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Volume 1, 2022

Frontiers of Mathematical Finance

Open Access Articles

Implied price processes anchored in statistical realizations
Dilip B. Madan and King Wang
2022 doi: 10.3934/fmf.2021008 +[Abstract](135) +[HTML](60) +[PDF](439.0KB)

It is observed that statistical and risk neutral densities of compound Poisson processes are unconstrained relative to each other. Continuous processes are too constrained and generally not consistent with market data. Pure jump limit laws deliver operational models simultaneously consistent with both data sets with the additional imposition of no measure change on the arbitrarily small moves. The measure change density must have a finite Hellinger distance from unity linking the two worlds. Models are constructed using the bilateral gamma and the CGMY models for the risk neutral specification. They are linked to the physical process by measure change models. The resulting models simultaneously calibrate statistical tail probabilities and option prices. The resulting models have up to eight or ten parameters permitting the study of risk reward relations at a finer level. Rewards measured by power variations of the up and down moves are observed to value negatively(positively) the even(odd) variations of their own side with the converse holding for the opposite side.

Quadratic variation, models, applications and lessons
Dilip B. Madan and King Wang
2022, 1(2): 189-217 doi: 10.3934/fmf.2021007 +[Abstract](503) +[HTML](241) +[PDF](1569.01KB)

Time changes of Brownian motion impose restrictive jump structures in the motion of asset prices. Quadratic variations also depart from time changes. Quadratic variation options are observed to have a nonlinear exposure to risk neutral skewness. Joint Laplace Fourier transforms for quadratic variation and the stock are developed. They are used to study the multiple of the cap strike over the variance swap quote attaining a given percentage price reduction for the capped variance swap. Market prices for out-of-the-money options on variance are observed to be above those delivered by the calibrated models. Bootstrapped data and simulated paths spaces are used to study the multiple of the dynamic hedge return desired by a quadratic variation contract. It is observed that the optimized hedge multiple in the bootstrapped data is near unity. Furthermore, more generally, it is exposures to negative cubic variations in path spaces that raise variance swap prices, lower hedge multiples, increase residual risk charges and gaps to the log contract hedge. A case can be made for both, the physical process being closer to a continuous time change of Brownian motion, while simultaneously risk neutrally this may not be the case. It is recognized that in the context of discrete time there are no issues related to equivalence of probabilities.

Asset price bubbles: Invariance theorems
Robert Jarrow, Philip Protter and Jaime San Martin
2022, 1(2): 161-188 doi: 10.3934/fmf.2021006 +[Abstract](769) +[HTML](387) +[PDF](477.47KB)

This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process' quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset's drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes' strict local martingale behavior can be characterized.

Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model
Michael C. Fu, Bingqing Li, Rongwen Wu and Tianqi Zhang
2022, 1(1): 137-160 doi: 10.3934/fmf.2021005 +[Abstract](1118) +[HTML](523) +[PDF](483.72KB)

We consider option pricing using a discrete-time Markov switching stochastic volatility with co-jump model, which can capture asset price features such as leptokurtosis, skewness, volatility clustering, and varying mean-reversion speed of volatility. For pricing European options, we develop a computationally efficient method for obtaining the probability distribution of average integrated variance (AIV), which is key to option pricing under stochastic-volatility-type models. Building upon the efficiency of the European option pricing approach, we are able to price an American-style option, by converting its pricing into the pricing of a portfolio of European options. Our work also provides constructive guidance for analyzing derivatives based on variance, e.g., the variance swap. Numerical results indicate our methods can be implemented very efficiently and accurately.

A rough SABR formula
Masaaki Fukasawa and Jim Gatheral
2022, 1(1): 81-97 doi: 10.3934/fmf.2021003 +[Abstract](2779) +[HTML](790) +[PDF](1277.5KB)

Following an approach originally suggested by Balland in the context of the SABR model, we derive an ODE that is satisfied by normalized volatility smiles for short maturities under a rough volatility extension of the SABR model that extends also the rough Bergomi model. We solve this ODE numerically and further present a very accurate approximation to the numerical solution that we dub the rough SABR formula.

Multilayer heat equations: Application to finance
Andrey Itkin, Alexander Lipton and Dmitry Muravey
2022, 1(1): 99-135 doi: 10.3934/fmf.2021004 +[Abstract](866) +[HTML](448) +[PDF](674.78KB)

In this paper, we develop a Multilayer (ML) method for solving one-factor parabolic equations. Our approach provides a powerful alternative to the well-known finite difference and Monte Carlo methods. We discuss various advantages of this approach, which judiciously combines semi-analytical and numerical techniques and provides a fast and accurate way of finding solutions to the corresponding equations. To introduce the core of the method, we consider multilayer heat equations, known in physics for a relatively long time but never used when solving financial problems. Thus, we expand the analytic machinery of quantitative finance by augmenting it with the ML method. We demonstrate how one can solve various problems of mathematical finance by using our approach. Specifically, we develop efficient algorithms for pricing barrier options for time-dependent one-factor short-rate models, such as Black-Karasinski and Verhulst. Besides, we show how to solve the well-known Dupire equation quickly and accurately. Numerical examples confirm that our approach is considerably more efficient for solving the corresponding partial differential equations than the conventional finite difference method by being much faster and more accurate than the known alternatives.

Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing
Walter Farkas and Ludovic Mathys
2022, 1(1): 1-51 doi: 10.3934/fmf.2021001 +[Abstract](1019) +[HTML](583) +[PDF](1805.98KB)

The present article studies geometric step options in exponential Lévy markets. Our contribution is manifold and extends several aspects of the geometric step option pricing literature. First, we provide symmetry and duality relations and derive various characterizations for both European-type and American-type geometric double barrier step options. In particular, we are able to obtain a jump-diffusion disentanglement for the early exercise premium of American-type geometric double barrier step contracts and its maturity-randomized equivalent as well as to characterize the diffusion and jump contributions to these early exercise premiums separately by means of partial integro-differential equations and ordinary integro-differential equations. As an application of our characterizations, we derive semi-analytical pricing results for (regular) European-type and American-type geometric down-and-out step call options under hyper-exponential jump-diffusion models. Lastly, we use the latter results to discuss the early exercise structure of geometric step options once jumps are added and to subsequently provide an analysis of the impact of jumps on the price and hedging parameters of (European-type and American-type) geometric step contracts.

Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit
Andrey Itkin and Dmitry Muravey
2022, 1(1): 53-79 doi: 10.3934/fmf.2021002 +[Abstract](923) +[HTML](488) +[PDF](663.19KB)

We continue a series of papers devoted to construction of semi-analytic solutions for barrier options. These options are written on underlying following some simple one-factor diffusion model, but all the parameters of the model as well as the barriers are time-dependent. We managed to show that these solutions are systematically more efficient for pricing and calibration than, e.g., the corresponding finite-difference solvers. In this paper we extend this technique to pricing double barrier options and present two approaches to solving it: the General Integral transform method and the Heat Potential method. Our results confirm that for double barrier options these semi-analytic techniques are also more efficient than the traditional numerical methods used to solve this type of problems.



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