Analytically pricing variance and volatility swaps under a Markov-modulated model with liquidity risks

This paper determines strike prices of discretely sampled variance/volatility swaps taking into account stochastic liquidity risks and the switching of economic conditions. We adopt nonlinear regime switching volatility to reflect how asset prices are affected by economic cycles, and market prices of assets are discounted according to the level of market liquidity. We then establish a risk-neutral measure under regime switching Esscher transform, so that analytical valuation of variance/volatility swaps can be completed based on the closed-form forward characteristic function. The limiting behavior of discretely sampled variance/volatility swaps is also considered through the investigation of pricing continuously sampled variance/volatility swaps. Finally, based on the results from numerical implementation, we confirm that the new model is very flexible in reflecting different influence associated with common real market observations.     

A closed-form exact solution for pricing fixed-income variance swaps with affine-jump model

Fixed-income variance swaps became popular for investors to trade and hedge the fluctuation of interest rates after the recent global financial crisis over the past few decades, however, their valuations and risk management have not been studied sufficiently. This paper presents an analytic approach for pricing some discretely sampled fixed-income variance swaps under an affine-jump model with stochastic mean, stochastic volatility, and jumps. We employ a generalized characteristic function to derive the closed-form pricing formulas of these swaps, including two kinds of zero-coupon bond variance swap, Libor variance swap, and bond yield variance swap, to be precise. We also perform some numerical studies based on these models, which suggest that the fair strike values of these variance swaps are within a reasonable range regardless of estimation risk with data dependence and near-zero short rate regime. Our numerics show that the influences of varying sampling frequency and time-to-maturity on the values of these swaps are significant, and highlight the risks of specifying short rate model. Furthermore, the sensitivity analysis on the key parameters finds that the risks of stochastic volatility and jumps play prominent roles in pricing these variance swaps under the near-zero short rate regime.     

A unified approach to Bermudan and barrier options under stochastic volatility models with jumps

Many financial assets, such as currencies, commodities, and equity stocks, exhibit both jumps and stochastic volatility, which are especially prominent in the market after the financial crisis. Some strategic decision making problems also involve American-style options. In this paper, we develop a novel, fast and accurate method for pricing American and barrier options in regime switching jump diffusion models. By blending regime switching models and Markov chain approximation techniques in the Fourier domain, we provide a unified approach to price Bermudan, American options and barrier options under general stochastic volatility models with jumps. The models considered include Heston, Hull–White, Stein–Stein, Scott, the 3/2 model, and the recently proposed 4/2 model and the α-Hypergeometric model with general jump amplitude distributions in the return process. Applications include the valuation of discretely monitored contracts as well as continuously monitored contracts common in the foreign exchange markets. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed method.     

On the pricing of real estate index linked swaps

In this paper we discuss the pricing of commercial real estate index linked swaps (CREILS). This particular pricing problem has been studied by Buttimer et al. in a previous paper in this journal (6 [1997]: 16). We show that their results are only approximately correct and that the true theoretical price of the swap is in fact equal to zero. This result is shown to hold regardless of the specific model chosen for the index process, the dividend process, and the interest rate term structure. We provide an intuitive economic argument as well as a full mathematical proof of our results. In particular we show that the nonzero result in the previous paper is due to two specific numerical approximations introduced in that paper, and we discuss these approximation errors from a theoretical as well as from a numerical point of view.     

An effective hybrid variance reduction method for pricing the Asian options and its variants

In this paper, we propose a variance reduction method that combines importance sampling and control variates to price European Arithmetic Asian options and its variants (i.e., Asian options plus knock-in or knock-out options) under the Black-Scholes model. The numerical results show that the proposed methods are especially efficient under the following scenarios: in the money, low volatility, more sampling dates, and higher barrier thresholds.     

A two-stage realized volatility approach to estimation of diffusion processes with discrete data

This paper motivates and introduces a two-stage method of estimating diffusion processes based on discretely sampled observations. In the first stage we make use of the feasible central limit theory for realized volatility, as developed in [Jacod, J., 1994. Limit of random measures associated with the increments of a Brownian semiartingal. Working paper, Laboratoire de Probabilities, Universite Pierre et Marie Curie, Paris] and [Barndorff-Nielsen, O., Shephard, N., 2002. Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society. Series B, 64, 253–280], to provide a regression model for estimating the parameters in the diffusion function. In the second stage, the in-fill likelihood function is derived by means of the Girsanov theorem and then used to estimate the parameters in the drift function. Consistency and asymptotic distribution theory for these estimates are established in various contexts. The finite sample performance of the proposed method is compared with that of the approximate maximum likelihood method of [Aït-Sahalia, Y., 2002. Maximum likelihood estimation of discretely sampled diffusion: A closed-form approximation approach. Econometrica. 70, 223–262].     

Testing for unit roots in time series models with non-stationary volatility

Many of the key macro-economic and financial variables in developed economies are characterized by permanent volatility shifts. It is known that conventional unit root tests are potentially unreliable in the presence of such behaviour, depending on a particular function (the variance profile) of the underlying volatility process. Somewhat surprisingly then, very little work has been undertaken to develop unit root tests which are robust to the presence of permanent volatility shifts. In this paper we fill this gap in the literature by proposing tests which are valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth-transition) volatility change processes as special cases. Our solution uses numerical methods to simulate the asymptotic null distribution of the statistics based on a consistent estimate of the variance profile which we also develop. The practitioner is not required to specify a parametric model for volatility. An empirical illustration using producer price inflation series from the Stock–Watson database is reported.     

Implied recovery

In the absence of forward-looking models for recovery rates, market participants tend to use exogenously assumed constant recovery rates in pricing models. We develop a flexible jump-to-default model that uses observables: the stock price and stock volatility in conjunction with credit spreads to identify implied, endogenous, dynamic functions of the recovery rate and default probability. The model in this paper is parsimonious and requires the calibration of only three parameters, enabling the identification of the risk-neutral term structures of forward default probabilities and recovery rates. Empirical application of the model shows that it is consistent with stylized features of recovery rates in the literature. The model is flexible, i.e. it may be used with different state variables, alternate recovery functional forms, and calibrated to multiple debt tranches of the same issuer. The model is robust, i.e. evidences parameter stability over time, is stable to changes in inputs, and provides similar recovery term structures for different functional specifications. Given that the model is easy to understand and calibrate, it may be used to further the development of credit derivatives indexed to recovery rates, such as recovery swaps and digital default swaps, as well as provide recovery rate inputs for the implementation of Basel II.     

Realized volatility forecasting and option pricing

A growing literature advocates the use of microstructure noise-contaminated high-frequency data for the purpose of volatility estimation. This paper evaluates and compares the quality of several recently-proposed estimators in the context of a relevant economic metric, i.e., profits from option pricing and trading. Using forecasts obtained by virtue of alternative volatility estimates, agents price short-term options on the S&P 500 index before trading with each other at average prices. The agents’ average profits and the Sharpe ratios of the profits constitute the criteria used to evaluate alternative volatility estimates and the corresponding forecasts. For our data, we find that estimators with superior finite sample Mean-squared-error properties generate higher average profits and higher Sharpe ratios, in general. We confirm that, even from a forecasting standpoint, there is scope for optimizing the finite sample properties of alternative volatility estimators as advocated by Bandi and Russell [Bandi, F.M., Russell, J.R., 2005. Market microstructure noise, integrated variance estimators, and the accuracy of asymptotic approximations. Working Paper; Bandi, F.M., Russell, J.R., 2008b. Microstructure noise, realized variance, and optimal sampling. Review of Economic Studies 75, 339–369] in recent work.     

Stock market reactions to upside and downside volatility of Bitcoin: A quantile analysis

We investigate how sensitive developed and emerging equity markets are to volatility dynamics of Bitcoin during tranquil, bear, and bull market regimes. Intraday price fluctuations of Bitcoin are represented by three measures of realized volatility, viz. total variance, upside semivariance, and downside semivariance. Our empirical analysis relies on a quantile regression framework, after orthogonalizing raw returns with respect to an array of relevant global factors and accounting for structural shifts in the series. The results suggest that developed-market returns are positively related to the realized variance proxy across various market conditions, while emerging-market returns are positively (negatively) correlated with realized variance during bear (normal and bull) market periods. The upside (downside) component of realized variance has a negative (positive) influence on returns of either market category, and the dependence structure is highly asymmetric across the return distribution. Additionally, we document that developed and emerging markets are more sensitive to downside volatility than to upside volatility when they enter tranquil or bull territory. Our results offer practical implications for policymakers and investors.     

Binomial valuation of lookback options

We present a straightforward and computationally efficient binomial approximation scheme for the valuation of lookback options. This enables us to value American lookback options. Previous research on lookback options has assumed that the contracts are based on the extrema of the continuously observed price of the underlying security; in practice, however, contracts are often based on the extrema of prices sampled at a finite set of fixed dates. We adapt our binomial scheme to investigate the impact on the value of the options.     

Econometric analysis of jump-driven stochastic volatility models

This paper introduces and studies the econometric properties of a general new class of models, which I refer to as jump-driven stochastic volatility models, in which the volatility is a moving average of past jumps. I focus attention on two particular semiparametric classes of jump-driven stochastic volatility models. In the first, the price has a continuous component with time-varying volatility and time-homogeneous jumps. The second jump-driven stochastic volatility model analyzed here has only jumps in the price, which have time-varying size. In the empirical application I model the memory of the stochastic variance with a CARMA(2,1) kernel and set the jumps in the variance to be proportional to the squared price jumps. The estimation, which is based on matching moments of certain realized power variation statistics calculated from high-frequency foreign exchange data, shows that the jump-driven stochastic volatility model containing continuous component in the price performs best. It outperforms a standard two-factor affine jump–diffusion model, but also the pure-jump jump-driven stochastic volatility model for the particular jump specification.     

Carbon option price forecasting based on modified fractional Brownian motion optimized by GARCH model in carbon emission trading

With the rapid growth of carbon trading, the development of carbon financial derivatives such as carbon options has become inevitable. This paper established a model based on GARCH and fractional Brownian motion (FBM), hoping to provide reference for China's upcoming carbon option trading through carbon option price forecasting research. The fractal characteristic of carbon option prices indicates that it is reasonable to use FBM to predict option prices. The GARCH model can make up for the lack of fixed FBM volatility. In this paper, the daily closing prices of EUA option contracts on the European Energy Exchange are selected as samples for price prediction. The GARCH model was used to determine the return volatility, and then the FBM was used to calculate the forecast price for the next 60 days. The results showed that the predicted price can better fit the actual price. This paper further compares the price prediction results of this model with the other three models through line graphs and error evaluation indicators such as MAPE, MAE and MSE. It is confirmed that the prediction results of the model in this paper is the closest to the actual price.     

An analytical approximation formula for European option pricing under a new stochastic volatility model with regime-switching

In this paper, an analytical approximation formula for pricing European options is obtained under a newly proposed hybrid model with the volatility of volatility in the Heston model following a Markov chain, the adoption of which is motivated by the empirical evidence of the existence of regime-switching in real markets. We first derive the coupled PDE (partial differential equation) system that governs the European option price, which is solved with the perturbation method. It should be noted that the newly derived formula is fast and easy to implement with only normal distribution function involved, and numerical experiments confirm that our formula could provide quite accurate option prices, especially for relatively short-tenor ones. Finally, empirical studies are carried out to show the superiority of our model based on S&P 500 returns and options with the time to expiry less than one month.     

On pricing lookback options under the CEV process

Abstract We consider the problem of pricing European lookback options when the underlying asset price is driven by a constant elasticity of variance (CEV) process. The evaluation model is based on the binomial approximation developed by Nelson and Ramaswamy (1990) and we show how to apply it in the case of such options. We develop simple pricing algorithms that compute accurate estimates of the option prices.     

The impacts of index options on the underlying stocks: The case of the S&P 100

Existing theories predict lower trading volume, but ambiguous changes in price, bid–ask spread, and volatility for the underlying stocks following the advent of index derivatives. We further test these predictions around the introduction of the S&P 100 options in March 1983. Controlling for known factors respectively, we find that the listing of the S&P 100 options results in lower volume, spread, and volatility, but no price change for the underlying stocks, contrasting with the existing U.S. evidence and supporting the notion that the arrival of index derivatives induces informed and speculative portfolio traders to migrate from the underlying market to the derivatives market.     

Spillover effects in oil-related CDS markets during and after the sub-prime crisis

This paper investigates the return and volatility spillover effects across oil-related credit default swaps (CDSs), the oil market, and financial market risks for the US during and after the subprime crises. The empirical analysis is based on monthly return and realized volatility data from February 2004 to April 2020. We estimate both static and dynamic generalized dynamic spillover measures based on vector autoregressive (VAR) models. Our full sample empirical findings show that the oil market is the primary source of risk transmission for all the oil-related credit default swaps, while the bond market is the highest source of risk transmission to the stock market and vice versa. We also provide evidence that the regulated monopoly US utility sector has the least role in volatility transmission. Furthermore, the bailout program conducted by the US Treasury and Federal Reserve helped stabilize the US financial market through the purchase of toxic assets after the subprime financial crisis. We find strong evidence that the federal funds rate hike cycles lessen total risk transmission throughout the US bond market. Finally, our findings assert that oil price shocks have a significant effect on the oil-related CDSs in some sub-periods via the demand and supply transmission channels.     

Modelling the conditional volatility of commodity index futures as a regime switching process

Commodity index futures offer a versatile tool for gaining different forms of exposure to commodity markets. Volatility is a critical input in many of these applications. This paper examines issues in modelling the conditional variance of futures returns based on the Goldman Sachs Commodity Index (GSCI). Given that commodity markets tend to be ‘choppy’ (Webb, 1987 ), a general econometric model is proposed that allows for abrupt changes or regime shifts in volatility, transition probabilities which vary explicitly with observable fundamentals such as the basis, GARCH dynamics, seasonal variations and conditional leptokurtosis. The model is applied to daily futures returns on the GSCI over 1992–1997. The results show clear evidence of regime shifts in conditional mean and volatility. Once regime shifts are accounted for, GARCH effects are minimal. Consistent with the theory of storage, returns are more likely to switch to the high‐variance state when the basis is negative than when the basis is positive. The regime switching model also performs well in forecasting the daily volatility compared to standard GARCH models without regime switches. The model should be of interest to sophisticated traders who base their trading strategies on short‐term volatility movements, managed commodity funds interested in hedging an underlying diversified portfolio of commodities and investors of options and other derivatives tied to GSCI futures contracts. Copyright © 2001 John Wiley & Sons, Ltd.     

The VIX,the variance premium and stock market volatility

We decompose the squared VIX index, derived from US S&P500 options prices, into the conditional variance of stock returns and the equity variance premium. We evaluate a plethora of state-of-the-art volatility forecasting models to produce an accurate measure of the conditional variance. We then examine the predictive power of the VIX and its two components for stock market returns, economic activity and financial instability. The variance premium predicts stock returns while the conditional stock market variance predicts economic activity and has a relatively higher predictive power for financial instability than does the variance premium.     

Ultra high frequency volatility estimation with dependent microstructure noise

We analyze the impact of time series dependence in market microstructure noise on the properties of estimators of the integrated volatility of an asset price based on data sampled at frequencies high enough for that noise to be a dominant consideration. We show that combining two time scales for that purpose will work even when the noise exhibits time series dependence, analyze in that context a refinement of this approach is based on multiple time scales, and compare empirically our different estimators to the standard realized volatility.