Pricing collateralized derivatives with an arbitrary numeraire

Since the 2008 crisis collateralized derivatives have become commonplace in the market. There have been many papers in recent years on pricing collateralized derivatives but the topic has been surrounded by confusion with debate focusing on whether or not a risk‐free rate needs to be assumed. In addition, as pointed out by Bielecki and Rutkowski, several authors do not pay enough attention to the pricing measure they are working in when setting up their models. The contribution of this paper is to show the pricing formula for a collateralized derivative can be derived under the usual assumptions of an arbitrage‐free economy starting from any equivalent martingale measure and associated numeraire.     

PROPERTIES OF OPTION PRICES IN MODELS WITH JUMPS

We study convexity and monotonicity properties of option prices in a model with jumps using the fact that these prices satisfy certain parabolic integro–differential equations. Conditions are provided under which preservation of convexity holds, i.e., under which the value, calculated under a chosen martingale measure, of an option with a convex contract function is convex as a function of the underlying stock price. The preservation of convexity is then used to derive monotonicity properties of the option value with respect to the different parameters of the model, such as the volatility, the jump size, and the jump intensity.     

A BENCHMARK APPROACH TO FINANCE

This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling, and risk measurement. It is based on the natural assumption that investors prefer more rather than less, in the sense that given two portfolios with the same diffusion coefficient value, the one with the higher drift is preferred. Each such investor is shown to hold an efficient portfolio in the sense of Markowitz with units in the market portfolio and the savings account. The market portfolio of investable wealth is shown to equal a combination of the growth optimal portfolio (GOP) and the savings account. In this setup the capital asset pricing model follows without the use of expected utility functions, Markovianity, or equilibrium assumptions. The expected increase of the discounted value of the GOP is shown to coincide with the expected increase of its discounted underlying value. The discounted GOP has the dynamics of a time transformed squared Bessel process of dimension four. The time transformation is given by the discounted underlying value of the GOP. The squared volatility of the GOP equals the discounted GOP drift, when expressed in units of the discounted GOP. Risk-neutral derivative pricing and actuarial pricing are generalized by the fair pricing concept, which uses the GOP as numeraire and the real-world probability measure as pricing measure. An equivalent risk-neutral martingale measure does not exist under the derived minimal market model.     

MINIMAL ENTROPY–HELLINGER MARTINGALE MEASURE IN INCOMPLETE MARKETS

This paper defines an optimization criterion for the set of all martingale measures for an incomplete market model when the discounted price process is bounded and quasi-left continuous. This criterion is based on the entropy–Hellinger process for a nonnegative Doléans–Dade exponential local martingale. We develop properties of this process and establish its relationship to the relative entropy "distance." We prove that the martingale measure, minimizing this entropy–Hellinger process, is unique. Furthermore, it exists and is explicitly determined under some mild conditions of integrability and no arbitrage. Different characterizations for this extremal risk-neutral measure as well as immediate application to the exponential hedging are given. If the discounted price process is continuous, the minimal entropy–Hellinger martingale measure simply is the minimal martingale measure of Föllmer and Schweizer. Finally, the relationship between the minimal entropy–Hellinger martingale measure (MHM) and the minimal entropy martingale measure (MEM) is provided. We also give an example showing that in contrast to the MHM measure, the MEM measure is not robust with respect to stopping.     

ARBITRAGE AND FREE LUNCH WITH BOUNDED RISK FOR UNBOUNDED CONTINUOUS PROCESSES

We give two examples showing that for unbounded continuous price processes, the no-free-lunch assumption and the existence of an equivalent martingale measure are not equivalent. In fact it turns out that the notion of an equivalent local martingale measure is natural in this context.     

CONTINGENT CLAIMS VALUED AND HEDGED BY PRICING AND INVESTING IN A BASIS

Contingent claims with payoffs depending on finitely many asset prices are modeled as elements of a separable Hilbert space. Under fairly general conditions, including market completeness, it is shown that one may change measure to a reference measure under which asset prices are Gaussian and for which the family of Hermite polynomials serves as an orthonormal basis. Basis pricing synthesizes claim valuation and basis investment provides static hedging opportunities. For claims written as functions of a single asset price we infer from observed option prices the implicit prices of basis elements and use these to construct the implied equivalent martingale measure density with respect to the reference measure, which in this case is the Black-Scholes geometric Brownian motion model. Data on S & P 500 options from the Wall Street Journal are used to illustrate the calculations involved. On this illustrative data set the equivalent martingale measure deviates from the Black-Scholes model by relatively discounting the larger price movements with a compensating premia placed on the smaller movements.     

Exponential Hedging and Entropic Penalties

We solve the problem of hedging a contingent claim B by maximizing the expected exponential utility of terminal net wealth for a locally bounded semimartingale X . We prove a duality relation between this problem and a dual problem for local martingale measures Q for X where we either minimize relative entropy minus a correction term involving B or maximize the Q -price of B subject to an entropic penalty term. Our result is robust in the sense that it holds for several choices of the space of hedging strategies. Applications include a new characterization of the minimal martingale measure and risk-averse asymptotics.     

A Closer Look at Barrier Exchange Options

A barrier exchange option is an exchange option that is knocked out the first time the prices of two underlying assets become equal. Lindset, S., & Persson, S.‐A. (2006) present a simple dynamic replication argument to show that, in the absence of arbitrage, the current value of the barrier exchange option is equal to the difference in the current prices of the underlying assets and that this pricing formula applies irrespective of whether the option is European or American. In this study, we take a closer look at barrier exchange options and show, despite the simplicity of the pricing formula presented by Lindset, S., & Persson, S.‐A. (2006), that the barrier exchange option in fact involves a surprising array of key concepts associated with the pricing of derivative securities including: put–call parity, barrier in–out parity, static vs. dynamic replication, martingale pricing, continuous vs. discontinuous price processes, and numeraires. We provide valuable intuition behind the pricing formula which explains its apparent simplicity. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:29–43, 2013     

Mean–variance hedging of contingent claims with random maturity

We study the mean–variance hedging of an American-type contingent claim that is exercised at a random time in a Markovian setting. This problem is motivated by applications in the areas of employee stock option valuation, credit risk, or equity-linked life insurance policies with an underlying risky asset value guarantee. Our analysis is based on dynamic programming and uses PDE techniques. In particular, we prove that the complete solution to the problem can be expressed in terms of the solution to a system of one quasi-linear parabolic PDE and two linear parabolic PDEs. Using a suitable iterative scheme involving linear parabolic PDEs and Schauder's interior estimates for parabolic PDEs, we show that each of these PDEs has a classical C^{1, 2} solution. Using these results, we express the claim's mean–variance hedging value that we derive as its expected discounted payoff with respect to an equivalent martingale measure that does not coincide with the minimal martingale measure, which, in the context that we consider, identifies with the minimum entropy martingale measure as well as the variance-optimal martingale measure. Furthermore, we present a numerical study that illustrates aspects of our theoretical results.     

Variance risk premiums and predictive power of alternative forward variances in the corn market

We propose a fear index for corn using the variance swap rate synthesized from out‐of‐the‐money call and put options as a measure of implied variance. We find negative and time‐varying variance risk premiums (realized variance minus implied variance) in the corn market from 1987 to 2009. Our results contrast with Egelkraut, Garcia, and Sherrick (2007), but are in line with the findings of Simon (2002). We conclude that our synthesized model‐free implied variance estimation procedure contains superior information about future realized variance relative to traditional model‐dependent estimating procedures: the implied variance model by Black (1976) and the seasonal GARCH(1, 1) forecasted variance model. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 32:587–608, 2012     

NO‐ARMAGEDDON MEASURE FOR ARBITRAGE‐FREE PRICING OF INDEX OPTIONS IN A CREDIT CRISIS

In this work, we consider three problems of the standard market approach to credit index options pricing: the definition of the index spread is not valid in general, the considered payoff leads to a pricing which is not always defined, and the candidate numeraire for defining a pricing measure is not strictly positive, which leads to a nonequivalent pricing measure. We give a solution to the three problems, based on modeling the flow of information through a suitable subfiltration. With this we consistently take into account the possibility of default of all names in the portfolio, that is neglected in the standard market approach. We show on market inputs that, while the pricing difference can be negligible in normal market conditions, it can become highly relevant in stressed market conditions, like the situation caused by the credit crunch.     

BLACK–SCHOLES REPRESENTATION FOR ASIAN OPTIONS

Asian options are securities with a payoff that depends on the average of the underlying stock price over a certain time interval. We identify three natural assets that appear in pricing of the Asian options, namely a stock S, a zero coupon bond B^T with maturity T, and an abstract asset A (an “average asset”) that pays off a weighted average of the stock price number of units of a dollar at time T. It turns out that each of these assets has its own martingale measure, allowing us to obtain Black–Scholes type formulas for the fixed strike and the floating strike Asian options. The model independent formulas are analogous to the Black–Scholes formula for the plain vanilla options; they are expressed in terms of probabilities under the corresponding martingale measures that the Asian option will end up in the money. Computation of these probabilities is relevant for hedging. In contrast to the plain vanilla options, the probabilities for the Asian options do not admit a simple closed form solution. However, we show that it is possible to obtain the numerical values in the geometric Brownian motion model efficiently, either by solving a partial differential equation numerically, or by computing the Laplace transform. Models with stochastic volatility or pure jump models can be also priced within the Black–Scholes framework for the Asian options.     

中国企业年金替代率水平研究——基于随机波动模型的分析

构建我国企业年金替代率现金流模型,把年金资产的投资收益率作为随机变量引入随机波动(Stochastic Volatility,SV)模型,以年金资产投资于股票和国债两类风险资产为例,对年金替代率进行模拟分析。研究表明:从供给角度讲,在当前条件下,制度能够实现的替代率水平远远低于制度的预设目标;在总缴费率较高,而其它因素又很难改变的情况下,提高年金替代率的关键是提高投资收益率水平。为此,要改善年金资产的投资环境,引入竞争机制,降低基金管理费率。     

The components of interest rate swap spreads: Theory and international evidence

This article contains both a theoretical and an empirical analysis of the components of interest rate swap spreads defined as the difference between the fixed swap rate and the risk‐free rate of equal maturity. The components are determined by expected LIBOR spreads, default risk, and market structure. A model of the swap market incorporating debt market imperfections and corporate financing choices is used to explain participation by both swap buyers and sellers. The model also motivates an empirical relationship between swap spreads and the slope of the risk‐free term structure. The article then provides empirical evidence on the cross‐sectional and time‐series variation of swap spreads in seven international markets. The evidence is consistent with the suggested components across both markets and swap maturities as well as over time. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:347–387, 2003     

Portfolio Optimization and Martingale Measures

The paper studies connections between risk aversion and martingale measures in a discrete-time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b ≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b = 1; that is, in the case of the log-utility, the optimal dynamic portfolio defines the numéraire portfolio.     

Equity swaps in a LIBOR market model

This study extends the BGM (A. Brace, D. Gatarek, & M. Musiela, 1997) interest rate model (the London Interbank Offered Rate [LIBOR] market model) by incorporating the stock price dynamics under the martingale measure. As compared with traditional interest rate models, the extended BGM model is both appropriate for pricing equity swaps and easy to calibrate. The general framework for pricing equity swaps is proposed and applied to the pricing of floating‐for‐equity swaps with either constant or variable notional principals. The calibration procedure and the practical implementation are also discussed. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:893–920, 2007     

An Analytic Solution for Interest Rate Swap Spreads

This paper argues that liquidity differences between government securities and short–term Eurodollar borrowings account for interest rate swap spreads. It then models the convenience of liquidity as a linear function of two mean–reverting state variables and values it. The interest rate swap spread for a swap of particular maturity is the annuitized equivalent of this value. It has a closed form solution: a simple integral. Special cases examined include the Vasicek and Cox–Ingersoll–Ross one–factor term structure models. Numerical values for the parameters in both special cases illustrate that many realistic ‘swap spread term structures’ can be replicated. Model parameters are estimated using weekly data on the term structure of swap spreads from several countries. The model fits the data well.     

OPTIMAL NUMERAIRES FOR RISK MEASURES

Can the usage of a risky numeraire with a greater than risk free expected return reduce the capital requirements in a solvency test? I will show that this is not the case. In fact, under a reasonable technical condition, there exists no optimal numeraire which yields smaller capital requirements than any other numeraire.     

The Relation between Physical and Risk‐neutral Cumulants

Variance swaps are natural instruments for investors taking directional bets on volatility and are often used for portfolio protection. The empirical observation on skewness research suggests that derivative professionals may also desire to hedge beyond volatility risk and there exists the need to hedge higher‐moment market risks, such as skewness and kurtosis risks. We study two derivative contracts – skewness swap and kurtosis swap – which trade the forward realized third and fourth cumulants. Using S&P 500 index options data from 1996 to 2005, we document the returns of these swap contracts, i.e., skewness risk premium and kurtosis risk premium. We find that the both skewness and kurtosis risk premiums are significantly negative.     

Martingale Measures For A Class of Right-Continuous Processes

The subject of the present paper is the following. Suppose that W is a class of adapted, right-continuous processes on the continuous time horizon [0, 1], and for every stopping time and W , () is bounded below. A necessary and sufficient condition will be given for the existence of a probability measure Q which is equivalent to the original measure and such that each process in W is a martingale under Q . If the processes in W represent the discounted prices of available securities, then the condition given here for the existence of a martingale measure can be interpreted as absence of "free lunch" in the securities market. This is a familiar kind of theorem from the finance literature; the novelty of this paper is that the security prices are not required to be in LP for some 1 p , nor are they assumed to be continuous. Also, the concept of free lunch is invariant under the substitution of the original probability measure by an equivalent probability measure. the assumption that () is bounded below for every W and stopping time is quite natural since prices are nonnegative.
We shall define a class of admissible subjective probability measures and assume that each agent in the economy has selected a subjective probability measure from (hat class. Subjective free lunch for an agent will be defined using his or her subjective probability measure. It will be shown that under an additional condition the existence of free lunch is equivalent to the existence of a common subjective free lunch simultaneously for all possible agents in the economy.