arXiv subjects

Dan Pirjol

Publications and source records attributed to Dan Pirjol.

At least 37 records · Page 2Linked to original sources

arXiv2020

Asymptotics of the time-discretized log-normal SABR model: The implied volatility surface

We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler-Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certain asymptotic regime which includes the case of finite maturity, small vol-of-vol and large initial volatility with fixed product of vol-of-vol and initial volatility. We derive an almost sure limit and a large deviations result for the log-asset price in the limit of large number of time steps. We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in this regime. Using this representation we obtain analytical expansions of the implied volatility for small maturity and extreme strikes, which reproduce at leading order known asymptotic results for the continuous time model.

q-fin.MF↗

arXiv2019

Small-noise limit of the quasi-Gaussian log-normal HJM model

Quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low dimensional Markovian representation, which greatly simplifies their numerical implementation. We present a qualitative study of the solutions of the quasi-Gaussian log-normal HJM model. Using a small-noise deterministic limit we show that the short rate may explode to infinity in finite time. This implies the explosion of the Eurodollar futures prices in this model. We derive explicit explosion criteria under mild assumptions on the shape of the yield curve.

q-fin.MF↗

arXiv2019

Explosion in the quasi-Gaussian HJM model

We study the explosion of the solutions of the SDE in the quasi-Gaussian HJM model with a CEV-type volatility. The quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low dimensional Markovian representation which simplifies their numerical implementation and simulation. We show rigorously that the short rate in these models explodes in finite time with positive probability, under certain assumptions for the model parameters, and that the explosion occurs in finite time with probability one under some stronger assumptions. We discuss the implications of these results for the pricing of the zero coupon bonds and Eurodollar futures under this model.

q-fin.MF↗

arXiv2017

Short Maturity Forward Start Asian Options in Local Volatility Models

We study the short maturity asymptotics for prices of forward start Asian options under the assumption that the underlying asset follows a local volatility model. We obtain asymptotics for the cases of out-of-the-money, in-the-money, and at-the-money, considering both fixed strike and floating Asian options. The exponential decay of the price of an out-of-the-money forward start Asian option is handled using large deviations theory, and is controlled by a rate function which is given by a double-layer optimization problem. In the Black-Scholes model, the calculation of the rate function is simplified further to the solution of a non-linear equation. We obtain closed form for the rate function, as well as its asymptotic behaviors when the strike is extremely large, small, or close to the initial price of the underlying asset.

q-fin.PR↗

arXiv2017

Asymptotics for the Euler-Discretized Hull-White Stochastic Volatility Model

We consider the stochastic volatility model $dS_t = σ_t S_t dW_t,dσ_t = ωσ_t dZ_t$, with $(W_t,Z_t)$ uncorrelated standard Brownian motions. This is a special case of the Hull-White and the $β=1$ (log-normal) SABR model, which are widely used in financial practice. We study the properties of this model, discretized in time under several applications of the Euler-Maruyama scheme, and point out that the resulting model has certain properties which are different from those of the continuous time model. We study the asymptotics of the time-discretized model in the $n\to \infty$ limit of a very large number of time steps of size $τ$, at fixed $β=\frac12ω^2τn^2$ and $ρ=σ_0^2τ$, and derive three results: i) almost sure limits, ii) fluctuation results, and iii) explicit expressions for growth rates (Lyapunov exponents) of the positive integer moments of $S_t$. Under the Euler-Maruyama discretization for $(S_t,\log σ_t)$, the Lyapunov exponents have a phase transition, which appears in numerical simulations of the model as a numerical explosion of the asset price moments. We derive criteria for the appearance of these explosions.

q-fin.MF↗

arXiv2017

Asymptotics for the Discrete-Time Average of the Geometric Brownian Motion and Asian Options

The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps. We derive almost sure limit, fluctuations, large deviations, and also the asymptotics of the moment generating function of the average. Based on these results, we derive the asymptotics for the price of Asian options with discrete-time averaging in the Black-Scholes model, with both fixed and floating strike.

q-fin.PR↗

arXiv2017

Phenomenology of stochastic exponential growth

Stochastic exponential growth is observed in a variety of contexts, including molecular autocatalysis, nuclear fission, population growth, inflation of the universe, viral social media posts, and financial markets. Yet literature on modeling the phenomenology of these stochastic dynamics has predominantly focused on one model, Geometric Brownian Motion (GBM), which can be described as the solution of a Langevin equation with linear drift and linear multiplicative noise. Using recent experimental results on stochastic exponential growth of individual bacterial cell sizes, we motivate the need for a more general class of phenomenological models of stochastic exponential growth, which are consistent with the observation that the mean-rescaled distributions are approximately stationary at long times. We show that this behavior is not consistent with GBM, instead it is consistent with power law multiplicative noise with positive fractional powers. Therefore, we consider this general class of phenomenological models for stochastic exponential growth, provide analytical solutions, and identify the important dimensionless combination of model parameters which determines the shape of the mean-rescaled distribution. We also provide a prescription for robustly inferring model parameters from experimentally observed stochastic growth trajectories.

cond-mat.stat-mech↗

arXiv2017

Reexamining the photon polarization in $B\to Kππγ$

We reexamine, update and extend a suggestion we made fifteen years ago for measuring the photon polarization in $b \to sγ$ by observing in $B \to Kππγ$ an asymmetry of the photon with respect to the $Kππ$ plane. Asymmetries are calculated for different charged final states due to intermediate $K_1(1400)$ and $K_1(1270)$ resonant states. Three distinct interference mechanisms are identified contributing to asymmetries at different levels for these two kaon resonances. For $K_1(1400)$ decays including a final state $π^0$ an asymmetry around $+30\%$ is calculated, dominated by interference of two intermediate $K^*π$ states, while an asymmetry around $+10\%$ in decays including final $π^+π^-$ is dominated by interference of $S$ and $D$ wave $K^*π$ amplitudes. In decays via $K_1(1270)$ to final states including a $π^0$ a negative asymmetry is favored up to $-10\%$ if one assumes $S$ wave dominance in decays to $K^*π$ and $Kρ$, while in decays involving $π^+π^-$ the asymmetry can vary anywhere in the range $-13\%$ to $+24\%$ depending on unknown phases. For more precise asymmetry predictions in the latter decays we propose studying phases in $K_1 \to K^*π, Kρ$ by performing dedicated amplitude analyses of $B\to J/ψ(ψ') Kππ$. In order to increase statistics in studies of $B\to Kππγ$ we suggest using isospin symmetry to combine in the same analysis samples of charged and neutral $B$ decays.

hep-ph↗

arXiv2017

Short Maturity Asian Options for the CEV Model

We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows the Constant Elasticity of Variance (CEV) model. We present an analytical approximation for the Asian options prices which has the appropriate short maturity asymptotics, and demonstrate good numerical agreement of the asymptotic results with the results of Monte Carlo simulations and benchmark test cases for option parameters relevant in practical applications.

q-fin.PR↗

arXiv2016

Discrete Sums of Geometric Brownian Motions, Annuities and Asian Options

The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of geometric Brownian motions with geometric stopping time, and the finite sum of the geometric Brownian motions. These results are extended to the discrete sum of the exponential Lévy process. We derive tail asymptotics and compute numerically the asymptotic distribution function. We compare the results against the known results for the continuous time integral of the geometric Brownian motion up to an exponentially distributed time. The results are illustrated with numerical examples for life annuities with discrete payments, and Asian options.

q-fin.PR↗

arXiv2016

Short Maturity Asian Options in Local Volatility Models

We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows a local volatility model. The asymptotics for out-of-the-money, in-the-money, and at-the-money cases are derived, considering both fixed strike and floating strike Asian options. The asymptotics for the out-of-the-money case involves a non-trivial variational problem which is solved completely. We present an analytical approximation for Asian options prices, and demonstrate good numerical agreement of the asymptotic results with the results of Monte Carlo simulations and benchmark test cases in the Black-Scholes model for option parameters relevant in practical applications.

q-fin.PR↗

arXiv2016

Thermodynamics of a one-dimensional self-gravitating gas with periodic boundary conditions

We study the thermodynamical properties of a one-dimensional gas with one-dimensional gravitational interactions, and placed in a uniform mass background. Periodic boundary conditions are implemented as a modification of the potential consisting of a sum over mirror images (Ewald sum), regularized with an exponential cut-off. The system has a phase transition at a critical temperature. Above the critical temperature the gas density is uniform, while below the critical point the system becomes inhomogeneous. Numerical simulations of the model confirms the existence of the phase transition, and are in good agreement with the theoretical results.

cond-mat.stat-mech↗

arXiv2015

On the growth rate of a linear stochastic recursion with Markovian dependence

We consider the linear stochastic recursion $x_{i+1} = a_{i}x_{i}+b_{i}$ where the multipliers $a_i$ are random and have Markovian dependence given by the exponential of a standard Brownian motion and $b_{i}$ are i.i.d. positive random noise independent of $a_{i}$. Using large deviations theory we study the growth rates (Lyapunov exponents) of the positive integer moments $λ_q = \lim_{n\to \infty} \frac{1}{n} \log\mathbb{E}[(x_n)^q]$ with $q\in \mathbb{Z}_+$. We show that the Lyapunov exponents $λ_q$ exist, under appropriate scaling of the model parameters, and have non-analytic behavior manifested as a phase transition. We study the properties of the phase transition and the critical exponents using both analytic and numerical methods.

math.PR↗

arXiv2013

Explosive behavior in a log-normal interest rate model

We consider an interest rate model with log-normally distributed rates in the terminal measure in discrete time. Such models are used in financial practice as parametric versions of the Markov functional model, or as approximations to the log-normal Libor market model. We show that the model has two distinct regimes, at high and low volatilities, with different qualitative behavior. The two regimes are separated by a sharp transition, which is similar to a phase transition in condensed matter physics. We study the behavior of the model in the large volatility phase, and discuss the implications of the phase transition for the pricing of interest rate derivatives. In the large volatility phase, certain expectation values and convexity adjustments have an explosive behavior. For sufficiently low volatilities the caplet smile is log-normal to a very good approximation, while in the large volatility phase the model develops a non-trivial caplet skew. The phenomenon discussed here imposes thus an upper limit on the volatilities for which the model behaves as intended.

q-fin.CP↗

arXiv2012

Equivalence of interest rate models and lattice gases

We consider the class of short rate interest rate models for which the short rate is proportional to the exponential of a Gaussian Markov process x(t) in the terminal measure r(t) = a(t) exp(x(t)). These models include the Black, Derman, Toy and Black, Karasinski models in the terminal measure. We show that such interest rate models are equivalent with lattice gases with attractive two-body interaction V(t1,t2)= -Cov(x(t1),x(t2)). We consider in some detail the Black, Karasinski model with x(t) an Ornstein, Uhlenbeck process, and show that it is similar with a lattice gas model considered by Kac and Helfand, with attractive long-range two-body interactions V(x,y) = -α(e^{-γ|x - y|} - e^{-γ(x + y)}). An explicit solution for the model is given as a sum over the states of the lattice gas, which is used to show that the model has a phase transition similar to that found previously in the Black, Derman, Toy model in the terminal measure.

q-fin.CP↗

arXiv2011

Asymptotic Expansion for the Normal Implied Volatility in Local Volatility Models

We study the dynamics of the normal implied volatility in a local volatility model, using a small-time expansion in powers of maturity T. At leading order in this expansion, the asymptotics of the normal implied volatility is similar, up to a different definition of the moneyness, to that of the log-normal volatility. This relation is preserved also to order O(T) in the small-time expansion, and differences with the log-normal case appear first at O(T^2). The results are illustrated on a few examples of local volatility models with analytical local volatility, finding generally good agreement with exact or numerical solutions. We point out that the asymptotic expansion can fail if applied naively for models with nonanalytical local volatility, for example which have discontinuous derivatives. Using perturbation theory methods, we show that the ATM normal implied volatility for such a model contains a term ~ \sqrt{T}, with a coefficient which is proportional with the jump of the derivative.

q-fin.CP↗

arXiv2011

Phase transition in a log-normal Markov functional model

We derive the exact solution of a one-dimensional Markov functional model with log-normally distributed interest rates in discrete time. The model is shown to have two distinct limiting states, corresponding to small and asymptotically large volatilities, respectively. These volatility regimes are separated by a phase transition at some critical value of the volatility. We investigate the conditions under which this phase transition occurs, and show that it is related to the position of the zeros of an appropriately defined generating function in the complex plane, in analogy with the Lee-Yang theory of the phase transitions in condensed matter physics.

q-fin.CP↗

arXiv2010

Matching the quark model to the 1/Nc expansion

We compute the coefficients of the effective mass operator of the 1/Nc expansion for negative parity L=1 excited baryons using the Isgur-Karl model in order to compare the general approach, where the coefficients are obtained by fitting to data, with a specific constituent quark model calculation. We discuss the physics behind the fitted coefficients for the scalar part of the most general two-body quark-quark interaction. We find that both pion exchange and gluon exchange lead to the dominance of the same operator at the level of the effective mass operator, which is also observed from data.

hep-ph↗