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.     

On pricing and hedging options in regime-switching models with feedback effect

We study the pricing and hedging of European-style derivative securities in a Markov, regime-switching, model with a feedback effect depending on the economic condition. We adopt a pricing kernel which prices both financial and economic risks explicitly in a dynamically incomplete market and we provide an equilibrium analysis. A martingale representation for a European-style index option's price is established based on the price kernel. The martingale representation is then used to construct the local risk-minimizing strategy explicitly and to characterize the corresponding pricing measure.     

Generalized affine transform on pricing quanto range accrual note

This paper was to price and hedge a quanto floating range accrual note (QFRAN) by an affine term structure model with affine-jump processes. We first generalized the affine transform proposed by Duffie et al. (2000) under both the domestic and foreign risk-neutral measures with a change of measure, which provides a flexible structure to value quanto derivatives. Then, we provided semi-analytic pricing and hedging solutions for QFRAN under a four-factor affine-jump model with the stochastic mean, stochastic volatility, and jumps. The numerical results demonstrated that both the common and local factors significantly affect the value and hedging strategy of QFRAN. Notably,  the factor of stochastic mean plays the most important role in either valuation or hedging. This study suggested that ignorance of these factors in a term-structure model will result in significant pricing and hedging errors in QFRAN. In summary, this study provided flexible and easily implementable solutions in valuing quanto derivatives.     

股指期货的最优套期保值率实证研究——基于沪深300指数期货仿真交易视角

在分析和比较常用的几种股指期货最优套期保值比率确定模型的基础上,基于风险最小化模型框架,利用沪深300指数期货合约模拟运行以来的样本数据,通过最小二乘回归模型、向量自回归模型、误差修正模型以及广义自回归条件异方差模型四种估计方法,对其最优套期保值比率进行了实证测算和绩效比较,提出了相应的政策建议和投资策略。     

Mean-variance hedging for interest rate models¶with stochastic volatility

The mean-variance hedging approach for pricing and hedging claims in incomplete markets was originally introduced for risky assets. The aim of this paper is to apply this approach to interest rate models in the presence of stochastic volatility, seen as a consequence of incomplete information. We fix a finite number of bonds such that the volatility matrix is invertible and provide an explicit formula for the density of the variance-optimal measure which is independent of the chosen times of maturity. Finally, we compute the mean-variance hedging strategy for a caplet and compare it with the optimal stategy according to the local risk minimizing approach. Received: 14 July 2000 / Accepted: 10 April 2001     

Option hedging theory under transaction costs

The problem of option hedging in the presence of proportional transaction costs can be formulated as a singular stochastic control problem. Hodges and Neuberger [1989. Optimal replication of contingent claims under transactions costs. Review of Futures Markets 8, 222–239] introduced an approach that is based on maximization of the expected utility of terminal wealth. We develop a new algorithm to solve the corresponding singular stochastic control problem and introduce a new approach to option hedging which is closer in spirit to the pathwise replication of Black and Scholes [1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–654]. This new approach is based on minimization of a Black–Scholes-type measure of pathwise risk, defined in terms of a market delta, subject to an upper bound on the hedging cost. We provide an efficient backward induction algorithm for the problem of cost-constrained risk minimization, whose associated singular stochastic control problem is shown to be equivalent to an optimal stopping problem. This algorithm is then modified to solve the singular stochastic control problem associated with utility maximization, which cannot be reduced to an optimal stopping problem. We propose to choose an optimal parameter (risk-aversion coefficient or Lagrange multiplier) in either approach by minimizing the mean squared hedging error and demonstrate that with this “best” choice of the parameter, both approaches have similar performance. We also discuss the different notions of risk in both approaches and propose a volatility adjustment for the risk-minimization approach, which is analogous to that introduced by Zakamouline [2006. European option pricing and hedging with both fixed and proportional transaction costs. Journal of Economic Dynamics and Control 30, 1–25] for the utility maximization approach, thereby providing a unified treatment of both approaches.     

Measure preserving derivatives and the pricing kernel puzzle

Recent empirical studies have found evidence of nonmonotonicity in the pricing kernels for a variety of market indices. This phenomenon is known as the pricing kernel puzzle. The payoff distribution pricing model of Dybvig predicts that the payoff distribution of a direct investment of $1 in a market index may be replicated by investing less than $1 in some derivative written on that market index whenever the associated pricing kernel is nondecreasing. Using the Hardy–Littlewood rearrangement inequality, we obtain an explicit solution for the cheapest replicating derivative, which we refer to as the optimal measure preserving derivative. The optimal measure preserving derivative is the permutation appearing in Ryff’s decomposition of the pricing kernel with respect to the market payoff measure. We compute optimal measure preserving derivatives corresponding to the estimated physical and risk neutral distributions in the paper by Jackwerth (2000) that first brought attention to the pricing kernel puzzle.     

Bivariate garch estimation of the optimal commodity futures Hedge

Six different commodities are examined using daily data over two futures contract periods. Cash and futures prices for all six commodities are found to be well described as martingales with near-integrated GARCH innovations. Bivariate GARCH models of cash and futures prices are estimated for the same six commodities. The optimal hedge ratio (OHR) is then calculated as a ratio of the conditional covariance between cash and futures to the conditional variance of futures. The estimated OHRs reveal that the standard assumption of a time-invariant OHR is inappropriate. For each commodity the estimated OHR path appears non-stationary, which has important implications for hedging strategies.     

EVT and tail-risk modelling: Evidence from market indices and volatility series

Value-at-Risk (VaR) has become the universally accepted risk metric adopted internationally under the Basel Accords for banking industry internal control, capital adequacy and regulatory reporting. The recent extreme financial market events such as the Global Financial Crisis (GFC) commencing in 2007 and the following developments in European markets mean that there is a great deal of attention paid to risk measurement and risk hedging. In particular, to risk indices and attached derivatives as hedges for equity market risk. The techniques used to model tail risk such as VaR have attracted criticism for their inability to model extreme market conditions. In this paper we discuss tail specific distribution based Extreme Value Theory (EVT) and evaluate different methods that may be used to calculate VaR ranging from well known econometrics models of GARCH and its variants to EVT based models which focus specifically on the tails of the distribution. We apply Univariate Extreme Value Theory to model extreme market risk for the FTSE100 UK Index and S&P-500 US markets indices plus their volatility indices. We show with empirical evidence that EVT can be successfully applied to financial market return series for predicting static VaR, CVaR or Expected Shortfall (ES) and also daily VaR and ES using a GARCH(1,1) and EVT based dynamic approach to these various indices. The behaviour of these indices in their tails have implications for hedging strategies in extreme market conditions.     

基于非线性相关的最小方差套期保值比率研究

在最小方差套期保值模型的基础上,提出了最小方差套期保值的期货与现货波动非线性对冲原理,利用非线性相关系数代替传统的线性相关系数,提高了套期保值比率的准确性。提出了套期保值的收益率波动预测原理,利用GARCH(1,1)预测期货收益率的方差,利用EWMA模型预测现货收益率的方差,解决了收益率在历史期和套期保值期间因收益率波动发生结构性变化所导致的套期保值效果失真的问题。实证研究结果表明,本研究的套期保值比率的有效性高于现有模型,应用本研究模型进行套期保值,可以有效规避现货价格风险。     

Inflation and interest rate derivatives for FX risk management: Implications for exporting firms under real wealth

Firms that export goods face risks such as product price, cost, and exchange rate risks. Price and cost risks can substantially reduce the FX hedging performance in real wealth. We thus investigate hedging strategies that are intended to improve the performance of the FX hedge in real terms using inflation and interest rate derivatives. The impact of these additional instruments is not clear and has only been briefly analyzed in the hedging literature so far. For this purpose, we derive variance-minimizing hedge positions of an exporting firm. A cointegrated VAR and bootstrap methods are used to evaluate the efficiencies of several hedging strategies. While inflation derivatives work better in the short run, interest rate derivatives perform better over longer hedge horizons.     

Weak convergence of tree methods,to price options on defaultable assets

Abstract We discuss a practical method to price and hedge European contingent claims on assets with price processes which follow a jump-diffusion. The method consists of a sequence of trinomial models for the asset price and option price processes which are shown to converge weakly to the corresponding continuous time jump-diffusion processes. The main difference with many existing methods is that our approach ensures that the intermediate discrete time approximations generate models which are themselves complete, just as in the Black-Scholes binomial approximations. This is only possible by dropping the assumption that the approximations of increments of the Wiener and Poisson processes on our trinomial tree are independent, but we show that the dependence between these processes disappears in the weak limit. The approximations thus define an easy and flexible method for pricing and hedging in jump-diffusion models using explicit trees for hedging and pricing. Mathematics Subject Classification (2000): 60B10, 60H35 Journal of Economic Literature Classification: G13     

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.     

Swap pricing and hedging of general DCFs

This paper shows that under payoff and/or interest rate uncertainty the splitting up of discounted cash flows (DCFs) into period addenda not only permits to quantify the contribution of each period to the total (random) DCF, but also allows us to price it and hedge the its risk. The contribution effect was already well known, while the pricing and hedging issue is new in this context. We first notice that—through the decomposition—each cash flow process can be interpreted as a swap one. We then resort to the risk-neutral pricing and hedging technique as applied to (exotic) swaps.     

Risk management of precious metals

This paper examines volatility and correlation dynamics in price returns of gold, silver, platinum and palladium, and explores the corresponding risk management implications for market risk and hedging. Value-at-Risk (VaR) is used to analyze the downside market risk associated with investments in precious metals, and to design optimal risk management strategies. We compute the VaR for major precious metals using the calibrated RiskMetrics, different GARCH models, and the semi-parametric Filtered Historical Simulation approach. The best approach for estimating VaR based on conditional and unconditional statistical tests is documented. The economic importance of the results is highlighted by assessing the daily capital charges from the estimated VaRs.     

Implied risk aversion and pricing kernel in the FTSE 100 index

This paper studies the estimation of the pricing kernel and explains the pricing kernel puzzle found in the FTSE 100 index. We use prices of options and futures on the FTSE 100 index to derive the risk neutral density (RND). The option-implied RND is inverted by using two nonparametric methods: the implied-volatility surface interpolation method and the positive convolution approximation (PCA) method. The actual density distribution is estimated from the historical data of the FTSE 100 index by using the threshold GARCH (TGARCH) model. The results show that the RNDs derived from the two methods above are relatively negatively skewed and fat-tailed, compared to the actual probability density, that is consistent with the phenomenon of “volatility smile.” The derived risk aversion is found to be locally increasing at the center, but decreasing at both tails asymmetrically. This is the so-called pricing kernel puzzle. The simulation results based on a representative agent model with two state variables show that the pricing kernel is locally increasing with the wealth at the level of 1 and is consistent with the empirical pricing kernel in shape and magnitude.     

Hedging local currency risk with precious metals

Precious metals are popular instruments for hedging local currency risk. Most precious metals are priced in US dollars, which is a single-currency numéraire. The numéraire of a precious metal can easily be changed, which allows investors to choose their own local currency as the precious metal’s numéraire and subsequently use it to hedge their own local currency risk. In this paper, we decompose the standard hedge ratio into two parts, namely, a precious-metal hedge ratio and a local-currency hedge ratio. We consider three main precious metals, namely, gold, silver and platinum, from the beginning of 1990 to the end of 2019 to hedge local currency risk for individual G10 currencies. Over the full sample, we find that all standard hedge ratios are negative for all combinations of currencies and precious metals. However, the negativity is driven by the precious metal’s numéraire, rather than the precious metal.     

Corporate hedging and input price risk

Input price variability is an important source of risk for corporations that process raw commodities. Models of optimal input hedging are developed in this paper based on the maximization of managerial expected utility. The relationship between hedging strategies and output decisions is examined to assess the impact of the ability to set output prices on futures market participation. As a firm's ability to set output prices diminishes in the short run, input futures positions increase although the optimal hedge ratio may either increase or decrease. For a perfectly competitive firm, however, shifts in output price caused by input price changes provide a natural cash market hedge of input price risk and reduce the firm's optimal input futures position.     

Crack spread hedging: accounting for time‐varying volatility spillovers in the energy futures markets

Crude oil, heating oil, and unleaded gasoline futures contracts are simultaneously analysed for their effectiveness in reducing price volatility for an energy trader. A conceptual model is developed for a trader hedging the ‘crack spread’. Various hedge ratio estimation techniques are compared to a Multivariate GARCH model that directly incorporates the time to maturity effect often found in futures markets. Modelling of the time‐variation in hedge ratios via the Multivariate GARCH methodology, and thus taking into account volatility spillovers between markets is shown to result in significant reductions in uncertainty even while accounting for trading costs. Copyright © 2002 John Wiley & Sons, Ltd.     

The valuation of American barrier options using the decomposition technique

In this paper, we propose an alternative approach for pricing and hedging American barrier options. Specifically, we obtain an analytic representation for the value and hedge parameters of barrier options, using the decomposition technique of separating the European option value from the early exercise premium. This allows us to identify some new put-call ‘symmetry’ relations and the homogeneity in price parameters of the optimal exercise boundary. These properties can be utilized to increase the computational efficiency of our method in pricing and hedging American options. Our implementation of the obtained solution indicates that the proposed approach is both efficient and accurate in computing option values and option hedge parameters. Our numerical results also demonstrate that the approach dominates the existing lattice methods in both accuracy and efficiency. In particular, the method is free of the difficulty that existing numerical methods have in dealing with spot prices in the proximity of the barrier, the case where the barrier options are most problematic.