A Triple Comparison between Anticipating Stochastic Integrals in Financial Modeling

Abstract

We consider a simplified version of the problem of insider trading in a financial market. We approach it by means of anticipating stochastic calculus and compare the use of the Hitsuda-Skorokhod, the Ayed-Kuo, and the Russo-Vallois forward integrals within this context. Our results give some indication that, while the forward integral yields results with a suitable financial meaning, the Hitsuda-Skorokhod and the Ayed-Kuo integrals do not provide an appropriate formulation of this problem. Further results regarding the use of the Ayed-Kuo integral in this context are also provided, including the proof of the fact that the expectation of a Russo-Vallois solution is strictly greater than that of an Ayed-Kuo solution. Finally, we conjecture the explicit solution of an Ayed-Kuo stochastic differential equation that possesses discontinuous sample paths with finite probability.2010 Mathematics Subject Classification. 60H05, 60H07, 60H10, 60H30, 91G80.

Optimal Portfolios for Different Anticipating Integrals under Insider Information

“…Indeed, from our present results along with those in [12] it follows that: E S (I) (T) = E M (AK) (T) = E M (HS) (T) < E M (RV) (T) , where we have chosen the optimal investment allocation in each case. If we selected the optimal Russo-Vallois strategy for all the three anticipating cases, the situation becomes even worse, see [12]; and it could be even more critical if we allowed different strategies, see [11]. All in all, our results suggest that while the Russo-Vallois forward integral allowed the insider to make full use of the privileged information, both the Ayed-Kuo and Hitsuda-Skorokhod integrals effectively transformed the insider into an uninformed trader.…”

“…Since the preeminent place for this debate on the noise interpretation has been the physics literature, the focus has been put on diffusion processes, perhaps due to the influence of the seminal works by Einstein [9] and Langevin [10]. Herein we move out of even the Markovian setting and, following the steps of [11,12], we concentrate on stochastic differential equations with non-adapted terms. Non-adaptedness arises in financial markets concomitantly to the presence of insider traders.…”

Chances for the honest in honest versus insider trading

“…For this reason it is not surprising that it has been frequently used to treat insider information ( [2], [3], [4], [6], [5], [10], [12]). As noted in the previous section, the forward integral is genuinely different from the Skorokhod one, and we recommend the former and not the latter that has to be used in the mathematical modeling of insider trading in view of previous results [7,1,8].…”

“…To be more precise, let us consider a filtered probability space (Ω, F, F s , P) in which a Brownian motion B s is defined; moreover assume F s = σ{B u , 0 ≤ u ≤ s}. We furthermore assume that the wealth of the honest investor evolves according to the stochastic differential equation (1) dM s = [(1 − π s )r s M s + π s µ s M s ]ds + π s σ s M s dB s , where M 0 is assumed to be a positive real number, r s is the risk-free rate, µ s the expected return of the stock market, σ s its volatility, and π s is the time-dependent investor strategy or, in other words, her portfolio. Of course, this means we are assuming a Black-Scholes market with two assets: one is risky and the other is riskless.…”

“…If the portfolio π s ∈ L 2 (Ω × [0, t]) is F s −adapted (what ultimately guarantees that our first trader is honest) then equation (1) admits the unique solution [13]…”