In this paper, with the assumptions of principle of noarbitrage and a complete market, based on the model of optimal consumption and portfolio strategy in a continuous time of Merton [
The complete market model assumes [
1) Investors exist in a time period [0,T], T < ∞Investors will consume and invest in any (consecutive) time points, and insurance at time 0.
2) Uncertainty of Market is described by a complete probability space {Ω,Φ,P}. Information structure is generated by the Brownian motion on the definition of the above probability space, is
where
.
3) Suppose there are two financial assets in the marketone is a risk-free asset, 
, the other is risk asset, its price S(t) follow the Brownian motion.
.
where the risk-free interest rate r(t), expected rate of return of risk assets
; and the transient volatility is adapted to
and uniformly bounded in 
; 
is a standard Wiener process, the process of discounted price is
.
4) The decision-making variables which investors control are two, one is the consumption for each moment, that is, the consumption process C(t), the other is the trading process or portfolio process 
(present the number of assets); the number of risk-free assets is
, the number of risk assets is
, and 
is a predictable two-dimensional random variable, where
.
5) The initial wealth of investor is
. 
is the trading strategy with initial wealth, where
, following by the process of wealth of investor
.
where
, the process of wealth generated by the trading strategy and consumption is:
The above formula demonstrates that the total wealth of the investor in the moment of t equals to the initial wealth plus payoff of risk assets and risk-free assets, and minus the total accumulating consumption. The proportions of investment in risk-free assets and risk assets are
, when the investment strategy is definited by using the relative proportion of the assets investing in the total personal wealth.
where 
is the portfolio process
, and
, i = 0, 1. is a self-financing strategyaccording (1), the simple process of wealth is achieved:
Given a positive initial wealth x, a self-financing strategy 
or 
produce a non-negative wealth process
, which is a admissible strategy. The entire set of feasible strategies 
based on x is denoted
.
6) The market price of risk in complete market 
has a unique solution which is bounded. There are Radon-Nikodym derivative
and the only measure of risk-neutral (martingale probability measure Q) according to Camron-Martin-Girsanov theorem [
, and the risk asset under the martingale probability measure Q is
after discounting, it becomes
then the price of risk asset 
turns into a martingale Q.
7) The state price density process
is positive, continuous and progressively predictable, and is the unique solution of the following stochastic differential equation.
where 
is the discount factor.
8) There are all kinds of options matching with risk asset in the market.
Theorem 1. Given the initial wealth + 
and a admissible self-financing strategy
, the corresponding process of wealth meets
M is a non-negative random variable and adapted to
, C(t) is a consumption process, 
,where
.
So there is a portfolio process
, where
, and the corresponding wealth process satisfies.
The proof of the theorem which can be found in literature [
The optimal model of portfolio insurance is expressed as to maximization of the total expected utility about consumption and the final wealth, that is,
Thus, an optimal strategy is solved.
where
, which demonstrates a weak and integrable condition.
The variables of dynamic trading strategies 
include the number of risk-free assets, risk assets and the put option whose underlying is the risk assets with investing in portfolio insurance.
In order to simplify the process of analysis, we assume that the consumption 
during the period of investment is zero, and the variables of trading strategy controlled is
. Because of dynamic replication strategy, the premium of options is zero. Based on these descriptions, portfolio insurance model is expressed as
According to (1), the process of wealth generated by the portfolio insurance trading strategy is
Investors hold one risk asset and one put option P (the exercise price is K which is equal to the initial value of underlying assets), since they invest in portfolio insurance, then
, (11) becomes
That is the total wealth of investors at the time t equaling to the initial wealth plus the payoff of the investment in risk-free assets and risk assets as well as the put option. If 
is self-financing, the change of wealth are from the change of prices and the risk-free rate, the process of wealth is expressed as
Under the condition of no-arbitrage, we can replicate a put option by investing in the long position of risk assets
and a short position of risk-free assets, then (13)
becomes
ordering
then, (14) becomes
is the set of investment strategy which includes ports of investing in risk-free assets and risk assets, according dynamic replication of options under the condition of no-arbitrary equilibrium. Comparing of (16) and (3), the model of portfolio insurance is similar to the general model of investment and consumption, because they have the same objective function which is maximizing the utility function of the wealth of investors. However the investment strategies 
which include investment in risk-free assets, risk assets as well as the put option, is changing into 
which only include investment in risk-free assets and risk assets. So the model of portfolio insurance is
where the process of wealth is (16). Similarly, according to Theorem 1, there is
, then
.
The most important difference between our model and the models of literatures [1,4,5] is that the strategy include trading of portfolio insurance by using replicating technology,, rather than adding a restriction to the final level of wealth.
of the investment-consumption model into 
of the portfolio insurance model. Then
is the set of all final wealth which is generated by a feasible trading strategy with the initial wealth
, and
. To optimize the model of portfolio insurance (17) and get the optimal final wealth
, we can maximize all the random variables of
, which is achieved with the martingale probability measure Q :
is the optimal solution to the equation (18), 
with 
has to be achieved. To find a trading strategy with
, we construct the Lagrange function:
in equation (19).
which makes the optimal level of final wealth is
, to the optimal final wealth
.
and
, which is the optimal solution of equation (18).
demonstrates the initial wealth, the investment strategy and consumption. To simplify the model, the consumption process is always 0.
is the investment strategy set investing in risk-free assets and risk assets using dynamic replicating options method under no-arbitrage equilibrium analysis. As the wealth process of the self-financing portfolio process 
is expressed as the formula (4), using the state price function, there is
.
is always positive, so
investing in risk-free assets and risk assets is
is the solution of
and satisfies the formula (27), the process of consumption 
is always 0,
.
with
, we can obtain the trading strategy 
of the portfolio insurance problem (10) which includes investments in risk-free assets, risk assets as well as options underlying risk assets.
in the optimal investment strategy of the portfolio insurance model includes two parts. One is the investment in risk-free assets without insurance (the first part of the right of formula (34)), the other is the investment in risk-free assets which replicates options with insurance (the latter part of the right of formula (34)). The sum of both is less than the share of investment in risk-free assets of the optimal investment and consumption model. The share of investment in risk assets is more than that of the optimal investment and consumption model by comparing the denominator of formula (33) whit that of formula (26), which the denominator of 
is smaller than of
. Because 
is always less than 1, which is the same to the result of replicating one option by investing in the long of risk assets 
and the short of risk-free assets in the analysis of no-arbitrage equilibrium. To implement the portfolio insurance strategy, That is, in order to the investor must invest more 
in the risk assets for risk assets.
, the expected return rate of risk assets 
and the instantaneous volatility rate 
are constant, the trading strategy 
of portfolio insurance model (10) after putting the formula (33) and (34) into the formula is
. And it is independent of the level of wealth, which is decided by the risk premium (the ratio of the difference between the average return rate of risk assets and the return rate of risk-free assets to volatility rate) and the amount 
of investing in risk assets to replicate portfolio insurance strategy.
risk-free in the strategy of dynamic investment portfolio insurance. That is investors must have more risk assets to insure.
is independent of the level of wealth. It indicates the demand of investors for insurance is independent of the wealth, but dependent on the market risk. In the other words, the higher of the market risk, the greater the demand for portfolio insurance. The conclusions of this paper reflect the function of avoiding the risk of the portfolio insurance strategy.