SEARCH · Search arXiv

Results for “q-fin.PR”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 235 records · Page 13Linked to original sources

arXiv2011

Two stock options at the races: Black-Scholes forecasts

Suppose one buys two very similar stocks and is curious about how much, after some time T, one of them will contribute to the overall asset, expecting, of course, that it should be around 1/2 of the sum. Here we examine this question within the classical Black and Scholes (BS) model, focusing on the evolution of the probability density function P(w) of a random variable w = a_T^{(1)}/(a_T^{(1)} + a_T^{(2)}) where a_T^{(1)} and a_T^{(2)} are the values of two (either European- or the Asian-style) options produced by two absolutely identical BS stochastic equations. We show that within the realm of the BS model the behavior of P(w) is surprisingly different from common-sense-based expectations. For the European-style options P(w) always undergoes a transition, (when T approaches a certain threshold value), from a unimodal to a bimodal form with the most probable values being close to 0 and 1, and, strikingly, w =1/2 being the least probable value. This signifies that the symmetry between two options spontaneously breaks and just one of them completely dominates the sum. For path-dependent Asian-style options we observe the same anomalous behavior, but only for a certain range of parameters. Outside of this range, P(w) is always a bell-shaped function with a maximum at w = 1/2.

q-fin.ST↗

arXiv2011

Quadratic Reflected BSDEs with Unbounded Obstacles

In this paper, we analyze a real-valued reflected backward stochastic differential equation (RBSDE) with an unbounded obstacle and an unbounded terminal condition when its generator $f$ has quadratic growth in the $z$-variable. In particular, we obtain existence, comparison, and stability results, and consider the optimal stopping for quadratic $g$-evaluations. As an application of our results we analyze the obstacle problem for semi-linear parabolic PDEs in which the non-linearity appears as the square of the gradient. Finally, we prove a comparison theorem for these obstacle problems when the generator is convex or concave in the $z$-variable.

math.PR↗

arXiv2010

Market Price of Risk and Random Field Driven Models of Term Structure: A Space-Time Change of Measure Look

No-arbitrage models of term structure have the feature that the return on zero-coupon bonds is the sum of the short rate and the product of volatility and market price of risk. Well known models restrict the behavior of the market price of risk so that it is not dependent on the type of asset being modeled. We show that the models recently proposed by Goldstein and Santa-Clara and Sornette, among others, allow the market price of risk to depend on characteristics of each asset, and we quantify this dependence. A key tool in our analysis is a very general space-time change of measure theorem, proved by the first author in earlier work, and covers continuous orthogonal local martingale measures including space-time white noise.

q-fin.PR↗

arXiv2011

Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management

In this paper we investigate novel applications of a new class of equations which we call time-delayed backward stochastic differential equations. Time-delayed BSDEs may arise in finance when we want to find an investment strategy and an investment portfolio which should replicate a liability or meet a target depending on the applied strategy or the past values of the portfolio. In this setting, a managed investment portfolio serves simultaneously as the underlying security on which the liability/target is contingent and as a replicating portfolio for that liability/target. This is usually the case for capital-protected investments and performance-linked pay-offs. We give examples of pricing, hedging and portfolio management problems (asset-liability management problems) which could be investigated in the framework of time-delayed BSDEs. Our motivation comes from life insurance and we focus on participating contracts and variable annuities. We derive the corresponding time-delayed BSDEs and solve them explicitly or at least provide hints how to solve them numerically. We give a financial interpretation of the theoretical fact that a time-delayed BSDE may not have a solution or may have multiple solutions.

q-fin.PR↗

arXiv2010

The Impact of Credit Risk and Implied Volatility on Stock Returns

This paper examines the possibility of using derivative-implied risk premia to explain stock returns. The rapid development of derivative markets has led to the possibility of trading various kinds of risks, such as credit and interest rate risk, separately from each other. This paper uses credit default swaps and equity options to determine risk premia which are then used to form portfolios that are regressed against the returns of stock portfolios. It turns out that both, credit risk and implied volatility, have high explanatory power in regard to stock returns. Especially the returns of distressed stocks are highly dependent on credit risk fluctuations. This finding leads to practical implications, such as cross-hedging opportunities between equity and credit instruments and potentially allows forecasting stock returns based on movements in the credit market.

q-fin.PR↗

arXiv2010

Functionals of Exponential Brownian Motion and Divided Differences

We provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between exponential Brownian motion and its time average, and we find the use of divided differences greatly elucidates formulae, providing a path to several new results. As applications, we find that this correlation coefficient is always at least $1/\sqrt{2}$ and, via the Hermite--Genocchi integral relation, demonstrate that all moments of the time average are certain divided differences of the exponential function. We also prove that these moments agree with the somewhat more complex formulae obtained by Oshanin and Yor.

math.NA↗

arXiv2013

Discrete tenor models for credit risky portfolios driven by time-inhomogeneous Lévy processes

The goal of this paper is to specify dynamic term structure models with discrete tenor structure for credit portfolios in a top-down setting driven by time-inhomogeneous Lévy processes. We provide a new framework, conditions for absence of arbitrage, explicit examples, an affine setup which includes contagion and pricing formulas for STCDOs and options on STCDOs. A calibration to iTraxx data with an extended Kalman filter shows an excellent fit over the full observation period. The calibration is done on a set of CDO tranche spreads ranging across six tranches and three maturities.

q-fin.PR↗

arXiv2011

Price as a matter of choice and nonstochastic randomness

A version of indifference valuation of a European call option is proposed that includes statistical regularities of nonstochastic randomness. Classical relations (forward contract value and Black-Scholes formula) are obtained as particular cases. We show that in the general case of nonstochastic randomness the minimal expected profit of uncovered European option position is always negative. A version of delta hedge is proposed.

q-fin.PR↗

arXiv2010

Bounds on Stock Price probability distributions in Local-Stochastic Volatility models

We show that in a large class of stochastic volatility models with additional skew-functions (local-stochastic volatility models) the tails of the cumulative distribution of the log-returns behave as exp(-c|y|), where c is a positive constant depending on time and on model parameters. We obtain this estimate proving a stronger result: using some estimates for the probability that Ito processes remain around a deterministic curve from Bally et al. '09, we lower bound the probability that the couple (X,V) remains around a two-dimensional curve up to a given maturity, X being the log-return process and V its instantaneous variance. Then we find the optimal curve leading to the bounds on the terminal cdf. The method we rely on does not require inversion of characteristic functions but works for general coefficients of the underlying SDE (in particular, no affine structure is needed). Even though the involved constants are less sharp than the ones derived for stochastic volatility models with a particular structure, our lower bounds entail moment explosion, thus implying that Black-Scholes implied volatility always displays wings in the considered class of models. In a second part of this paper, using Malliavin calculus techniques, we show that an analogous estimate holds for the density of the log-returns as well.

q-fin.PR↗

arXiv2010

Interest-Rate Modeling with Multiple Yield Curves

The crisis that affected financial markets in the last years leaded market practitioners to revise well known basic concepts like the ones of discount factors and forward rates. A single yield curve is not sufficient any longer to describe the market of interest rate products. On the other hand, using different yield curves at the same time requires a reformulation of most of the basic assumptions made in interest rate models. In this paper we discuss market evidences that led to the introduction of a series of different yield curves. We then define a HJM framework based on a multi-curve approach, presenting also a bootstrapping algorithm used to fit these different yield curves to market prices of plain-vanilla contracts such as basic Interest Rate Swaps (IRS) and Forward Rate Agreements (FRA). We then show how our approach can be used in practice when pricing other interest rate products, such as forward starting IRS, plain-vanilla European Swaptions, Constant Maturity Swaps (CMS) and CMS spread options, with the final goal to investigate whether the market is actually using a multi-curve approach or not. We finally present some numerical examples for a simple formulation of the framework which embeds by construction the multi-curve structure; once the model is calibrated to market prices of plain-vanilla options, it can be used via a Monte Carlo simulation to price more complicated exotic options.

q-fin.PR↗

arXiv2010

The fractional volatility model: No-arbitrage, leverage and risk measures

Based on a criterium of mathematical simplicity and consistency with empirical market data, a stochastic volatility model has been obtained with the volatility process driven by fractional noise. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behavior. Here, the no-arbitrage and incompleteness properties of the model are studied. Some risk measures are also discussed in this framework.

q-fin.PR↗

arXiv2010

Pricing in an equilibrium based model for a large investor

We study a financial model with a non-trivial price impact effect. In this model we consider the interaction of a large investor trading in an illiquid security, and a market maker who is quoting prices for this security. We assume that the market maker quotes the prices such that by taking the other side of the investor's demand, the market maker will arrive at maturity with the maximal expected utility of the terminal wealth. Within this model we provide an explicit recursive pricing formula for an exponential utility function, as well as an asymptotic expansion for the price for a "small" simple demand.

q-fin.PR↗

arXiv2011

Picard approximation of stochastic differential equations and application to LIBOR models

The aim of this work is to provide fast and accurate approximation schemes for the Monte Carlo pricing of derivatives in LIBOR market models. Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. Our contribution is twofold. Firstly, we propose an alternative approximation scheme based on Picard iterations. This approach is similar in accuracy to the Euler discretization, but with the feature that each rate is evolved independently of the other rates in the term structure. This enables simultaneous calculation of derivative prices of different maturities using parallel computing. Secondly, the product terms occurring in the drift of a LIBOR market model driven by a jump process grow exponentially as a function of the number of rates, quickly rendering the model intractable. We reduce this growth from exponential to quadratic using truncated expansions of the product terms. We include numerical illustrations of the accuracy and speed of our method pricing caplets, swaptions and forward rate agreements.

q-fin.CP↗

arXiv2011

Approximations and asymptotics of upper hedging prices in multinomial models

We give an exposition and numerical studies of upper hedging prices in multinomial models from the viewpoint of linear programming and the game-theoretic probability of Shafer and Vovk. We also show that, as the number of rounds goes to infinity, the upper hedging price of a European option converges to the solution of the Black-Scholes-Barenblatt equation.

q-fin.PR↗

arXiv2010

Asymptotic equivalence in Lee's moment formulas for the implied volatility and Piterbarg's conjecture

The asymptotic behavior of the implied volatility associated with a general call pricing function has been extensively studied in the last decade. The main topics discussed in this paper are Lee's moment formulas for the implied volatility, and Piterbarg's conjecture, describing how the implied volatility behaves in the case where all the moments of the stock price are finite. We find various conditions guaranteeing the existence of the limit in Lee's moment formulas. We also prove a modified version of Piterbarg's conjecture and provide a non-restrictive sufficient condition for the validity of this conjecture in its original form. The asymptotic formulas obtained in the paper are applied to the implied volatility in the CEV model and in the Heston model perturbed by a compound Poisson process with double exponential law for jump sizes.

q-fin.PR↗

arXiv2010

Moment Explosion in the LIBOR Market Model

In the LIBOR market model, forward interest rates are log-normal under their respective forward measures. This note shows that their distributions under the other forward measures of the tenor structure have approximately log-normal tails.

q-fin.PR↗

arXiv2010

Non-existence of Markovian time dynamics for graphical models of correlated default

Filiz et al. (2008) proposed a model for the pattern of defaults seen among a group of firms at the end of a given time period. The ingredients in the model are a graph, where the vertices correspond to the firms and the edges describe the network of interdependencies between the firms, a parameter for each vertex that captures the individual propensity of that firm to default, and a parameter for each edge that captures the joint propensity of the two connected firms to default. The correlated default model can be re-rewritten as a standard Ising model on the graph by identifying the set of defaulting firms in the default model with the set of sites in the Ising model for which the spin is +1. We ask whether there is a suitable continuous time Markov chain taking values in the subsets of the vertex set such that the initial state of the chain is the empty set, each jump of the chain involves the inclusion of a single extra vertex, the distribution of the chain at some fixed time horizon time is the one given by the default model, and the distribution of the chain for other times is described by a probability distribution in the same family as the default model. We show for three simple but financially natural special cases that this is not possible outside of the trivial case where there is complete independence between the firms.

q-fin.CP↗

arXiv2010

Sibuya copulas

The standard intensity-based approach for modeling defaults is generalized by making the deterministic term structure of the survival probability stochastic via a common jump process. The survival copula of the vector of default times is derived and it is shown to be explicit and of the functional form as dealt with in the work of Sibuya. Besides the parameters of the jump process, the marginal survival functions of the default times appear in the copula. Sibuya copulas therefore allow for functional parameters and asymmetries. Due to the jump process in the construction, they allow for a singular component. Depending on the parameters, they may also be extreme-value copulas or Levy-frailty copulas. Further, Sibuya copulas are easy to sample in any dimension. Properties of Sibuya copulas including positive lower orthant dependence, tail dependence, and extremal dependence are investigated. An application to pricing first-to-default contracts is outlined and further generalizations of this copula class are addressed.

math.PR↗