The evaluation of European compound option prices under stochastic volatility using Fourier transform techniques

Compound options are not only sensitive to future movements of the underlying asset price, but also to future changes in volatility levels. Because the Black–Scholes analytical valuation formula for compound options is not able to incorporate the sensitivity to volatility, the aim of this paper is to develop a numerical pricing procedure for this type of option in stochastic volatility models, specifically focusing on the model of Heston. For this, the compound option value is represented as the difference of its exercise probabilities, which depend on three random variables through a complex functional form. Then the joint distribution of these random variables is uniquely determined by their characteristic function and therefore the probabilities can each be expressed as a multiple inverse Fourier transform. Solving the inverse Fourier transform with respect to volatility, we can reduce the pricing problem from three to two dimensions. This reduced dimensionality simplifies the application of the fast Fourier transform (FFT) method developed by Dempster and Hong when transferred to our stochastic volatility framework. After combining their approach with a new extension of the fractional FFT technique for option pricing to the two-dimensional case, it is possible to obtain good approximations to the exercise probabilities. The resulting upper and lower bounds are then compared with other numerical methods such as Monte Carlo simulations and show promising results.     

Approximate option valuation for arbitrary stochastic processes

We show how a given probability distribution can be approximated by an arbitrary distribution in terms of a series expansion involving second and higher moments. This theoretical development is specialized to the problem of option valuation where the underlying security distribution, if not lognormal, can be approximated by a lognormally distributed random variable. The resulting option price is expressed as the sum of a Black-Scholes price plus adjustment terms which depend on the second and higher moments of the underlying security stochastic process. This approach permits the impact on the option price of skewness and kurtosis of the underlying stock's distribution to be evaluated.     

The effect of sequential information releases on trading volume and price behaviour

This paper examines a two-period setting in which each trader receives a private signal, possibly different, in each period before he trades. The principal objectives are threefold. First, we describe the risky asset demands and price reactions in a noisy rational expectations equilibrium where the time 1 average private signal is not revealed by the price sequence but the time 2 average private signal is. Secondly, we analyse how informed trading volume is affected by the revealed information and supply shocks when pure noise trading volume is uncorrected with observable market variables. Our result indicates that no trade occurs for informed traders when net supply remains fixed across rounds of trade. And, when supply shocks are random, trading volume is induced by the informed and the noise traders, but noise trading is not predictable. Finally, we investigate these properties in the case when pure noise trading volume is correlated with observable market variables. It is shown that no informed trading takes place when there is no supply shock. However, when net supply contains random shocks, trading volume consists of noise and informed trading, both of which can be estimated.     

Strategic trading and learning about liquidity

Many practitioners point out that the speculative profits of institutional traders are eroded by the difficulty in gauging the price impact of their trades. In this paper, we develop a model of strategic trading where speculators face such a dilemma because of incomplete information about time-varying market liquidity. Unlike the competitive market makers that they trade against, informed traders do not know the distribution of liquidity (“noise”) trades. Instead, they have to learn about liquidity from past prices and trading volume. This learning implies that strategic trades and market statistics such as informational efficiency are path-dependent on past market outcomes. Our paper also has normative implications for practitioners.     

A Characterization of the Daily and Intraday Behavior of Returns on Options

The daily and intraday behavior of returns on Chicago Board Options Exchange options is examined. Option returns contain systematic patterns even after adjusting for patterns in the means and variances of the underlying assets. This is consistent with the hypothesis that informed trading in options can make the order flow in the options market informative about the value of the underlying asset, making options nonredundant. The intraday patterns in adjusted option return variances are further consistent with a model of strategic trading by informed and discretionary liquidity traders.     

An empirical investigation of large trader market manipulation in derivatives markets

Using account-level transaction data in options and futures markets, we investigate the existence of market manipulation, which is the ability of large traders to trade strategically, impacting prices and making abnormal profits. First, large trader’s option positions have a quantity impact on the underlying asset’s price. Second, large traders generate significantly positive alphas from trading options and futures. Among the different investor types, proprietary dealers generate the largest positive alphas. Third, these abnormal returns are consistent with strategic trading and cross-market manipulation. The evidence supports market manipulation across the options and futures markets, but not within the futures market itself.     

Is high-frequency trading tiering the financial markets?

Some academic research has identified the possibility of High-Frequency Trading (HFT) creating a two tier market, in which the fast traders mostly deal with each other at most favourable prices and spread, leaving the slower investors to share the least profitable deals. Yet, although intriguing, this view has been seldom quantitatively investigated − and that is the gap found in previous research. A computer simulation has been produced to mimic the behaviour of both slow and fast traders, each category showing characteristics consistent with their behaviour on the markets. In particular, High-Frequency (HF) traders show their preference for aggressive orders when the bid-ask spread is tight and are less aggressive when spread is wide. The Low-Frequency (LF) traders are then forced to live with the remaining deals, hoping to profit from longer term price movements. The purpose of this piece of research is to verify whether HF traders (HFTs) tend to deal with each other and, something not investigated by previous studies, if LF traders also mainly restrict their trading with other slow traders.The results show some tendency toward markets actually splitting into two-tiers: they confirm previous findings of HF traders' tendency to deal with their similar counterparts in case of thin bid-ask spread (which means higher probability of profits for the aggressive side), leaving mostly LF traders to deal amongst themselves when the spread is wider, so aiming at long-term gain (which makes trading inherently riskier).     

Multivariate risk aversion and uninsurable risks: Theory and applications

The objective of this paper is to develop conditions for global multivariate comparative risk aversion in the presence of uninsurable, or background, risks, and thus generalize Kihlstrom and Mirman [1974] and Karni [1979,1989]. We analyze von Neumann-Morgenstern (VNM) utility functionsas well as smooth preference functionals which are nonlinear in distribution but locally linear in probabilities. In each case we provide an economic application which illustrates how our theorems can be used. We analyze a risk sharing, a portfolio choice, and a labor supply problem for VNM utility functions, and the optimal allocation of effort to risky technologies in the presence of a random supply (or quality) of a public good for nonlinear preference functionals. We consider thecase where the random variables are mean-independent as well as the case where they are independent. In the labor supply application for VNM utility functions, we show that if the two risks are independent, the comparative statics effect of greater risk aversion on labor supply in the presence of a background non-wage income risk is determined by a monotonic relationship between labor supply and the wage rate under certainty. That is, we extend the applicability of the Diamond-Stiglitz [1974]-Kihlstrom-Mirman [1974]single-crossing property to the case where an independent background risk is present.     

Capital Allocation for a Sum of Dependent Compound Mixed Poisson Variables: A Recursive Algorithm Approach

The sum of independent compound Poisson random variables is a widely used stochastic model in many economic applications, including non-life insurance, credit and operational risk management, and environmental sciences. In this article we generalize this model by introducing dependence among Poisson frequency variables through a latent random variable in a linear fashion, which can be translated as a common underlying risk factors affecting the frequencies of individual compound Poisson variables. Despite its natural interpretation, this generalization leads to a highly complicated model with no closed-form distribution function. For this dependent compound mixed Poisson sum with an arbitrary severity distribution, we obtain the Laplace transform and further develop a new recursive algorithm to efficiently compute the probability mass function, extending the well-known Panjer recursion. Furthermore, based on this recursion, we derive another recursive scheme to determine the capital allocation associated with the Conditional Tail Expectation, a popular risk management exercise. A numerical example is presented for the illustration of our findings.     

On moments of order statistics from the pareto distribution

Abstract

The evaluation of multiple integrals which occur in order statistics distribution theory is involved due to the fact that the integration is to be carried on over an ordered range of variables of integration. This difficulty is sometimes completely obviated by transforming the ordered variates to the unordered ones. Several such transformations are available in the Theory of Multiple Integrals. In previous papers [2, 3] the author used one such transformation, and gave alternative simplified proofs of several known results in the distribution theory of order statistics from the exponential and the power function distributions. In this paper we use such a known transformation to derive moments (and distributions if necessary) of order statistics from the Pareto distribution. Malik [4] has derived moments of order statistics from this distribution without the transformation of the ordered variates to the unordered ones. The process of direct integration used by Malik becomes complicated for dealing with the moments of more than two ordered variates. Further, the method which we use here is unformly applicable to derive the moments or the distributions of one or more ordered variables, and gives the distributions and moments without any complicated steps in integration. The transformation used by us considerably simplifies the manipulations necessary for the derivation of moments or the Mellin transforms, and thus we hope that our paper would at least be of Pedagogical interest.     

Correlation expansions for CDO pricing

This paper develops numerical approximations for pricing collateralized debt obligations (CDOs) and other portfolio credit derivatives in the multifactor Normal Copula model. A key aspect of pricing portfolio credit derivatives is capturing dependence between the defaults of the elements of the portfolio. But, compared with an independent-obligor model, pricing in a model with correlated defaults is more challenging. Our approach strikes a balance by reducing the problem of pricing in a model with correlated defaults to calculations involving only independent defaults. We develop approximations based on power series expansions in a parameter that scales the underlying correlations. These expansions express a CDO tranche price in a multifactor model as a series of prices in independent-obligor models, which are easy to compute. The approach builds on a classical approximation for multivariate Gaussian probabilities; we introduce an alternative representation that greatly reduces the number of terms required to evaluate the coefficients in the expansion. We also apply this method to the underlying problem of computing joint probabilities of multivariate normal random variables for which the correlation matrix has a factor structure.     

Limit Order Book as a Market for Liquidity

We develop a dynamic model of a limit order market populatedby strategic liquidity traders of varying impatience. In equilibrium,patient traders tend to submit limit orders, whereas impatienttraders submit market orders. Two variables are the key determinantsof the limit order book dynamics in equilibrium: the proportionof patient traders and the order arrival rate. We offer severaltestable implications for various market quality measures suchas spread, trading frequency, market resiliency, and time toexecution for limit orders. Finally, we show the effect of imposinga minimal price variation on these measures.     

Noise and competition in strategic oligopoly

In this paper, we propose a model where N strategic informed traders who are endowed with heterogeneous noisy signals with different precisions compete in a market with a single risky asset. We explicitly describe the unique linear equilibrium that exists in this setup and derive its properties. Moreover, we focus on the effects of noise on the competition between traders. We show that noise softens the competition between traders. In particular, for N exceeding three and for certain sets of noise in traders' signals, each trader's individual profit is greater than the one obtained in the case of perfect information.     

Bounds for Actuarial Present Values Under the Fractional Independence Assumption

We consider the fractional independence (FI) survival model, studied by Willmot (1997), for which the curtate future lifetime and the fractional part of it satisfy the statistical independence assumption, called the fractional independence assumption.

The ordering of risks of the FI survival model is analyzed, and its consequences for the evaluation of actuarial present values in life insurance is discussed. Our main fractional reduction (FR) theorem states that two FI future lifetime random variables with identical distributed curtate future lifetime are stochastically ordered (stop-loss ordered) if, and only if, their fractional parts are stochastically ordered (stop-loss ordered).

The well-known properties of these stochastic orders allow to find lower and upper bounds for different types of actuarial present values, for example when the random payoff functions of the considered continuous life insurances are convex (concave), or decreasing (increasing), or convex not decreasing (concave not increasing) in the future lifetime as argument. These bounds are obtained under the assumption that some information concerning the moments of the fractional part is given. A distinction is made according to whether the fractional remaining lifetime has a fixed mean or a fixed mean and variance. In the former case, simple unique optimal bounds are obtained in case of a convex (concave) present value function.

The obtained results are illustrated at the most important life insurance quantities in a continuous random environment, which include bounds for net single premiums, net level annual premiums and prospective net reserves.     

On the asymptotic distribution of weighted least squares estimators

Abstract

In the ELB (Empirical Linear Bayes)-approach to credibility, the unknown structural parameters are substituted by a set of parameter estimates. The weighted least squares estimators are known to be asymptotically normally distributed when the design variables are independent and identically distributed random variables. It is demonstrated that, with probability one, the conditional asymptotic distribution, given the design, is the same as the unconditional distribution. Estimation of the asymptotic covariance matrix will also be considered.     

Analysis of a Generalized Penalty Function in a Semi-Markovian Risk Model

Abstract

In this paper an extension of the semi-Markovian risk model studied by Albrecher and Boxma (2005) is considered by allowing for general interclaim times. In such a model, we follow the ideas of Cheung et al. (2010b) and consider a generalization of the Gerber-Shiu function by incorporating two more random variables in the traditional penalty function, namely, the minimum surplus level before ruin and the surplus level immediately after the second last claim prior to ruin. It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation. Detailed examples are also considered when either the interclaim times or the claim sizes are exponentially distributed. Finally, we also consider the case where the claim arrival process follows a Markovian arrival process. Probabilistic arguments are used to derive the discounted joint distribution of four random variables of interest in this risk model by capitalizing on an existing connection with a particular fluid flow process.     

Equilibrium moment restrictions on asset returns: normal and crisis periods

Empirically, the covariance between stock returns varies with their volatility. We seek a robust theoretical explanation of this. With minimal assumptions, we model stochastic properties of equilibrium returns which result from the interaction between inter-temporal traders and noisy, price-sensitive short-term traders. The inter-temporal traders can have arbitrary investment rules, preferences and information. In all cases we find a set of restrictions between second moments of equilibrium returns. With two assets there is also a bound on the correlation between asset returns. Estimation with second moments of global stock returns supports our theoretical framework. Higher volatility in at least one market can increase comovement among markets. With globalization, covariances between two stock markets can also affect covariances between two other stock markets. We also find that the changes in trader behavior between normal and crisis periods lead to changes in the moment restrictions between asset returns.     

Information flow between the stock and option markets: Where do informed traders trade?

This paper investigates the flow of information between the equity and options markets. We argue that informed traders, in deciding where to place their trades, are not entirely indifferent to option moneyness, degree of information asymmetry, and option liquidity. Unlike some previous studies that find information to flow unilaterally from equity to options markets, we control for the above factors and discover feedback relations between trades in out-of-the-money (OTM) options and the underlying equities. The finding is consistent with the pooling equilibrium hypothesis, which asserts that informed traders trade in both the equity and options markets. Some informed traders are probably attracted to the out-of-the money options because of their higher liquidity, lower premiums, and higher delta-to-premium ratios, hence, lending support to the liquidity and leverage hypothesis.     

Transform analysis for Hawkes processes with applications in dark pool trading

Hawkes processes are a class of simple point processes that are self-exciting and have a clustering effect, with wide applications in finance, social networks and many other fields. This paper considers a self-exciting Hawkes process where the baseline intensity is time-dependent, the exciting function is a general function and the jump sizes of the intensity process are independent and identically distributed nonnegative random variables. This Hawkes model is non-Markovian in general. We obtain closed-form formulas for the Laplace transform, moments and the distribution of the Hawkes process. To illustrate the applications of our results, we use the Hawkes process to model the clustered arrival of trades in a dark pool and analyse various performance metrics including time-to-first-fill, time-to-complete-fill and the expected fill rate of a resting dark order.     

The Tail Stein's Identity with Applications to Risk Measures

In this article, we examine a generalized version of an identity made famous by Stein, who constructed the so-called Stein's Lemma in the special case of a normal distribution. Other works followed to extend the lemma to the larger class of elliptical distributions. The lemma has had many applications in statistics, finance, insurance, and actuarial science. In an attempt to broaden the application of this generalized identity, we consider the version in the case where we investigate only the tail portion of the distribution of a random variable. Understanding the tails of a distribution is very important in actuarial science and insurance. Our article therefore introduces the concept of the “tail Stein's identity” to the case of any random variable defined on an appropriate probability space with a Lebesgue density function satisfying certain regularity conditions. We also examine this “tail Stein's identity” to the class of discrete distributions. This extended identity allows us to develop recursive formulas for generating tail conditional moments. As examples and illustrations, we consider several classes of distributions including the exponential family, and we apply this result to demonstrate how to generate tail conditional moments. This holds a large promise for applications in risk management.