Unfortunately, the previous study ignored a real problem in operation: the continuous adjustment of assets in theory leads to some gap risks. Under continuous time frame, CPPI portfolio value at any moment is shown in formula (1) , in which [V.sub.t] is portfolio value at t; [F.sub.t] is bottom-line value at t; [S.sub.t] is risky assets value at t; m is multiple in CPPI strategy; r is risk-free interest rate; [sigma] is volatility of risk assets value; 0 [less than or equal to] t [less than or equal to] T; [V.sub.0], [F.sub.0], and [S.sub.0] denote initial values (t = 0) of variables.
Therefore, 1/m provides the largest drop of risky assets in an adjustment period ([t.sub.k],[t.sub.k+1]); we take the probability of risky assets value loss more than 1/m in adjustment period as a measurement of gap risks; the probability of risky assets value loss more than 1/m should be very small when setting m in the management of this risk.
We are interested in extreme risk of gap risks in CPPI strategy, that is, high score sites of loss distribution.
Figure 1 shows the situation in which gap risk happens between two adjustment points [t.sub.k] and [t.sub.k+1] possibly because risky assets value falls sharply before CPPI investors rebalance assets.
defined discrete CPPI strategy and its gap risk which is more congruent to the market environment .
Although the study above considered gap risk in CPPI strategy, it mainly concentrated on extracting the administrative cost and managing the gap risk by option hedging without any attention to the setting of the multiple.
Formula (3) demonstrates the relationship between multiple and the change of risky assets value under the requirement of gap risk management; [alpha] is given based on the requirement of CPPI investors.
According to formula (3), we control gap risk less than a, namely, guarantee portfolio insurance with confidence level 1 - [alpha], equivalent to
It is suitable for the description of extreme risk of gap risk in CPPI strategy.
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