Detecting Jump Risk and Jump-Diffusion Model for Bitcoin Options Pricing and Hedging

In this paper, we conduct a fast calibration in the jump-diffusion model to capture the Bitcoin price dynamics, as well as the behavior of some components affecting the price itself, such as the risk of pitfalls and its ambiguous effect on the evolution of Bitcoin’s price. In addition, in our study of the Bitcoin option pricing, we find that the inclusion of jumps in returns and volatilities are significant in the historical time series of Bitcoin prices. The benefits of incorporating these jumps flow over into option pricing, as well as adequately capture the volatility smile in option prices. To the best of our knowledge, this is the first work to analyze the phenomenon of price jump risk and to interpret Bitcoin option valuation as “exceptionally ambiguous”. Crucially, using hedging options for the Bitcoin market, we also prove some important properties: Bitcoin options follow a convex, but not strictly convex function. This property provides adequate risk assessment for convex risk measure.

Implied volatility estimation of bitcoin options and the stylized facts of option pricing

The recently developed Bitcoin futures and options contracts in cryptocurrency derivatives exchanges mark the beginning of a new era in Bitcoin price risk hedging. The need for these tools dates back to the market crash of 1987, when investors needed better ways to protect their portfolios through option insurance. These tools provide greater flexibility to trade and hedge volatile swings in Bitcoin prices effectively. The violation of constant volatility and the log-normality assumption of the Black–Scholes option pricing model led to the discovery of the volatility smile, smirk, or skew in options markets. These stylized facts; that is, the volatility smile and implied volatilities implied by the option prices, are well documented in the option literature for almost all financial markets. These are expected to be true for Bitcoin options as well. The data sets for the study are based on short-dated Bitcoin options (14-day maturity) of two time periods traded on Deribit Bitcoin Futures and Options Exchange, a Netherlands-based cryptocurrency derivative exchange. The estimated results are compared with benchmark Black–Scholes implied volatility values for accuracy and efficiency analysis. This study has two aims: (1) to provide insights into the volatility smile in Bitcoin options and (2) to estimate the implied volatility of Bitcoin options through numerical approximation techniques, specifically the Newton Raphson and Bisection methods. The experimental results show that Bitcoin options belong to the commodity class of assets based on the presence of a volatility forward skew in Bitcoin option data. Moreover, the Newton Raphson and Bisection methods are effective in estimating the implied volatility of Bitcoin options. However, the Newton Raphson forecasting technique converges faster than does the Bisection method.

Pricing and hedging Asian options under Levy processes and robust long-term investing with learning about stock returns

In this work we propose a parametric model using the techniques of time-changed subordination that captures the implied volatility smile. We demonstrate that the Fourier-Cosine method can be used in a semi-static way to hedge for quadratic, VaR and AVaR risk. We also observe that investors looking to hedge VaR can simply hold the amount in a portfolio of mostly cash, whereas an investor hedging AVaR will need to hold more risky assets. We also extend ES risk to a robust framework. A conditional calibration method to calibrate the bivariate model is proposed. For a robust long-term investor who maximizes their recursive utility and learns about the stock returns, as the willingness to substitute over time increases, the equity demand decreases and consumption-wealth ratio increases. As the preference for robustness increases the demand for risk decreases. For a positive correlation, we observe that learning about returns encourages the investor to short the bond at all levels of u and vice versa

Pricing and hedging Asian options under Levy processes and robust long-term investing with learning about stock returns

In this work we propose a parametric model using the techniques of time-changed subordination that captures the implied volatility smile. We demonstrate that the Fourier-Cosine method can be used in a semi-static way to hedge for quadratic, VaR and AVaR risk. We also observe that investors looking to hedge VaR can simply hold the amount in a portfolio of mostly cash, whereas an investor hedging AVaR will need to hold more risky assets. We also extend ES risk to a robust framework. A conditional calibration method to calibrate the bivariate model is proposed. For a robust long-term investor who maximizes their recursive utility and learns about the stock returns, as the willingness to substitute over time increases, the equity demand decreases and consumption-wealth ratio increases. As the preference for robustness increases the demand for risk decreases. For a positive correlation, we observe that learning about returns encourages the investor to short the bond at all levels of u and vice versa

The Heston Model with Time-Dependent Correlation Driven by Isospectral Flows

In this work, we extend the Heston stochastic volatility model by including a time-dependent correlation that is driven by isospectral flows instead of a constant correlation, being motivated by the fact that the correlation between, e.g., financial products and financial institutions is hardly a fixed constant. We apply different numerical methods, including the method for backward stochastic differential equations (BSDEs) for a fast computation of the extended Heston model. An example of calibration to market data illustrates that our extended Heston model can provide a better volatility smile than the Heston model with other considered extensions.

Patterns of 50 ETF Options Implied Volatility in China: On Implied Volatility Functions

The aim of this study is to examine the volatility smile based on the European options on Shanghai stock exchange 50 ETF. The data gives evidence of the existence of a well-known U-shaped implied volatility smile for the SSE 50 ETF options market in China. For those near-month options, the implied volatility smirk is also observed. And the implied volatility remains high for the short maturity and decreases as the maturity increases. The patterns of the implied volatility of SSE 50 ETF options indicate that in-the-money options and out-of-the-money options are more expensive relative to at-the-money options. This makes the use of at-the-money implied volatility for pricing out-of- or in-the-money options questionable. In order to investigate the implied volatility, the regression-based implied volatility functions model is considered employed to study the implied volatility in this study as this method is simple and easy to apply in practice. Several classical implied volatility functions are investigated in this paper to find whether some kind of implied volatility functions could lead to more accurate options pricing values. The potential determinants of implied volatility are the degree of moneyness and days left to expiration. The empirical work has been expressed by means of simple ordinary least squares framework. As the study shows, when valuing options, the results of using volatility functions are mixed. For far-month options, using at-the-money implied volatility performs better than other volatility functions in option valuation. For near-month options, the use of volatility functions can improve the valuation accuracy for deep in-the-money options or deep out-of-the-money options. However, no particular implied volatility function performs very well for options of all moneyness level and time to maturity.

The impact of the leverage effect on the implied volatility smile: evidence for the German option market

It is a widely known theoretical derivation, that the firm’s leverage is negatively related to volatility of stock returns, although the empirical evidence is still outstanding. To empirically evaluate the leverage we first complement previous simulation studies by deriving theoretical predictions of leverage changes on the volatility smile. Even more important, we empirically test these predictions with an event study using intra-day Eurex option data and a unique data set of 138 ad-hoc news. For our theoretically derived predictions we observe that changes in leverage of DAX companies from 1999 to 2014 cause significant changes to the implied volatility smile.