Complete-market models of stochastic volatility

In the Black–Scholes option-pricing theory, asset prices are modelled as geometric Brownian motion with a fixed volatility parameter σ, and option prices are deter-mined as functions of the underlying asset price. Options are in principle redundant in that their exercise values can be replicated by trading in the underlying. However, it is an empirical fact that the prices of exchange-traded options do not correspond to a fixed value of σ as the theory requires. This paper proposes a modelling framework in which certain options are non-redundant: these options and the underlying are modelled as autonomous financial assets, linked only by the boundary condition at exercise. A geometric condition is given, under which a complete market is obtained in this way, giving a consistent theory under which traded options as well as the underlying asset are used as hedging instruments

Local Volatility Calibration by Optimal Transport

The calibration of volatility models from observable option prices is a fundamental problem in quantitative finance. The most common approach among industry practitioners is based on the celebrated Dupire's formula [6], which requires the knowledge of vanilla option prices for a continuum of strikes and maturities that can only be obtained via some form of price interpolation. In this paper, we propose a new local volatility calibration technique using the theory of optimal transport. We formulate a time continuous martingale optimal transport problem, which seeks a martingale diffusion process that matches the known densities of an asset price at two different dates, while minimizing a chosen cost function. Inspired by the seminal work of Benamou and Brenier [1], we formulate the problem as a convex optimization problem, derive its dual formulation, and solve it numerically via an augmented Lagrangian method and the alternative direction method of multipliers (ADMM) algorithm. The solution effectively reconstructs the dynamic of the asset price between the two dates by recovering the optimal local volatility function, without requiring any time interpolation of the option prices

Root to Kellerer

We revisit Kellerer's Theorem, that is, we show that for a family of real probability distributions $(\mu_t)_{t\in [0,1]}$ which increases in convex order there exists a Markov martingale $(S_t)_{t\in[0,1]}$ s.t.\ $S_t\sim \mu_t$. To establish the result, we observe that the set of martingale measures with given marginals carries a natural compact Polish topology. Based on a particular property of the martingale coupling associated to Root's embedding this allows for a relatively concise proof of Kellerer's theorem. We emphasize that many of our arguments are borrowed from Kellerer \cite{Ke72}, Lowther \cite{Lo07}, and Hirsch-Roynette-Profeta-Yor \cite{HiPr11,HiRo12}.Comment: 8 pages, 1 figur

Adiabaticity Conditions for Volatility Smile in Black-Scholes Pricing Model

Our derivation of the distribution function for future returns is based on the risk neutral approach which gives a functional dependence for the European call (put) option price, C(K), given the strike price, K, and the distribution function of the returns. We derive this distribution function using for C(K) a Black-Scholes (BS) expression with volatility in the form of a volatility smile. We show that this approach based on a volatility smile leads to relative minima for the distribution function ("bad" probabilities) never observed in real data and, in the worst cases, negative probabilities. We show that these undesirable effects can be eliminated by requiring "adiabatic" conditions on the volatility smile

Modification terms to the Black-Scholes model in a realistic hedging strategy with discrete temporal steps

Option pricing models generally require the assumption that stock prices are described by continuous-time stochastic processes. Although the time-continuous trading is easy to conceive theoretically, it is practically impossible to execute in real markets. One reason is because real markets are not perfectly liquid and purchase or sell any amount of an asset would change the asset price drastically. A realistic hedging strategy needs to consider trading that happens at discrete instants of time. This paper focuses on the impact and effect due to temporal discretisation on the pricing partial differential equation (PDE) for European options. Two different aspects of temporal discretisation are considered and used to derive the modification or correction source terms to the continuous pricing PDE. First the finite difference discretisation of the standard Black-Scholes PDE and its modification due to discrete trading. Second the discrete trading leads to a discrete time re-balancing strategy that only cancels risks on average by using a discrete analogy of the stochastic process of the underlying asset. In both cases high order terms in the Taylor series expansion are used and the respective correction source terms are derived

Local time and the pricing of time-dependent barrier options

A time-dependent double-barrier option is a derivative security that delivers the terminal value $\phi(S_T)$ at expiry $T$ if neither of the continuous time-dependent barriers b_\pm:[0,T]\to \RR_+ have been hit during the time interval $[0,T]$. Using a probabilistic approach we obtain a decomposition of the barrier option price into the corresponding European option price minus the barrier premium for a wide class of payoff functions $\phi$, barrier functions $b_\pm$ and linear diffusions $(S_t)_{t\in[0,T]}$. We show that the barrier premium can be expressed as a sum of integrals along the barriers $b_\pm$ of the option's deltas \Delta_\pm:[0,T]\to\RR at the barriers and that the pair of functions $(\Delta_+,\Delta_-)$ solves a system of Volterra integral equations of the first kind. We find a semi-analytic solution for this system in the case of constant double barriers and briefly discus a numerical algorithm for the time-dependent case.Comment: 32 pages, to appear in Finance and Stochastic

Advances on antiviral activity of Morus spp. plant extracts: Human coronavirus and virus-related respiratory tract infections in the spotlight

(1) Background: Viral respiratory infections cause life-threatening diseases in millions of people worldwide every year. Human coronavirus and several picornaviruses are responsible for worldwide epidemic outbreaks, thus representing a heavy burden to their hosts. In the absence of specific treatments for human viral infections, natural products offer an alternative in terms of innovative drug therapies. (2) Methods: We analyzed the antiviral properties of the leaves and stem bark of the mulberry tree (Morus spp.). We compared the antiviral activity of Morus spp. on enveloped and nonenveloped viral pathogens, such as human coronavirus (HCoV 229E) and different members of the Picornaviridae family—human poliovirus 1, human parechovirus 1 and 3, and human echovirus 11. The antiviral activity of 12 water and water–alcohol plant extracts of the leaves and stem bark of three different species of mulberry—Morus alba var. alba, Morus alba var. rosa, and Morus rubra—were evaluated. We also evaluated the antiviral activities of kuwanon G against HCoV-229E. (3) Results: Our results showed that several extracts reduced the viral titer and cytopathogenic effects (CPE). Leaves’ water-alcohol extracts exhibited maximum antiviral activity on human coronavirus, while stem bark and leaves’ water and water-alcohol extracts were the most effective on picornaviruses. (4) Conclusions: The analysis of the antiviral activities of Morus spp. offer promising applications in antiviral strategies