Optimization of Dynamic Portfolio Insurance Model *

This paper establishes a dynamic portfolio insurance model under the condition of continuous time, based on Meton’s optimal investment-consumption model, which combined the method of replicating dynamic synthetic put option using risk-free and risk assets. And it transfers the problem of investor’s individual inter-temporal dynamic portfolio insurance decision into a problem of static utility maximization under condition of continuous time, and give the optimal capital combination strategies corresponding to the optimal wealth level of the portfolio insurers, and compares the difference of strategies between this model and Merton model. The conclusions show that investors’ optimal strategies of portfolio insurance are not dependent on their wealth, but market risk. That is to say, the higher the risk is, the more the demand of portfolio insurance is.


Introduction
In the past two decades, with the development of the globalization of international finance，there is the wave of financial innovation that financial institutions aid to avoid financial risks and improve their competitiveness.The early 80s of last century, a new hedging tool based financial derivatives-the portfolio insurance strategyis very popular in the financial markets.Investors used this new type of hedging tools largely, and there were excessive trades, once a crisis, it may affect the entire financial system stability because of a series of events of default.There were two cases, one is the chain reaction of the collapse of Barings Bank in 1995, the other is the outbreak of the sub-prime crisis in 2007 that led to the global proliferation and amplification of financial risk, whose root cause is the credit risk of the excessive use of financial derivatives.
Earliestly, Rubinstein, etc. [1] put forward the concept of portfolio insurance, Grossman, etc. [2] gave a specific definition of that is a technology of hedging and shifting the risk using the financial derivative instruments such as options, futures or simulated options.The portfolio insurance developed very rapidly before the crash of stock market in 1987.At this stage, the appraisal of portfolio insurance was more affirmative, and the research focused on how to avoid the risk of stock market and how to util-ize hedge.Literatures [1,3,4] proposed the model of portfolio insurance which is only deformation of optimal control model of Merton [5], and the ultimate goal of which is to control the end of the wealth of not less than a certain pre-set values.After the crashing of US stock market in October 1987, the attentions of scholars were affected by the problems that whether the implementation of portfolio insurance strategy would increase market volatility and how to use the portfolio insurance strategy effectively.The results of literatures [6][7][8][9][10] showed that the implementation of portfolio insurance using options do have influence on the market liquidity increase transaction costs, and then increase market volatility; the studies of literatures [11,12] showed that the increasing market volatility is not included by using of options to avoid risks, but by the uncertainty of the model itself; literature [13] researched the relations between portfolio insurance, benefits and risk of investors in high-risk market conditions; literatures [14,15] compared the different portfolio insurance strategies with different trading; literature [16,17] analyzed the problem of the minimum cost of portfolio insurance strategies, as well as the application of lattice theory in portfolio insurance; literature [18] analyzed portfolio insurance strategies under the transferring of the market volatility and structural changes; literature [19] studied the optimization of the CPPI strategy; literatures [20,21] proved the portfolio insurance is a stochastic dominance strategy with limitation using random process theory; literature [22] showed that portfolio insurance strategies contribute to the allocation of financial risks; literature [23] studied the CPPI strategy in the conditions of discrete-time.
The financial derivatives in China and relatively research are few lacking.Literature [24] showed the portfolio insurance strategy can be created by dynamical replicating futures options; literature [25] analyzed the portfolio insurance strategy by a empirical study of Monte Carlo, on; In literature [26] studied the problem of portfolio insurance using VaR-based futures markets; literature [27] analyzed not only the relations between the price volatility of risk asset and the value of portfolio insurance, but also between revenue and its cost, etc; literature [28] analyzed the dynamic insurance with the constraints of EaR.
Because of the long history of the development of foreign financial market, and more varieties of financial derivatives, the study on theory or empirical analysis of portfolio insurance focused on how to use portfolio insurance strategies to avoid risk and how to affect on the market volatility [1,29,30].The financial market in China is just starting, derivative instruments is lacking, and options market does not exist currently, so the study also is limited to how to use the dynamic portfolio insurance strategies.
In this paper, we try to use stock and risk-free assets to replicate options to achieve the portfolio insurance, and establish a new dynamic model different from the existing model to analyze the implementation of the dynamic process of the strategy and the investors' actions using dynamic replication strategy.Firstly, this paper established a dynamic portfolio insurance model in the conditions of continuous-time model by using the technology of dynamic replication options, which is not like the literature [1,3,4], which only added a lower limit in the final level of wealth; Secondly, the investor's personal inter-phase dynamic portfolio insurance decision-making problem is changed into a static utility maximization problem; Finally, this paper achieves the best optimal portfolio strategy in the level of participants' optimal asset, and compares similarities and differences between the optimal investment portfolio insurance model and the optimal investment strategy of the optimal investment and consumption model of Merton [5].

Model of Portfolio Insurance
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 [5], a model of dynamic portfolio insurance is created using dynamic replication of put option.

Complete Market Hypothesis
The complete market model assumes [29] under the con-dition of continuous-time is as follows.
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 3) Suppose there are two financial assets in the market, one is a risk-free asset,   dW t 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 5) The initial wealth of investor is is the trading strategy with initial wealth, where 0 x  owing by the process of wealth of investor where The above formula demonstrates that the total wealth of by the t and consumption is: 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 , w t w t , when the investment strategy is definited relative proportion of the assets investing in by using the the total personal wealth.
where is the portfolio process he simple process of wealth is achieved: Given a positive initial wealth x, a self-financing strategy , which is a admissible strategy.The ble strategies and the only measure of risk-neutral (martingale probability measure Q) according to Camron-Martin-Girsanov theorem [30], where   , and the risk dP t obability mea asset under the martingale pr sure Q is after discounting, it becomes then the price of risk asset turns into a martingale Q. S  p 7) The state price density rocess is positive, continuous and progressively predi  d ctable, and is the unique solution of the following stochastic differential equation.

 
where s is the discount factor.
ere are all kinds of o k the market.

8) Th ptions matching with ris asset in
Theorem 1.Given the initial wealth + 0 x  and a admissible self-financing strategy M is a non-negative random variable and apted to ad , C(t) is a consumption process, ,where , where So there is a portfolio process , and the c wealth The proof of the t ilar to the process of optimal investm Model pressed as about con-orresponding cess satisfies.
heorem which can be found in literature [31] is sim ent and consumption, and omitted.

Portfolio Insurance Market
The optimal model of portfolio insurance is ex to maximization of the total expected utility sumption and the final wealth, that is, Thus, an optimal strategy is solved.
, which demonstrates a weak variables of dyna The mic trading strategies   t  include the number of risk-free assets, risk a pu ssets and the t 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   C t during the period of investment is zero, and the variables of trading strategy controlled is   t  .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 wea the portfolio insurance trading strategy is , lth generated by Copyright © 2012 SciRes.JMF 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 portfo ance, then , (11) becomes lio insur- 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 p If is self-financing, the change of wealth are from of prices and the risk-free rate, the pr ut option.

   
,0 t  the change ocess of wealth is expressed as Und h condition of no-arbitrage, we can replicate a put option by investing in the long position of risk assets er t e P S     and a short position of risk-free assets, then (13) becomes then, (14) becomes is the set of investment strategy of investing in risk-free assets an ic replication of options under the ary equilibrium.Comparing of ( 16 ,0 v t 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) ly, according to Theorem 1, there is st important our model and the models of literatures [1,4,5] is t clude trading of portfolio insurance by using replicating

3.
cision-making problem of the investor's anged optimal wealth level of the investor; the second step is that to of the portdifference between The mo hat the strategy intechnology,, rather than adding a restriction to the final level of wealth.

Optimization of Portfolio Insurance Model
In the continuous-time conditions, the optimization of portfolio insurance model includes two steps: the first step is that the de personal inter-phase dynamic portfolio insurance is ch into a static utility maximization problem, then the the corresponding optimal strategy is obtained with martingale representation theorem.

Deformation of Model
Portfolio insurance model ( 17) is similar to the investment-consumption model in micro-finance, and the investment strategy is changed from , we definite As a r the portfolio insurance model (17) becomes the static optimization of model (18).esult,

Optimization of Model
If M is the optimal solution to the Equation ( 18), has to be achieved.To find a trading strategy with   T X M  , we construct the Lagrange function: We calculate derivatives of , M  in Equation (19).
Because the utility function is monotonic strictly, there is an inverse function To substitute Equation ( 22) into ( 21). ( Ordering According to Equation ( 22), the optimal solution is Because of In the optimization of Equation ( 17), if There is a self-financing portfolio process   v t and , which is the optimal olution of v    Theorem 1 is similar, the proof is omitted.Superscript s Equation (18).
Theorem 2 and , ,0 x v demonstrates e initial wealth, the investment strategy and consumption.To simplify th ysis of the Optimal Portfolio Insurance of optim s th e model, the consumption process is always 0.

Anal Strategy
To find the solution al portfolio insurance strat-egy, from (2) the value of risk-free assets and risk assets which the investors invest in portfolio insurance i where is the investment strategy set investing in risk-free assets and risk assets using dynamic replicating options method under no-arbitrage equilibrium analysis.As t process of the self-financing portfolio pro he wealth cess   v t the state price function, the is expressed as the formula re is (4), using The integr indicated the consumption process, that is the consumption is 0.t The opti al 0 mal portfolio process is As the ea x, the formula ( 28) is investor participates in insurance, the smallest rning of investor in the end is After dynamic replicating option, we can o investment strategy set of the model (18).And the inve rategy set btain the stment st -free assets and risk assets is ortfolio l of portfolio insurance using the way of dynamic replication options, the investor's proportion of investing the risk assets is simila result of the optimal investment consumption model (where the consumption is 0).

w S t B t
Formula (30) is the same to the result of literatures [5,31] and satisfies the formula (27), the process of consump- , we can obtain the trading rtfolio insurance problem (10) nts in ris assets, risk assets as well as options underlying risk sets.

s  
 is the process of wealth accompanied by the optimal trading ategy Comparing ( 33), ( 34) with (31), there are similarities and differences between the optimal model and the optimum investment strategy of the optimum investment consumption model.
Firstly, the former set of optimal strategies is divided into three parts: investment in risk-free assets, ri and options; the latter is divided into the two parts: inve e is the investa (34)).The portfolio insurance sk assets stments in risk-free assets and risk assets.Secondly, the share of investment in risk-free assets   0 t  in the optimal investment strategy of the portfolio insurance model includes two parts.On me su risk-fr nt 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 formul m of both is less than the share of investment in ee 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   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.
(36) 0 t  Therefore, to achieve the positive initia investor of portfolio insurance investing in risk assets and options are both l wealth x, the  .And it is independent of the level of wealth, which is decided b risk premium (the ratio of the difference between the av y the erage 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.

Conclusions
The portfolio insurance model of this paper is similar to the investment consumption model which has the same function, and is the maximization of the investor' utility function.The only difference is that the other models have a restriction of the final wealth.This paper shows ce strategy by replicating put option and risk assets in the entire investlong position of risk assets and th the portfolio insuran with risk-free assets ment process, so we may clearly analyze the dynamic behavior of investors.
Through the analysis of the optimization of dynamic portfolio insurance model, we find that the share of investment in risk-free assets of the optimal investment strategy is less than the share in the general model.However, the share of investment in risk-free asset is more than the share in the general model, because the option is copied by the e short position of  risk-free in the strategy of dynamic investment portfolio insurance.That is investors must have more risk assets to insure.
The proportion investing in risk assets and options of portfolio insurance which is determined by the risk premium and  is independent of the level of wealth.It indicates the demand of investors for insurance is independent of the wealth, ut dependent on the market risk.In the other words, the higher of the market risk, the greater the demand for portfolio insu The implementation of portfolio insurance strategy is a complicated systems engineering.It is used to avoid and manage the risk of market, and faces the potential risks because of the special character of the strategy itself.Based on dynamic portfolio insurance model and its optimal strategy, the study of this paper is a basis for the further research of the portfolio insurance in China and give some conclusions that provides a reference for the th eoretical research and practical investment operations of financial market.Meanwhile, it also is a theoretical foundation for China's development of financial derivatives market and provides some technical supports for the investors to avoid market risk using the portfolio insurance strategy.
sets, according dynam condition of no-arbitr similar to the 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 risk-free assets, risk assets as well as the put option, is changing into     the set of all final wealth which is generated ble trading strategy with the initial wealth by a feasi we can maximize all the random variables of   M x , which is achieved with the martingale probability measure Q : b rance.The conclusio risk of the portfolio insurance strategy.ns of this paper reflect the function of avoiding the Copyright © 2012 SciRes.JMF