数理金融

We take a look the changes of different asset prices over variable periods, using both traditional and spectral methods, and discover universality phenomena which hold (in some cases) across asset classes.

The literature provides strong evidence that stock prices can be predicted from past price data. Principal component analysis (PCA) is a widely used mathematical technique for dimensionality reduction and analysis of data by identifying a…

In this paper, we investigate the non-linear Black--Scholes equation: $$u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\quad a,b>0,\ c\geq0.$$ and show that the one can be reduced to the equation $$u_t+(u_{xx}+u_x)^2=0$$ by an appropriate point…

We suggest that the broad distribution of time scales in financial markets could be a crucial ingredient to reproduce realistic price dynamics in stylised Agent-Based Models. We propose a fractional reaction-diffusion model for the dynamics…

Long maturity options or a wide class of hybrid products are evaluated using a local volatility type modelling for the asset price S(t) with a stochastic interest rate r(t). The calibration of the local volatility function is usually…

In this study, we extend the optimal execution problem with convex market impact function studied in Kato (2014) to the case where the market impact function is S-shaped, that is, concave on $[0, \bar {x}_0]$ and convex on $[\bar {x}_0,…

The quotient of random variables with normal distributions is examined and proven to have have power law decay, with density $f\left( x\right) \simeq f_{0}x^{-2}$, with the coefficient depending on the means and variances of the numerator…

In this paper we derive a second order approximation for an infinite dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator…

We provide a lean, non-technical exposition on the pricing of path-dependent and European-style derivatives in the Cox-Ross-Rubinstein (CRR) pricing model. The main tool used in the paper for cleaning up the reasoning is applying static…

Cryptocurrencies are examined through the asset flow equations and experimental asset markets. Since tangible value of a typical cryptocurrency is non-existent, the theory suggests that price will gravitate toward liquidity value, i.e., the…

This paper is concerned with the MAXVAR risk measure on L^2 space. We present an elementary and direct proof of its coherency and averseness. Based on the observation that the MAXVAR measure is a continuous convex combination of the CVaR…

We prove the existence of a Radner equilibrium in a model with proportional transaction costs on an infinite time horizon and analyze the effect of transaction costs on the endogenously determined interest rate. Two agents receive…

In commodity markets the convergence of futures towards spot prices, at the expiration of the contract, is usually justified by no-arbitrage arguments. In this article, we propose an alternative approach that relies on the expected profit…

The paper studies sub and super-replication price bounds for contingent claims defined on general trajectory based market models. No prior probabilistic or topological assumptions are placed on the trajectory space, trading is assumed to…

We provide a dual characterisation of the weak$^*$-closure of a finite sum of cones in $L^\infty$ adapted to a discrete time filtration $\mathcal{F}_t$: the $t^{th}$ cone in the sum contains bounded random variables that are…

The general problem of asset pricing when the discount rate differs from the rate at which an asset's cash flows accrue is considered. A pricing kernel framework is used to model an economy that is segmented into distinct markets, each…

We consider the framework proposed by Burgard and Kjaer (2011) that derives the PDE which governs the price of an option including bilateral counterparty risk and funding. We extend this work by relaxing the assumption of absence of…

This paper corrects an error in [Keller-Ressel, M. and Steiner T. "Yield curve shapes and the asymptotic short rate distribution in affine one-factor models." Finance and Stochastics 12.2 (2008): 149-172]. The error concerns the correct…

We consider the stochastic volatility model $dS_t = \sigma_t S_t dW_t,d\sigma_t = \omega \sigma_t dZ_t$, with $(W_t,Z_t)$ uncorrelated standard Brownian motions. This is a special case of the Hull-White and the $\beta=1$ (log-normal) SABR…

We develop a robust framework for pricing and hedging of derivative securities in discrete-time financial markets. We consider markets with both dynamically and statically traded assets and make minimal measurability assumptions. We obtain…