The Optimal Hedging Ratio for Contingent Claims Based on Different Risk Aversions

Based on utility theory, this paper firstly combined different utility functions with risk aversion coefficient and constructed different kinds of optimizing problems for hedgers to hedge for stochastic-payment-typed contingent claim, and then, by the aid of dynamic programming theory, effective multi-stage hedging strategy is proposed for different risk-averse hedgers. Lastly, the research results that the optimal hedging ratios for three kinds of utility functions are equivalent and do not relate to the risk aversion coefficient.

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As far as I know that most of those presented documents about hedging did not take the risk preference of investors into account, however, different investors have different risk preference and different ability of foreseeing financial risk, thus, for different investors, their hedging strategies must be not compatible with each other. In this paper, first off, we combine different utility functions with risk aversion coefficient and construct different kinds of optimizing problems for hedgers to hedge for stochastic-payment-typed contingent claim, and then, by the aid of dynamic programming theory, effective multi-stage hedging strategy is proposed for different risk-averse hedgers 2. Models

Utility Function
First, confirm that you have the correct template for your paper size. This template has been tailored for output on the custom paper size (21 cm * 28.5 cm). Risk aversion or risk preference is scale to measure investors' attitude to financial risk, Arrow (1971) [9] pointed out in his reference about financial risk that there were absolute risk aversion coefficient and relative risk aversion coefficient. Assuming that WN denotes the value of the terminal wealth while U(WN) denotes a utility function which is continuous and is two-order differentiable, we 2) Power utility function: which is also called as relative risk-hedging utility function. When W N is a random variable, the expected utility value can be denoted as 3) negative exponential utility function: which is also called as absolute risk-hedging utility function and the expected utility value can be denoted as

The Hedging Model
Assuming one people, such as an executive chief of a corporation, acquired a copy of contingent claim that will be executed at the terminal moment T according to his or her work rate, which is called as a kind of Stochastic-Payment-Typed Contingent Claim, in order to acquire the maximal profit, the executive chief, as a hedger, shall use the underlying asset related to the contingent claim to hedge the potential risk at discrete moments ∆ and r denotes the risk-free rate and.
Under the constraint of self-financing as (1), we can construct the hedging model just as following expressions (2).

The Basic Theory
According to the Bellman principal [12], the expressions (2) can be rewritten Open Journal of Business and Management as (3) as following, and the optimal hedging ratios at each moment t n t = ∆ may be acquired by using the backward recursion method.

The Optimal Hedging Ratio
Proposition 1: In non-arbitrage market, the optimal hedging ratio of hedging Proof: according to the dynamic programming principal, we can acquire the expressions (4) as following steps.
Step 1, At the moment ( ) Only need to make derivative calculation on (6), and let the differential coefficient equal to zero, we can acquire the optimal hedging ratio * − ∆ as following expressions (7).
When in non-arbitrage market, there ( ) , and because the basic difference between the spot(contingent claim) and the future equals to zero, i.e., , the (7) may be expressed as (8).
Similarly, if put (9) Only need to make derivative calculation on 2 N ϑ − in (10), and let the differential coefficient equal to zero, we can acquire the optimal hedging ratio * − ∆ as following expressions (11).
Step 2, assume at any moment Step 3, we only need to prove that there is * In fact, according to (1), we can get the recursion expressions as following (12).
( ) Now, only need to put all ( ) (12), there is the optimizing problem (13), and in non-arbitrage market, we can solve (13) and acquire the optimal hedging ratio at moment t n t = ∆ as in (14).
Assuming the market is complete, i.e.,  In fact, according to (1), we can get the recursion expressions as following There is *  The prove is similar to proposition 1.

Conclusion
In this paper, based on utility theory, we researched the hedging problem for stochastic-payment-typed contingent claim. Firstly, we combined different utility functions with risk aversion coefficient and constructed different kinds of optimizing problems for hedgers to hedge for stochastic-payment-typed contingent claim, and then, by the aid of dynamic programming theory, effective multi-stage hedging strategy is proposed for different risk-averse hedgers. Lastly, the research results that the optimal hedging ratios for different kinds of utility functions are equivalent and do not relate to the risk aversion coefficient.

Fund
This research is supported by the Social Science Planning Fund Program of Hunan Province of China (Project Number: 17YBA354).

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.