Estimating state price densities by Hermite polynomials: theory and application to the Italian derivatives market

We study the problem of extracting the state price densities from the market prices of listed options. Adapting a model of Madan and Milne to a multiple expiration setting, we present an estimation method for the risk-neutral probability at a moving horizon of fixed length. With the exception of volatility, all model parameters can be estimated by linear regression and their number can be chosen arbitrarily, depending on the size of the dataset. We discuss empirical issues related to the application of this model to real data and show results on listed options on the Italian MIB30 equity index.option pricing, state-price densities, orthogonal polynomials, risk-neutral valuation, calibration

Hedging, arbitrage and optimality with superlinear frictions

In a continuous-time model with multiple assets described by c\`{a}dl\`{a}g processes, this paper characterizes superhedging prices, absence of arbitrage, and utility maximizing strategies, under general frictions that make execution prices arbitrarily unfavorable for high trading intensity. Such frictions induce a duality between feasible trading strategies and shadow execution prices with a martingale measure. Utility maximizing strategies exist even if arbitrage is present, because it is not scalable at will.Comment: Published at http://dx.doi.org/10.1214/14-AAP1043 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robust Portfolios and Weak Incentives in Long-Run Investments

When the planning horizon is long, and the safe asset grows indefinitely, isoelastic portfolios are nearly optimal for investors who are close to isoelastic for high wealth, and not too risk averse for low wealth. We prove this result in a general arbitrage-free, frictionless, semimartingale model. As a consequence, optimal portfolios are robust to the perturbations in preferences induced by common option compensation schemes, and such incentives are weaker when their horizon is longer. Robust option incentives are possible, but require several, arbitrarily large exercise prices, and are not always convex

Consistent price systems and face-lifting pricing under transaction costs

In markets with transaction costs, consistent price systems play the same role as martingale measures in frictionless markets. We prove that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs. This result applies to a large class of Markovian and non-Markovian models, including geometric fractional Brownian motion. Using the constructed price systems, we show, under very general assumptions, the following ``face-lifting'' result: the asymptotic superreplication price of a European contingent claim $g(S_T)$ equals $\hat{g}(S_0)$, where $\hat{g}$ is the concave envelope of $g$ and $S_t$ is the price of the asset at time $t$. This theorem generalizes similar results obtained for diffusion processes to processes with conditional full support.Comment: Published in at http://dx.doi.org/10.1214/07-AAP461 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fragility of arbitrage and bubbles in local martingale diffusion models

For any positive diffusion with minimal regularity, there exists a semimartingale with uniformly close paths that is a martingale under an equivalent probability. As a result, in models of asset prices based on such diffusions, arbitrage and bubbles alike disappear under proportional transaction costs or under small model mis-specifications. Thus, local martingale diffusion models of arbitrage and bubbles are not robust to small trading and monitoring frictions. © 2015, Springer-Verlag Berlin Heidelberg

Transaction Costs, Trading Volume, and the Liquidity Premium

In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities, and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. Results are robust to consumption and finite horizons. We exploit the equivalence of the transaction cost market to another frictionless market, with a shadow risky asset, in which investment opportunities are stochastic. The shadow price is also found explicitly.Comment: 29 pages, 5 figures, to appear in "Finance and Stochastics". arXiv admin note: text overlap with arXiv:1207.733