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.     

Approximation and Calibration of Short-Term Implied Volatilities Under Jump-Diffusion Stochastic Volatility

We derive an asymptotic expansion formula for option impliedvolatility under a two-factor jump-diffusion stochastic volatilitymodel when time-to-maturity is small. We further propose a simplecalibration procedure of an arbitrary parametric model to short-termnear-the-money implied volatilities. An important advantageof our approximation is that it is free of the unobserved spotvolatility. Therefore, the model can be calibrated on optiondata pooled across different calendar dates to extract informationfrom the dynamics of the implied volatility smile. An exampleof calibration to a sample of S&P 500 option prices is provided.(JEL G12)     

An endogenous volatility approach to pricing and hedging call options with transaction costs

Standard delta hedging fails to exactly replicate a European call option in the presence of transaction costs. We study a pricing and hedging model similar to the delta hedging strategy with an endogenous volatility parameter for the calculation of delta over time. The endogenous volatility depends on both the transaction costs and the option strike prices. The optimal hedging volatility is calculated using the criterion of minimizing the weighted upside and downside replication errors. The endogenous volatility model with equal weights on the up and down replication errors yields an option premium close to the Leland [J. Finance, 1985 Leland, HE. 1985. Option pricing and replication with transaction costs. J. Finance, 40: 1283–1301. [Crossref], [Web of Science ®] , [Google Scholar], 40, 1283–1301] heuristic approach. The model with weights being the probabilities of the option's moneyness provides option prices closest to the actual prices. Option prices from the model are identical to the Black–Scholes option prices when transaction costs are zero. Data on S&P 500 index cash options from January to June 2008 illustrate the model.     

Generic pricing of FX,inflation and stock options under stochastic interest rates and stochastic volatility

We consider the pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, for which we use a generic multi-currency framework. We allow for a general correlation structure between the drivers of the volatility, the inflation index, the domestic (nominal) and the foreign (real) rates. Having the flexibility to correlate the underlying FX/inflation/stock index with both stochastic volatility and stochastic interest rates yields a realistic model that is of practical importance for the pricing and hedging of options with a long-term exposure. We derive explicit valuation formulas for various securities, such as vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. These vanilla derivatives can be valued in closed form under Schöbel and Zhu [Eur. Finance Rev., 1999, 4, 23–46] stochastic volatility, whereas we devise an (Monte Carlo) approximation in the form of a very effective control variate for the general Heston [Rev. Financial Stud., 1993, 6, 327–343] model. Finally, we investigate the quality of this approximation numerically and consider a calibration example to FX and inflation market data.     

Reading the smile: the message conveyed by methods which infer risk neutral densities

In this study we compare the quality and information content of risk neutral densities obtained by various methods. We consider a non-parametric method based on a mixture of log–normal densities, the semi-parametric ones based on an Hermite approximation or based on an Edgeworth expansion, the parametric approach of Malz which assumes a jump-diffusion for the underlying process, and Heston's approach assuming a stochastic volatility model. We apply those models on FF/DM exchange rate options for two dates. Models differ when important news hits the market (here anticipated elections). The non-parametric model provides a good fit to options prices but is unable to provide as much information about market participants expectations than the jump-diffusion model.     

Asymptotic replication with modified volatility under small transaction costs

We consider the dynamic hedging of a European option under a general local volatility model with small proportional transaction costs. Extending the approach of Leland, we introduce a class of continuous strategies of finite cost that asymptotically (super-)replicate the payoff. An associated central limit theorem for the hedging error is proved. We also obtain an explicit trading strategy minimizing the asymptotic error variance.     

Pricing methods and hedging strategies for volatility derivatives

We explore the valuation and hedging of discretely observed volatility derivatives using three different models for the price of the underlying asset: Geometric Brownian motion with constant volatility, a local volatility surface, and jump-diffusion. We begin by comparing the effects on valuation of variations in contract design, such as the differences between specifying log returns or actual returns and incorporating caps on the level of realized volatility. We then focus on the difficulties associated with hedging these products. Delta hedging strategies are ineffective for hedging volatility derivatives since they require very frequent rebalancing. Moreover, they provide limited protection in the jump-diffusion context. We study the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in vanilla options at each volatility observation.     

Robustly Hedging Variable Annuities With Guarantees Under Jump and Volatility Risks

Recent variable annuities offer participation in the equity market and attractive protection against downside movements. Accurately quantifying this additional equity market risk and robustly hedging options embedded in the guarantees of variable annuities are new challenges for insurance companies. Due to sensitivities of the benefits to tails of the account value distribution, a simple Black–Scholes model is inadequate in preventing excessive liabilities. A model which realistically describes the real world price dynamics over a long time horizon is essential for the risk management of the variable annuities. In this article, both jump risk and volatility risk are considered for risk management of lookback options embedded in guarantees with a ratchet feature. We evaluate relative performances of delta hedging and dynamic discrete risk minimization hedging strategies. Using the underlying as the hedging instrument, we show that, under a Black–Scholes model, local risk minimization hedging can be significantly better than delta hedging. In addition, we compare risk minimization hedging using the underlying with that of using standard options. We demonstrate that, under a Merton's jump diffusion model, hedging using standard options is superior to hedging using the underlying in terms of the risk reduction. Finally, we consider a market model for volatility risks in which the at‐the‐money implied volatility is a state variable. We compute risk minimization hedging by modeling at‐the‐money Black–Scholes implied volatility explicitly; the hedging effectiveness is evaluated, however, under a joint model for the underlying price and implied volatility. Our computational results suggest that, when implied volatility risk is suitably modeled, risk minimization hedging using standard options, compared to hedging using the underlying, can potentially be more effective in risk reduction under both jump and volatility risks.     

Optimal hedging in an extended binomial market under transaction costs

We develop an approach to optimal hedging of a contingent claim under proportional transaction costs in a discrete time financial market model which extends the binomial market model with transaction costs. Our model relaxes the binomial assumption on the stock price ratios to the case where the stock price ratio distribution has bounded support. Non-self-financing hedging strategies are studied to construct an optimal hedge for an investor who takes a short position in a European contingent claim settled by delivery. We develop the theoretical basis for our optimal hedging approach, extending results obtained in our previous work. Specifically, we derive a no-arbitrage option price interval and establish properties of the non-self-financing strategies and their residuals. Based on the theoretical foundation, we develop a computational algorithm for optimizing an investor relevant criterion over the set of admissible non-self-financing hedging strategies. We demonstrate the applicability of our approach using both simulated data and real market data.     

(S,s)-adjustment Strategies and Hedging under Markovian Dynamics

We study the destabilizing effect of hedging strategies under Markovian dynamics with transaction costs. Once transaction costs are taken into account, continuous portfolio rehedging is no longer an optimal strategy. Using a non-optimizing (local in time) strategy for portfolio rebalancing, explicit dynamics for the price of the underlying asset are derived, focusing in particular on excess volatility and feedback effects of these portfolio insurance strategies. Moreover, it is shown how these latter depend on the heterogeneity of the insured payoffs. Finally, conditions are derived under which it may be still reasonable, from a practical viewpoint, to implement Black–Scholes strategies.     

THE ECONOMIC VALUE OF USING REALIZED VOLATILITY IN FORECASTING FUTURE IMPLIED VOLATILITY

We examine the economic benefits of using realized volatility to forecast future implied volatility for pricing, trading, and hedging in the S&P 500 index options market. We propose an encompassing regression approach to forecast future implied volatility, and hence future option prices, by combining historical realized volatility and current implied volatility. Although the use of realized volatility results in superior performance in the encompassing regressions and out-of-sample option pricing tests, we do not find any significant economic gains in option trading and hedging strategies in the presence of transaction costs.     

A new computational tool for analysing dynamic hedging under transaction costs

This paper highlights a framework for analysing dynamic hedging strategies under transaction costs. First, self-financing portfolio dynamics under transaction costs are modelled as being portfolio affine. An algorithm for computing the moments of the hedging error on a lattice under portfolio affine dynamics is then presented. In a number of circumstances, this provides an efficient approach to analysing the performance of hedging strategies under transaction costs through moments. As an example, this approach is applied to the hedging of a European call option with a Black–Scholes delta hedge and Leland's adjustment for transaction costs. Results are presented that demonstrate the range of analysis possible within the presented framework.     

Deep hedging

We present a framework for hedging a portfolio of derivatives in the presence of market frictions such as transaction costs, liquidity constraints or risk limits using modern deep reinforcement machine learning methods. We discuss how standard reinforcement learning methods can be applied to non-linear reward structures, i.e. in our case convex risk measures. As a general contribution to the use of deep learning for stochastic processes, we also show in Section 4 that the set of constrained trading strategies used by our algorithm is large enough to ε-approximate any optimal solution. Our algorithm can be implemented efficiently even in high-dimensional situations using modern machine learning tools. Its structure does not depend on specific market dynamics, and generalizes across hedging instruments including the use of liquid derivatives. Its computational performance is largely invariant in the size of the portfolio as it depends mainly on the number of hedging instruments available. We illustrate our approach by an experiment on the S&P500 index and by showing the effect on hedging under transaction costs in a synthetic market driven by the Heston model, where we outperform the standard ‘complete-market’ solution.     

Option valuation and hedging using an asymmetric risk function: asymptotic optimality through fully nonlinear partial differential equations

Finance and Stochastics - Discrete-time hedging produces a residual P&L, namely the tracking error. The major problem is to get valuation/hedging policies minimising this error. We evaluate...     

Market liquidity,closeout procedures and initial margin for CCPs*

ABSTRACT

Closeout procedures enable central counterparties (CCPs) to respond to events that challenge the continuity of their normal operations, most frequently triggered by the default of one or more clearing members. The procedures typically entail three main phases: splitting, hedging, and liquidation. Together, these ensure the regularity of the settlement process through the prudent and orderly liquidation of the defaulters’ portfolios. Traditional approaches to CCPs’ margin requirements typically assume a simple closeout profile, not accounting for the ‘real life’ constraints embedded in the management of a default. The paper proposes an approach to assess how distinct closeout strategies may expose a CCP to different sets of risks and costs taking into account real-life frictions. The proposed approach enables the evaluation of a full spectrum of hedging strategies and the assessment of the trade-offs between the risk-reducing benefits of hedging and the transaction costs associated with it. Using an unexplored set of transactional level data, the proposed framework is evaluated assuming the hypothetical default of a real CCP clearing member. We consider the worst-case loss of a large interest rate swap portfolio observed over the past 10 years (i.e. 2005–2015) and show that an efficient hedging strategy which minimises risk may not be optimal when transaction costs are taken into account. The empirical analysis suggests that transaction costs are a significant factor and should be accounted for when designing a hedging strategy. Specifically, it is shown that the risk-reducing benefits arising from more tailored hedging strategies may introduce higher transaction costs, and therefore may change the effectiveness of the strategies.     

Efficient analytic approximation of the optimal hedging strategy for a European call option with transaction costs

One of the most successful approaches to option hedging with transaction costs is the utility-based approach, pioneered by Hodges and Neuberger [Rev. Futures Markets, 1989, 8, 222–239]. Judging against the best possible trade-off between the risk and the costs of a hedging strategy, this approach seems to achieve excellent empirical performance. However, this approach has one major drawback that prevents the broad application of this approach in practice: the lack of a closed-form solution. We overcome this drawback by presenting a simple yet efficient analytic approximation of the solution. We provide an empirical testing of our approximation strategy against the asymptotic and some other well-known strategies and find that our strategy outperforms all the others.     

Hedging errors with Leland's option model in the presence of transaction costs

Nonzero transaction costs invalidate the Black–Scholes [1973. Journal of Political Economy 81, 637–654] arbitrage argument based on continuous trading. Leland [1985. Journal of Finance 40, 1283–1301] developed a hedging strategy which modifies the Black–Scholes hedging strategy with a volatility adjusted by the length of the rebalance interval and the rate of the proportional transaction cost. Kabanov and Safarian [1997. Finance and Stochastics 1, 239–250] calculated the limiting hedging error of the Leland strategy and pointed out that it is nonzero for the approximate pricing of an European call option, in contradiction to Leland's claim. As a further contribution, we first identify the mathematical flaw in the argument of Leland's claim and then quantify the expected percentage of hedging losses in terms of the hedging frequency and the level of the option strike price.     

Model risk of contingent claims

Paralleling regulatory developments, we devise value-at-risk and expected shortfall type risk measures for the potential losses arising from using misspecified models when pricing and hedging contingent claims. Essentially, P&L from model risk corresponds to P&L realized on a perfectly hedged position. Model uncertainty is expressed by a set of pricing models, each of which represents alternative asset price dynamics to the model used for pricing. P&L from model risk is determined relative to each of these models. Using market data, a unified loss distribution is attained by weighing models according to a likelihood criterion involving both calibration quality and model parsimony. Examples demonstrate the magnitude of model risk and corresponding capital buffers necessary to sufficiently protect trading book positions against unexpected losses from model risk. A further application of the model risk framework demonstrates the calculation of gap risk of a barrier option when employing a semi-static hedging strategy.     

Enhancing hedging performance with the spanning polynomial projection

Statistical time-series approaches to hedging are difficult to beat, especially out-of-sample, and are capable of out-performing many theory-based derivative pricing model approaches to hedging commodity price risks using futures contracts. However, the vast majority of time-series approaches to hedging discussed in the literature are essentially linear statistical projections, whether univariate or multivariate. Little is known about the potential hedging capabilities of nonlinear methods. This study describes how least-squares orthogonal polynomial approximation methods based on the spanning polynomial projection (SPP) can be used to enhance standard (linear) optimal hedging methods and improve hedging performance for a hedger with a mean–variance objective. Empirical analyses show that the SPP can be used effectively for hedging and gives better out-of-sample hedging performance than the benchmark VEC-GARCH hedging model. Results are robust to the inclusion of transaction costs and risk-aversion assumptions.     

The valuation of catastrophe bonds with exposure to currency exchange risk

In this paper, we present a new model that takes an arbitrage approach to the valuation of catastrophic risk bonds (CAT bonds). The model considers the sponsor's exposure to currency exchange risk and the risk of catastrophic events. We use a jump-diffusion process for catastrophic events, a three-dimensional stochastic process for the exchange rate and domestic and foreign interest rates, and a hedging cost for the currency risk to derive a semi-closed-form formula for the CAT bond price. We also extend to three factors Joshi and Leung's (2007) Monte Carlo simulation approach to obtain numerical results showing the following: in addition to catastrophic risk, the CAT bond price is affected mainly by the volatility of the exchange rate and its correlations with domestic and foreign interest rates. The first two factors have a negative impact while the third has a positive impact.