Non-linear Expectation Theory And Model Uncertainty In Financial Market

The dissertation mainly studies the non-linear expectation theory and model uncertainty in financial market.There are four chapters,the former two of which are basically theoretical researches.The first chapter delves into Product Space under Non-linear Expectation,the Regularity of Product Space under Nonlinear Expectation and the Product Space of Processes with Independent Increments under the Nonlinear Expectation,which is perfect and supplement to the Non-linear Expectation Theory.The second chapter discusses Maximum principle for forward-backward SDEs with a general cost functional and Pricing of contingent claims in an incomplete market with finite state stochastic processes in discrete time.The latter two chapters are basically applied researches,which delves into the Model Uncertainty and the application of the Non-linear Expectation Theory in Financial Market.The third chapter introduces Asset Pricing Theory under the Non-linear Expectation,improves the current international most common SPAN margin system and SPAN calculation principle and obtains SPAN Margin System under Mean-Variance Uncertainty by using the Non-linear Expectation Theory,which can measure the risk more effectively and accurately.And the empirical test is carried out with S&P500 index option data.The fourth chapter discusses the Model Uncertainty in Financial Market,explains the uncertainty of the dis-tribution of financial data and parameter uncertainty in Financial Market.It also studies the Mean Uncertainty and Variance Uncertainty,and build an investment strategy by using the Mean Uncertainty.The main contents of each chapter are as follows.(?)The product space theory under nonlinear expecta-tionThis chapter studies the product space theory under nonlinear expectation.The main result of thischapter comes from:Gao Q,Hu M,Ji X,Liu G.Product space for two processes with indepen-dent increments under nonlinear expectations.Electronic Communications in Probability 22(2017).This chapter has the following two parts:1.The regularity of product space under nonlinear expectationWe study whether the product space of the regular sublinear expectation space still satisfies the regularity.First,we study the regularity of two product spaces under sublinear expectation.By extending the structure of the classical finite product probability space to the sublinear expectation space,we can proof that the product space of the two regular sublinear expectation spaces issatisfies the regularity.To further extend this result to the finite dimension case,we can get conclusions as follows:Given a finite number of regular sublinear expectation spaces(?i,Hi,(?)_i),i =1,2,…n,Their product space((?))still satisfies the regularity.And then by reductio,the conclusions can be extended to the countable sublinear expectations of spaces case.Further study whether the complete product space under the sublinear ex-pectation still satisfies the regularity.In this case,by using the notion of weak compactness in completely separable distance space,we finally prove that the complete product space under sublinear expectation still maintain regularity.Given a finite regular sublinear expectation spaces(Qi,Hi,(?)_i),i=1,2,…,n.Denote((?))as the complete space of((?)).We can get that((?))si still a regular sublinear expectationspace,(?)and there is a series of weakly compact probabilities(?)(?)(?)(?)(?)(?)(?)Then we can proof the main conclusion as follows:Given a series of regular sublinear expectation spaces(?i,Hi,(?)_i),i?1,(?)is completely separable distance space,Hi= Cb.Lip(?i)Denote ?=(?)is complete space of(?,H,E),we can(?)proof(?,L'(?),E)is still a regular sublinear expectation spaces and satisfies that.Cb(?)(?)L'(?).2.The product space for processes with independent increments under nonlinear expectationsPeng[5,6]introduced the notions of distribution and independence under nonlinear expectation spaces.Under sublinear case,Peng[9]obtained the corre-sponding central limit theorem for a sequence of i.i.d.random vectors.The limit distribution is called G-normal distribution.Based on this distribution,Peng[7,8]gave the definition of G-Brownian motion,which is a kind of processes with stationary and independent increments,and then discussed the Ito stochastic analysis with respect to G-Brownian motion.It is well-known that the existence for a sequence of i.i.d.random vectors is important for central limit theorem.In the nonlinear case,Peng[10]introduced the product space technique to construct a sequence of i.i.d.random vectors.But this product space technique does not hold in the continuous time case.More precisely,let(Mt)t?0 and(Nt)t?0 be two d-dimensional processes with indepen-dent increments defined respectively on nonlinear expectation spaces(?i,Hi,(?)_i),and(?2,H2,E2),we want to construct a 2d-dimensional process(Mt,Nt)t?0 with independent increments defined on a nonlinear expectation spaces(?,H,E)such that(?)and(?)Usually,set ?=?1×?2,Mt(?)= Mt(?1),Nt(?)=Nt(?2)for each ?=(?1,?2)??,t?0.If we use Peng's product space technique,then we can only get a 2d-dimensional process(Mt,,Nt)t?0 such that(Mt)t?0 is independent from(Nt)t?0 or((Nt)t?0 is indepen-dent from(Mt)t?0 Different from linear expectation case,the independence is not mutual under nonlinear case(see[3]).So this(Mt,Nt)t?0 is not a process with independent increments.In this paper,we introduce a discretization method,which can overcome the problem of independence.More precisely,for each given Dn={i2-n:i?0},we can construct a nonlinear expectation En under which(Mt,Nt)t?Dn possesses independent increments.But En,n ? 1,are not consistent,i.e.,the values of the same random variable under En are not equal.Fortunately,we can prove that the limit of En exisits by using the notion of tightness,which was introduced by Peng in[11]to prove central limit theorem under sublinear case.Denote the limit of En by E,we show that(Mt,Nt)t?0 is the process with independent increments under E.First we consider the product space for two processes with independent increments under nonlinear expectations,the main theorem is an follows:Theorem 0.1.Let(Mt)t?0 and(Nt)t?0 be two d-dimensional processes with independent increments defined respectively on nonlinear(resp.sublinear)expec-tation spaces(?1,H1,E1)and(?2,H2,E2)satisfying the assumption(A).Then there exists a 2d-dimensional process(Mt,Nt)t?0with independent increments defined on a nonlinear(resp.sublinear)expectation space(?,H,E)such that(?)and(?).Furthermore,(Mt,Nt)t?0 is a pro-cess with stationary and independent increments if(Mt)t?0 and(Nt)t?0 are two processes with stationary and independent increments.In the following,we only prove the sublinear expectation case.The nonlinear expectation case can be proved by the same method.Moreover,the following lemma shows that we only need to prove the theorem for t ?[0,1].In order to construct E,we set,for each fixed n>1,where ?0 is obtained backwardly by Step 1 in the following sense:Lemma 0.1.Let(?,Hn,En)be defined as above.ThenThus(?,Hn,En)is the sublinear expactation space satisfies the theorem 0.1 in Dn={i2-n:0?i?2n},then we extend the conclusion to the continuous case.Obviously,Hn(?)Hn+1 for each n ? 1.We set(?).It is easily seen(?)tha L is a subspace of H such that if Y1,...,Ym?L,then ?(Y1,...,Ym)?L for each ??Cb.Lip(Rm)In the following,we want to define a sublinear expectation E:L?R.Un-fortunately,En+1[·]?En[·]on Hn,because the order of independence under sublinear expectation space is unchangeable.But the following lemma will allow us to construct E.Lemma 0.2.For each fixed n ? 1,let Fkn,k ? n,be the distribution of(X?n,X2?n-X?n,...,X2n?n-X(2n-1)?n)under Ek.Then {Fkn:k ? n} is tight.Now we will use this lemma to construct a sublinear expectation E:L?R.Lemma 0.3.Set D ={i2-n:n? 1,0 ? i ? 2n?.Then there exists a sublinear expectation E:L?R satisfying the following properties:(1)For each 0 ? t1<…<tn with ti?D,i?n,Xtn-Xtn-1 is independent from(Xt1,...,Xtn-1);(2)For each 0 ? t1<…<tn with(?)(?)and(?).Then we can finally proof the theorem 0.1 by using those lemma before we extend the conclusion to the countable case.Theorem 0.2.Let(Mti)t?0,i ? 1 be a series of countable d-dimensional processes with independent increments defined respectively on nonlinear(resp.sublinear)expectation spaces(Qi,Hi,(?)_i),i ? 1.Then there exists a process(Mt1,Mt2,...,Mti,...)t?0 with independent increments defined on a nonlinear(resp.sublinear)expectation space(Q,H,E)such that(?)Furthermore,(Mt1,Mt2,...,Mti,...)t?0 is a process with stationary and inde-pendent increments if(Mti)t?0,i?1 are a series of countable d-dimensional processes with stationary and independent increments.Then we extend the conclusion to the uncountable case.Give the defination of up-independence increment process as follows:Let(?,H,E)be a nonlinear expectation space.A m-dimensional random vector YY is said to be up-independence increment from another n-dimensional random vector X under E[·]if,for each test function ??Cb.Lip(Rm+d),we haveE[?(X,Y)]?E[E[?(x,Y)]x=X),A d-dimensional process(Xt)t?0 with X0=0 on a nonlinear expectation space(?,H,E)is said to have up-independent increments if,for each 0 ?t1<…tn,Xtn-Xtn-1 is up-independent from(Xt1,...Xtn-1).Furthermore,If the process(Xt)t?0 also staisfies that for each t,s ? 0,(?)then(Xt)? is said to have stationary up-independent increments.Give the main result in uncountable case as follows:Theorem 0.3.Let(Mt?)t?0,?? I(I is an uncountable set)be a series of uncountable 1-dimensional processes with independent increments defined respec-tively on nonlinear(resp.sublinear)expectation spaces(??,H?,E?).Then there exists a process(Mt?,??I)t?0 with up-independent increments defined on a nonlinear(resp.sublinear)expectation space(?,H,E)such that(?)(?).Furthermore,(Mt?)?0,??I is a process with stationary and up-independent increments if(Mti)t?0,i?1 are a series of uncountable 1-dimensional processes with stationary and independent increments.(?)BSDE stochastic control problem and Pricing of con-tingent claims in an incomplete marketThis chapter mainly studies maximum principle for forward-backward SDEs with a general cost functional and pricing of contingent claims in an incomplete market with finite state stochastic processes in discrete time.The main result of this chapter comes from:1)Gao Q,Yang S.Maximum principle for forward-backward SDEs with a general cost functional.International journal of control(2016):1-7.2)Gao Q,Yang S.Pricing of contingent claims in an incomplete market with finite state stochastic processes in discrete time Completed Manuscript,1-10.This chapter has the following two parts:1.Maximum principle for forward-backward SDEsThe optimal control problems derived by backward stochastic differential equa-tiorns(BSDEs)or forward-backward stochastic differential equations(FBSDEs)were first introduced by Peng[53]and Peng[29],which have been generalized bymany researchers(Xu[57],Lim and Zhou[24],Shi and Wu[54],and so on).In[29],Peng first studied the stochastic optimal cont,rol problem whose state vari-ables are described by the system of forward and backward stochastic differential equations,with the following cost functionalIndeed,the functions h(·)and ?(·)of equation(0.5)may not only depend on the terminal and initial values but also rely on the system state variables among the global time t ?[0,T].Therefore,in this study,we investigate the stochastic maximum principle for forward backward SDEs with a general cost functionalwhich relies on the global terminal condition,where Notice that equation(0.5)is a special case of equation(0.6).To the best knowl-edge of the author,the maximum principle of the control problem with the cost functional(0.6)is still unconsidered.The difficulty of this problem mainly relies on the adjoint equation.Thanks to the Riesz representation theorem,the Frechet derivatives Dxh(x[o,T)and Dx?(y[o,T)can be described by finite measures ? and ?.After decomposing the measures ? and ? as continuous part and jumps part,we derive the corre-sponding maximum principle via constructing a series of adjoint equations which need to be solved step by step.In more details,we verify the optimal control and maximum principle stepwise with the forward and backward equations on global time.Since it is different from the classical optimal control problems,we show an example to describe our model.The brief procedure is shown as follows:Let U be a nonempty convex sudset of R.We set U = {u(·)? M 2(R)|u(t)?U,a.e.,a.s.}.An element of U is called an admissible control.Let u(·)be anoptimal control and let(x(·),y(·),z(·))be the corresponding trajectory.Let u(·)be such that u(·)+ u(·)?u.Since u is convex,then,we have 0???1,up=u(·)+?u(·)=(1-?)u(·)+?(u(·)+u(·))+?u.Let(?(·),?(·),?(·))be the so-lution of variational equations.We denote by(x?(·),y?(·),z?(·))the trajectory corresponding to up.Then we can show their convergence result.Then we give the definition of Frechet derivative in C([0,T]).In the Frame-work of Frechet derivative,for h(x[0,T])and-?(y[0,T]),by the Riesz representation theorem,there are unique finite Borel measures ? and ? on[0,T]such that(?)?[0,T]?C([0,T])Since ? and ? are finite measures on[0,T],there are at most countable points with positive measure.Denote as(?)and(?)with 0<…<l2<l1=T and 0 = s1<s2<...<T.Suppose that(?)and(?)each has one cluster point and denote as l0 and s0 with s0<l0.For deriving the maximum principle,we introduce the following series adjoint equations,where ?'(t)is the derivative of ?(t)and li+ is the right limit of li,define p(l1+)= 0,andwhere ?'(t)is the derivative of ?(t)and si-is the left limit of si,define q(s1-)= 0.We can find a triple(p(·),?(·),q(·))which can solve(2.20)and(2.21).Since u(·)is an optimal control,then,?-1[J(u(·)+?u(·))-J(u(·))]? 0.then thefollowing variational inequality holds,We define the Hamiltonian function H:R × R× R × R ×[0,T]? R as follows:H(x,y,2,u,p,k,q,t)= pb(x,u,t)+ k?{x,u,t)+ qg(x,y,z,u,t)+ f(x,y,z,u,t).Coinsequently,we can rewrite the series adjoint equations We now show the main resulrts.Theorem 0.4.Assume(?)-(?).Letu(·)be an optimal control and let(x(·),y(·),z(·))be the corresponding trajectory.Then we have(?)where(?)is the solution of the series adjoint equations.2.Pricing of contingent claims in an incomplete marketIn this study,we consider the pricing contingent claims or options from the price dynamic of certain securities with finite state processes in discrete time.When the market is complete,prices can be derived from the absence of arbitrage.Since the price is not possible to replicate the payoff of a given contingent claim by a controlled portfolio of the basic securities in incomplete market,we investigate the maximum and minimum prices by stochastic control methods.Any probability measure is called a P-martingale measure,if it is equivalent to P on F? and is such that the discounted price processes are martingales.We denote P the set of all P-martingale measures.Notice that,if the market is complete,P = {Q}.If the market is incomplete,there are several P-martingale measures.As in the complete market,for given contingent claim U,there exists some y ? 0 and some portfolio process ?d such that(?)(?)(?)and x is the arbitrage free price at t = 0.In the incomplete market,there are maybe several prices for U and U cannot be priced by arbitrage.Thus,it seems interesting to determine the bounds of the set of possible prices for U.That is,at t=0,the minimum price for U is in fP?PEP(Ud)and the maximum price for U is supP?PEP(Ud)By optimal control techniques,we shall study dynamically those minimum and maximum prices.The minimum of the possible prices for U at time t is given by(?)where Pa denotes the probability measure that admits the following kind of Gir-sanov transformation with respect to P:(?)(?)Then we turn to the study of the essential infimum of the possible prices for Ud.Let K(t)be the process satisfying K(t)=essinfv??EPv[Ud|F(t)].We can derived the properties as follows:There exit a portfolio process ?(t)and a right continuous increasing process g(t)with g(0)= 0 such that(?)(?)(?)(?)The SPAN margin system under the non-linear expectationThis chapter discusses the application of the Nonlinear Expectation Theory in margin calculation,the result comes from??,????.?????????????????,????????????????????(?????),1-46,2014.This chapter first introduces the margin system and the international main-stream margin calculation system and discusses Margin Management System SPAN.It also introduces the calculation principle of the SPAN margin:the core price detection risk module is based on situational simulation estimating changes in future prices and volatility,which divides the furture market into 16 possible scenarios,calculates possible losses in 16 cases separately and take the maximum as the maximum expected loss to develop the corresponding margin standards.Besides,SPAN margin also includes the cross-month spread risk,the delivery month risk value,the Interval between goods,and the short-term option minimum risk value,ect.The disadvantage of the SPAN margin is that it only calculate 16 possible scenarios,which cant cover a variety of possibilities for future markets.Furthermore,the theoretical basis is the Black-Scholes formula and its assuming volatilityis a constant,so the risk of volatility uncertainty can not be estimated.Having analyzed principles of other SPAN improvement sys-tems and compared SPAN16 with SPAN-44 and SPAN-93 by using SP500 stock option data,more possible scenarios are discover divided by SPAN margin sys-tem,which can measure the risk more accurately in a way,but in the meantime increase the amount of calculation and can't solve the risk caused by the real market volatility uncertainty.Then this chapter introduces three important distributions of the non-linear expectation theory:the Maximum distribution,G-normal distribution and G-distribution.It also introduces three important Corresponding stochastic pro-cesses:G-Brownian movement,quadratic variation G-Brownian motion and G-generalized Brownian motion,the incremental process of which follows the pre-vious three distributions.For instance,the G-Brownian motion is related to variance uncertainty(volatility uncertainty)directly.The G-normal distribution random variable can be denoted by(?),(?)describing the vari-ance uncertainty of X.In a one-dimensional situation,(?),(?).Then the variance(volatility)uncertainty in-terval is[?2,?2].The maximum distribution quadratic variation G-Brownian rrmotion is related to mean(return)uncertainty,and the maximum distribution random variablecan be denoted by(?).(?)describes the degree of mean uncertainty of Y.In a one-dimensional situation,(?),(?),and the mean uncertainty interval is[?,?].The two distributions above can make up a new distribution called G-distribution,which describes the mean-variance uncertainty uncertainty,that is,the return-volatility uncertainty or price uncertainty in the financial market.Then we can draw the geometrical G-Brownian motion as follows:Supposing the stock price satisfy the next stochastic differential equation under G-expectation:dXs = uXsd?s + ?XsdBs?Xt=x,where ? is the maximal distribution,and Bt is the G-Brownian motion,(?);(?).The payoff of the European option price is?(XT).Define the risk is u(t,Xt):=E[-?(XT)].u(t,Xt)is the solution(viscosity solution)of the following PDE:(?)(?)Equivalent,we consider the bounded boundary value problem,The equation can be discretized by a standard finite difference method,and we can prove the convergence of the Newton iteration for full implicit schemes.We improve the SPAN system with nonlinear expectation theory,give the SPAN method under volatility uncertainty:Supposing the stock price satisfy the next stochastic differential equation under G-expectation:dXs = ?Xsds + ?XsdBt?Xt=x Bt is the G-Brownian motion,(?)The payoff of the European option price is?(XT).Define the risk is u(t,Xt):?E[-?(XT)].u(t,Xt)is the solution(viscosity solution)of the following PDE:We can get 9-situation under SPAN method:the volatility is in[?t-????t + ??],the value of the margin is:We can give the SPAN method under mean uncertainty or mean-volatility un-certainty in the same way.The SPAN method under mean-volatility uncertainty as follow?where ? is the maximal distribution,and Bt is the G-Brownian motion.The payoff of the European option price is?(XT).Define the risk is u(t,Xt):=E[-?(XT)].u(t,Xt)is the solution(viscosity solution)of the following PDE:where There is only one situation with this method:[Xt-?x,Xt + ?x]×[?-??,?+ ??],where ?x = PSR,??= VSR?the value of the margin is:Based on SP500 index options empirical Analysis,performance of G-SPAN method is superior to the classical SPAN method,especially in this risk aversion,Consider the situation of the domestic financial market,the options and futures markets in our country are still in their infancy.Their complexity,incompleteness and uncertainty result in great market risk.G-SPAN method is particularly suitable for our current financial market.(?)Financial market under model uncertaintyThe model uncertainty in financial market mainly reflect in:the uncertainty of financial data distribution,the uncertainty of Financial Data Characterization Parameter,and the model uncertainty of financial data.First we need to verify the uncertainty of financial data distribution.The normal distribution is one of the most important distributions in financial markets and many financial studies are based on the assumption of normal distribution.In the financial data analysis,it is often assumed that financial data within a certain period of time are subject to the same distribution.For instance assuming that the return on assets is subject to normal distribution,now we choose to test the Shanghai and Shenzhen 300 index and the corresponding CSI 300 stock index futures data.By the empirical test,if the length of the window is one day,the number of days to obey the normality hypothesis is few.Stock is less than 20%and stock index futures is only less than 10%.If the length of the window is one week,the number of weeks to obey the normality hypothesis is less than 1%.Therefore there is a big problem of the normal distribution hypothesis in financial market.In fact,not only is the normal distribution hypothesis difficult to set up,it's also hard to find one or more different distributions to describe the economic and financial data distribution accurately.Different financial data show different data characteristics.Even the same financial data may comes from the common effect of different economic,financial and social principles.Usually,even the data from the same economic,financial principles,we still find it difficult to find one or more distributions to describe the specific characteristics of the data accurately.Therefore,the uncertainty of distribution does exist in finance,which can be seen in the ormalized daily data distribution histogram.The uncertainty also exists in mean(first order moment)and variance(volatility,second moment)as well as the rate of return and volatility.Analyzing the average and variance of daily returns of CSI 300 stock index and the Shanghai and Shenzhen 300 stock index futures,we can find that the uncertainty exists in its mean and variance.The changes of Stock index futures are more intense compared with the changes of the stock,which has greater uncertainty.In a period of time,mean and vari-ance vary within a range.When the amount of data is large enough,it can be said that the mean and variance continuously change in a range.Thus we can see uncertainty in the distribution of financial data and the uncertainty of charac-teristic parameters.In the same time period,The data from the same economic phenomenon doesn't come from the same distribution but different distributions or an indefinite distribution family.Its characteristic parameters,like the mean and variance,are not certain values.They continuously change in a range.We study the uncertainty of the mean,calculate the range of mean uncertainty and mean regression model of mean uncertainty in view of the important mean re-gression in the financial market.The true mean return isn't revolving around an average,but returning in a mean uncertainty interval.The mean uncertainty interval can be considered as reasonable price range.The price fluctuating in this interval is considered reasonable.When the price deviates from the upper bound or the lower bound,the price will have a trend to return to a reasonable price range.Let's set the asset price to X,its mean is ?,mean uncert.ainty interval is[?,?].Under the classical probability system,when X<? or X>?,,the price will return to ? However,there's only one parameter ?,we can't determine the specific regression point.While under the mean uncertainty,the price changes around the mean ?.Any volatility in the interval[?,?]will be considered as unbiased mean,which is reasonable.When X<? or X>?,the price will be considered unbiased and return to the mean.Therefore we build an investment strategy,choose contract of this month and the next of CSI 300 stock index fu-tures to carry out intertemporal arbitrage.The investment strategy:when spread is over ?,we open short positions;otherwise,we open long positions.Besides,when each loss exceeds the stop line,we close out in advance and at the end of the day,we close out forcibly.We analyze the data from Jan.1st,,2015 to Dec.31st,2016 and evaluate the strategy with five indicators:cumulative rate of return,a,nnualized rate of return,volatility,maximum retracement and sharp rate.We study the actual situation of financial market.and consider Financial market liquidity,limited policy,transaction fee,transaction delay,stopping loss and margin.In the relatively actual financial market parameter settings(the fee is 0.O01%,Quantity 10 per transaction,the stop loss line is 10%),the cumulative income of strategy is about 4 times,the maximum retracement is about 4%and the sharp rate is near 6,which is excellent.After analyzing the main developing stages of Shanghai and Shenzhen 300 financial markets in China and the fea-sibility,applicability and stability of the market situation analysis strategy for different stages,we find that the strategy has a good performance in most mar-ket stages.In a word,the mean regression strategy under mean uncertainty is theoretically more reasonable,works well in the actual simulation and provides investors with more investment strategies,which is reasonable,stable and flexible.