Stochastic volatility and stochastic leverage

This paper proposes the new concept of stochastic leverage in stochastic volatility models. Stochastic leverage refers to a stochastic process which replaces the classical constant correlation parameter between the asset return and the stochastic volatility process. We provide a systematic treatment of stochastic leverage and propose to model the stochastic leverage effect explicitly, e.g. by means of a linear transformation of a Jacobi process. Such models are both analytically tractable and allow for a direct economic interpretation. In particular, we propose two new stochastic volatility models which allow for a stochastic leverage effect: the generalised Heston model and the generalised Barndorff-Nielsen & Shephard model. We investigate the impact of a stochastic leverage effect in the risk neutral world by focusing on implied volatilities generated by option prices derived from our new models. Furthermore, we give a detailed account on statistical properties of the new models.     

The leverage effect puzzle: Disentangling sources of bias at high frequency

The leverage effect refers to the generally negative correlation between an asset return and its changes of volatility. A natural estimate consists in using the empirical correlation between the daily returns and the changes of daily volatility estimated from high frequency data. The puzzle lies in the fact that such an intuitively natural estimate yields nearly zero correlation for most assets tested, despite the many economic reasons for expecting the estimated correlation to be negative. To better understand the sources of the puzzle, we analyze the different asymptotic biases that are involved in high frequency estimation of the leverage effect, including biases due to discretization errors, to smoothing errors in estimating spot volatilities, to estimation error, and to market microstructure noise. This decomposition enables us to propose novel bias correction methods for estimating the leverage effect.     

A Better Asymmetric Model of Changing Volatility in Stock and Exchange Rate Returns: Trend-GARCH

Abstract

The impact of short run price trending on the conditional volatility is tested empirically. A new family of conditionally heteroscedastic models with a trend-dependent conditional variance equation: The Trend-GARCH model is described. Modern microeconomic theory often suggests the connection between the past behaviour of time series, the subsequent reaction of market individuals, and thereon changes in the future characteristics of the time series. Results reveal important properties of these models, which are consistent with stylized facts found in financial data sets. They can also be employed for model identification, estimation, and testing. The empirical analysis supports the existence of trend effects. The Trend-GARCH model proves to be superior to alternative models such as EGARCH, AGARCH, TGARCH OR GARCH-in-Mean in replicating the leverage effect in the conditional variance, in fitting the news impact curve and in fitting the volatility estimates from high frequency data. In addition, we show that the leverage effect is dependent on the current trend, i.e. it differentiates between bullish and bearish markets. Furthermore, trend effects can account for a significant part of the long memory property of asset price volatilities.     

Double leverage cycle,interest rate,and financial crisis

We view mortgage as a risky derivative of its underlying house collateral and combine no-arbitrage valuation with equilibrium valuation approaches to develop a dynamic model of leverage cycle and interest rate. This model provides a unified explanation to pro-cyclical optimism, asset prices, and leverage, and counter-cyclical volatility and interest rate. In addition, the model shows that tightening funding margin in the mortgage securities market dampens optimism, asset prices, and leverage, whereas it raises volatility and interest rate in the housing market. A double leverage cycle leads to more volatile markets and a severe leverage cycle, thus resulting in worse financial crises.     

The Jacobi stochastic volatility model

We introduce a novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process. It contains the Heston model as a limit case. We show that the joint density of any finite sequence of log-returns admits a Gram–Charlier A expansion with closed-form coefficients. We derive closed-form series representations for option prices whose discounted payoffs are functions of the asset price trajectory at finitely many time points. This includes European call, put and digital options, forward start options, and can be applied to discretely monitored Asian options. In a numerical study, we show that option prices can be accurately and efficiently approximated by truncating their series representations.     

Portfolio choices and VaR constraint with a defaultable asset

We consider a Constant Elasticity of Variance (CEV) model for the asset price of a defaultable asset showing the so-called leverage effect (high volatility when the asset price is low). We show that a VaR constraint re-evaluated over time induces an agent more risk averse than a logarithmic utility to take more risk than in the unconstrained setting.     

Algorithmic market making for options

In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model—e.g. the Heston model—the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided.     

Pricing average options under time-changed Lévy processes

This paper presents an approximate formula for pricing average options when the underlying asset price is driven by time-changed Lévy processes. Time-changed Lévy processes are attractive to use for a driving factor of underlying prices because the processes provide a flexible framework for generating jumps, capturing stochastic volatility as the random time change, and introducing the leverage effect. There have been very few studies dealing with pricing problems of exotic derivatives on time-changed Lévy processes in contrast to standard European derivatives. Our pricing formula is based on the Gram–Charlier expansion and the key of the formula is to find analytic treatments for computing the moments of the normalized average asset price. In numerical examples, we demonstrate that our formula give accurate values of average call options when adopting Heston’s stochastic volatility model, VG-CIR, and NIG-CIR models.     

Pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes

We derive efficient and accurate analytical pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes. By extending the conditioning variable approach, we derive the lower bound on the Asian option price and construct an upper bound based on the sharp lower bound. We also consider the general partially exact and bounded (PEB) approximations, which include the sharp lower bound and partially conditional moment matching approximation as special cases. The PEB approximations are known to lie between a sharp lower bound and an upper bound. Our numerical tests show that the PEB approximations to discrete arithmetic Asian option prices can produce highly accurate approximations when compared to other approximation methods. Our proposed approximation methods can be readily applied to pricing Asian options under most common types of underlying asset price processes, like the Heston stochastic volatility model nested in the class of time-changed Lévy processes with the leverage effect.     

Application of Heston’s Model to the Chinese Stock Market

This article applies Heston’s (1993) stochastic volatility model to the Chinese stock market indices and subsequently assesses its pricing performance. A two-step estimation procedure is adopted to calibrate Heston’s model. First, we find that the option price is affected by both the moneyness and the maturity. Second, Heston’s model is more likely to overprice options, whereas the BS model tends to underestimate options. Finally, Heston’s model, by employing volatility as a random process, significantly improves the pricing accuracy compared to the BS model. Therefore, Heston’s model is tractable to analyze the Chinese stock market indices, and there is volatility risk that must not be overlooked in the Chinese stock market.     

The volatility target effect in structured investment products with capital protection

Designing a structured investment product with capital protection which would be characterized by high capital protection level as well as high equity participation rate is a challenging task in the current market environment. Low interest rates and high volatility levels negatively affect the above key parameters of such investment products. One way to increase the participation rate of a structured investment product with a fixed capital protection level is to use a volatility target (VolTarget) strategy as an underlying asset for a financial option embedded in such a product. We introduce an extended VolTarget mechanism with interest rate dependent volatility target levels and provide a detailed comparative numerical study of European options linked to VolTarget strategies within a hybrid Heston–Vasi?ec model with stochastic volatility and stochastic interest rate.     

Leverage and the Cross‐Section of Equity Returns

Building on theoretical asset pricing literature, we examine the role of market risk and the size, book‐to‐market (BTM), and volatility anomalies in the cross‐section of unlevered equity returns. Compared with levered (stock) returns, unlevered market beta plays a more important role in explaining the cross‐section of unlevered equity returns, even after controlling for size and BTM. The size effect is weakened, while the value premium and the volatility puzzle virtually disappear for unlevered returns. We show that leverage induces heteroskedasticity in returns. Unlevering returns removes this pattern, which is otherwise difficult to address by controlling for leverage in regressions.     

A comparison principle between rough and non-rough Heston models—with applications to the volatility surface

We present a number of related comparison results, which allow one to compare moment explosion times, moment generating functions and critical moments between rough and non-rough Heston models of stochastic volatility. All results are based on a comparison principle for certain non-linear Volterra integral equations. Our upper bound for the moment explosion time is different from the bound introduced by Gerhold, Gerstenecker and Pinter [Moment explosions in the rough Heston model. Decisions in Economics and Finance, 2019, 42, 575–608] and tighter for typical parameter values. The results can be directly transferred to a comparison principle for the asymptotic slope of implied variance between rough and non-rough Heston models. This principle shows that the ratio of implied variance slopes in the rough versus non-rough Heston model increases at least with power-law behavior for small maturities.     

Informed traders and limit order markets

We consider a dynamic limit order market in which traders optimally choose whether to acquire information about the asset and the type of order to submit. We numerically solve for the equilibrium and demonstrate that the market is a “volatility multiplier”: prices are more volatile than the fundamental value of the asset. This effect increases when the fundamental value has high volatility and with asymmetric information across traders. Changes in the microstructure noise are negatively correlated with changes in the estimated fundamental value, implying that asset betas estimated from high-frequency data will be incorrect.     

Leverage causes fat tails and clustered volatility

We build a simple model of leveraged asset purchases with margin calls. Investment funds use what is perhaps the most basic financial strategy, called ‘value investing’, i.e. systematically attempting to buy underpriced assets. When funds do not borrow, the price fluctuations of the asset are approximately normally distributed and uncorrelated across time. This changes when the funds are allowed to leverage, i.e. borrow from a bank, which allows them to purchase more assets than their wealth would otherwise permit. During good times, funds that use more leverage have higher profits, increasing their wealth and making them dominant in the market. However, if a downward price fluctuation occurs while one or more funds is fully leveraged, the resulting margin call causes them to sell into an already falling market, amplifying the downward price movement. If the funds hold large positions in the asset, this can cause substantial losses. This in turn leads to clustered volatility: before a crash, when the value funds are dominant, they damp volatility, and after the crash, when they suffer severe losses, volatility is high. This leads to power-law tails, which are both due to the leverage-induced crashes and due to the clustered volatility induced by the wealth dynamics. This is in contrast to previous explanations of fat tails and clustered volatility, which depended on ‘irrational behavior’, such as trend following. A standard (supposedly more sophisticated) risk control policy in which individual banks base leverage limits on volatility causes leverage to rise during periods of low volatility, and to contract more quickly when volatility becomes high, making these extreme fluctuations even worse.     

ARE PRICE LIMITS EFFECTIVE? EVIDENCE FROM THE ISTANBUL STOCK EXCHANGE

There is considerable discussion about controlling volatility by imposing price limits on asset prices. We examine the effects of price limits on a stock market by testing the volatility spillover, delayed price discovery, and trading interference hypotheses in a leading emerging market, the Istanbul Stock Exchange, which has a unique market microstructure as related to price limits. Our results support the volatility spillover, delayed price discovery, and trading interference hypotheses. We also show price locks at limits provide significantly stronger evidence regarding the effects of price limits than limit moves only. Finally, price limits have a significant effect on the stock market, casting doubt on their effectiveness.     

Leverage effect breakdowns and flight from risky assets

Though part of ‘market lore’, in 1976 Black first reported the inverse relationship between price and volatility, calling it the ‘leverage effect’. Without providing evidence, in 1988 Black claimed that in the months leading up to the October 1987 crash the relationship changed: price and volatility both rose. Using daily data for the Old VIX, derived from S&P 100 Index option market prices, to estimate intra-quarterly regressions of implied volatility against price from Q2 1986 to Q1 2012, the author verifies Black’s claim for the October 1987 crash, and interestingly, for subsequent periods of crisis. He then analyses several constant-elasticity-of-variance optimal portfolio rules, which include the leverage effect, to show the elasticity sign switch implies that investors reduce their risky asset holdings to zero.     

American option pricing under the double Heston model based on asymptotic expansion

This paper focuses on pricing American put options under the double Heston model proposed by Christoffersen et al. By introducing an explicit exercise rule, we obtain the asymptotic expansion of the solution to the partial differential equation for pricing American put options. We calculate American option price by the sum of the European option price and the early exercise premium. The early exercise premium is calculated by the difference between the American and European option prices based on asymptotic expansions. The European option price is obtained by the efficient COS method. Based on the obtained American option price, the double Heston model is calibrated by minimizing the distance between model and market prices, which yields an optimization problem that is solved by a differential evolution algorithm combined with the Matlab function fmincon.m. Numerical results show that the pricing approach is fast and accurate. Empirical results show that the double Heston model has better performance in pricing short-maturity American put options and capturing the volatility term structure of American put options than the Heston model.     

Stock market volatility and equity returns: Evidence from a two-state Markov-switching model with regressors

This paper proposes a two-state Markov-switching model for stock market returns in which the state-dependent expected returns, their variance and associated regime-switching dynamics are allowed to respond to market information. More specifically, we apply this model to examine the explanatory and predictive power of price range and trading volume for return volatility. Our findings indicate that a negative relation between equity market returns and volatility prevails even after having controlled for the time-varying determinants of conditional volatility within each regime. We also find an asymmetry in the effect of price range on intra- and inter-regime return volatility. While price range has a stronger effect in the high volatility state, it appears to significantly affect only the transition probabilities when the stock market is in the low volatility state but not in the high volatility state. Finally, we provide evidence consistent with the ‘rebound’ model of asset returns proposed by Samuelson (1991), suggesting that long-horizon investors are expected to invest more in risky assets than short-horizon investors.     

Investigating the determinants of banking coexceedances in Europe in the summer of 2008

We examine the nature, extent and possible causes of bank contagion in a high frequency setting. Looking at six major European banks in the summer and autumn of 2008, we model the lower coexceedances of these banks returns. We find that market microstructure, volatility (measured by range based measures) and limited general market conditions are key determinants of these coexceedances. We find some evidence that herding occurred.