How can one relate stock fluctuations and information-based human activities? We present a model of an incomplete market by adjoining the Black-Scholes exponential Brownian motion model for stock fluctuations with a hidden Markov process, which represents the state of information in the investors' community. The drift and volatility parameters take different values depending on the state of this hidden Markov process. Standard option pricing procedure under this model becomes problematic. Yet, with an additional economic assumption, we provide an explicit closed-form formula for the arbitrage-free price of the European call option. Our model can be discretized via a Skorohod embedding technique. We conclude with an example of a simulation of IBM stock, which shows that, not surprisingly, information does affect the market.

Information and option pricings

In theory, an investor can make infinite profits by taking unlimited positions in an arbitrage. In reality, however, investors must satisfy margin requirements which completely change the economics of arbitrage. We derive the optimal investment policy for a risk-averse investor in a market where there are arbitrage opportunities. We show that it is often optimal to underinvest in the arbitrage by taking a smaller position than margin constraints allow. In some cases, it is actually optimal for an investor to walk away from a pure arbitrage opportunity. Even when the optimal policy is followed, the arbitrage strategy may underperform the riskless asset or have an unimpressive Sharpe ratio. Furthermore, the arbitrage portfolio typically experiences losses at some point before the final convergence date. These results have important implications for the role of arbitrageurs in financial markets.

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Losing Money on Arbitrage: Optimal Dynamic Portfolio Choice in Markets with Arbitrage Opportunities

https://people.math.ethz.ch/~mschweiz/Files/ais_sfb.pdf

Additional logarithmic utility of an insider

The Kyle (1985) and Back (1992) model of continuous-time asset pricing with asymmetric information is studied. A larger class of price processes is considered, namely price processes that allow the price to depend in a certain way on the path of the market order. A no expected (or inconspicuous insider) trade theorem\/ is satisfied regardless of how much the informed agent is sensitive to the risk. When the informed agent is risk-neutral, the price pressure is constant over time and the price depends only on the cumulative market order. As a corollary, this paper extends Kyle's (1985) and Back's (1992) uniqueness result by showing that the equilibrium is also unique on this larger class of price processes. When the informed agent is risk-averse, in contrast, the price pressure decreases over time and the price depends on the entire path. The price pressure with risk aversion converges uniformly to the risk-neutral price pressure as the informed agent becomes less and less risk-averse; similarly, the equilibrium with risk aversion converges to the risk-neutral equilibrium.

Continuous auctions and insider trading: uniqueness and risk aversion

This paper uses the enlargement of Brownian filtrations and a probability change for modelling the observation of a financial market by an insider trader. A characterization of admissible strategies and a criterion for optimization are given. Then a statistical test is proposed to test whether or not the trader is an insider.

Insider Trading in a Continuous Time Market Model

We study a classical stochastic control problem arising in financial economics: to maximize expected logarithmic utility from terminal wealth and/or consumption. The novel feature of our work is that the portfoilo is allowed to anticipate the future, i.e. the terminal values of the prices, or of the driving Brownian motion, are known to the investor, either exactly or with some uncertainty. Results on the finiteness of the value of the control problem are obtained in various setups, using techniques from the so-called enlargement of filtrations. When the value of the problem is finite, we compute it explicitly and exhibit an optimal portfolio in closed form.

Igor Pikovsky

Papers

Anticipative portfolio optimization