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Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the…
We consider the class of self-similar Gaussian stochastic volatility models, and compute the small-time (near-maturity) asymptotics for the corresponding asset price density, the call and put pricing functions, and the implied volatilities.…
We investigate qualitative and quantitative behavior of a solution of the mathematical model for pricing American style of perpetual put options. We assume the option price is a solution to the stationary generalized Black-Scholes equation…
It is well-known that the Black-Scholes formula has been derived under the assumption of constant volatility in stocks. In spite of evidence that this parameter is not constant, this formula is widely used by financial markets. This paper…
Barrieu, Rouault, and Yor [J. Appl. Probab. 41 (2004)] determined asymptotics for the logarithm of the distribution function of the Hartman-Watson distribution. We determine the asymptotics of the density. This refinement can be applied to…
In this paper we generalize and analyze the model for pricing American-style Asian options due to (Hansen and Jorgensen 2000) by including a continuous dividend rate $q$ and a general method of averaging of the floating strike. We focus on…
We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price process $S$ follows a general martingale. This is equivalent to studying the first centered absolute moment of $S$. We show that…
We present an algorithm for the calibration of local volatility from market option prices through deep self-consistent learning, by approximating both market option prices and local volatility using deep neural networks. Our method uses the…
This paper investigates asymptotically optimal importance sampling (IS) schemes for pricing European call options under the Heston stochastic volatility model. We focus on two distinct rare-event regimes where standard Monte Carlo methods…
We study the pricing of European-style options written on forward contracts within function-valued infinite-dimensional affine stochastic volatility models. The dynamics of the underlying forward price curves are modeled within the…
We present closed analytical approximations for the pricing of basket options, also applicable to Asian options with discrete averaging under the Black-Scholes model with time-dependent parameters. The formulae are obtained by using a…
We present a numerical approach for solving the free boundary problem for the Black-Scholes equation for pricing American style of floating strike Asian options. A fixed domain transformation of the free boundary problem into a parabolic…
We consider the pricing problem related to payoffs that can have discontinuities of polynomial growth. The asset price dynamic is modeled within the Black and Scholes framework characterized by a stochastic volatility term driven by a…
Exponential L\'evy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such…
We present a differential machine learning method for zero-days-to-expiry (0DTE) options under a stochastic-volatility jump-diffusion model. To handle the ultra-short-maturity regime, we express the option price in Black-Scholes form with a…
We derive an explicit asymptotic approximation for the implied volatilities of Call options written on bonds assuming the short-rate is described by an affine short-rate model. For specific affine short-rate models, we perform numerical…
In this note, Black--Scholes implied volatility is expressed in terms of various optimisation problems. From these representations, upper and lower bounds are derived which hold uniformly across moneyness and call price. Various symmetries…
When trading American and Asian options in the FX derivatives market, banks must calculate prices using a complex mathematical model. It is often observed that different models produce varying prices for the same exotic option, which…
We present closed analytical approximations for the pricing of Asian basket spread options under the Black-Scholes model. The formulae are obtained by using a stochastic Taylor expansion around a log-normal proxy model and are found to be…
We provide a short-time large deviation principle (LDP) for stochastic volatility models, where the volatility is expressed as a function of a Volterra process. This LDP does not require strict self-similarity assumptions on the Volterra…