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At least 595 records · Page 33Linked to original sources
arXiv2013

Hedging under multiple risk constraints

Motivated by the asset-liability management of a nuclear power plant operator, we consider the problem of finding the least expensive portfolio, which outperforms a given set of stochastic benchmarks. For a specified loss function, the expected shortfall with respect to each of the benchmarks weighted by this loss function must remain bounded by a given threshold. We consider different alternative formulations of this problem in a complete market setting, establish the relationship between these formulations, present a general resolution methodology via dynamic programming in a non-Markovian context and give explicit solutions in special cases.

q-fin.RM↗
arXiv2013

Pricing and Hedging Derivative Securities with Unknown Local Volatilities

A common assumption in financial engineering is that the market price for any derivative coincides with an objectively defined risk-neutral price - a plausible assumption only if traders collectively possess objective knowledge about the price dynamics of the underlying security over short time scales. Here we assume that traders have an objective knowledge about the underlying security's price trajectories only for large time scales. We show that avoidance of arbitrage that is still feasible uniquely determines the prices of options with large expiration times, and we derive limit theorems useful for estimation of model parameters and present-value analysis of derivative portfolios.

q-fin.PR↗
arXiv2013

When to sell a Markov chain asset?

This paper is concerned with an optimal stock selling rule under a Markov chain model. The objective is to find an optimal stopping time to sell the stock so as to maximize an expected return. Solutions to the associated variational inequalities are obtained. Closed-form solutions are given in terms of a set of threshold levels. Verification theorems are provided to justify their optimality. Finally, numerical examples are reported to illustrate the results.

q-fin.PR↗
arXiv2013

When roll-overs do not qualify as numéraire: bond markets beyond short rate paradigms

We investigate default-free bond markets where the standard relationship between a possibly existing bank account process and the term structure of bond prices is broken, i.e. the bank account process is not a valid numéraire. We argue that this feature is not the exception but rather the rule in bond markets when starting with, e.g., terminal bonds as numéraires. Our setting are general càdlàg processes as bond prices, where we employ directly methods from large financial markets. Moreover, we do not restrict price process to be semimartingales, which allows for example to consider markets driven by fractional Brownian motion. In the core of the article we relate the appropriate no arbitrage assumptions (NAFL), i.e. no asymptotic free lunch, to the existence of an equivalent local martingale measure with respect to the terminal bond as numéraire, and no arbitrage opportunities of the first kind (NAA1) to the existence of a supermartingale deflator, respectively. In all settings we obtain existence of a generalized bank account as a limit of convex combinations of roll-over bonds. Additionally we provide an alternative definition of the concept of a numéraire, leading to a possibly interesting connection to bubbles. If we can construct a bank account process through roll-overs, we can relate the impossibility of taking the bank account as numéraire to liquidity effects. Here we enter endogenously the arena of multiple yield curves. The theory is illustrated by several examples.

q-fin.PR↗
arXiv2014

On the martingale property in stochastic volatility models based on time-homogeneous diffusions

Lions and Musiela (2007) give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert (2009) and Mijatović and Urusov (2012c) give necessary and sufficient conditions in the case of perfect correlation (ρ=1). For financial applications, such as checking the martingale property of the stock price process in correlated stochastic volatility models, we extend their work to the arbitrary correlation case (-1<=ρ<=1). We give a complete classification of the convergence properties of integral functionals of time-homogeneous diffusions and generalize results in Mijatović and Urusov (2012b) (2012c) with alternate proofs avoiding the use of separating times (concept introduced by Cherny and Urusov (2004) and extensively used in the proofs of Mijatović and Urusov (2012c)).

math.PR↗
arXiv2013

Convergence of the discrete variance swap in time-homogeneous diffusion models

In stochastic volatility models based on time-homogeneous diffusions, we provide a simple necessary and sufficient condition for the discretely sampled fair strike of a variance swap to converge to the continuously sampled fair strike. It extends Theorem 3.8 of Jarrow, Kchia, Larsson and Protter (2013) and gives an affirmative answer to a problem posed in this paper in the case of 3/2 stochastic volatility model. We also give precise conditions (not based on asymptotics) when the discrete fair strike of the variance swap is higher than the continuous one and discuss the convex order conjecture proposed by Keller-Ressel and Griessler (2012) in this context.

q-fin.PR↗
arXiv2017

Shapes of implied volatility with positive mass at zero

We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can--theoretically--distinguish between mass at the origin and a heavy-left-tailed distribution. We numerically test our model-free results in stochastic models with absorption at the boundary, such as the CEV process, and in jump-to-default models. Note that while Lee's moment formula tells that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as Benaim and Friz (09) or Gulisashvili (10) do not apply in this setting--essentially due to the breakdown of Put-Call duality.

q-fin.PR↗
arXiv2013

Arbitrage-Free Pricing Before and Beyond Probabilities

"Fundamental theorem of asset pricing" roughly states that absence of arbitrage opportunity in a market is equivalent to the existence of a risk-neutral probability. We give a simple counterexample to this oversimplified statement. Prices are given by linear forms which do not always correspond to probabilities. We give examples of such cases. We also show that arbitrage freedom is equivalent to the continuity of the pricing linear form in the relevant topology. Finally we analyze the possible loss of martingality of asset prices with lognormal stochastic volatility. For positive correlation martingality is lost when the financial process is modelled through standard probability theory. We show how to recover martingality using the appropriate mathematical tools.

q-fin.PR↗
arXiv2014

Non-Arbitrage up to Random Horizon for Semimartingale Models

This paper addresses the question of how an arbitrage-free semimartingale model is affected when stopped at a random horizon. We focus on No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) concept, which is also known in the literature as the first kind of non-arbitrage. For this non-arbitrage notion, we obtain two principal results. The first result lies in describing the pairs of market model and random time for which the resulting stopped model fulfills NUPBR condition. The second main result characterises the random time models that preserve the NUPBR property after stopping for any market model. These results are elaborated in a very general market model, and we also pay attention to some particular and practical models. The analysis that drives these results is based on new stochastic developments in semimartingale theory with progressive enlargement. Furthermore, we construct explicit martingale densities (deflators) for some classes of local martingales when stopped at random time.

q-fin.PR↗
arXiv2017

Implicit transaction costs and the fundamental theorems of asset pricing

This paper studies arbitrage pricing theory in financial markets with implicit transaction costs. We extend the existing theory to include the more realistic possibility that the price at which the investors trade is dependent on the traded volume. The investors in the market always buy at the ask and sell at the bid price. Implicit transaction costs are composed of two terms, one is able to capture the bid-ask spread, and the second the price impact. Moreover, a new definition of a self-financing portfolio is obtained. The self-financing condition suggests that continuous trading is possible, but is restricted to predictable trading strategies having cádlág (right-continuous with left limits) and cáglád (left-continuous with right limits) paths of bounded quadratic variation and of finitely many jumps. That is, cádlág and cáglád predictable trading strategies of infinite variation, with finitely many jumps and of finite quadratic variation are allowed in our setting. Restricting ourselves to cáglád predictable trading strategies, we show that the existence of an equivalent probability measure is equivalent to the absence of arbitrage opportunities, so that the first fundamental theorem of asset pricing (FFTAP) holds. It is also shown that the use of continuous and bounded variation trading strategies can improve the efficiency of hedging in a market with implicit transaction costs. To better understand how to apply the theory proposed we provide an example of an implicit transaction cost economy that is linear and non-linear in the order size.

q-fin.PR↗
arXiv2014

Taylor approximation of incomplete Radner equilibrium models

In the setting of exponential investors and uncertainty governed by Brownian motions we first prove the existence of an incomplete equilibrium for a general class of models. We then introduce a tractable class of exponential-quadratic models and prove that the corresponding incomplete equilibrium is characterized by a coupled set of Riccati equations. Finally, we prove that these exponential-quadratic models can be used to approximate the incomplete models we studied in the first part.

q-fin.PM↗
arXiv2014

Power identities for Lévy risk models under taxation and capital injections

In this paper we study a spectrally negative Lévy process which is refracted at its running maximum and at the same time reflected from below at a certain level. Such a process can for instance be used to model an insurance surplus process subject to tax payments according to a loss-carry-forward scheme together with the flow of minimal capital injections required to keep the surplus process non-negative. We characterize the first passage time over an arbitrary level and the cumulative amount of injected capital up to this time by their joint Laplace transform, and show that it satisfies a simple power relation to the case without refraction. It turns out that this identity can also be extended to a certain type of refraction from below. The net present value of tax collected before the cumulative injected capital exceeds a certain amount is determined, and a numerical illustration is provided.

q-fin.PR↗
arXiv2016

Small-maturity asymptotics for the at-the-money implied volatility slope in Lévy models

We consider the at-the-money strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the behavior of the slope for infinite activity exponential Lévy models including a Brownian component. As auxiliary results, we obtain asymptotic expansions of short maturity at-the-money digital call options, using Mellin transform asymptotics. Finally, we discuss when the at-the-money slope is consistent with the steepness of the smile wings, as given by Lee's moment formula.

q-fin.PR↗
arXiv2013

Superreplication when trading at market indifference prices

We study superreplication of European contingent claims in discrete time in a large trader model with market indifference prices recently proposed by Bank and Kramkov. We introduce a suitable notion of efficient friction in this framework, adopting a terminology introduced by Kabanov, Rasonyi, and Stricker in the context of models with proportional transaction costs. In our framework, efficient friction ensures that large positions of the investor may lead to large losses, a fact from which we derive the existence of superreplicating strategies. We illustrate that without this condition there may be no superreplicating strategy with minimal costs. In our main result, we establish efficient friction under a tail condition on the conditional distributions of the traded securities and under an asymptotic criterion on risk aversions of the market makers. Another result asserts that strict monotonicity of the conditional essential infima and suprema of the security prices is sufficient for efficient friction. We give examples that satisfy the assumptions in our conditions, which include non-degenerate finite sample space models as well as Levy processes and an affine stochastic volatility model of Barndorff-Nielsen-Shepard type.

q-fin.PR↗
arXiv2014

Regulatory-Optimal Funding

Funding is a cost to trading desks that they see as an input. Current FVA-related literature reflects this by also taking funding costs as an input, usually constant, and always risk-neutral. However, this funding curve is the output from a Treasury point of view. Treasury must consider Regulatory-required liquidity buffers, and both risk-neutral (Q) and physical measures (P). We describe the Treasury funding problem and optimize against both measures, using the Regulatory requirement as a constraint. We develop theoretically optimal strategies for Q and P, then demonstrate a combined approach in four markets (USD, JPY, EUR, GBP). Since we deal with physical measures we develop appropriate statistical tests, and demonstrate highly significant (p<0.00001), out-of-sample, improvements on hedged funding with a combined approach achieving 44% to 71% of a perfect information criterion. Thus regulatory liquidity requirements change both the funding problem and funding costs.

q-fin.PR↗
arXiv2014

Regulatory-Compliant Derivatives Pricing is Not Risk-Neutral

Regulations impose idiosyncratic capital and funding costs for holding derivatives. Capital requirements are costly because derivatives desks are risky businesses; funding is costly in part because regulations increase the minimum funding tenor. Idiosyncratic costs mean no single measure makes derivatives martingales for all market participants. Hence Regulatory-compliant pricing is not risk-neutral. This has implications for exit prices and mark-to-market.

q-fin.PR↗