A liquidity-based model for asset price bubbles

We provide a new liquidity-based model for financial asset price bubbles that explains bubble formation and bubble bursting. The martingale approach to modeling price bubbles assumes that the asset's market price process is exogenous and the fundamental price, the expected future cash flows under a martingale measure, is endogenous. In contrast, we define the asset's fundamental price process exogenously and asset price bubbles are endogenously determined by market trading activity. This enables us to generate a model that explains both bubble formation and bubble bursting. In our model, the quantity impact of trading activity on the fundamental price process—liquidity risk—is what generates price bubbles. We study the conditions under which asset price bubbles are consistent with no arbitrage opportunities and we relate our definition of the fundamental price process to the classical definition.     

Foreign currency bubbles

This paper develops a new model for studying foreign currency exchange rate bubbles. The model constructed is a modification of the martingale based bubble approach of Jarrow et al. (Adv Math Finance 105–130, 2006; Math Finance 20(2):145–185, 2008). This model generates some new insights into our understanding of exchange rate bubbles and it can be utilized empirically to test for their existence. The new insights are: (1) exchange rate bubbles can be negative, in contrast to asset price bubbles, (2) exchange rate bubbles are caused by price level bubbles in either or both of the relevant countries’ currencies, and (3) price level bubbles decrease the expected inflation rate in the domestic economy.     

Understanding the Implied Volatility Surface for Options on a Diversified Index

This paper describes a two-factor model for a diversified index that attempts to explain both the leverage effect and the implied volatility skews that are characteristic of index options. Our formulation is based on an analysis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed squared Bessel process of dimension four. It turns out that for this index model an equivalent risk neutral martingale measure does not exist because the corresponding Radon-Nikodym derivative process is a strict local martingale. However, a consistent pricing and hedging framework is established by using the benchmark approach. The proposed model, which includes a random initial condition for market activity, generates implied volatility surfaces for European call and put options that are typically observed in real markets. The paper also examines the price differences of binary options for the proposed model and their Black-Scholes counterparts. Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

On the multiplicity of option prices under CEV with positive elasticity of variance

The discounted stock price under the Constant Elasticity of Variance model is not a martingale when the elasticity of variance is positive. Two expressions for the European call price then arise, namely the price for which put-call parity holds and the price that represents the lowest cost of replicating the call option’s payoffs. The greeks of European put and call prices are derived and it is shown that the greeks of the risk-neutral call can substantially differ from standard results. For instance, the relation between the call price and variance may become non-monotonic. Such unfamiliar behavior then might yield option-based tests for the potential presence of a bubble in the underlying stock price.     

On the law of one price

We consider the standard discrete-time model of a frictionless financial market and show that the law of one price holds if and only if there exists a martingale density process with strictly positive initial value. In contrast to the classical no-arbitrage criteria, this density process may change its sign. We also give an application to the CAPM.Received: November 2003, Mathematics Subject Classification (2000): 60G44JEL Classification: G13, G11Freddy Delbaen: This research was done during the stay of the author at Université de Franche-Comté.     

Stochastic orders in dynamic reinsurance markets

We consider a dynamic reinsurance market, where the traded risk process is driven by a compound Poisson process and where claim amounts are unbounded. These markets are known to be incomplete, and there are typically infinitely many martingale measures. In this case, no-arbitrage pricing theory can typically only provide wide bounds on prices of reinsurance claims. Optimal martingale measures such as the minimal martingale measure and the minimal entropy martingale measure are determined, and some comparison results for prices under different martingale measures are provided. This leads to a simple stochastic ordering result for the optimal martingale measures. Moreover, these optimal martingale measures are compared with other martingale measures that have been suggested in the literature on dynamic reinsurance markets.Received: March 2004, Mathematics Subject Classification (2000): 62P05, 60J75, 60G44JEL Classification: G10     

Lévy term structure models: No-arbitrage and completeness

The Lévy term structure model due to Eberlein and Raible is extended to non-homogeneous driving processes. The classes of equivalent martingale and local martingale measures for various filtrations are characterized. It turns out that in a number of standard situations the martingale measure is unique.Received: May 2004, Mathematics Subject Classification (2000): 60H30, 91B28, 60G51JEL Classification: E43, G13Work supported in part by the European Communitys Human Potential Programme under contract HPRN-CT-2000-00100, DYNSTOCH.     

Valuation of American options in the presence of event risk

This paper studies the valuation of American options in the presence of external/non-hedgeable event risk. When the event occurs, the American option is terminated and a rebate is paid instead of the promised pay-off profile. Consequently, the presence of event risk influences the exercise strategy of the option holder. For the financial market in a diffusion setting, the probabilistic structure in terms of equivalent martingale measures is briefly analysed. Then, for a given equivalent martingale measure the optimal stopping problem of the American option is solved. As a main result, no-arbitrage bounds for American option values in the presence of event risk are derived, as well as hedging strategies corresponding to the no-arbitrage bounds.Received: May 2004, Mathematics Subject Classification: 90C47, 60H30, 60G40JEL Classification: G13, D52, D81The author thanks John Gould and Ross Maller for useful discussions. The author is also grateful to a referee for helpful comments. This research was partially supported by University of Western Australia Research Grant RA/1/485.     

Optimal Insurance Design Under a Value-at-Risk Framework

This study designs an optimal insurance policy form endogenously, assuming the objective of the insured is to maximize expected final wealth under the Value-at-Risk (VaR) constraint. The optimal insurance policy can be replicated using three options, including a long call option with a small strike price, a short call option with a large strike price, and a short cash-or-nothing call option. Additionally, this study also calculates the optimal insurance levels for these models when we restrict the indemnity to be one of three common forms: a deductible policy, an upper-limit policy, or a policy with proportional coinsurance. JEL Classification No: G22     

An extension of mean-variance hedging to the discontinuous case

Our goal in this paper is to give a representation of the mean-variance hedging strategy for models whose asset price process is discontinuous as an extension of Gouriéroux, Laurent and Pham (1998) and Rheinländer and Schweizer (1997). However, we have to impose some additional assumptions related to the variance-optimal martingale measure.Received: April 2004, Mathematics Subject Classification (2000): 91B28, 60G48, 60H05JEL Classification: G10I would like to express my gratitude to Martin Schweizer and referees for their much valuable advice. I also would like to express my gratitude to Tsukasa Fujiwara, Hideo Nagai and Jun Sekine for many helpful comments.     

An example of indifference prices under exponential preferences

The aim herein is to analyze utility-based prices and hedging strategies. The analysis is based on an explicitly solved example of a European claim written on a nontraded asset, in a model where risk preferences are exponential, and the traded and nontraded asset are diffusion processes with, respectively, lognormal and arbitrary dynamics. Our results show that a nonlinear pricing rule emerges with certainty equivalent characteristics, yielding the price as a nonlinear expectation of the derivatives payoff under the appropriate pricing measure. The latter is a martingale measure that minimizes its relative to the historical measure entropy.Received: July 2003, Mathematics Subject Classification: 93E20, 60G40, 60J75JEL Classification: C61, G11, G13The second author acknowledges partial support from NSF Grants DMS-0102909 and DMS-0091946. We have received valuable comments from the participants at the Conferences in Paris IX, Dauphine (2000), ICBI Barcelona (2001) and 14th Annual Conference of FORC Warwick (2001). While revising this work, we came across the paper by Henderson (2002) in which a special case of our model is investigated     

On the Upper Bound of a Call Option

Using only a weak set of assumptions, Merton (1973) shows that the upper bound of a European or American call option on a non-dividend paying stock is the underlying stock price: a result which is often extended to options on dividend paying stocks. In this short technical piece we show that the underlying stock price is in fact not the least upper bound of either a European or an American call option on a stock that pays one or more known dividends prior to maturity. Based on Merton's (1973) original framework, new upper bounds are established which depend on the size(s) of the dividend(s) compared to the size of the strike. JEL Classification: G12, G13