The characteristic function of rough Heston models

It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non‐Markovian nature of the fractional Brownian motion, they raise new issues when it comes to derivatives pricing. Using an original link between nearly unstable Hawkes processes and fractional volatility models, we compute the characteristic function of the log‐price in rough Heston models. In the classical Heston model, the characteristic function is expressed in terms of the solution of a Riccati equation. Here, we show that rough Heston models exhibit quite a similar structure, the Riccati equation being replaced by a fractional Riccati equation.     

No‐arbitrage implies power‐law market impact and rough volatility

Market impact is the link between the volume of a (large) order and the price move during and after the execution of this order. We show that in a quite general framework, under no‐arbitrage assumption, the market impact function can only be of power‐law type. Furthermore, we prove this implies that the macroscopic price is diffusive with rough volatility, with a one‐to‐one correspondence between the exponent of the impact function and the Hurst parameter of the volatility. Hence, we simply explain the universal rough behavior of the volatility as a consequence of the no‐arbitrage property. From a mathematical viewpoint, our study relies, in particular, on new results about hyper‐rough stochastic Volterra equations.     

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.     

Optimal portfolios and Heston's stochastic volatility model: an explicit solution for power utility

Given an investor maximizing utility from terminal wealth with respect to a power utility function, we present a verification result for portfolio problems with stochastic volatility. Applying this result, we solve the portfolio problem for Heston's stochastic volatility model. We find that only under a specific condition on the model parameters does the problem possess a unique solution leading to a partial equilibrium. Finally, it is demonstrated that the results critically hinge upon the specification of the market price of risk. We conclude that, in applications, one has to be very careful when exogenously specifying the form of the market price of risk.     

Moment explosions in stochastic volatility models

In this paper, we demonstrate that many stochastic volatility models have the undesirable property that moments of order higher than 1 can become infinite in finite time. As arbitrage-free price computation for a number of important fixed income products involves forming expectations of functions with super-linear growth, such lack of moment stability is of significant practical importance. For instance, we demonstrate that reasonably parametrized models can produce infinite prices for Eurodollar futures and for swaps with floating legs paying either Libor-in-arrears or a constant maturity swap rate. We systematically examine the moment explosion property across a spectrum of stochastic volatility models. We show that lognormal and displaced-diffusion type models are easily prone to moment explosions, whereas CEV-type models (including the so-called SABR model) are not. Related properties such as the failure of the martingale property are also considered.

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A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets

This paper provides comparative theoretical and numerical results on risks, values, and hedging strategies for local risk-minimization versus mean-variance hedging in a class of stochastic volatility models. We explain the theory for both hedging approaches in a general framework, specialize to a Markovian situation, and analyze in detail variants of the well-known Heston (1993) and Stein and Stein (1991) stochastic volatility models. Numerical results are obtained mainly by PDE and simulation methods. In addition, we take special care to check that all of our examples do satisfy the conditions required by the general theory.     

INDIFFERENCE PRICES AND IMPLIED VOLATILITIES

We consider a general local‐stochastic volatility model and an investor with exponential utility. For a European‐style contingent claim, whose payoff may depend on either a traded or nontraded asset, we derive an explicit approximation for both the buyer's and seller's indifference prices. For European calls on a traded asset, we translate indifference prices into an explicit approximation of the buyer's and seller's implied volatility surfaces. For European claims on a nontraded asset, we establish rigorous error bounds for the indifference price approximation. Finally, we implement our indifference price and implied volatility approximations in two examples.     

The implied Sharpe ratio

In an incomplete market, including liquidly traded European options in an investment portfolio could potentially improve the expected terminal utility for a risk-averse investor. However, unlike the Sharpe ratio, which provides a concise measure of the relative investment attractiveness of different underlying risky assets, there is no such measure available to help investors choose among the different European options. We introduce a new concept—the implied Sharpe ratio—which allows investors to make such a comparison in an incomplete financial market. Specifically, when comparing various European options, it is the option with the highest implied Sharpe ratio that, if included in an investor's portfolio, will improve his expected utility the most. Through the method of Taylor series expansion of the state-dependent coefficients in a nonlinear partial differential equation, we also establish the behaviour of the implied Sharpe ratio with respect to an investor's risk-aversion parameter. In a series of numerical studies, we compare the investment attractiveness of different European options by studying their implied Sharpe ratio.     

Regime switching rough Heston model

This model combines two important stylized features of volatility, the rough behavior consistent with a Hurst parameter less than , and the regime switching property consistent with more long-term economic considerations. It is nevertheless highly tractable in the sense of semianalytic formulae for European options, and permits a partial Monte Carlo method of similar computational speed as the semianalytic formula (at an appropriate number of Monte Carlo simulations). While option prices are relatively insensitive to the choice of Hurst parameter, introducing rough volatility allows for a better fit to the at-the-money skew.     

The affine Heston model with correlated Gaussian interest rates for pricing hybrid derivatives

In this article we define a multi-factor equity–interest rate hybrid model with non-zero correlation between the stock and interest rate. The equity part is modeled by the Heston model and we use a Gaussian multi-factor short-rate process. By construction, the model fits in the framework of affine diffusion processes, allowing fast calibration to plain vanilla options. We also provide an efficient Monte Carlo simulation scheme.