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Mathematical Finance's Blog

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Semi‐efficient valuations and put‐call parity

November 28, 2017 Comments (0)

Abstract
We propose an approach to the valuation of payoffs in general semimartingale models of financial markets where prices are nonnegative. Each asset price can hit 0; we only exclude that this ever happens simultaneously for all assets. We start from two simple, economically motivated axioms, namely, absence of arbitrage (in the sense of NUPBR) and absence of relative arbitrage among all buy‐and‐hold strategies (called static efficiency). A valuation process for a payoff is then called...

Risk management with weighted VaR

November 28, 2017 Comments (0)

Abstract
This article studies the optimal portfolio selection of expected utility‐maximizing investors who must also manage their market‐risk exposures. The risk is measured by a so‐called weighted value‐at‐risk (WVaR) risk measure, which is a generalization of both value‐at‐risk (VaR) and expected shortfall (ES). The feasibility, well‐posedness, and existence of the optimal solution are examined. We obtain the optimal solution (when it exists) and show how risk measures change asset allocation...

The valuation of American options in a multidimensional exponential Lévy model

November 28, 2017 Comments (0)

Abstract
We consider the problem of valuation of American options written on dividend‐paying assets whose price dynamics follow a multidimensional exponential Lévy model. We carefully examine the relation between the option prices, related partial integro‐differential variational inequalities, and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for a broad class of payoff...

Error analysis of finite difference and Markov chain approximations for option pricing

November 28, 2017 Comments (0)

Abstract
Mijatović and Pistorius proposed an efficient Markov chain approximation method for pricing European and barrier options in general one‐dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are nonsmooth, are rarely available. In this paper, we solve this problem for general one‐dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is...

Option pricing in the moderate deviations regime

November 28, 2017 Comments (0)

Abstract
We consider call option prices close to expiry in diffusion models, in an asymptotic regime (“moderately out of the money”) that interpolates between the well‐studied cases of at‐the‐money and out‐of‐the‐money regimes. First and higher order small‐time moderate deviation estimates of call prices and implied volatilities are obtained. The expansions involve only simple expressions of the model parameters, and we show how to calculate them for generic local and stochastic volatility...

The optimal method for pricing Bermudan options by simulation

November 28, 2017 Comments (0)

Abstract
Least‐squares methods enable us to price Bermudan‐style options by Monte Carlo simulation. They are based on estimating the option continuation value by least‐squares. We show that the Bermudan price is maximized when this continuation value is estimated near the exercise boundary, which is equivalent to implicitly estimating the optimal exercise boundary by using the value‐matching condition. Localization is the key difference with respect to global regression methods, but is...

Consistent recalibration of yield curve models

November 28, 2017 Comments (0)

Abstract
The analytical tractability of affine (short rate) models, such as the Vasiček and the Cox–Ingersoll–Ross (CIR) models, has made them a popular choice for modeling the dynamics of interest rates. However, in order to properly account for the dynamics of real data, these models must exhibit time‐dependent or even stochastic parameters. This breaks their tractability, and modeling and simulating become an arduous task. We introduce a new class of Heath–Jarrow–Morton (HJM) models that...

Liquidity effects of trading frequency

November 28, 2017 Comments (0)

Abstract
In this paper, we present a discrete‐time modeling framework, in which the shape and dynamics of a Limit Order Book (LOB) arise endogenously from an equilibrium between multiple market participants (agents). We use the proposed modeling framework to analyze the effects of trading frequency on market liquidity in a very general setting. In particular, we demonstrate the dual effect of high trading frequency. On the one hand, the higher frequency increases market efficiency, if the...

Analytical approximations of local‐Heston volatility model and error analysis

November 28, 2017 Comments (0)

Abstract
This paper studies the expansion of an option price (with bounded Lipschitz payoff) in a stochastic volatility model including a local volatility component. The stochastic volatility is a square root process, which is widely used for modeling the behavior of the variance process (Heston model). The local volatility part is of general form, requiring only appropriate growth and boundedness assumptions. We rigorously establish tight error estimates of our expansions, using Malliavin...

On the market viability under proportional transaction costs

November 28, 2017 Comments (0)

Abstract
This paper studies the market viability with proportional transaction costs. Instead of requiring the existence of strictly consistent price systems as in the literature, we show that strictly consistent local martingale systems (SCLMS) can successfully serve as the dual elements such that the market viability can be verified. We introduce two weaker notions of no arbitrage conditions on market models named no unbounded profit with bounded risk (NUPBR) and no local arbitrage with...