Pricing under rough volatility

From an analysis of the time series of realized variance using recent high-frequency data, Gatheral et al. [Volatility is rough, 2014] previously showed that the logarithm of realized variance behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable timescale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated variance. We analyse in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.     

A forward started jump-diffusion model and pricing of cliquet style exotics

In this paper we present an alternative model for pricing exotic options and structured products with forward-starting components. As presented in the recent study by Eberlein and Madan (Quantitative Finance 9(1):27–42, 2009), the pricing of such exotic products (which consist primarily of different variations of locally/globally, capped/floored, arithmetic/geometric etc. cliquets) depends critically on the modeling of the forward–return distributions. Therefore, in our approach, we directly take up the modeling of forward variances corresponding to the tenor structure of the product to be priced. We propose a two factor forward variance market model with jumps in returns and volatility. It allows the model user to directly control the behavior of future smiles and hence properly price forward smile risk of cliquet-style exotic products. The key idea, in order to achieve consistency between the dynamics of forward variance swaps and the underlying stock, is to adopt a forward starting model for the stock dynamics over each reset period of the tenor structure. We also present in detail the calibration steps for our proposed model.     

The Components of the Bid‐Ask Spread: the Case of the Athens Stock Exchange

We analyse the components of the bid‐ask spread in the Athens Stock Exchange (ASE), which was recently characterised as a developed market. For large and medium capitalisation stocks, we estimate the adverse selection and the order handling component of the spreads as well as the probability of a trade continuation on the same side of either the bid or the ask price, using the Madhavan et al. (1997) model. We extend it by incorporating the traded volume and we find that the adverse selection component exhibits U‐shape patterns, while the cost component pattern depends on the stock price. For high priced stocks, the usual U‐shape applies, while for low‐priced ones, it is an increasing function of time, mainly due to the order handling spread component. Furthermore, the expected price change and the liquidity adjustment to Value‐at‐Risk that is needed are higher in the low capitalisation stocks, while the most liquid stocks are the high priced ones. Moreover, by estimating the Madhavan et al. (1997) model for two distinct periods we explain why there are differences in the components of the bid‐ask spread.     

Regime-Switching Periodic Models For Claim Counts

Abstract

We study a Cox risk model that accounts for both seasonal variations and random fluctuations in the claims intensity. This occurs with natural phenomena that evolve in a seasonal environment and affect insurance claims, such as hurricanes.

More precisely, we define an intensity process governed by a periodic function with a random peak level. The periodic intensity function follows a deterministic pattern in each short-term period and is illustrated by a beta-type function. A Markov chain with m states, corresponding to different risk levels, is chosen for the level process, yielding a so-called regime-switching process.

The properties of the corresponding claim-counting process are discussed in detail. By properly defining a Lundberg-type coefficient, we derive upper bounds for finite time ruin probabilities in a two-state case.     

Quadratic Hawkes processes for financial prices

We introduce and establish the main properties of QHawkes (‘Quadratic’ Hawkes) models. QHawkes models generalize the Hawkes price models introduced in Bacry and Muzy [Quant. Finance, 2014, 14(7), 1147–1166], by allowing feedback effects in the jump intensity that are linear and quadratic in past returns. Our model exhibits two main properties that we believe are crucial in the modelling and the understanding of the volatility process: first, the model is time-reversal asymmetric, similar to financial markets whose time evolution has a preferred direction. Second, it generates a multiplicative, fat-tailed volatility process, that we characterize in detail in the case of exponentially decaying kernels, and which is linked to Pearson diffusions in the continuous limit. Several other interesting properties of QHawkes processes are discussed, in particular the fact that they can generate long memory without necessarily being at the critical point. A non-parametric fit of the QHawkes model on NYSE stock data shows that the off-diagonal component of the quadratic kernel indeed has a structure that standard Hawkes models fail to reproduce. We provide numerical simulations of our calibrated QHawkes model which is indeed seen to reproduce, with only a small amount of quadratic non-linearity, the correct magnitude of fat-tails and time reversal asymmetry seen in empirical time series.     

The immediate price impact of trades on the Australian Stock Exchange

We study the immediate price impact of a single trade executed in the Australian Stock Exchange (ASX). By ordering the top 300 stocks on the ASX in order of their free float market capitalization, a clear pattern emerges, with higher cap stocks experiencing lower price impact than lower cap stocks for the same traded volume. We investigate this relationship in detail, and show that the price impact and liquidity scale as a power of the market capitalization. This relationship is used to obtain a single market impact curve which shows average price shift as a function of volume traded. We obtain similar results for every year from 2001 to 2004.     

Credit portfolios: What defines risk horizons and risk measurement?

The strong autocorrelation between economic cycles demands that we analyze credit portfolio risk in a multiperiod setup. We embed a standard one-factor model in such a setup. We discuss the calibration of the model to Standard & Poor’s ratings data in detail. But because single-period risk measures cannot capture the cumulative effects of systematic shocks over several periods, we define an alternative risk measure, which we call the time-conditional expected shortfall (TES), to quantify credit portfolio risk over a multiperiod horizon.     

Higher moments and beta asymmetry: evidence from Australia

We examine whether systematic higher moments capture beta asymmetry in an asset pricing model whereby the conditional beta of a risky asset increases (decreases) during a bear (bull) market state. We first provide a simple conceptual outline from the microeconomic literature to show that beta asymmetry is driven by time‐varying higher‐order risk preferences (prudence and temperance) across different market states. We then empirically relate these higher‐order risk preferences to systematic skewness and systematic kurtosis. We find that beta asymmetry in Australian stock returns cannot be explained by Carhart (1997) 4‐factor model but is subsumed by systematic higher moments.     

The Impact of Financing Surpluses and Large Financing Deficits on Tests of the Pecking Order Theory

This paper extends the basic pecking order model of Shyam-Sunder and Myers by separating the effects of financing surpluses, normal deficits, and large deficits. Using a panel of US firms over the period 1971-2005, we find that the estimated pecking order coefficient is highest for surpluses (0.90), lower for normal deficits (0.74), and lowest when firms have large financing deficits (0.09). These findings shed light on two empirical puzzles: 1) small firms, although having the highest potential for asymmetric information, do not behave according to the pecking order theory, and 2) the pecking order theory has lost explanatory power over time. We provide a solution to these puzzles by demonstrating that the frequency of large deficits is higher in smaller firms and increasing over time. We argue that our results are consistent with the debt capacity in the pecking order model.     

Debtor-in-possession financing

Several recent papers have documented the benefits of debtor-in-possession (DIP) financing in the restructuring of firms in Chapter 11. However, the view on benefits is not unanimous and some legal scholars have raised doubts about DIP financing's effects on debt-holders and the possibility of expropriative wealth transfers. In this paper we address this issue by analyzing both stock and bond price data for a comprehensive sample of DIP loans and find significant positive abnormal stock and bond returns at the announcement of DIP loans. Also, we do not find evidence of wealth transfers from junior to senior debt-holders. Further, we examine the DIP loan process in detail and we document important institutional features of DIP loans such as maturity, covenants, fees and interest charges. We find evidence of intense monitoring using covenants. We also find higher fees and charges associated with DIP loans. We argue that overall the results are consistent with the information processing role of financial intermediaries.     

A cross-currency Lévy market model

The Lévy Libor or market model which was introduced in Eberlein and Özkan (The Lévy Libor model. Financ. Stochast., 2005, 9, 327–348) is extended to a multi-currency setting. As an application we derive closed form pricing formulas for cross-currency derivatives. Foreign caps and floors and cross-currency swaps are studied in detail. Numerically efficient pricing algorithms based on bilateral Laplace transforms are derived. A calibration example is given for a two-currency setting (EUR, USD).     

The residual market factor,the APT,and mean-variance efficiency

The rapidly increasing volume of both published and unpublished work on the arbitrage pricing theory (APT) of Ross (1976) has given rise to a number of misunderstandings at the interface of theoretical and econometric work. In this article we extend the theoretical structure of our previous work (McElroy and Burmeister, 1985, 1988; Burmeister and McElroy, 1987, 1988) to provide a broad yet rigorous framework both for econometric estimation and for better economic interpretation of new empirical results. We begin with the case where allK factors are observed, and then present the second case ofK−1≡J observed APT factors and one unobserved factor, theresidual market factor introduced in McElroy and Burmeister (1985). The economic interpretations for equivalent specifications of this model are discussed, and we enumerate several immediate payoffs to these specifications. The main new results are concerned with the sometimes intricate relationships among APT models withK factors and APT models withK factors that are constrained to satisfy mean-variance efficiency restrictions. These results are not only of theoretical interest, but more importantly they provide the basis for econometric estimation and testing of nested hypotheses. These econometric issues are discussed in detail.     

The price impact of order book events: market orders,limit orders and cancellations

While the long-ranged correlation of market orders and their impact on prices has been relatively well studied in the literature, the corresponding studies of limit orders and cancellations are scarce. We provide here an empirical study of the cross-correlation between all these different events, and their respective impact on future price changes. We define and extract from the data the ‘bare’ impact these events would have if they were to happen in isolation. For large tick stocks, we show that a model where the bare impact of all events is permanent and non-fluctuating is in good agreement with the data. For small tick stocks, however, bare impacts must contain a history-dependent part, reflecting the internal fluctuations of the order book. We show that this effect can be accurately described by an autoregressive model of the past order flow. This framework allows us to decompose the impact of an event into three parts: an instantaneous jump component, the modification of the future rates of the different events, and the modification of the jump sizes of future events. We compare in detail the present formalism with the temporary impact model that was proposed earlier to describe the impact of market orders when other types of events are not observed. Finally, we extend the model to describe the dynamics of the bid–ask spread.     

Optimal order placement in limit order markets

To execute a trade, participants in electronic equity markets may choose to submit limit orders or market orders across various exchanges where a stock is traded. This decision is influenced by characteristics of the order flows and queue sizes in each limit order book, as well as the structure of transaction fees and rebates across exchanges. We propose a quantitative framework for studying this order placement problem by formulating it as a convex optimization problem. This formulation allows the study of how the optimal order placement decision depends on the interplay between the state of order books, the fee structure, order flow properties and the aversion to execution risk. In the case of a single exchange, we derive an explicit solution for the optimal split between limit and market orders. For the general case of order placement across multiple exchanges, we propose a stochastic algorithm that computes the optimal routing policy and study the sensitivity of the solution to various parameters. Our algorithm does not require an explicit statistical model of order flow but exploits data on recent order fills across exchanges in the numerical implementation of the algorithm to acquire this information through a supervised learning procedure.     

Costly contracting in a long‐term relationship

We examine a model of contracting where parties interact repeatedly and can contract at any point in time, but writing formal contracts is costly. A contract can describe the external environment and the parties' behavior in a more or less detailed way, and the cost of writing a contract is proportional to the amount of detail. We consider both formal (externally enforced) and informal (self‐enforcing) contracts. The presence of writing costs has important implications both for the optimal structure of formal contracts, particularly the tradeoff between contingent and spot contracting, and for the interaction between formal and informal contracting. Our model sheds light on these implications and generates a rich set of predictions about the determinants of the optimal mode of contracting.     

Risky asset pricing based on safety first fund management

We study a portfolio selection model based on Kataoka's safety-first criterion (KSF model in short). We assume that the market is complete but without risk-free asset, and that the returns are jointly elliptically distributed. With these assumptions, we provide an explicit analytical optimal solution for the KSF model and obtain some geometrical properties of the efficient frontier in the plane of probability risk degree z _α and target return r _α. We further prove a two-fund separation and tangency portfolio theorem in the spirit of the traditional mean-variance analysis. We also establish a risky asset pricing model based on risky funds that is similar to Black's zero-beta capital asset pricing model (CAPM, for short). Moreover, we simplify our risky asset pricing model using a derivative risky fund as a reference for market evaluation.     

On the Equivalence between the APV and the wacc Approach in a Growing Leveraged Firm

While in a steady state framework the choice between the wacc approach ( Modigliani‐Miller, 1963 ) and the adjusted present value (APV) approach ( Myers, 1974 ) is irrelevant since the two approaches provide the same result, however, in a growing firm context the wacc equation seems to be inconsistent with the APV result. In this paper we propose a simple model to evaluate the tax savings in a growing firm in order to show under which assumptions the two approaches lead to the same results. We demonstrate that the use of the wacc model in a steady‐growth scenario gives rise to some unusual assumptions with regard to the discount rates to be used in calculating tax shields. We show that the widely used wacc formula, if used, as it is in most cases, in a growth context, implies that a) debt tax shield related to already existing debt are discounted using k_d; b) debt tax shield related to new debt, due to company's growth, are discounted, according to a mixed procedure, using both k_u and k_d. We discuss the inconsistency of such a discounting procedure and the preferred features of the APV approach.     

What Constitutes Evidence of Discrimination in Lending?

We analyze a simple model of bank lending in order to ascertain what can be inferred from relative denial and default rates about lending discrimination. We show that if minority applicants are of lower average creditworthiness than majority applicants, then, contrary to a popular argument, a uniform, nondiscriminatory credit policy cannot simultaneously produce (i) higher denial rates for minority applicants, and (ii) equal default rates for minority and majority applicants. Moreover, we show that equality of denial or default rates always implies discrimination. In particular, equal denial (default) rates imply discrimination against majority (minority) applicants.     

Modelling spikes and pricing swing options in electricity markets

Most electricity markets exhibit high volatilities and occasional distinctive price spikes, which result in demand for derivative products which protect the holder against high prices. In this paper we examine a simple spot price model that is the exponential of the sum of an Ornstein–Uhlenbeck and an independent mean-reverting pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of the spot price process at maturity T. Hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulae for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilized which in turn uses approximations to the conditional density of the spot process.