The market for crash risk

This paper examines the equilibrium when stock market crashes can occur and investors have heterogeneous attitudes towards crash risk. The less crash averse insure the more crash averse through options markets that dynamically complete the economy. The resulting equilibrium is compared with various option pricing anomalies: the tendency of stock index options to overpredict volatility and jump risk, the Jackwerth [Recovering risk aversion from option prices and realized returns. Review of Financial Studies 13, 433–451] implicit pricing kernel puzzle, and the stochastic evolution of option prices. Crash aversion is compatible with some static option pricing puzzles, while heterogeneity partially explains dynamic puzzles. Heterogeneity also magnifies substantially the stock market impact of adverse news about fundamentals.     

Pricing vulnerable options with stochastic liquidity risk

In this paper, we consider vulnerable options with stochastic liquidity risk. We employ liquidity-adjusted pricing models to describe the underlying stock price and option issuer’s assets. In addition, the correlation between these assets is stochastic, depending on the market liquidity measures. In the proposed framework, we derive closed forms of vulnerable European options with stochastic liquidity risk and then use them to illustrate the effects of stochastic liquidity risk on vulnerable option prices. Numerical results show that the effects of liquidity risk on the prices of out-of-the-money options or the options with a short maturity are not negligible.     

Option Pricing under the Double Exponential Jump-Diffusion Model with Stochastic Volatility and Interest Rate

This paper proposes an efficient option pricing model that incorporates stochastic interest rate (SIR), stochastic volatility (SV), and double exponential jump into the jump-diffusion settings. The model comprehensively considers the leptokurtosis and heteroscedasticity of the underlying asset’s returns, rare events, and an SIR. Using the model, we deduce the pricing characteristic function and pricing formula of a European option. Then, we develop the Markov chain Monte Carlo method with latent variable to solve the problem of parameter estimation under the double exponential jump-diffusion model with SIR and SV. For verification purposes, we conduct time efficiency analysis, goodness of fit analysis, and jump/drift term analysis of the proposed model. In addition, we compare the pricing accuracy of the proposed model with those of the Black–Scholes and the Kou (2002) models. The empirical results show that the proposed option pricing model has high time efficiency, and the goodness of fit and pricing accuracy are significantly higher than those of the other two models.     

用模拟退火算法寻找Heston期权定价模型参数

随机波动率模型由于放松了Black-Sholes模型的假定而更符合市场情况,因此成为研究金融衍生品定价的热点。Heston随机波动率不同于其他随机波动率模型之处在于其存在闭形式解。Heston期权定价模型在应用中需要确定五个待估参数,此问题通常比较困难。本文采用模拟退火算法并利用最小化残差平方和来估算,该算法以一定概率跳出局部极小值,从而以概率1收敛到全局极小值,最终得到Heston模型的待估参数。在实证研究中,本文利用香港恒生股票指数期权在2010年10月15日交易的数据,得到待估参数,并用该参数对2010年10月18日期权进行了模拟定价。     

Asymmetry in the jump-size distribution of the S&P 500: Evidence from equity and option markets

This paper studies alternative distributions for the size of price jumps in the S&P 500 index. We introduce a range of new jump-diffusion models and extend popular double-jump specifications that have become ubiquitous in the finance literature. The dynamic properties of these models are tested on both a long time series of S&P 500 returns and a large sample of European vanilla option prices. We discuss the in- and out-of-sample option pricing performance and provide detailed evidence of jump risk premia. Models with double-gamma jump size distributions are found to outperform benchmark models with normally distributed jump sizes.     

Pure jump models for pricing and hedging VIX derivatives

Recent non-parametric statistical analysis of high-frequency VIX data (Todorov and Tauchen, 2011) reveals that VIX dynamics is a pure jump semimartingale with infinite jump activity and infinite variation. To our best knowledge, existing models in the literature for pricing and hedging VIX derivatives do not have these features. This paper fills this gap by developing a novel class of parsimonious pure jump models with such features for VIX based on the additive time change technique proposed in Li et al., 2016a, Li et al., 2016b. We time change the 3/2 diffusion by a class of additive subordinators with infinite activity, yielding pure jump Markov semimartingales with infinite activity and infinite variation. These processes have time and state dependent jumps that are mean reverting and are able to capture stylized features of VIX. Our models take the initial term structure of VIX futures as input and are analytically tractable for pricing VIX futures and European options via eigenfunction expansions. Through calibration exercises, we show that our model is able to achieve excellent fit for the VIX implied volatility surface which typically exhibits very steep skews. Comparison to two other models in terms of calibration reveals that our model performs better both in-sample and out-of-sample. We explain the ability of our model to fit the volatility surface by evaluating the matching of moments implied from market VIX option prices. To hedge VIX options, we develop a dynamic strategy which minimizes instantaneous jump risk at each rebalancing time while controlling transaction cost. Its effectiveness is demonstrated through a simulation study on hedging Bermudan style VIX options.     

带Poisson跳的Black-Scholes模型的期权定价

主要讨论欧式期权的定价公式。首先给出一个B-S期权定价公式的简化方法,使具有一般微积分知识的读者就能理解;并假定股票价格过程遵循带Poisson跳的扩散过程,在股票预期收益率、波动率和无风险利率均为时间函数的情况下,得到欧式期权定价公式和买权与卖权之间的平价关系。     

Multi-step barrier products and static hedging

This paper examines multi-step barrier options with an arbitrary payoff function using extended static hedging methods. Although there have been studies using extended reflection principles to obtain joint distribution functions for barrier options with complex barrier conditions, and static hedging methods to evaluate limited barrier options with well-known payoff functions, we obtain an explicit expression of barrier option price which has a general payoff function under the Black–Scholes framework assumption. The explicit multi-step barrier options prices we discuss in this paper are not only useful in that they can handle different levels and time steps barrier and all types of payoff functions, but can also extend to pricing of barrier options under finite discrete jump–diffusion models with a simple barrier. In the last part, we supplement the theory with numerical examples of various multi-step barrier options under the Black–Scholes or discrete jump–diffusion model for comparison purposes.     

Exchange options for catastrophe risk management

In this paper, we investigate the pricing issue and catastrophe risk management of exchange options. Exchange options allow the holder to exchange its stocks for another at maturity and can be seen as an extended version of catastrophe equity put options with another traded asset price as strike prices. Since option holders have to issue new shares to exercise the option, we illustrate the differences between option prices calculated using pre-exercise and post-exercise share prices. The effects of default risk on option prices and risk management are also considered. Finally, risk management analysis shows that exchange options can effectively hedge catastrophe risk.     

Do interest rate options contain information about excess returns?

There is strong empirical evidence that long-term interest rates contain a time-varying risk premium. Options may contain valuable information about this risk premium because their prices are sensitive to the underlying interest rates. We use the joint time series of swap rates and interest rate option prices to estimate dynamic term structure models. The risk premiums that we estimate using option prices are better able to predict excess returns for long-term swaps over short-term swaps. Moreover, in contrast to the previous literature, the most successful models for predicting excess returns have risk factors with stochastic volatility. We also show that the stochastic volatility models we estimate using option prices match the failure of the expectations hypothesis.     

Minute by minute: Efficiency,normality, and randomness in intra-daily asset prices

In this study we test the efficiency of asset markets at intervals as short as 30 seconds. We also describe the properties of a simple new stochastic process as a potential model of the behaviour of asset prices and test it on intra-daily Deutsche Mark futures prices. According to this process, asset prices are constant between economically relevant events, which occur at the random times generated by a Poisson process. At the moments of these events, prices jump to new values; the size of the jump is drawn from a normal distribution. Tests of this process indicate that it cannot be rejected for most of the days in the sample.     

Min–max multi-step barrier options and their variants

This paper studies a new type of barrier option, min–max multi-step barrier options with diverse multiple up or down barrier levels placed in the sub-periods of the option’s lifetime. We develop the explicit pricing formula of this type of option under the Black–Scholes model and explore its applications and possible extensions. In particular, the min–max multi-step barrier option pricing formula can be used to approximate double barrier option prices and compute prices of complex barrier options such as discrete geometric Asian barrier options. As a practical example of directly applying the pricing formula, we introduce and evaluate a re-bouncing equity-linked security. The main theorem of this work is capable of handling the general payoff function, from which we obtain the pricing formulas of various min–max multi-step barrier options. The min–max multi-step reflection principle, the boundary-crossing probability of min–max multi-step barriers with icicles, is also derived.     

期权“隐含波动率微笑”成因分析

Black-Scholes期权定价模型低估深实值和深虚值期权的现象称为“波动率微笑”。其主要原因是资产价格过程假设和市场机制因素给期权卖方的△套期保值带来了额外风险和成本。确定波动率和随机波动率研究都对BS模型做出了修正。     

A new class of multidimensional Wishart-based hybrid models

In this article, we present a new class of pricing models that extend the application of Wishart processes to the so-called stochastic local volatility (or hybrid) pricing paradigm. This approach combines the advantages of local and stochastic volatility models. Despite the growing interest on the topic, however, it seems that no particular attention has been paid to the use of multidimensional specifications for the stochastic volatility component. Our work tries to fill the gap: we introduce two hybrid models in which the stochastic volatility dynamics is described by means of a Wishart process. The proposed parametrizations not only preserve the desirable features of existing Wishart-based models but significantly enhance the ability of reproducing market prices of vanilla options.

     

Option pricing with discrete time jump processes

In this paper we propose new option pricing models based on class of models with jumps contained in the Lévy-type based models (NIG-Lévy, Schoutens, 2003, Merton-jump, Merton, 1976 and Duan based model, Duan et al., 2007). By combining these different classes of models with several volatility dynamics of the GARCH type, we aim at taking into account the dynamics of financial returns in a realistic way. The associated risk neutral dynamics of the time series models is obtained through two different specifications for the pricing kernel: we provide a characterization of the change in the probability measure using the Esscher transform and the Minimal Entropy Martingale Measure. We finally assess empirically the performance of this modelling approach, using a dataset of European options based on the S&P 500 and on the CAC 40 indices. Our results show that models involving jumps and a time varying volatility provide realistic pricing and hedging results for options with different kinds of time to maturities and moneyness. These results are supportive of the idea that a realistic time series model can provide realistic option prices making the approach developed here interesting to price options when option markets are illiquid or when such markets simply do not exist.     

State Prices of Conditional Quantiles: New Evidence on Time Variation in the Pricing Kernel

We develop a set of statistics to represent the option‐implied stochastic discount factor and we apply them to S&P 500 returns between 1990 and 2012. Our statistics, which we call state prices of conditional quantiles (SPOCQ), estimate the market's willingness to pay for insurance against outcomes in various quantiles of the return distribution. By estimating state prices at conditional quantiles, we separate variation in the shape of the pricing kernel from variation in the probability of a particular event. Thus, without imposing strong assumptions about the distribution of returns, we obtain a novel view of pricing‐kernel dynamics. We document six features of SPOCQ for the S&P 500. Most notably, and in contrast to recent studies, we find that the price of downside risk decreases when volatility increases. Under a standard asset pricing model, this result implies that most changes in volatility stem from fluctuations in idiosyncratic risk. Consistent with this interpretation, no known systematic risk factors such as consumer sentiment, liquidity or macroeconomic risk can account for the negative relationship between the price of downside risk and volatility. Copyright © 2016 John Wiley & Sons, Ltd.     

Valuation of double trigger catastrophe options with counterparty risk

This study presents a novel catastrophe option pricing model that considers counterparty risk. Asset prices are modeled through a jump-diffusion process which is correlated to counterparty loss process and collateral assets. Because of the long term of catastrophe options, this study also examines the model in the stochastic interest rate environment. The numerical results indicate that counterparty risk significantly affects the value of options. Recently, numerous serious financial events have demonstrated the importance of counterparty risk when valuing financial products.     

Hermite polynomial based expansion of European option prices

We seek a closed-form series approximation of European option prices under a variety of diffusion models. The proposed convergent series are derived using the Hermite polynomial approach. Departing from the usual option pricing routine in the literature, our model assumptions have no requirements for affine dynamics or explicit characteristic functions. Moreover, convergent expansions provide a distinct insight into how and on which order the model parameters affect option prices, in contrast with small-time asymptotic expansions in the literature. With closed-form expansions, we explicitly translate model features into option prices, such as mean-reverting drift and self-exciting or skewed jumps. Numerical examples illustrate the accuracy of this approach and its advantage over alternative expansion methods.     

Dynamic advertising and pricing with constant demand elasticities

In this paper we analyze a stochastic dynamic advertising and pricing model with isoelastic demand. The state space is discrete, time is continuous and the planing horizon is allowed to be finite or infinite. A dynamic version of the Dorfman–Steiner identity will be derived. Explicit expressions of the optimal advertising and pricing policies, of the value function and of the optimal advertising expenditures will be given. The general results will be used to analyze the case of impatient customers. Furthermore, particular time inhomogeneous models and homogeneous ones with and without discounting will be examined. We will study the social efficiency of a monopolist's optimal policies and the consequences of specific subsidies. From a buyer's perspective, our analysis reveals that waiting – when looking at (immediate) expected prices – is never profitable should two or more units be available. But we will also prove that the sequence of average sales prices is monotone decreasing. Moreover, the techniques applied to solve the discrete stochastic advertising and pricing problem will be used to solve a related deterministic control problem with continuous state space.     

Real options and contingent convertibles with regime switching

We consider a firm with no assets in place but an option to invest in a project. The investment is irreversible but delayable in a regime-switching economy. The firm issues equity, straight bonds (SBs) and contingent convertibles (CoCos). We provide the closed-form prices for the firm׳s securities and the pricing and timing of the option. Our numerical analyses discover that issuing CoCos instead of SBs induces much less agency cost of debt. The agency cost is higher in a boom economy than in recession but the difference is small. There is a unique CoCos׳ conversion ratio such that the agency cost arrives at the minimum value zero. The inefficiencies arising from asset substitution and debt overhang are much more significant in recession than in boom. Only if the conversion ratio is not too small, the two inefficiencies disappear during boom periods. While the effects of the conversion rate on optimal capital structure and firm value and those of supervision and jump intensity on optimal CoCos׳ coupon are ambiguous and weak, the stricter the supervision or the longer the economy remains in recession, the less the option value and the optimal SBs׳ coupon.