Optimal Hedging Strategies of Stock Index Futures Based on the Perspective of Information Asymmetry

Based on two different risk measurement criteria, this article studied the optimal hedging strategies of stock index futures in the case of asymmetric information, and discussed the influence of insider information on the hedging effect. Through simulation analysis, it can be shown that hedging people with insider information can save hedging costs to a certain extent, which also ex-plains the reason why investors try to obtain corporate information in actual investment activities.


Introduction
Information asymmetry means some investors can own insider information while others can't. If one can acquire insider information about some enterprises, she will win a penny; thus, striving to acquire insider information becomes many investors' investment focus; accordingly, hedging with insider information also receives attention from researchers. For example, Anna A. et al. [1] studied superhedging for contingent claims under insider information situation, in their research; Anna A. et al. considered two agents, one who only observes the stock prices and another with some additional information, and investigated when the pricing-hedging duality for the former extends to the latter. Eyraud-Loisel A. and Royer-Carenzi M. [2] studied the American option hedging by an insider by using the backward stochastic differential equations; they proved the existence and uniqueness of backward stochastic differential equations solutions, when terminal time was random, under an initially enlarged filtration. Klusik P. et al. [3] considered the problem of the quantile hedging from the point of view of a better informed agent acting on the market. Schweizer M. et al. [4] solved the problems of mean-variance hedging (MVH) and mean-variance portfolio selection (MVPS) under restricted information. Biagini and Oksendal [5] studied insider's variance-minimizing hedging under diffusion settings. Subsequently, Biagini and Oksendal [6] also investigated the same problem with stochastic integral method. Lee K. and Song S. [7] studied locally risk-minimizing hedging problem under jump-diffusion process, in which the Poisson's intensity is driven by insider information. Yan H.F. et al. [8] assumed that there is additional market information in the financial market and considered the mixed hedging problem. Yang et al. [9] [10] respectively studied the shortfall-risk minimizing hedging and quadratic hedging for contingent claims under insider information situation and respectively acquired hedging strategy expressions.
However, throughout the above-mentioned literatures, though research on insider's hedging has been deeply studied and many research findings have been acquired, most are restricted to theoretical research and the optimal hedging strategy expressions comprise components hard to be measured in practical hedging.
In this paper, assuming the underlying asset price evolving as a jump-diffusion process, we aim to study the hedging strategies under asymmetric information situations and try to give an explicit solution to the optimal hedging strategy of insiders; then, we also discuss the effect of insider information to the hedging effect.
The remainder of this paper is organized as follows. In the next section, we simply introduce the market model and some definition and signs. In Section 3, we will discuss the hedging questions with different risk-measuring criteria for insiders and general hedgers and empirical analysis based on simulation method is proposed in Section 4, while discussion and conclusion in Section 5.

Some Preliminaries
Hypothesis: In this paper, assuming the underlying asset price evolving as a jump-diffusion process, we study hedging strategies under asymmetric information situations and discuss the effect of insider information to hedging effect.
In order to measure the asymmetry of the financial market information, in this paper, we assume that T F -measurable random variable t l denotes insider information, such as risky asset's price or its fluctuation range, or an impulse to risky asset's price. Thus, as for insiders, their acquired information set is enlarged information flow There, we assume the insider information t l obey to some certain distribution,

( )
N t is a Poisson process with Poisson's intensity λ . About Equation (1), it can be solved as is a standard normal variable. In fact, Equation (1) also denotes the risky asset's price process of general investors if let 0 t l ≡ .
Thus, we can call a 2-dimension stochastic process an investment strategy, which follows as where ( ) (where, r is riskless interest rate). Furthermore, we call it a self-financing strategy when ( ) As a matter of convenience, we equidistantly divide the hedging horizon [ ]

Quadratic Hedging Strategies of Insider and General Hedger
Suppose an investor has initially written a share of Stock Index Future with striking price K and T horizon, in order to minimize the terminal squared error between value of hedging portfolio and contingent claim, she hedges by , thus, the hedging model is as following . Assume all assets' prices are discounted. According to dynamic programming and with backward recursion method, we can solve (4) as following At time and solving (5), then let Substituting (6) into (7), there is Just as acquiring ( ) In fact, if let 0 l ≡ , then, (8) denote the quadratic hedging strategy of general hedger.

Risk Minimizing Hedging Strategies of Insider and General Hedger
The quadratic hedging are restricted with self-financing, in this subsection, we relax self-financing constraint and study risk-minimizing hedging problem, which try to minimize the terminal squared error with following optimizing model (9) Open Journal of Applied Sciences In fact, because where, there is ( ) Next, making use of the martingale character of hedging cost process with risk minimizing optimizing object, at 2, ,1,0 n N = −  , we can recursively solve (12) and there are Open Journal of Applied Sciences In fact, if let 0 l ≡ , (14) denotes the quadratic hedging strategy of general hedger.

Numerical Example
In this section, taking example for hedging for Stock Index Future with 3-month maturity, we compare and analyze the hedging effect of insiders and general hedgers. Let underlying asset's initial price 0 100 S = , striking price of Stock In- Furthermore, we respectively assume the insider information As to two different hedgers, we denote Strategy Outsider: general hedger who hedges respectively according to (8) sand (14) with 0 l ≡ . Strategy Insider: insider who hedges respectively according to (8) and (14). Then, we can respectively calculate the total hedging costs and total hedging error for insider and general hedger just following (15) and (16) There, 0.0005 f = , denoting the transaction commission rate of riskless asset.  Table 1 and Table 2 respectively show the total hedging cost and total hedging error of insider and general hedger with quadratic hedging criteria, when Table 3 shows the total hedging cost of insider and general hedger with risk-minimizing hedging criteria, because of the constraint of ( )

Conclusions
In this paper, we studied the hedging problems with two different risk-measuring criteria for insiders and general hedgers. First, based on the analysis of insider market information, we construct a jump-diffusion model to depict risky asset price process, then, in virtue of dynamic programming, and backward recursive method, we solve hedging problems for insiders and analyze the influence of inside information on hedging effect. Table 1 and Table 3 indicate that insiders can save hedging cost to a certain extent for owning some insider information than those investors who are outsiders. From Table 1 and Table 3, we can detect that, for quadratic hedging and risk-minimizing hedging, insiders can save hedging cost about 0.21% -1.17% than outsiders. At the same time, we can also see from Table 2 that the total hedging error of insider with risk-minimizing hedging criteria is small than that of general hedger. By this token, whether owning additional insider information is a crucial factor for investors to succeed or not, which is one reason why many people strive to defraud enterprise information.
So, to supervisory board, to reinforce insider information management is a necessary and important method to ensure fair transaction and domestic financial market's healthy development. For example, the legislative branch can restrict the leakage of inside information and prevent insider trading through legislation. Law enforcement or regulatory authorities may impose corresponding penalties on insider traders. For traders, they must consciously abide by the rules of the transaction and jointly maintain the fairness of the transaction.
All in all, this paper studied the hedging problem under the condition of asymmetric information, and gave the analytical formula of the optimal hedging strategy under the square hedging criterion and the risk minimization hedging criterion for inside information hedgers and general hedgers. However, because the minimum loss hedging criterion is non-differentiable, the limitation of this research is that this article has not conducted research on the optimal hedging strategy under the minimum loss criterion, which is also one of our future research directions.