Primitive Assets Research Articles

Insurance risk analysis of financial networks vulnerable to a shock

We conduct a risk analysis of non-core insurance business of selling protection to financial firms against investment losses due to a shock. A static structural model is constructed, composed of a network of firms who cross-hold each other, a financial market consisting of multiple primitive assets that are vulnerable to a shock, and an insurer who resides external to the network and assesses the opportunity to sell protection to the financial firms. Assume that each firm in the network is rational and able to decide how much protection to purchase to optimize its portfolio according to the mean-variance principle. As a result, the shock may impact on the insurer but indirectly through the network. In view of the robust-yet-fragile nature of financial networks that has been discovered, both empirically and theoretically, by various recent works, one expects that the network integration and the shock play an intertwined role in the insurance risk. Our study forms a theoretical confirmation of this surmise: Depending on the shock size, there are three mutually exclusive scenarios in which an increase in the network integration can either reduce or amplify the impact of the shock on the insurance risk.

Effect of Variance Swap in Hedging Volatility Risk

This paper studies the effect of variance swap in hedging volatility risk under the mean-variance criterion. We consider two mean-variance portfolio selection problems under Heston’s stochastic volatility model. In the first problem, the financial market is complete and contains three primitive assets: a bank account, a stock and a variance swap, where the variance swap can be used to hedge against the volatility risk. In the second problem, only the bank account and the stock can be traded in the market, which is incomplete since the idiosyncratic volatility risk is unhedgeable. Under an exponential integrability assumption, we use a linear-quadratic control approach in conjunction with backward stochastic differential equations to solve the two problems. Efficient portfolio strategies and efficient frontiers are derived in closed-form and represented in terms of the unique solutions to backward stochastic differential equations. Numerical examples are provided to compare the solutions to the two problems. It is found that adding the variance swap in the portfolio can remarkably reduce the portfolio risk.

Market completion with derivative securities

Let S^{F} be a ℙ-martingale representing the price of a primitive asset in an incomplete market framework. We present easily verifiable conditions on the model coefficients which guarantee the completeness of the market in which in addition to the primitive asset, one may also trade a derivative contract S^{B}. Both S^{F} and S^{B} are defined in terms of the solution X to a two-dimensional stochastic differential equation: S^{F}_{t} = f(X_{t}) and S^{B}_{t}:=mathbb{E}[g(X_{1}) | mathcal{F}_{t}]. From a purely mathematical point of view, we prove that every local martingale under ℙ can be represented as a stochastic integral with respect to the ℙ-martingale S :=(S^{F}, S^{B}). Notably, in contrast to recent results on the endogenous completeness of equilibria markets, our conditions allow the Jacobian matrix of (f,g) to be singular everywhere on mathbb{R}^{2}. Hence they cover as a special case the prominent example of a stochastic volatility model being completed with a European call (or put) option.

Using Weighted Shapley Values to Measure the Systemic Risk of Interconnected Banks

AbstractA macro‐prudential approach to financial regulation should reasonably evaluate the systemic risk of individual institutions. We introduce a measure of the systemic risk of interconnected banks using the weighted Shapley value, which characterizes banks' systemic risk by capturing two characteristics: (i) a bank's risk exposure of the primitive asset structure; and (ii) its position in the interbank network. An empirical implementation reveals that the primitive asset structure and interbank connections are main drivers of each bank's systemic risk. In addition, different types of interbank linkages (debt or share) make significant differences in affecting an individual bank's systemic risk.

On the equivalence of financial structures with long-term assets

In a stochastic financial exchange economy, two financial structures are equivalent if, for each given state price, regardless the associated arbitrage free price, the marketable payoffs are identical. The key property of two equivalent financial structures is that, when associated with any standard exchange economy, they lead to the same financial equilibrium. We exhibit a sufficient condition for the equivalence of two financial structures with possibly long-term assets. We then apply this result to financial structures built upon primitive assets and their re-trading to get a necessary and sufficient condition. We also borrow an assumption from [7] to prove the equivalence between a financial structure and its reduced forms and between to complete financial structures.

Why not use SDF rather than beta models in performance measurement?

This paper analyzes performance measurement based on stochastic dis- countfactors,comparedtobetamodelstraditionallyusedincomputingfunds'(Jensen) alphas. From a theoretical point of view, standard alphas suffer from several limita- tions. Our paper addresses this issue from an empirical point of view using a sample of Swiss mutual funds from 2000 to 2011. Our results suggest that the key for a fair comparison between stochastic discount function (SDF) and beta models is the spec- ification of the set of primitive assets used to calibrate the SDF function. Once this is established, the size of (absolute) performance differences considerably decreases between the two model families. However, there are sizeable performance deviations in the cross-section of funds if conditioning information is incorporated in the tests, up to some 20 basis points per month, or about 2.3% per year. In almost all cases, the SDF-alphas arelowerthanthestandard(Jensen)alphas.Inabsoluteterms,theaverage SDF-basedunderperformanceofthefundsiswaylargerthantheaveragetotalexpense ratio (TER) of the funds, both in a conditional and unconditional setting.

Equilibrium Theory with Link Portfolios

Portfolio constraints often prevent financial derivatives from being synthetically created by primitive assets and thus, open a way for the 'redundant' assets to participate in expanding risk-sharing opportunities. They bring about peculiar portfolios, called 'link portfolios,' at an aggregate level. Link portfolios make no monetary payoff in equilibrium but may account for feasibility of optimal portfolios in constrained asset markets. They are abundant enough to form a linear subspace. This fact has two implications. First, they have null market value in equilibrium because they are marginally useless once the feasibility of optimal portfolios is met. Second, they are free but may have impact on the welfare of agents in equilibrium. Such elusive behavior of link portfolios blocks extending the conventional approach to equilibrium analysis to constrained asset markets. It has remained unresolved for a long time to verify the existence of equilibrium with general forms of convex portfolio constraint. This is because the literature fails to recognize the presence of link portfolios in constrained asset markets and thus, do justice to the analysis of their equilibrium behavior. The paper shows the existence of equilibrium in two-period constrained asset markets with link portfolios. This result is built on the fundamental theorem of portfolio decomposition which allows us to explicate the behavior of equilibrium asset prices and capture the risk-sharing role of link portfolios in equilibrium.

Why Not Use SDF - Rather than Beta Models in Performance Measurement?

This paper analyzes performance measurement based on stochastic discount factors, compared to beta models traditionally used in computing funds’ (Jensen) alphas. From a theoretical point of view, standard alphas suffer from several limitations. Our paper addresses this issue from an empirical point of view using a sample of Swiss mutual funds from 2000 to 2011. Our results suggest that the key for a “fair” comparison between SDF and beta models is the specification of the set of primitive assets used to calibrate the SDF function. Once this is established, the size of (absolute) performance differences considerably decreases between the two model families. However, there are sizeable performance deviations in the cross-section of funds if conditioning information is incorporated in the tests, up to some 20 basis points per month, or about 2.3% per year. In almost all cases, the SDF-alphas are lower than the standard (Jensen) alphas. In absolute terms, the average SDF-based underperformance of the funds is way larger than the average expense ratio of the funds, both in a conditional and unconditional setting.

Growth to value: Option exercise and the cross section of equity returns

We propose a general equilibrium model to study the link between the cross section of expected returns and book-to-market characteristics. We model two primitive assets: value assets and growth assets that are options on assets in place. The cost of option exercise, which is endogenously determined in equilibrium, is highly procyclical and acts as a hedge against risks in assets in place. Consequently, growth options are less risky than value assets, and the model features a value premium. Our model incorporates long-run risks in aggregate consumption and replicates the empirical failure of the conditional capital asset pricing model (CAPM) prediction. The model also quantitatively accounts for the pattern in mean returns on book-to-market sorted portfolios, the magnitude of the CAPM-alphas, and other stylized features of the cross-sectional data.

Computational methods for option replication

A computational method is described for option replication. In particular, a procedure is provided for computing the projection basis that corresponds to a positive basis of ℝ m . Application of this procedure in order to compute maximal submarkets that replicate any option is demonstrated. Specifically, we provide a computational study for the replication of options in security markets with a finite number of states and a finite number of primitive assets with payoffs given by linearly independent vectors of ℝ m . The theoretical background of this work follows the results in Polyrakis and Xanthos [Maximal submarkets that replicate any option, Ann. Finance, DOI: 10.1007/s10436-009-0143-9]. Our goal is to make option replication computationally tractable and hence more viable as a financial tool.

Conditional performance evaluation for German equity mutual funds

We investigate the conditional performance of a sample of German equity mutual funds over the period from 1994 to 2003 using both the beta-pricing approach and the stochastic discount factor (SDF) framework. On average, mutual funds cannot generate excess returns relative to their benchmark that are large enough to cover their total expenses. Compared to unconditional alphas, fund performance sharply deteriorates when we measure conditional alphas. Given that stock returns are to some extent predictable based on publicly available information, conditional performance evaluation raises the benchmark for active fund managers because it gives them no credit for exploiting readily available information. Underperformance is more pronounced in the SDF framework than in beta-pricing models. The fund performance measures derived from alternative model specifications differ depending on the number of primitive assets taken to calibrate the SDF as well as the number of instrument variables used to scale assets and/or factors.

Pricing a European Basket Option in the Presence of Proportional Transaction Costs

A crucial assumption in the Black–Scholes theory of options pricing is the no transaction costs assumption. However, following such a strategy in the presence of transaction costs would lead to immediate ruin. This paper presents a stochastic control approach to the pricing and hedging of a European basket option, dependent on primitive assets whose prices are modelled as lognormal diffusions, in the presence of costs proportional to the size of the transaction. Under certain assumptions on the individual preferences, it is able to reduce the dimensionality of the resulting control problem. This facilitates considerably the study of the value function and the characterisation of the optimal trading policy. For solution of the problem a perturbation analysis scheme is utilized to derive a non‐trivial, asymptotically optimal result. The findings reveal that this result can be expressed by means of a small correction to the corresponding solution of the frictionless Black–Scholes type problem, resembling a multi‐dimensional ‘bandwidth’ around the vanilla case, which, moreover, is readily tractable.

From measure changes to time changes in asset pricing

The goal of the paper is to review the last 35 years of continuous-time finance by focusing on two major advances: (i) The powerful elegance of the martingale representation for primitive assets and attainable contingent claims in more and more general settings, thanks to the probabilistic tool of probability change and the economic flexibility in the choice of the numéraire relative to which prices are expressed. This numéraire evolved over time from the money market account to a zero-coupon bond or a stock price, lastly to strictly positive quantities involved in the Libor or swap market models and making the pricing of caps or swaptions quite efficient. (ii) The persistent central role of Brownian motion in finance across the 20th century: even when the underlying asset price is a very general semi-martingale, the no-arbitrage assumption and Monroe theorem [Monroe, I., 1978. Processes that can be embedded in Brownian motion. Annals of Probability 6, 42–56] allow us to write it as Brownian motion as long as we are willing to change the time. The appropriate stochastic clock can be shown empirically to be driven by the cumulative number of trades, hence by market activity. Consequently, starting with a general multidimensional stochastic process S defined on a probability space ( Ω, F , F t , P) and representing the prices of primitive securities, the no-arbitrage assumption allows, for any chosen numéraire, to obtain a martingale representation for S under a probability measure Q S equivalent to P. This route will be particularly beneficiary for the pricing of complex contingent claims. Alternatively, changing the clock, i.e., changing the filtration ( F t ), we can recover the Brownian motion and normality of returns. In all cases martingales appear as the central representation of asset prices, either through a measure change or through a time change.

Endogenous collateral

We study an economy where there are two types of assets. Consumers’ promises are the primitive defaultable assets secured by collateral chosen by the consumers themselves. These personalized assets are purchased by financial intermediaries who finance these purchases by selling back derivatives to consumers. We show that non-arbitrage prices of primitive assets are strict submartingales, whereas non-arbitrage prices of derivatives are supermartingales. Next we establish existence of equilibrium, without imposing bounds on short-sales. The nonconvexity of the budget set is overcome by considering a continuum of agents.

Equilibrium Pricing in the Presence of Cumulative Dividends Following a Diffusion

The paper presents some security market pricing results in the setting of a security market equilibrium in continuous time. The theme of the paper is financial valuation theory when the primitive assets pay out real dividends represented by processes of unbounded variation. In continuous time, when the models are also continuous, this is the most general representation of real dividends, and it can be of practical interest to analyze such models. Taking as the starting point an extension to continuous time of the Lucas consumption‐based model, we derive the equilibrium short‐term interest rate, present a new derivation of the consumption‐based capital asset pricing model, demonstrate how equilibrium forward and futures prices can be derived, including several examples, and finally we derive the equilibrium price of a European call option in a situation where the underlying asset pays dividends according to an Itô process of unbounded variation. In the latter case we demonstrate how this pricing formula simplifies to known results in special cases, among them the famous Black–Scholes formula and the Merton formula for a special dividend rate process.

A direct test for the mean variance efficiency of a portfolio

We develop a direct test for examining the mean-variance efficiency of a given bench-mark asset return. Unlike traditional tests for mean-variance efficiency, this test allows for the possibility that short positions in the primitive assets may not be possible. Using this test, we cannot reject the hypothesis that the value weighted return on exchange traded stocks is mean-variance efficient with reference to the mean-variance frontier generated by the 25 stock portfolios constructed by Fama and French (J. Financial Econom. 33 (1993) 3), when short selling is not allowed.

A solution approach to valuation with unhedgeable risks

We study a class of stochastic optimization models of expected utility in markets with stochastically changing investment opportunities. The prices of the primitive assets are modelled as diffusion processes whose coefficients evolve according to correlated diffusion factors. Under certain assumptions on the individual preferences, we are able to produce reduced form solutions. Employing a power transformation, we express the value function in terms of the solution of a linear parabolic equation, with the power exponent depending only on the coefficients of correlation and risk aversion. This reduction facilitates considerably the study of the value function and the characterization of the optimal hedging demand. The new results demonstrate an interesting connection with valuation techniques using stochastic differential utilities and also, with distorted measures in a dynamic setting.

Bernoulli speculator and trading strategy risk

In a continuous-time model of a complete information economy, we examine the case of a “pure” speculator who chooses to trade only on forward or futures contracts written on interest-rate-sensitive instruments. Assuming logarithmic utility, we assess whether his strategy exhibits the same structure as when he uses primitive assets only. It turns out that when interest rates follow stochastic processes, as in the model of Heath, Jarrow, and Morton (1992), where the instantaneous forward rate is driven by an arbitrary number of factors, the speculative trading strategy involving forwards exhibits an extra term vis-a-vis the one using futures or primitive assets. This extra term, different from a Merton–Breeden dynamic hedge, is novel and can be interpreted as a hedge against an “endogenous risk,” namely the interest-rate risk brought about by the optimal trading strategy itself. Thus, only the strategy using futures (or the cash assets themselves) involves a single speculative term, even for the Bernoulli speculator. This result illustrates another major aspect of the marking to market feature that differentiates futures and forwards, and thus has some bearing on the issue of the optimal design of financial contracts. Real financial markets being, in fact, incomplete, the additional “endogenous” risk associated with forwards cannot be hedged perfectly. Since using futures eliminates the latter, risk-averse agents will find them attractive in relation to forward contracts, other things being equal. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20: 507–523, 2000

Pricing of Non-Redundant Derivatives in a Complete Market

We consider a complete financial market with primitive assets and derivatives on these primitive assets. Nevertheless, the derivative assets are non-redundant in the market, in the sense that the market is complete, only with their existence. In such a framework, we derive an equilibrium restriction on the admissible prices of derivatives assets. The equilibrium condition imposes a well-ordering principle equivalent martingale measures. This restriction is preference free and applies whenever the utility functions belong to the general class of Von-Neumann Morgenstern functions. We provide numerical examples that show the applicability of restriction for the computation of option prices.

Pricing of non-redundant derivatives in a complete market

We consider a complete financial market with primitive assets and derivatives on these primitive assets. Nevertheless, the derivative assets are non-redundant in the market, in the sense that the market is complete,only with their existence. In such a framework, we derive an equilibrium restriction on the admissible prices of derivative assets. The equilibrium condition imposes a well-ordering principle restricting the set of probability measures that qualify as candidate equivalent martingale measures. This restriction is preference free and applies whenever the utility functions belong to the general class of Von-Neumann Morgenstern functions. We provide numerical examples that show the applicability of the restriction for the computation of option prices.