Past Nomura Seminar

We consider the market for a risky asset for which agents have interdependent private valuations. We study competitive rational expectations equilibria under the standard CARA-normal assumptions. Equilibrium is partially revealing even though there are no noise traders. Complementarities in information acquisition arise naturally in this setting. We characterize stable equilibria with endogenous information acquisition. Our framework encompasses the classical REE models in the CARA-normal tradition.

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We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. Using a very recent theory of singular changes of measures for Ito processes, we formulate the principal-agent problem in this context, and solve it in the case of CARA preferences. In that case, the optimal contract is linear in these factors: the contractible sources of risk, including the output, the quadratic variation of the output and the cross-variations between the output and the contractible risk sources. Thus, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. This is a joint work with Nizar Touzi (CMAP, Ecole Polytechnique) and Jaksa Cvitanic (Caltech).

A sharp increase in the popularity of commodity investing in the past decade has triggered an unprecedented inflow of institutional funds into commodity futures markets. Such financialization of commodities coincided with significant booms and busts in commodity markets, raising concerns of policymakers. In this paper, we explore the effects of financialization in a model that features institutional investors alongside traditional futures markets participants. The institutional investors care about their performance relative to a commodity index. We find that if a commodity futures is included in the index, supply and demand shocks specific to that commodity spill over to all other commodity futures markets. In contrast, supply and demand shocks to a nonindex commodity affect just that commodity market alone. Moreover, prices and volatilities of all commodity futures go up, but more so for the index futures than for nonindex ones. Furthermore, financialization — the presence of institutional investors — leads to an increase in correlations amongst commodity futures as well as in equity-commodity correlations. Consistent with empirical evidence, the increases in the correlations between index commodities exceed those for nonindex ones. We model explicitly demand shocks which allows us to disentangle the effects of financialization from the effects of demand and supply (fundamentals). We perform a simple calibration and find that financialization accounts for 11% to 17% of commodity futures prices and the rest is attributable to fundamentals.

We consider evaluation methods for payoffs with an inherent

financial risk as encountered for instance for portfolios held

by pension funds and insurance companies. Pricing such payoffs

in a way consistent to market prices typically involves

combining actuarial techniques with methods from mathematical

finance. We propose to extend standard actuarial principles by

a new market-consistent evaluation procedure which we call `two

step market evaluation.' This procedure preserves the structure

of standard evaluation techniques and has many other appealing

properties. We give a complete axiomatic characterization for

two step market evaluations. We show further that in a dynamic

setting with continuous stock prices every evaluation which is

time-consistent and market-consistent is a two step market

evaluation. We also give characterization results and examples

in terms of $g$-expectations in a Brownian-Poisson setting.

We present a new model of financial markets that studies the evolution of wealth

among investment strategies. An investment strategy can be generated by maximizing utility

given some expectations or by behavioral rules. The only requirement is that any investment strategy

is adapted to the information filtration. The model has the mathematical structure of a random dynamical system.

We solve the model by characterizing evolutionary properties of investment strategies (survival, evolutionary stability, dominance).

It turns out that only a fundamental strategy investing according to expected relative dividends satisfies these evolutionary criteria.

We study lower- and dual upper-bounds for Bermudan options in a MonteCarlo/MC setting and provide four contributions. 1) We introduce a local least-squares MC method, based on maximizing the Bermudan price and which provides a lower-bound, which "also" minimizes (not the dual upper-bound itself, but) the gap between these two bounds; where both bounds are specified recursively. 2) We confirm that this method is near optimal, for both lower- and upper-bounds, by pricing Bermudan max-call options subject to an up-and-out barrier; state-of-the-art methods including Longstaff-Schwartz produce a large gap of 100--200 basis points/bps (Desai et al. (2012)), which we reduce to just 5--15 bps (using the same linear basis of functions). 3) For dual upper-bounds based on continuation values (more biased but less time intensive), it works best to reestimate the continuation value in the continuation region only. And 4) the difference between the Bermudan option Delta and the intrinsic value slope at the exercise boundary gives the sensitivity to suboptimal exercise (up to a 2nd-order Taylor approximation). The up-and-out feature flattens the Bermudan price, lowering the Bermudan Delta well below one when the call-payoff slope is equal to one, which implies that optimal exercise "really" matters.

We propose a model where an algorithmic trader takes a view on the distribution of prices at a future date and then decides how to trade in the direction of her predictions using the optimal mix of market and limit orders. As time goes by, the trader learns from changes in prices and updates her predictions to tweak her strategy. Compared to a trader that cannot learn from market dynamics or form a view of the market, the algorithmic trader's profits are higher and more certain. Even though the trader executes a strategy based on a directional view, the sources of profits are both from making the spread as well as capital appreciation of inventories. Higher volatility of prices considerably impairs the trader's ability to learn from price innovations, but this adverse effect can be circumvented by learning from a collection of assets that co-move.

The risk of a financial position is usually summarized by a risk measure.

As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In

statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile based risk

measures such as value-at-risk are elicitable. However, the coherent risk measure expected shortfall is not elicitable. Hence, it is unclear how to perform

forecast verification or comparison. We address the question whether coherent and elicitable risk measures exist (other than minus the expected value).

We show that one positive answer are expectiles, and that they play a special role amongst all elicitable law-invariant coherent risk measures.

In this work, we want to construct the solution $(Y,Z,K)$ to the following BSDE

$$\begin{array}{l}

Y_t=\xi+\int_t^Tf(s,Y_s,Z_s)ds-\int_t^TZ_sdB_s+K_T-K_t, \quad 0\le t\le T, \\

{\mathbf E}[l(t, Y_t)]\ge 0, \quad 0\le t\le T,\\

\int_0^T{\mathbf E}[l(t, Y_t)]dK_t=0, \\

\end{array}

$$

where $x\mapsto l(t, x)$ is non-decreasing and the terminal condition $\xi$

is such that ${\mathbf E}[l(T,\xi)]\ge 0$.

This equation is different from the (classical) reflected BSDE. In particular, for a solution $(Y,Z,K)$,

we require that $K$ is deterministic. We will first study the case when $l$ is linear, and then general cases.

We also give some application to mathematical finance. This is a joint work with Philippe Briand and Romuald Elie.

We construct and study market models admitting optimal arbitrage. We say that a model admits optimal arbitrage if it is possible, in a zero-interest rate setting, starting with an initial wealth of 1 and using only positive portfolios, to superreplicate a constant c>1. The optimal arbitrage strategy is the strategy for which this constant has the highest possible value. Our definition of optimal arbitrage is similar to the one in Fenrholz and Karatzas (2010), where optimal relative arbitrage with respect to the market portfolio is studied. In this work we present a systematic method to construct market models where the optimal arbitrage strategy exists and is known explicitly. We then develop several new examples of market models with arbitrage, which are based on economic agents' views concerning the impossibility of certain events rather than ad hoc constructions. We also explore the concept of fragility of arbitrage introduced in Guasoni and Rasonyi (2012), and provide new examples of arbitrage models which are not fragile in this sense.

References:

Fernholz, D. and Karatzas, I. (2010). On optimal arbitrage. The Annals of Applied Probability, 20(4):1179–1204.

Guasoni, P. and Rasonyi, M. (2012). Fragility of arbitrage and bubbles in diffusion models. preprint.

There are many financial models used in practice (CIR/Heston, Vasicek,

Stein-Stein, quadratic normal) whose popularity is due, in part, to their

analytically tractable asset pricing. In this talk we will show that it is

possible to generalise these models in various ways while maintaining

tractability. Conversely, we will also characterise the family of models

which admit this type of tractability, in the spirit of the classification

of polynomial term structure models.

The finance literature documents a relation between labor income and

the cross-section of stock returns. One possible explanation for this

is the hedging decisions of investors with relative wealth concerns.

This implies a negative risk premium associated with stock returns

correlated with local undiversifiable wealth, since investors are

willing to pay more for stocks that help their hedging goals. We find

evidence that is consistent with these regularities. In addition, we

show that the effect varies across geographic areas depending on the

size and variability of undiversifiable wealth, proxied by labor income.

An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth.

In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market,

cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to

optimal execution problems and to other market frictions such as proportional and fixed transaction costs.

Worst-case portfolio optimization has been introduced in Korn and Wilmott

(2002) and is based on distinguishing between random stock price

fluctuations and market crashes which are subject to Knightian

uncertainty. Due to the absence of full probabilistic information, a

worst-case portfolio problem is considered that will be solved completely.

The corresponding optimal strategy is of a multi-part type and makes an

investor indifferent between the occurrence of the worst possible crash

and no crash at all.

We will consider various generalizations of this setting and - as a very

recent result - will in particular answer the question "Is it good to save

for bad times or should one consume more as long as one is still rich?"

We construct a model for asset price in a limit order book, which captures on one hand main stylized facts of microstructure effects, and on the other hand is tractable for dealing with optimal high frequency trading by stochastic control methods. For this purpose, we introduce a model for describing the fluctuations of a tick-by-tick single asset price, based on Markov renewal process.

We consider a point process associated to the timestamps of the price jumps, and marks associated to price increments. By modeling the marks with a suitable Markov chain, we can reproduce the strong mean-reversion of price returns known as microstructure noise. Moreover, by using Markov renewal process, we can model the presence of spikes in intensity of market activity, i.e. the volatility clustering. We also provide simple parametric and nonparametric statistical procedures for the estimation of our model. We obtain closed-form formulae for the mean signature plot, and show the diffusive behavior of our model at large scale limit. We illustrate our results by numerical simulations, and find that our model is consistent with empirical data on futures Euribor and Eurostoxx. In a second part, we use a dynamic programming approach to our semi Markov model applied to the problem of optimal high frequency trading with a suitable modeling of market order flow correlated with the stock price, and taking into account in particular the adverse selection risk. We show a reduced-form for the value function of the associated control problem, and provide a convergent and computational scheme for solving the problem. Numerical tests display the shape of optimal policies for the market making problem.

This talk is based on joint works with Pietro Fodra.

We extend Kyle's (1985) model of insider trading to the case where liquidity provided

by noise traders follows a general stochastic process. Even though the level of noise

trading volatility is observable, in equilibrium, measured price impact is stochastic.

If noise trading volatility is mean-reverting, then the equilibrium price follows a

multivariate stochastic volatility `bridge' process. More private information is revealed

when volatility is higher. This is because insiders choose to optimally wait to trade

more aggressively when noise trading volatility is higher. In equilibrium, market makers

anticipate this, and adjust prices accordingly. In time series, insiders trade more

aggressively, when measured price impact is lower. Therefore, aggregate execution costs

to uninformed traders can be higher when price impact is lower

We study optimal portfolio strategies in a market

where the drift is driven by an unobserved Markov chain. Information on

the state of this chain is obtained from stock prices and from expert

opinions in the form of signals at random discrete time points. We use

stochastic filtering to transform the original problem into an

optimization problem under full information where the state variable is

the filter for the Markov chain. This problem is studied with dynamic

programming techniques and with regularization arguments. Finally we

discuss a number of numerical experiments

We consider over-the-counter (OTC) transactions with bilateral default risk, and study the optimal design of the Credit Support Annex (CSA). In a setting where agents have access to a trading technology, default penalties and collateral costs arise endogenously as a result of foregone investment opportunities. We show how the optimal CSA trades off the costs of the collateralization procedure against the reduction in exposure to counterparty risk and expected default losses. The results are used to provide insights on the drivers of different collateral rules, including hedging motives, re-hypothecation of collateral, and close-out conventions. We show that standardized collateral rules can have a detrimental impact on risk sharing, which should be taken into account when assessing the merits of standardized vs. bespoke CSAs in non-centrally cleared OTC instruments. This is joint work with D. Bauer and L.R. Sotomayor (GSU).

We analyse the impact of market makers' risk aversion on the equilibrium in a speculative market consisting of a risk neutral informed trader and noise traders. The unwillingness of market makers to bear risk causes the informed trader to absorb large shocks in their inventories. The informed trader's optimal strategy is to drive the market price to its fundamental value while disguising her trades as the ones of an uninformed strategic trader. This results in a mean reverting demand, price reversal, and systematic changes in the market depth. We also find that an increase in risk aversion leads to lower market depth, less efficient prices, stronger price reversal and slower convergence to fundamental value. The endogenous value of private information, however, is non-monotonic in risk aversion. We will mainly concentrate on the case when the private signal of the informed is static. If time permits, the implications of a dynamic signal will be discussed as well.

Based on a joint work with Albina Danilova.

The performance of the shares of a closed end bond fund is based on the returns of an underlying portfolio of bonds. The returns on closed end bond funds are typically higher than those of comparable open ended bond funds and this result is attributed to the use of leverage by closed end bond funds. This talk develops a simple model to assess the impact of leverage on the expected return and riskiness of a closed end bond fund. We illustrate the model with some examples

Advanced models such as Lévy models require advanced numerical methods for developing efficient pricing algorithms. Here we focus on PIDE based methods. There is a large arsenal of numerical methods for solving parabolic equations that arise in this context. Especially Galerkin and Galerkin inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.

We therefore classify Lévy processes according to the solution spaces of the associated parabolic PIDEs. We define the Sobolev index of a Lévy process by a certain growth condition on the symbol. It follows that for Lévy processes with a certain Sobolev index b the corresponding evolution problem has a unique weak solution in the Sobolev-Slobodeckii space with index b/2. We show that this classification applies to a wide range of processes. Examples are the Brownian motion with or without drift, generalised hyperbolic (GH), CGMY and (semi) stable Lévy processes.

A comparison of the Sobolev index with the Blumenthal-Getoor index sheds light on the structural implication of the classification. More precisely, we discuss the Sobolev index as an indicator of the smoothness of the distribution and of the variation of the paths of the process.

An application to financial models requires in particular to admit pure jump processes as well as unbounded domains of the equation. In order to deal at the same time with the typical payoffs which can arise, the weak formulation of the equation has to be based on exponentially weighted Sobolev-Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.

The talk is based on joint work with Ernst Eberlein.

The martingale optimal transportation problem is motivated by

model-independent bounds for the pricing and hedging exotic options in

financial mathematics.

In the simplest one-period model, the dual formulation of the robust

superhedging cost differs from the standard optimal transport problem by

the presence of a martingale constraint on the set of coupling measures.

The one-dimensional Brenier theorem has a natural extension. However, in

the present martingale version, the optimal coupling measure is

concentrated on a pair of graphs which can be obtained in explicit form.

These explicit extremal probability measures are also characterized as

the unique left and right monotone martingale transference plans, and

induce an optimal solution of the kantorovitch dual, which coincides

with our original robust hedging problem.

By iterating the above construction over n steps, we define a Markov

process whose distribution is optimal for the n-periods martingale

transport problem corresponding to a convenient class of cost functions.

Similarly, the optimal solution of the corresponding robust hedging

problem is deduced in explicit form. Finally, by sending the time step

to zero, this leads to a continuous-time version of the one-dimensional

Brenier theorem in the present martingale context, thus providing a new

remarkable example of Peacock, i.e. Processus Croissant pour l'Ordre

Convexe. Here again, the corresponding robust hedging strategy is

obtained in explicit form.