We study zero-sum optimal stopping games (Dynkin games) between two players who disagree about the underlying model. In a Markovian setting, a verification result is established showing that if a pair of functions can be found that satisfies some natural conditions then a Nash equilibrium of stopping times is obtained, with the given functions as the corresponding value functions. In general, however, there is no uniqueness of Nash equilibria, and different equilibria give rise to different value functions. As an example, we provide a thorough study of the game version of the American call option under heterogeneous beliefs. Finally, we also study equilibria in randomized stopping times.

© 2015 Elsevier Inc. In this paper we use a recently developed econometric test to identify bubble-like price behaviour in the gold market. We find that the price of gold followed an explosive price process between 2002 and 2012 and exhibited super-exponential growth between 2002 and 2008, indicating excessive speculative trading and exuberance in the gold market. We also provide a theoretical foundation for such bubble tests based on a behavioural model in which chartists can cause episodes of explosive price dynamics.The identification strategy yields economically intuitive results and is a simple alternative to using more complex estimation techniques commonly used in the heterogeneous agents literature.

Gold has been a store of value for centuries and a safe haven for investors in the past decades. However, the increased investment in gold for speculative or hedging purposes has changed the safe haven property. We demonstrate theoretically and empirically that investor behaviour has the potential to destroy the safe haven property of gold. The results suggest that an asset cannot be both an investment asset and an effective safe haven asset. This finding has important implications for financial stability since assets are more likely to exhibit excess comovement and volatility in the absence of a safe haven.

Following the economic rationale of the British put and call option, we present a new class of lookback options (by first studying the canonical 'Russian' variant) where the holder enjoys the early exercise feature of American options, whereupon his payoff (deliverable immediately) is the 'best prediction' of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature which is key to the British Russian option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimize his losses. The practical implications of this protection feature are most remarkable as not only is the option holder afforded a unique protection against unfavourable stock price movements (covering the ability to sell in a liquid option market completely endogenously), but also when the stock price movements are favourable he will generally receive high returns. We derive a closed-form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results, we perform a financial analysis of the British Russian option that leads to the conclusions above and shows that with the contract drift properly selected, the British Russian option becomes a very attractive alternative to the classic European/American Russian option.

The recent financial crisis and related liquidity issues have illuminated an urgent need for a better understanding of the effects of limited liquidity on all aspects of the financial system. This paper considers such effects on the BlackScholesMerton financial model, which for the most part result in highly nonlinear partial differential equations (PDEs). We investigate in detail a model studied by Schönbucher and Wilmott (2000) which incorporates the price impact of option hedging strategies. First, we consider a first-order feedback model, which leads to the exceptional case of a linear PDE. Numerical results, and more particularly an asymptotic approach close to option expiry, reveal subtle differences from the BlackScholesMerton model. Second, we go on to consider a full-feedback model in which price impact is fully incorporated into the model. Here, standard numerical techniques lead to spurious results in even the simplest cases. An asymptotic approach, valid close to expiry, is mounted, and a robust numerical procedure, valid for all times, is developed, revealing two distinct classes of behavior. The first may be attributed to the infinite second derivative associated with standard option payoff conditions, for which it is necessary to admit solutions with discontinuous first derivatives; perhaps even more disturbingly, negative option values are a frequent occurrence. The second failure (applicable to smoothed payoff functions) is caused by a singularity in the coefficient of the diffusion term in the option-pricing equation. Our conclusion is that several classes of model in the literature involving permanent price impact irretrievably break down (i.e., there is insufficient financial modeling in the pricing equation). Our analysis should provide the information necessary to avoid such pitfalls in the future.

Following the economic rationale of Peskir and Samee (2008a,b), we present a new class of Asian options where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the `best prediction of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature that is key to the British Asian option. Should the option holder believe the true drift of the stock price to be unfavorable (based upon the observed price movements), he can substitute the true drift with the contract drift and minimize his losses. The practical implications of this protection feature are most remarkable, as not only is the option holder afforded a unique protection against unfavorable stock price movements (covering the ability to sell in a liquid option market completely endogenously), but also when the stock price movements are favorable he will generally receive high returns. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British Asian option that leads to the conclusions above and shows that with the contract drift properly selected the British Asian option becomes a very attractive alternative to the classic (European) Asian option.

We formulate a general model of a commercial bank and its regulator where the bank's loans are exposed to default risk. The bank's objective is to maximise equity value by appropriately controlling the rate at which new loans are issued, early clo- sure, and dividend payments. The regulator's objective is to reduce the probability of the bank's early closure, which they achieve by appropriately controlling the bank's minimum capital requirements. We show that the regulator can in fact minimise this probability of closure, which is achieved via suitably balancing the risk of insolvency (associated with lower capital requirements) and the risk of endogenous closure (as- sociated with higher capital requirements). Both analytic and numerical results are presented, thus allowing for the full non-linearity of the model to be understood.

We examine the British payoff mechanism (introduced in Peskir and Samee, 2008) in the context of path dependent options. In particular, we focus on the âBritish Asianâ and the âBritish Russianâ option. Such options provide their holder with an endogenous protection against unfavourable stock price movements. The price of such options can be characterised as the unique solution to a parabolic free-boundary problem, whose properties and solution we investigate. Finally, we provide a preliminary financial analysis of both options and conclude that in many circumstances these options can be considered an attractive alternative to existing path dependent options.

In this paper we use a test developed by Phillips et al. (2011) to identify a bubble in the gold market. We find that the price of gold followed an explosive price process between 2002 and 2012 interrupted only briefly by the
subprime crisis in 2008. We also provide a theoretical foundation for such bubble tests based on a behavioural model of heterogeneous agents and demonstrate that periods of explosive price behaviour are consistent with increased
chartist activity in the gold market. The identification strategy yields economically intuitive results and is a simple alternative to using more complex estimation techniques commonly used in the heterogeneous agents literature.

We analyse the effect of differing uncertainty assumptions on the costs of shareholder-bondholder conflicts arising from partially debt-financed investments. A partial equilibrium model, valid for a large class of diffusion
processes, is developed and then applied to the specific cases of a geometric Brownian motion (GBM) and a mean-reverting (MR) process. This allows for the comparison of the two scenarios and contributes to the ongoing discussion on the
effects of mean reversion on investment and financing behaviour. We find that agency costs are much lower under MR dynamics and, through the application of a novel agency cost decomposition, we show that for a high expected growth in
future profits (high growth GBM) agency costs are driven mainly by suboptimal financing decisions, as opposed to suboptimal (default and investment) timing decisions. The situation is reversed for lower growth assumptions and for an
increase in the speed of mean reversion. Our results on the components and drivers of agency costs are valuable to both policy makers and regulators alike.

Research Paper Number 269 Abstract: Following the economic rationale of [10] and [11] we present a new class of lookback options (by first studying the canonical âRussianâ variant) where the holder enjoys the early exercise feature of American options where upon his payoff (deliverable immediately) is the âbest predictionâ of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature which is key to the British Russian option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimise his losses. The practical implications of this protection feature are most remarkable as not only is the option holder afforded a unique protection against unfavourable stock price movements (covering thea bility to sell in a liquid market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British Russian option that leads to the conclusions above and shows that with the contract drift properly selected the British Russian option becomes a very attractive alternative to the classic European/American Russian option.

Research Paper Number: 249 Abstract: Following the economic rationale of [7] and [8] we present a new class of Asian options where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the âbest predictionâ of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature which is key to the British Asian option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimise his losses. The practical implications of this protection feature are most remarkable as not only is the option holder afforded a unique protection against unfavourable stock price movements (covering the ability to sell in a liquid market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British Asian option that leads to the conclusions above and shows that with the contract drift properly selected the British Asian option becomes a very attractive alternative to the classic (European) Asian option.