We present three models of stock price with time-dependent interest rate, dividend yield, and volatility, respectively, that allow for explicit forms of the optimal exercise boundary of the American put option. The optimal exercise boundary satisfies nonlinear integral equation of Volterra type. We choose time-dependent parameters of the model so that the integral equation for the exercise boundary can be solved in the closed form. We also define the contracts of put type with time-dependent strike price that support the explicit optimal exercise boundary. All these results can be used as approximations to the standard model with constant parameters, i.e., geometric Brownian motion process.

We theoretically study a general family of economic geography models that features endogenous agglomeration. In many-region settings, the spatial scale---global or local---of the dispersion force(s) in a model plays a key role in determining the resulting endogenous spatial patterns and comparative statics. A global dispersion force accrues from competition between different locations and leads to the formation of multiple economic clusters, or cities. A local dispersion force is caused by crowding effects within each location and induces the flattening of each city. By distinguishing local and global dispersion forces, we can reduce a wide variety of extant models into only three prototypical classes that are qualitatively different in implications. Our framework adds consistent interpretations to the empirical literature and also provides general predictions on treatment effects in structural economic geography models.

Machine learning (especially reinforcement learning) methods for trading are increasingly reliant on simulation for agent training and testing. Furthermore, simulation is important for validation of hand-coded trading strategies and for testing hypotheses about market structure. A challenge, however, concerns the robustness of policies validated in simulation because the simulations lack fidelity. In fact, researchers have shown that many market simulation approaches fail to reproduce statistics and stylized facts seen in real markets. As a step towards addressing this we surveyed the literature to collect a set of reference metrics and applied them to real market data and simulation output. Our paper provides a comprehensive catalog of these metrics including mathematical formulations where appropriate. Our results show that there are still significant discrepancies between simulated markets and real ones. However, this work serves as a benchmark against which we can measure future improvement.

When the in-sample Sharpe ratio is obtained by optimizing over a k-dimensional parameter space, it is a biased estimator for what can be expected on unseen data (out-of-sample). We derive (1) an unbiased estimator adjusting for both sources of bias: noise fit and estimation error. We then show (2) how to use the adjusted Sharpe ratio as model selection criterion analogously to the Akaike Information Criterion (AIC). Selecting a model with the highest adjusted Sharpe ratio selects the model with the highest estimated out-of-sample Sharpe ratio in the same way as selection by AIC does for the log-likelihood as measure of fit.

In this paper, Malliavin calculus is applied to arrive at exact formulas for the difference between the volatility swap strike and the zero vanna implied volatility for volatilities driven by fractional noise. To the best of our knowledge, our estimate is the rst to show the rigorous relationship between the zero vanna implied volatility and the volatility swap strike. In particular, we will see that the zero vanna implied volatility has a higher rate of convergence than the at-the-money (ATM) implied volatility for both zero and non-zero correlation and for all values of the Hurst parameter.

We examine how extreme market risks are priced in the cross-section of asset returns at various horizons. Based on the decomposition of covariance between indicator functions capturing fluctuations of different parts of return distributions over various frequencies, we define a \textit{quantile spectral} beta representation that characterizes asset's risk generally. Nesting the traditional frameworks, the new representation explains \textit{tail}-specific as well as horizon-, or frequency-specific \textit{spectral} risks. Further, we work with two notions of frequency-specific extreme market risks. First, we define tail market risk that captures dependence between extremely low market and asset returns. Second, extreme market volatility risk is characterized by dependence between extremely high increments of market volatility and extremely low asset return. Empirical findings based on the datasets with long enough history, 30 Fama-French Industry portfolios, and 25 Fama-French portfolios sorted on size and book-to-market support our intuition. We reach the same conclusion using stock-level data as well as daily data. These results suggest that both frequency-specific tail market risk and extreme volatility risk are priced and our final model provides significant improvement over specifications considered by previous literature.

At the zero lower bound, the New Keynesian model predicts that output and inflation collapse to implausibly low levels, and that government spending and forward guidance have implausibly large effects. To resolve these anomalies, we introduce wealth into the utility function; the justification is that wealth is a marker of social status, and people value status. Since people partly save to accrue social status, the Euler equation is modified. As a result, when the marginal utility of wealth is sufficiently large, the dynamical system representing the zero-lower-bound equilibrium transforms from a saddle to a source---which resolves all the anomalies.

Among all the emerging markets, the cryptocurrency market is considered the most controversial and simultaneously the most interesting one. The visibly significant market capitalization of cryptos motivates modern financial instruments such as futures and options. Those will depend on the dynamics, volatility, or even the jumps of cryptos. In this paper, the risk characteristics for Bitcoin are analyzed from a realized volatility dynamics view. The realized variance is estimated with the corrected threshold jump components, realized semi-variance, and signed jumps. Our empirical results show that the BTC is far riskier than any of the other developed financial markets. Up to 68% of the days are identified to be entangled with jumps. However, the discontinuities do not contribute to the variance significantly. The full-sample fitting suggests that future realized variance has a positive relationship with downside risk and a negative relationship with the positive jump. The rolling-window out-of-sample forecasting results reveal that the forecasting horizon plays an important role in choosing forecasting models. For the long horizon risk forecast, explicitly modeling jumps and signed estimators improve forecasting accuracy and give extra utility up to 19 bps annually, while the HAR model without accounting jumps or signed estimators suits the short horizon case best. Lastly, a simple equal-weighted portfolio of BTC not only significantly reduces the size and quantity of jumps but also gives investors higher utility in short horizon case.

Computing risk measures of a financial portfolio comprising thousands of options is a challenging problem because (a) it involves a nested expectation requiring multiple evaluations of the loss of the financial portfolio for different risk scenarios and (b) evaluating the loss of the portfolio is expensive and the cost increases with its size. In this work, we look at applying Multilevel Monte Carlo (MLMC) with adaptive inner sampling to this problem and discuss several practical considerations. In particular, we discuss a sub-sampling strategy that results in a method whose computational complexity does not increase with the size of the portfolio. We also discuss several control variates that significantly improve the efficiency of MLMC in our setting.

The latest financial crisis has painfully revealed the dangers arising from a globally interconnected financial system. Conventional approaches based on the notion of the existence of equilibrium and those which rely on statistical forecasting have seen to be inadequate to describe financial systems in any reasonable way. A more natural approach is to treat financial systems as complex networks of claims and obligations between various financial institutions present in an economy. The generic framework of complex networks has been successfully applied across several disciplines, e.g., explaining cascading failures in power transmission systems and epidemic spreading. Here we review various network models addressing financial contagion via direct inter-bank contracts and indirectly via overlapping portfolios of financial institutions. In particular, we discuss the implications of the "robust-yet-fragile" nature of financial networks for cost-effective regulation of systemic risk.

We compare the revenue of the optimal third-degree price discrimination policy against a uniform pricing policy. A uniform pricing policy offers the same price to all segments of the market. Our main result establishes that for a broad class of third-degree price discrimination problems with concave revenue functions and common support, a uniform price is guaranteed to achieve one half of the optimal monopoly profits. This revenue bound obtains for any arbitrary number of segments and prices that the seller would use in case he would engage in third-degree price discrimination. We further establish that these conditions are tight, and that a weakening of common support or concavity leads to arbitrarily poor revenue comparisons.