Dynamic Delta Hedging Research Articles

Fractional Brownian motion in option pricing and dynamic delta hedging: Experimental simulations

This research examines the impact of fractional Brownian motion (fBm) on option pricing and dynamic delta hedging. Through experimental simulations, we analyze the influence of the Hurst exponent on option price prediction. Our findings highlight the necessity for continuous calibration of the Hurst exponent for a specific market dataset. By estimating option prices using fBm, we evaluate price prediction accuracy and explore fBm’s benefits in option pricing models. We also investigate dynamic delta hedging strategies for call options within the fBm framework, providing an algorithm and code that consider the Hurst exponent. The study’s insights contribute to advancing financial modeling and risk management practices, illuminating the dynamic nature of market phenomena and underscoring calibration’s significance in capturing market dynamics. The findings emphasize the dynamic interplay between the Hurst exponent and option pricing, offering valuable implications for effective risk management strategies.

Limits of Arbitrage and Primary Risk-Taking in Derivative Securities

Abstract Classic option pricing theory values a derivative contract via dynamic delta hedging and treating the contract as redundant relative to the underlying security. Dynamic delta hedging proves highly effective in practice, but the remaining risk is still large because of the practical limits of arbitrage. Derivatives can play primary roles in risk allocation. This paper quantifies the percentage variance reduction of delta hedging on U.S. stock options, proposes a top-down return attribution framework to identify the remaining risk sources of the delta-hedged option investment, and constructs a statistical return factor model to explain the variations of the delta-hedged option returns. (JEL C13, C51, G12, G13)

Bias Correction for Bond Option Greeks via Jackknife

The underlying models for bond options are often based on some linear drift functions such that the option Greeks depend crucially on the mean reversion parameters. Substantial estimation bias may arise when these parameters are estimated using standard methods such as maximum likelihood estimation, leading to a bias in estimating the Greeks. To address this issue, following Phillips and Yu (2005), a jackknife method is adopted in this article. In particular, we apply the method directly to the estimation of option Greeks, rather than the estimation of parameters. This approach is general and computationally inexpensive; hence, it is convenient in practice. The finite-sample performance is investigated in several Monte Carlo studies. At last, in dynamic Delta hedging, we show that the bias reduction in the estimation of option Greeks using the proposed method can achieve some economic value. <b>TOPICS:</b>Derivatives, options, fixed income and structured finance, quantitative methods, simulations <b>Key Findings</b> ▪ Like option pricing, the bias problem is still serious in estimating the Greeks of the options. ▪ The jackknife method is a good method to reduce the estimation bias in estimating the Greeks. ▪ The bias reduction in the estimation of option Greeks using the proposed method can achieve some economic values in option markets around the world.

Analytic formula for option margin with liquidity costs under dynamic delta hedging

ABSTRACT This study derives the expected liquidity cost when performing the delta hedging process of a European option. This cost is represented by an integration formula that includes European option prices and a certain function depending on the delta process. We first define a unit liquidity cost and then show that the liquidity cost is a multiplication of the unit liquidity cost, stock price, supply curve parameter, and the square of the number of options. Using this formula, the expected liquidity cost before hedging can be calculated much faster than when using a Monte Carlo simulation. Numerically computed distributions of liquidity costs in special cases are also provided.

Delta hedging bitcoin options with a smile

We analyse robust dynamic delta hedging of bitcoin options using a set of smile-implied and other smile-adjusted deltas that are either model-free, in the sense that they are the same for every scale-invariant stochastic and/or local volatility model, or they are based on simple regime-dependent parameterisations of local volatility. These deltas are popular with option market makers in traditional assets because they are very easy to implement. Previous empirical research on dynamic delta hedging is based solely on equity index options, but analysis of our unique data on hourly historical bitcoin option prices reveals that bitcoin implied volatility curves behave very differently from those of equity index options. For call and put options with a wide range of moneyness and with synthetic constant maturities of 10, 20 and 30 days, we compare the dynamic hedging performance of different smile-adjusted deltas over two one-year periods. We also examine the use of the perpetual contract rather than the standard futures as hedging instrument because the basis risk for the perpetual is very much smaller than it is for calendar futures. Results are presented as testable statistics of hedging error variance ratios. In certain periods the use of smile-implied hedge ratios can significantly out-perform the simple Black–Scholes delta hedge, especially when using the perpetual swap as hedging instrument, where efficiency gains can exceed 30% for out-of-the-money puts, and reach an average of 15% when hedging short-term out-of-the money calls during periods when the implied volatility curve slopes upwards. The advantage of using the perpetual contract is especially evident during 2021, for the longer-term contracts for which the basis is still rather large.

A new investment method with AutoEncoder: Applications to crypto currencies

This paper proposes a novel approach to the portfolio management using an AutoEncoder. In particular, features learned by an AutoEncoder with ReLU are directly exploited to portfolio constructions. Since the AutoEncoder extracts characteristics of data through a non-linear activation function ReLU, its realization is generally difficult due to the non-linear transformation procedure. In the current paper, we solve this problem by taking full advantage of the similarity of ReLU and an option payoff. Especially, this paper shows that the features are successfully replicated by applying so-called dynamic delta hedging strategy. An out of sample simulation with crypto currency dataset shows the effectiveness of our proposed strategy.

Exploring Option Pricing and Hedging via Volatility Asymmetry

This paper evaluates the application of two well-known asymmetric stochastic volatility (ASV) models to option price forecasting and dynamic delta hedging. They are specified in discrete time in contrast to the classical stochastic volatility models used in option pricing. There is some related literature, but little is known about the empirical implications of volatility asymmetry on option pricing. The objectives of this paper are to estimate ASV option pricing models using a Bayesian approach unknown in this type of literature, and to investigate the effect of volatility asymmetry on option pricing for different size equity sectors and periods of volatility. Using the S&P MidCap 400 and S&P 500 European call option quotes, results show that volatility asymmetry benefits the accuracy of option price forecasting and hedging cost effectiveness in the large-cap equity sector. However, ASV models do not improve the option price forecasting and dynamic hedging in the mid-cap equity sector.

A New Investment Method with Autoencoder: Applications to Cryptocurrencies

This paper proposes a novel approach to the portfolio management using an AutoEncoder. In particular, the features learned by an AutoEncoder with ReLU are directly exploited to the portfolio construction. Since the AutoEncoder extracts the characteristics of the data through the non-linear activation function ReLU, its realization is generally difficult due to the non-linear transformation procedure. In the current paper, we solve this problem by taking full advantage of the similarity of the ReLU and the option payoff. Especially, this paper shows that the features are successfully replicated by applying so-called the dynamic delta hedging strategy. An out of sample simulation with crypto currency dataset shows the effectiveness of our proposed strategy. Furthermore, we investigate the background of our proposed methodology, which suggests that the first principal component is quite important.

SIMPLIFIED HEDGE FOR PATH-DEPENDENT DERIVATIVES

Path-dependent derivatives are typically difficult to hedge. Traditional dynamic delta hedging does not perform well because of the difficulty to evaluate the Greeks and the high cost of constantly rebalancing. We propose to price and hedge path-dependent derivatives by constructing simplified alternatives that preserve certain distributional properties of their terminal payoffs, and that can be hedged by semi-static replication. The method is illustrated by a geometric Asian option and by a lookback option in the Black–Scholes setting, for which explicit forms of the simplified alternatives exist. Extensions to a Lévy market and to a Heston stochastic volatility model are discussed as well.

Rainbow trend options: valuation and applications

Asset selection and timing decisions are major investment concerns. To resolve these issues simultaneously, a new class of rainbow trend options is proposed. The diversification effect of rainbow options can reduce the importance of asset selection decisions and trend options can mitigate unfavorable effects on market entry and exit decisions. We consider a general framework to facilitate the derivation of analytic pricing formulas for simple, pure, and Asian rainbow trend options using the martingale pricing method. The properties of these options and their Greeks are analyzed. We also investigate the performance of the dynamic delta hedging strategy for issuers of rainbow trend options. Last, this paper explores the applications of rainbow trend options for hedging price risks, designing executive stock options, modifying countercyclical capital buffer proposed by Basel Committee, and acting as control variates of the Monte Carlo simulation.

Simple Robust Hedging with Nearby Contracts

This paper proposes a new hedging strategy based on approximate matching of contract characteristics instead of risk sensitivities. The strategy hedges an option with three options at different maturities and strikes by matching the option function expansion along maturity and strike rather than risk factors. Its hedging effectiveness varies with the maturity and strike distance between the target and the hedge options, but is robust to variations in the underlying risk dynamics. Simulation analysis under different risk environments and historical analysis on S&P 500 index options both show that a wide spectrum of strike-maturity combinations can outperform dynamic delta hedging.

Exploring Option Pricing and Hedging via Volatility Asymmetry

This paper evaluates the application of two well-known asymmetric stochastic volatility (ASV) models to option price forecasting and dynamic delta hedging. They are specified in discrete time in contrast to the classical stochastic volatility (SV) models used in option pricing. There is some related literature, but little is known about the empirical implications of volatility asymmetry on option pricing. The objectives of this paper are to estimate ASV option pricing models using a Bayesian approach unknown in this type of literature, and to investigate the effect of volatility asymmetry on option pricing for different size equity sectors and periods of volatility. Using the S&P MidCap 400 and S&P 500 European call option quotes, results show that volatility asymmetry benefits the accuracy of option price forecasting and hedging cost effectiveness in the large-cap equity sector. However, ASV models do not improve the option price forecasting and dynamic hedging in the mid-cap equity sector.

인공신경망을 이용한 주식워런트증권(ELW)의 헤징 방안

From the perspective of risk management, financial organization that have issued ELW require an efficient hedging methodology due to recently increased trade volume of ELW. This study presents an ELW hedging methodology using artificial neural network(ANN) to minimize hedging costs. The performance of the presented methodology in this study is examined by analysis utilizing the prices and volatilities of underlying assets, risk free interest rates, and maturities and computational experiments show that the proposed method is superior to existing dynamic delta hedging(DDH) technique in terms of hedging costs ranged from 25% to 250%.

기계학습과 동적델타헤징을 이용한 옵션 헤지 전략

동적 델타 헤징(Dynamic Delta Hedging)이란 옵션 발행자가 옵션의 만기정산금액(payoff)을 지급하기 위해 주기적으로 델타에 근거한 헤지 포지션을 조절함으로써 옵션의 payoff를 복제하고 옵션 가치변화에 따른 위험을 회피하는 방법이다. 본 연구에서는 헤지에 있어서 주요 변수인 블랙-숄즈의 모형에 의해 산출된 델타의 대체 값을 찾기 위해 기계학습의 일종인 인공신경망 학습을 적용하여 옵션의 만기 시 헤지 비용의 최소화 및 차익 실현을 위한 방법론을 제시하고자 한다. 기초자산의 현재가격, 변동성, 무위험이자율, 만기 등의 시장 상황 변화에 따른 다양한 시나리오에 대한 실험을 통해 본 연구에서 제시하는 방법론의 성능을 분석하고 그 우수성을 보인다. Option issuers generally utilize Dynamic Delta Hedging(DDH) technique to avoid the risk resulting from continuously changing option value. DDH duplicates payoff of option position by adjusting hedge position according to the delta value from Black-Scholes(BS) model in order to maintain risk neutral state. DDH, however, is not able to guarantee optimal hedging performance because of the weaknesses caused by impractical assumptions inherent in BS model. Therefore, this study presents a methodology for dynamic option hedge using artificial neural network(ANN) to enhance hedging performance and show the superiority of the proposed method using various computational experiments.

Simple Robust Hedging with Nearby Contracts

Most existing hedging theories are derived under strong, idealistic assumptions on both the underlying security price dynamics and the trading environments. Practical concerns such as contract availability, transaction cost, and uncertainty regarding the security price dynamics impose severe limitations on the actual application of these theories. This paper proposes a new, simple, and robust hedging theory that overcomes all these limitations. While most existing hedging methods are based on neutralizing risk exposures defined under a pre-specified model, our hedge portfolio is formed based on the affinity of the derivative contracts. Specifically, in hedging a target option, we form hedge portfolios using three options at three different strikes and two different maturities that form a stable maturity-strike triangle, and we derive the portfolio weights for the hedge portfolio as a function of the strike and maturity spacing of the triangle relative to the target option. Our hedging portfolio does not depend on any assumptions on the underlying risk dynamics, as it is derived purely as a function of the contract characteristics. The portfolio overcomes transaction cost concerns as it only holds three options in static positions. The strikes and maturities of the three hedging options can be flexibly chosen to balance between contract availability, order flow, transaction cost, and hedging effectiveness. Simulation analysis under commonly proposed security price dynamics shows that the hedging performance of our maturity-strike triangles compares favorably against the dynamic delta hedging strategy with daily rebalancing. Although delta hedging performance deteriorates dramatically in the presence of large jumps or stochastic volatility, the hedging performance of our maturity-strike triangles is very stable across different model environments. A historical hedging exercise on S&P 500 index option further highlights the superior performance of our strategies.

Strips of hourly power options—Approximate hedging using average-based forward contracts

We study approximate hedging strategies for a contingent claim consisting of a strip of independent hourly power options. The payoff of the contingent claim is a sum of the contributing hourly payoffs. As there is no forward market for specific hours, the fundamental problem is to find a reasonable hedge using exchange-traded forward contracts, e.g. average-based monthly contracts. The main result is a simple dynamic hedging strategy that reduces a significant part of the variance. The idea is to decompose the contingent claim into mathematically tractable components and to use empirical estimations to derive hedging deltas. Two benefits of the method are that the technique easily extends to more complex power derivatives and that only a few parameters need to be estimated. The hedging strategy based on the decomposition technique is compared with dynamic delta hedging strategies based on local minimum variance hedging, using a correlated traded asset.

Delta hedging strategies comparison

In this paper we implement dynamic delta hedging strategies based on several option pricing models. We analyze different subordinated option pricing models and we examine delta hedging costs using ex-post daily prices of S&P 500. Furthermore, we compare the performance of each subordinated model with the Black–Scholes model.

Pricing and Hedging of Discrete Dynamic Guaranteed Funds

AbstractWe derive a risk‐neutral pricing model for discrete dynamic guaranteed funds with geometric Gaussian underlying security price process. We propose a dynamic hedging strategy by adding a gamma factor to the conventional delta. Simulation results demonstrate that, when hedging discretely, the risk‐neutral gamma‐adjusted‐delta strategy outperforms the dynamic delta hedging strategy by reducing the expected hedging error, lowering the hedging error variability, and improving the self‐financing possibility. The discrete dynamic delta‐only hedging not only causes potential overcharge to clients but also could be costly to the issuers. We show that a naive application of continuous‐time hedging formula to a discrete‐time hedging setting tends to worsen these possibilities.

Principal Component Analysis of Volatility Smiles and Skews

This paper develops a model for volatility sensitivity to the underlying asset price. It has applications to option pricing and dynamic delta hedging under stochastic volatility. The model allows at-the-money volatility sensitivity to change continuously with S and this corresponds to a quadratic parameterization to the volatility surface. The extension to fixed strike volatility sensitivities is achieved using a principal component analysis on the deviation of fixed strike volatilities from at-the-money volatility.