Hedging Of Options Research Articles

An iterated Azéma–Yor type embedding for finitely many marginals

We solve the $n$-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures $\mu_{1},\ldots,\mu_{n}$ which are in convex order and satisfy an additional technical assumption. Our construction is explicit and is a multiple marginal generalization of the Azema and Yor [In Seminaire de Probabilites, XIII (Univ. Strasbourg, Strasbourg, 1977/78) (1979) 90–115 Springer] solution. In particular, we recover the stopping boundaries obtained by Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558–578] and Madan and Yor [Bernoulli 8 (2002) 509–536]. Our technical assumption is necessary for the explicit embedding, as demonstrated with a counterexample. We discuss extensions to the general case giving details when $n=3$. In our analysis we compute the law of the maximum at each of the $n$ stopping times. This is used in Henry-Labordere et al. [Ann. Appl. Probab. 26 (2016) 1–44] to show that the construction maximizes the distribution of the maximum among all solutions to the $n$-marginal Skorokhod embedding problem. The result has direct implications for robust pricing and hedging of Lookback options.

Hedging and pricing illiquid options with market impacts

In this paper, we consider hedging and pricing of illiquid options on an untradable underlying asset, where an alternative asset is used as a hedging instrument. Particularly, we consider the situation where the trade price of the hedging instrument is subject to market impacts caused by the hedger and the liquidity costs paid as a spread from the mid price. Pricing illiquid options, which often appears in trading of structured products, is a critical issue in practice because of its difficulties in hedging mainly due to untradability of the underlying asset as well as the liquidity costs and market impacts of the hedging instrument. First, by setting the problem under a discrete time model, where the optimal hedging strategy is defined by the local risk-minimization, we present algorithms to obtain the option price along with the hedging strategy by an asymptotic expansion. Moreover, we provide numerical examples. This model enables the estimation of the effect of both the market impacts and the liquidity costs on option prices, which is important in practice.

Errata for the article “Pricing and static hedging of American-style options under the jump to default extended CEV model”

Abstract This errata corrects an error in Ruas et al. (2013, Equation 27) and updates the numerical results contained in Ruas et al. (2013, Tables 4 and 5). The material provided here is meant to be read strictly in conjunction with Ruas et al. (2013).

Optimal hedging with basis risk under mean–variance criterion

Basis risk occurs naturally in a number of financial and insurance risk management problems. A notable example is in the context of hedging a derivative where the underlying security is either non-tradable or not sufficiently liquid. Other examples include hedging longevity risk using index-based longevity instrument and hedging crop yields using weather derivatives. These applications give rise to basis risk and it is imperative that such a risk needs to be taken into consideration for the adopted hedging strategy. In this paper, we consider the problem of hedging a European option using another correlated and liquidly traded asset and we investigate an optimal construction of hedging portfolio involving such an asset. The mean–variance criterion is adopted to evaluate the hedging performance, and a subgame Nash equilibrium is used to define the optimal solution. The problem is solved by resorting to a dynamic programming procedure and a change-of-measure technique. A closed-form optimal control process is obtained under a diffusion model setup. The solution we obtain is highly tractable and to the best of our knowledge, this is the first time the analytical solution exists for dynamic hedging of general European options with basis risk under the mean–variance criterion. Examples on hedging European call options are presented to foster the feasibility and importance of our optimal hedging strategy in the presence of basis risk.

Research on the indexed stock options’ quadratic hedging under jump-diffusion settings

This paper studied the hedging of European indexed stock options, whose exercise price is a variable varying with the market index. Firstly, using the expectation of the squared error between option’s terminal value and hedging portfolio to measure market risk , and with the constraint of self-financing, a squared hedge decision model was built; Then, by using the Dynamic Programming Principle and backward recursion method, the optimal hedging positions’ analytical expressions were acquired; Lastly, all hedging results with different option maturities and different position rebalancing frequencies show that there is dependent relationship between hedging positions and option’s maturity time limit; hedging positions also rely on the relative variation of the underlying asset price.

ROBUST TRADING OF IMPLIED SKEW

In this paper, we present a method for constructing a (static) portfolio of co-maturing European options whose price sign is determined by the skewness level of the associated implied volatility. This property holds regardless of the validity of a specific model — i.e. the method is robust. The strategy is given explicitly and depends only on one’s beliefs about the future values of implied skewness, which is an observable market indicator. As such, our method allows the use of existing statistical tools to formulate the beliefs, providing a practical interpretation of the more abstract mathematical setting, in which the beliefs are understood as a family of probability measures. One of the applications of the results established herein is a method for trading one’s views on the future changes in implied skew, largely independently of other market factors. Another application of our results provides a concrete improvement of the model-independent super-replication and sub-replication strategies for barrier options proposed in [H. Brown, D. Hobson & L. C. G. Rogers (2001) Robust hedging of barrier options, Mathematical Finance 11 (3), 285–314.], which exploits the given beliefs on the implied skew. Our theoretical results are tested empirically, using the historical prices of S&P 500 options.

Optimal Hedging with Basis Risk under Mean-Variance Criterion

Basis risk occurs naturally in a number of financial and insurance risk management problems. A notable example is in the context of hedging a derivative where the underlying security is either non-tradable or not sufficiently liquid. Other examples include hedging longevity risk using index-based longevity instrument and hedging crop yields using weather derivatives. These applications give rise to basis risk and it is imperative that such a risk needs to be taken into consideration for the adopted hedging strategy. In this paper, we consider the problem of hedging a European option using another correlated and liquidly traded asset and we investigate an optimal construction of hedging portfolio involving such an asset. The mean-variance criterion is adopted to evaluate the hedging performance, and a subgame Nash equilibrium is used to define the optimal solution. The problem is solved by resorting to a dynamic programming procedure and a change-of-measure technique. A closed-form optimal control process is obtained under a diffusion model setup. The solution we obtain is highly tractable and to the best of our knowledge, this is the first time the analytical solution exists for dynamic hedging of general European options with basis risk under the mean-variance criterion. Examples on hedging European call options are presented to foster the feasibility and importance of our optimal hedging strategy in the presence of basis risk.

ON THE NUMERICAL ASPECTS OF OPTIMAL OPTION HEDGING WITH TRANSACTION COSTS

We present a numerical study of non-self-financing hedging of European options under proportional transaction costs. We describe an algorithmic approach based on a discrete time financial market model that extends the classical binomial model. We review the analytical basis for our algorithm and present a variety of empirical results using real market data. The performance of the algorithm is evaluated by comparing to a Black–Scholes delta hedge with transaction costs incorporated. We also evaluate the impact of recalibrating the hedging strategy one or more times during the life of the option using the most recent market data. These results are compared to a recalibrated Black–Scholes delta hedge modified for transaction costs.

Optimal Hedging of Basket Barrier Options with Additive Models and Its Application to Equity Value Separation Problem

At the heart of optimal hedging with additive models in Yamada (Recent advances in financial engineering: proceedings of the KIER-TMU international workshop on financial engineering, World Scientific, pp 225–245, 2010; Proceedings of the 2011 American control conference, pp 3856–3861, 2011; Asia-Pac Financ Mark 19(2):149–179, 2012) is to replicate the payoff of European basket options using separate options as close as possible. In this paper, we extend their technique for the case of path-dependent barrier options, where the mean square error of the payoffs between the basket barrier option and the sum of options on the individual assets is minimized over any smooth payoff functions. To this end, we propose to represent the underlying assets using the Brownian bride decomposition and show that computations involving conditional expectations of basket barrier options boil down to those of unconditional expectations. This procedure enables us to provide an algorithm to compute the necessary and sufficient condition for the optimal hedging problem based on the Monte Carlo method. Then, we consider to apply our methodology to the Black–Cox type first passage time structural model, where a defaultable company possesses/runs multiple assets/projects and the default may occur the first time the asset value hits a certain lower threshold before the maturity. We formulate the equity value separation problem using additive models, in which individual equity values are introduced so that their sum approximates the total equity value as close as possible. It is also shown that any portion of total equity value may be assigned as an initial value of each individual equity when using the optimal smooth functions. Finally, we examine the contributions of individual equity values to default or survival by applying a certain normalization for conditional expectations via numerical experiments to illustrate our proposed methodology.

Distributed Calibration of Option Pricing Models with Multiple Contracts Written on Different Underlying Assets

Think about a situation, where a financial institution has multiple option positions, each written on a different underlying asset, and the unexpected arrival of market-wide news shakes the markets. In the case of such a market-wide news arrival, all the volatility models on different underlyings must be immediately re-calibrated for robust option pricing and hedging. Unfortunately, the calibration of models using data on multiple underlying securities can take too long, especially, if advanced non-affine models without analytical solutions are used. This work demonstrates how multiple independent calibration tasks (on different underlyings) can be efficiently accelerated by Techila, reducing the length of the wall-clock computation time from 7 hours to a few minutes.

Analysis of Risk Management Strategies for Contingent Convertible Bonds

The contingent convertible bond (CoCo) is a structured instrument that emerged at the end of 2009. This paper explores the CoCo risk management strategy from the standpoint of investors. Taking the Equity Derivation Law as its framework, this study analyzes the hedging performance based on the static hedging of options and then introduces jumps risk to allow sudden bank defaults, observing the changes in hedging performance. By scenario analysis, this study finds that CoCo can control its investment risks via equity derivatives and that static hedging can effectively reduce the standard deviation and value-at-risk (VaR). Key words: Contingent convertible bonds, static hedge, hedging performance, value at risk

Analysis of Risk Management Strategies for Contingent Convertible Bonds=或有可轉債之風險管理策略分析

The contingent convertible bond (CoCo) is a structured instrument that emerged at the end of 2009. This paper explores the CoCo risk management strategy from the standpoint of investors. Taking the Equity Derivation Law as its framework, this study analyzes the hedging performance based on the static hedging of options and then introduces jumps risk to allow sudden bank defaults, observing the changes in hedging performance. By scenario analysis, this study finds that CoCo can control its investment risks via equity derivatives and that static hedging can effectively reduce the standard deviation and value-at-risk (VaR). Key words: Contingent convertible bonds, static hedge, hedging performance, value at risk

PRICING AND HEDGING GAME OPTIONS IN CURRENCY MODELS WITH PROPORTIONAL TRANSACTION COSTS

The pricing, hedging, optimal exercise and optimal cancellation of game or Israeli options are considered in a multi-currency model with proportional transaction costs. Efficient constructions for optimal hedging, cancellation and exercise strategies are presented, together with numerical examples, as well as probabilistic dual representations for the bid and ask price of a game option.

Weather Risk Management in Agriculture

The paper focuses on valuation of a weather derivative with payoffs depending on temperature. We use historical data from the weather station in the Slovak town Košice to obtain unique prices of option contracts in an incomplete market. Numerical examples of prices of some contracts are presented, using the Burn analysis. We provide an example of how a weather contract can be designed to hedge the financial risk of a suboptimal temperature condition. The comparative comparison of the selected option hedging strategies has shown the best results for the producers in agricultural industries who hedges against an unfavourable weather conditions. The results of analysis proved that by buying put option or call option, the farmer establishes the highest payoff in the case of temperature decrease or increase. The Long Straddle Strategy is the most expensive but is available to the farmer who hedges against a high volatility in temperature movement. We conclude with the findings that weather derivatives could be useful tools to diminish the financial losses for agricultural industries highly dependent for temperature.

Move-based hedging of variable annuities: A semi-analytic approach

In this paper, we propose a semi-analytic algorithm for measuring the mean and variance of the cost associated with a two-sided move-based hedging of options written on an underlying asset whose price follows a geometric Brownian motion. Numerical examples are presented to illustrate the computational accuracy and efficiency of the algorithm. We then apply the technique to a structured product-based variable annuity with buffered protection and an annual ratchet variable annuity.

Hedging an option portfolio with minimum transaction lots: A fuzzy goal programming problem

Options are designed to hedge against risks to their underlying assets such as stocks. One method of forming option-hedging portfolios is using stochastic programming models. Stochastic programming models depend heavily on scenario generation, a challenging task. Another method is neutralizing the Greek risks derived from the Black–Scholes formula for pricing options. The formula expresses the option price as a function of the stock price, strike price, volatility, risk-free interest rate, and time to maturity. Greek risks are the derivatives of the option price with respect to these variables. Hedging Greek risks requires no human intervention for generating scenarios. Linear programming models have been proposed for constructing option portfolios with neutralized risks and maximized investment profit. However, problems with these models exist. First, feasible solutions that can perfectly neutralize the Greek risks might not exist. Second, models that involve multiple assets and their derivatives were incorrectly formulated. Finally, these models lack practicability because they consider no minimum transaction lots. Considering minimum transaction lots can exacerbate the infeasibility problem. These problems must be resolved before option hedging models can be applied further. This study presents a revised linear programming model for option portfolios with multiple underlying assets, and extends the model by incorporating it with a fuzzy goal programming method for considering minimum transaction lots. Numerical examples show that current models failed to obtain feasible solutions when minimum transaction lots were considered. By contrast, while the proposed model solved the problems efficiently.

Option Hedging with Smooth Market Impact

We consider intraday hedging of an option position, for a large trader who experiences temporary and permanent market impact. We formulate the general model including overnight risk, and solve explicitly in two cases which we believe are representative. The first case is an option with approximately constant gamma: the optimal hedge trades smoothly towards the classical Black–Scholes delta, with trading intensity proportional to instantaneous mishedge and inversely proportional to illiquidity. The second case is an arbitrary non-linear option structure but with no permanent impact: the optimal hedge trades toward a value offset from the Black–Scholes delta. We estimate the effects produced on the public markets if a large collection of traders all hedge similar positions. We construct a stable hedge strategy with discrete time steps.

The Binomial CEV Model and the Greeks

AbstractThis article compares alternative binomial approximation schemes for computing the option hedge ratios studied by Chung and Shackleton (2002), Chung, Hung, Lee, and Shih (2011), and Pelsser and Vorst (1994) under the lognormal assumption, but now considering the constant elasticity of variance (CEV) process proposed by Cox (1975) and using the continuous‐time analytical Greeks recently offered by Larguinho, Dias, and Braumann (2013) as the benchmarks. Among all the binomial models considered in this study, we conclude that an extended tree binomial CEV model with the smooth and monotonic convergence property is the most efficient method for computing Greeks under the CEV diffusion process because one can apply the two‐point extrapolation formula suggested by Chung et al. (2011). © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:90–104, 2017

Pricing and Hedging of European Plain Vanilla Options under Jump Uncertainty

This paper studies the pricing and hedging problem of European plain vanilla options in a modified Black–Scholes market. That is the price of the risky asset is allowed to jump, where the timing and the size of the jump is unknown (with no jump being possible as well). Using a superhedging approach, worst case pricing formulae, Greeks, and superhedging strategy for call and put options will be given in closed form (where the closed form is of the same level as the Black–Scholes solution) and will be discussed. Moreover, the worst case prices explain the volatility smile which can be observed in market data. Finally, the model is calibrated to market data.

Risk-minimizing pricing and hedging foreign currency options under regime-switching jump-diffusion models

ABSTRACTThis article mainly investigates risk-minimizing European currency option pricing and hedging strategy when the spot foreign exchange rate is driven by a Markov-modulated jump-diffusion model. We suppose the domestic and foreign money market floating interest rates, the drift, and the volatility of the exchange rate dynamics all depend on the state of the economy, which is modeled by a continuous-time hidden Markov chain. The model considered in this article will provide market practitioners with flexibility in characterizing the dynamics of the spot foreign exchange rate. Using the minimal martingale measure, we obtain a system of coupled partial-differential-integral equations satisfied by the currency option price and find the corresponding hedging strategies and the residual risk. According to simulation of currency option prices in the special case of double exponential jump-diffusion regime-switching model, we further discuss and show the effects of the parameters on the prices.