship between the daily change in model-free volatility and the daily change in index level is stronger in the near term than in the far term; and (iii) the slope of the term structure is positively associated with the index level, having a tendency to display a negative slope during bear markets and a positive slope during bull markets. These significant results have important implications for pricing and hedging index derivatives and portfolios.