Delta Hedging Research Articles

Holes in Black-Scholes?

In this article, we provide a model-free test for whether the dynamic hedging argument actually works in practice and a quantification of how much jumps influence the delta hedge.

Research on the Applications of Neural Network Algorithms in Deep Hedging

Under market completeness assumptions, hedging a portfolio of derivatives is straightforward. In view of friction, transaction costs, liquidity and other factors, a framework is presented to extend the pricing and hedging with the hedging strategy treated as a neural network. We study the deep hedging model under incomplete market constraints such as frictions, traction cost, permanent impacts on the market and illiquidity. We discuss the limitations of certain models concerning the applications in deep hedging with constraints. After which, we analyse the advantages of different models and their joint models and find that the hedging strategy is close to the Black-Scholes delta hedging strategy. An example is also given when training after designing two hedging models. The Black-Scholes delta hedging is indeed approximated by unsupervised learning.

Conditional expectation strategy under the long memory Heston stochastic volatility model

This article deals with an European option pricing via proportional transaction costs in the incomplete environment with and without arbitrage opportunities under two long memory versions of the Heston model. Observing and introducing a traded proxy for the volatility in the modern market, we use the conditional expectation and the delta hedging strategies and present the generalized fractional Ito formula to obtain the option price partial differential equations (PDEs). To solve these PDEs, we apply the finite difference method and employ the K-antithetic variates algorithm based on the Monte-Carlo simulation as a benchmark for this method. Finally, we provide numerical results to illustrate the effectiveness of the proposed model.

Bruno: A Julia package for simulation, financial asset pricing and delta hedging

Bruno: A Julia package for simulation, financial asset pricing and delta hedging

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)

Relationship between deep hedging and delta hedging: leveraging a statistical arbitrage strategy

Relationship between deep hedging and delta hedging: leveraging a statistical arbitrage strategy

Psychological barriers and option pricing in a local volatility model

In this paper, we investigate the possible existence of psychological barriers in the SSE 50 Index and find that the volatility of the index returns and the implied volatility around psychological barriers are significantly different. To capture this special behavior, a local volatility model (LVM) is introduced, and the price of the European call option is derived under this framework with the help of the Laplace transform approach. We also report the LVM’s empirical and delta hedging performance relative to the Black–Scholes (BS) model, constant elasticity of variance (CEV) model and Jang et al.’s threshold model and find that for 50ETF and S&P 500 option pricing, the LVM is superior to the other three models.

Performance of Delta-Hedging Strategy on TQQQ Stock Options

This paper studies the performance of the same incremental hedging strategy of stock options. Especially when the stock market fluctuates greatly, this delta hedging strategy can always ensure investors’ fund always keep in a relative safe level. Thus, it can help individuals and institutional investors to build their portfolios and choose hedging strategies. This paper use the data of ten options of TQQQ stock to calibrate the implied volatility of the stock. Then, by calculating the implied volatility, a hedge portfolio is formed, including a specific option unit of the company and the incremental shares of the underlying stock. Finally, this paper evaluate the performance of the hedged portfolio. The results show that, the hedging strategy of TQQQ stock option performs well. The results of this paper are helpful for individual and institutional investors to choose the most suitable hedging strategies according to the nature of the underlying assets.

Option pricing under mixed hedging strategy in time-changed mixed fractional Brownian model

In this paper, we obtain a mixed hedging strategy and a pricing formula in a discrete time setting for a European call option in the time-changed mixed fractional Brownian model. In this manner, we generalize the mixed hedging and the pricing formula in Brownian motion to the time-changed mixed fractional Brownian motion. Finally, through some numerical experiments and empirical analysis, we show that our mixed hedging is better than delta hedging on the hedging error ratio in some cases.

The impact of option hedging on the spot market volatility

We theoretically model and empirically quantify the feedback effect of delta hedging for the spot market volatility of the forex market. We start from an economy with two types of traders, an aggregated option market maker (OMM) and an aggregated option market taker (OMT), whose exposures reflect the total outstanding positions of all option traders in the market. A different hedge ratio of the OMM and OMT leads to a net delta hedge activity, which introduces market friction. We represent this friction by a simple linear permanent impact model. This approach allows us to derive the dependence of the spot market volatility on the gamma exposure of the traders. Our theoretical model shows that the spot market volatility is increased (decreased) by a negative (positive) gamma exposure of the OMM, whereby the amount of the increase depends on the net delta hedge amount executed in the spot market. To empirically test this model, we first reconstruct the aggregated OMM’s gamma exposure by using trade repository data and find that it is negative. The empirical analysis performed over the period from 21st October 2017 to 30th June 2018 using our reconstructed OMM data then strongly supports our theoretical model: The gamma exposure of the OMM turns out to be highly significant for the spot market volatility and, as expected, the spot market volatility is increased with the OMM’s short gamma exposure. Quantitatively, a negative gamma exposure of the OMM of approximately −1000 billion USD (which is around what we observe from our reconstructed OMM data) leads to an absolute increase in volatility of 0.7% in EURUSD and 0.9% in USDJPY.

Financial market disruption and investor awareness: the case of implied volatility skew

<abstract> <p>The crash of 1987 is considered one of the most significant events in the history of financial markets due to the severity and swiftness of market declines worldwide. In the aftermath of the crash, a permanent change in options market occurred; implied volatility skew started appearing in options markets worldwide. In this article, we argue that the emergence of the implied volatility skew can be understood as arising from increased investor awareness about the stock price process and its implications for delta hedging. Delta-hedging aims to eliminate the directional risk associated with price movements in the underlying asset. Before the crash, investors were unaware of the proposition that "a delta-hedged portfolio is risky". That is, they implicitly believed in the proposition that "a delta-hedged portfolio is risk-free". The crash caused "portfolio insurance delta-hedges" to fail spectacularly. The resulting visceral shock drove home the lesson that "a delta-hedged portfolio is risky", thus, increasing investor awareness. We show that this sudden realization that a delta-hedged portfolio is risky is sufficient to generate the implied volatility skew and is equivalent to replacing the risk-free rate with a higher rate in the European call option formula. It follows that investor awareness (beyond asymmetric information) is an important consideration that matters for financial market behavior.</p> </abstract>

Delta Hedging Bitcoin Options with a Smile

Delta Hedging Bitcoin Options with a Smile

Pricing American Put Option using RBF-NN: New Simulation of Black-Scholes

Abstract The present work proposes an Artificial Neural Network framework for calculating the price and delta hedging of American put option. We consider a sequence of Radial Basis function Neural Network, where each network learns the difference of the price function according to the Gaussian basis function. Based on Black Scholes partial differential equation, we improve the superiority of Artificial Neural Network by comparing the performance and the results achieved used in classic Monte Carlo Least Square simulation with those obtained by Neural networks in one dimension. Thus, numerical result shows that the Artificial Neural Network solver can reduce the computing time significantly as well as the error training.

A Leland Model for Delta Hedging in Central Risk Books

A Leland Model for Delta Hedging in Central Risk Books

Study of European style options under Itô-McKean Brownian motion with Azzalini skew-normal distribution

In this article, we deal European style option, with arbitrary payoff which includes both put and call options, on an asset whose price evolves as It?-McKean skew Brownian motion with Azzalini skew-normal distribution. Initially, we investigate a condition which leads the It?-McKean skew Brownian motion to be a martingale. Next, we price the option and show that if the payoff function is convex then so is the price function. After this, we show if the payoff is finite then the price function satisfies a partial differential equation with respect to time. Further, we provide a necessary and sufficient condition for the price function to satisfy Feymann-Kac type equation. Next, we study Black-Scholes type equation and give expressions for the delta hedge. Finally, we study the particular case of an European call option in order to compare some of our results with the existing literature. Our results can be used to investigate the optimal exercise boundary, discrete time hedging strategies etc. of the option.

Pricing equity warrants under the sub-mixed fractional Brownian motion regime with stochastic interest rate

<abstract> <p>This paper proposes a pricing model for equity warrants under the sub-mixed fractional Brownian motion regime with the interest rate following the Merton short rate model. By using the delta hedging strategy, the corresponding partial differential equations for equity warrants are obtained. Moreover, the explicit pricing formula for equity warrants and some numerical results are given.</p> </abstract>

Empirical deep hedging

Existing hedging strategies are typically based on specific financial models: either the strategies are directly based on a given option pricing model or stock price and volatility models are used indirectly by generating synthetic data on which an agent is trained with reinforcement learning. In this paper, we train an agent in a pure data-driven manner. Particularly, we do not need any specifications on volatility or jump dynamics but use large empirical intra-day data from actual stock and option markets. The agent is trained for the hedging of derivative securities using deep reinforcement learning (DRL) with continuous actions. The training data consists of intra-day option price observations on S&P500 index over 6 years, and top of that, we use other data periods for validation and testing. We have two important empirical results. First, a DRL agent trained using synthetic data generated from a calibrated stochastic volatility model outperforms the classic Black–Scholes delta hedging strategy. Second, and more importantly, we find that a DRL agent, which is empirically trained directly using actual intra-day stock and option prices without the prior specification of the underlying volatility or jump processes, has superior performance compared with the use of synthetic data. This implies that DRL can capture the dynamics of S&P500 from the actual intra-day data and to self-learn how to hedge actual options efficiently.

Higher-Order Error Estimates of the Discrete-Time Clark–Ocone Formula

In this article, we investigate the convergence rate of the discrete-time Clark–Ocone formula provided by Akahori–Amaba–Okuma (J Theor Probab 30: 932–960, 2017). In that paper, they mainly focus on the \(L_{2}\)-convergence rate of the first-order error estimate related to the tracking error of the delta hedge in mathematical finance. Here, as two extensions, we estimate “the higher order error” for Wiener functionals with an integrability index 2 and “an arbitrary differentiability index.”

KrigHedge: Gaussian Process Surrogates for Delta Hedging

We investigate a machine learning approach to option Greeks approximation based on Gaussian Process (GP) surrogates. Our motivation is to implement Delta hedging in cases where direct computation is expensive, such as in local volatility models, or can only ever be done approximately. The proposed method takes in noisily observed option prices, fits a non-parametric input-output map and then analytically differentiates the latter to obtain the various price sensitivities. Thus, a single surrogate yields multiple self-consistent Greeks. We provide a detailed analysis of numerous aspects of GP surrogates, including choice of kernel family, simulation design, choice of trend function and impact of noise. We moreover connect the quality of the Delta approximation to the resulting discrete-time hedging loss. Results are illustrated with two extensive case studies that consider estimation of Delta, Theta and Gamma and benchmark approximation quality and uncertainty quantification using a variety of statistical metrics. Among our key take-aways are the recommendation to use Matérn kernels, the benefit of including virtual training points to capture boundary conditions, and the significant loss of fidelity when training on stock-path-based datasets.

European option under a skew version of the GBM model with transaction costs by an RBF method

The principal aim of this paper is to calculate the price of the European option under a skew version of the geometric Brownian motion model. To do this, we use the Leland and Kabanov strategy to remove arbitrage opportunities by the delta hedging strategy and adding transaction costs to a portfolio that contains an option and a share of the related stock. This idea generates a partial differential equation (PDE) problem to calculate the option price. To solve the obtained PDE, we use a radial basis functions (RBFs) method and apply the operational matrix of derivative for multiquadric-RBF to reduce the problem to a set of algebraic equations. Finally, we present some numerical results to show the efficiency of the model and the method by considering the Newton–Raphson method and the Bayes information criterion.