Risk Measurement And Asset Pricing Under Model Uncertainty

As is well-known, the three subjects of financial industry are:(I) optimal investment;(II) pricing and hedging;(III) risk measurement and management. The marks of the "first revolution in finance" are Markowitz’s mean-variance analysis and Sharpe’s capital asset pricing model (CAPM for short); the "second revolution in finance" begins after Black-Scholes-Merton’s pricing formula; In1997, Artzner, Delbaen, Eber and Heath [2,3] introduced a new way of risk measuring, called coherent risk measure, which is later weakened as convex risk measure[55,56]. These landmarks have been termed as the "third revolution in finance"[153].This paper is related to the second and the third problems. Along with the rapid development of financial markets, there is an increasing challenge on pricing and risk measuring. So we have to seek for new ways and adopt advanced theory of mathematics when modeling the market. Sometimes we have to go about our work from different points of view, sometimes we have to make robust decisions whatever the scenario is. The present paper will study multidimensional risk measurement and asset pricing un-der model uncertainty adopting the theory of BSDEs and G-expectation. There are four chapters in this paper:Chapter1is the introduction of the paper; Chapter2studies multidimensional risk measures via conditional g-expectation; Chapter3deals with as-set pricing under model uncertainty in which there are a family of mutually singular probability measures; Chapter4is related to path-dependent Hamilton-Jacobi-Bellman equations, we provide a probabilistic interpretation via the approach of G-expectation. The following are the main results:(I) In Chapter1we give the introduction of the whole paper. (II) In Chapter2we study multidimensional dynamic risk measures in-duced by conditional g-expectations. A notion of multidimensional g-expectation is proposed to provide a multidimensional version of nonlinear expectation-s. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem and viability on a rectangle of solutions to multidimen-sional backward stochastic differential equations, some necessary and suf-ficient conditions are given for the constancy, monotonicity, positivity and translatability properties of multidimensional conditional9-expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g-risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi-monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g-risk mea-sure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacted subsidiaries; optimal risk sharing for7-tolerant g-risk measures and risk contribution for coherent g-risk measures are investigated. Insurance g-risk measure and other ways to induce g-risk measures are also studied at the end of this chapter.The main results of this chapter comes fromXu Yuhong, Multidimensional dynamic risk measure via conditional g-expectation, submitted to Mathematical Finance,1-32, Minor Revision.We first present an explicit expression for multidimensional comparison theorem given by Hu and Peng [76]. This criterion, of simpler form than Hu and Peng’s condition has been exploited as a sufficient condition to prove the multidimensional comparison theorem by Zhou H.[162] and Zhou S.[163]. By a short and forward calculus, we show that Hu and Peng’s criterion coincides with the condition in Zhou H.[162].Theorem2.1Suppose that g1, g2satisfy (H1)-(H3). Then the following are equiv-alent:(i) For any t∈[0, T].(?)ξ1,ξ2∈L2(Ft;Rn), such that ξ1≥ξ2, the unique solutions (Y1,Z1) and (Y2, Z2) in S2F(0,t;Rn)×L2F(0,t;Rn×d) to BSDEs (2.4) over time interval [0, t] satisfy: Y1s≥Y2s,s∈[0,t], (ii)For any k=1,2,…,n,(?)t∈[O,T],(?)y1∈Rn,91k (t,δky+y1,z)≥g2k(t,y1,z1),(26) for any δky∈Rn such that δky≥0,(δky)k=0,z,z1∈Rn×d and(z)k=(z1)k.We have the following uniqueness result for generators of multidimensional BSDEs (2.4).But it is not a direct consequence of Theorem2.1.Theorem2.2For any t∈[O,T],(?)ξ1=ξ2∈L2(Ft；Rn), Y1s-Y2S,(?)s∈[O,t], if and only if for any K=1,2,…,n,(?)(t,y,z)∈[O,T]×Rn×Rn×d, g1k(t,y,z)=g2k(t,y,z).Rectangles are special closed convex subset of Rn.We first give explicit expressions for necessary and sufficient conditions for non-negative and non-positive solutions of BSDEs.Theorem2.3Suppose that g satisfies(H1)～(H3).Then the following are equivalent:(i) For any t∈[O,T],(?)ξ∈L2(Ft；Rn),such that ξ≥0,(ξ≤0),the unique solution(Y,Z)over time interval[O,t]satisfies, Ys≥0,(Ys≤0),s∈[O,t],(ii)For any k=1,2…,n,(?)t∈[O,T], gk(t,δky,δkz)≥0,(gk(t,-δky,δkz)≤0)(27) for any δky∈Rn such that δky≥0,(δky)k=0,and δkz∈Rn×d,(δkz)k=0. Then we have:Corollary2.2Suppose that g satisfies(H1)～(H3).For any t∈[O,T],(i)Assume C∈Rn.(?)ξ∈L2(Ft,Rn),such that ξ≥C,(ξ≤C),then Ys≥C,(Ys≤C),(?)s∈[O,t], if and only if for any k=1,2,…,n,(?)t∈[O,T], gk(t,δky+C,δkz)≥0,(gk(t,-δky+C,δkz)≤0)(28) for any δky∈Rn such that δky≥0,(δky)k=0,and δkz∈Rn×d,(δkz)k=0. (ii) Assume condition (2.9) and CllC2G Rn.(?)ξ∈[C1:C2], Ys∈[C1:C2],(?)s∈[0,t], if and only if for any k=1,2,..., n,(?)t∈[O, T], gk(t, δky+C1, zk=0)≥0≥gk(t,-aky1+C2, z1k=0),(29) for any δky, δky1∈Rn such that δky, δky1∈[0, C2-C1],(δky)k=(δky1)k=0.□A BSDE is in fact a dynamical mechanism of nonlinear expectation. It is natural to define multidimensional g-expectation εtg[ξ|Fs]=Ys by multidimensional BSDEs. We give the necessary and sufficient condition for Jensen’s inequality of a multidimensional conditional g-expectation.Definition2.2Jensen’s inequality for multidimensional conditional g-expectation hold-s, if for any t∈[O, T], for all ξ∈L2(Ft;Rn), convex function (?): Rn�'Rn such that (?)(ξ)∈L2(Ft;Rn), and for any k:=1,2,…,n, a.s.,(?)s∈[O,t], εtgk[(?)k(ξk)|Fs]≥(?)k(εtgk[ξk|Fs]).(30)Theorem2.5Suppose that g satisfies (HI)～(H3) and condition (2.9). Jensen’s in--equality for a multidimensional conditional g-expectation holds if and only if(i) for any k=1,2,…, n, gk does not depend on y and (?)(t, Zk)∈[O, T)×Rd,9k(t, λzk)=λgk(t, zk),(?)λ≥0; gk(t, λzk)≥λgk(t, zk),(?)λ <0.The multidimensional p-risk measure is induced by multidimensional g-expectation.Definition2.4Suppose that the risk mechanism g satisfies (H1)～(H3). For any t∈[O, T] and the risk position ξ∈L2(Ft; Rn), we define pgO,t[ξ]=εtg[-ξ], pts,t[ξ]=εtg[-ξ]|Fs], s∈[O,t] as the multidimensional static g-risk measure and multidimensional dynamic g-risk measure respectively. Without ambiguity, we denote p9=ptO,T,,Pgt=pgt,TDefinition2.5A multidimensional dynamic g-risk measure pgs,t[·] is convex if for any t∈[O.T], and for all risk positions ξ2,ξ2∈L2(Ft;Rn),(?)λ∈[0,1],(?)s∈[O,t],pgs,t[λξ1+(1-λ)ξ2]≤λpgs,t[ξ1]+(1-λ)pgs,t[ξ2]. For a convex g-risk measure, we have:Theorem2.7Suppose that the risk mechanism g satisfies (H1)～(H3)。For any t∈[O, T], for all risk positions ξa,ξ2∈L2(Ft;Rn), the multidimensional dynamic g-risk measure pgs,t[·] is nonincreasing and convex if and only if (?)t∈[O, T],(?)(y1, z1)∈Rn×Rn×d, ε=1.2, for all k=1,2,…, n, gk does not depend on (zj)j≠k and (?)λ∈[O,1], gk(t. λy1+(1-λ)y2-δky, zk)≤λgk(t, y1, z1k)+(1-λgk(t, y2z2k),(31) for any δky∈Rn uch that δky≥0,(δky)k=0and zk=入z1k+(1-λ)z2k.Definition2.6A multidimensional dynamic g一risk measure pgs,t[·]is coherent if it sat-isfies:(i)Positive homogeneity.(?)ξ∈L2(Ft；Rn),(?)α∈R+,pg8,t[αξ]=αpgs,t[ξ](?)s∈[0,t]；(ii)Subadditivity.For all risk positionsξ1,∈2.∈L2(Ft；Rn),(?)s∈[0,t],pgs,t[ξ1+ξ2]≤pgs,t[ξ1]+pgs,[ξ2].Corollary2.3Suppose that the risk mechanism g satisfies(H1)～(H3).The multi-dimensional dynamic risk measure pgs,t[·]is nonincreasing and coherent if and only if for allκ=1,2,...，n,(？)t∈[0:T],(?)y∈Rn,gk does not depend on(zj)j≠k,(?)α∈R+, gk(t,αy,αzk)=αgk(t,y,zk),(32) and (?)(yi,z2k)∈Rn×Rd,i=1,2, gk(t,y1+y2-δky,z1k十z2k)≤gk(t,y1,z1k)+gk(t,y2,z2k),(33) for any δky∈Rn such that δky≥0,(δky)k=0.The convex g-risk measure has the following representation:Theorem2.8Suppose that the risk mechanism g satisfies(H1)～(H3)and condition (2.9).For each k=1,…,n,let pgt[·]k be the k’th component of pgt[·].Then the multidi-mensional dynamic convex risk measure pgt[·] over [0,T]has the following representation: For a risk position ξ∈L2(FT；Rn), where Qr is the probability measure with density process dLt=LtγtdBt,L0=1(n×n unit matrix) and is called the penalty term.In Section2.4Theorem2.9,we show that a multidimensional g-risk measure satis-fying the cash additive axiom and the monotonicity axiom is in fact n one dimensional g-risk measures put together.In the application part:we first show that how our approach applied to measure the the insolveney risk of a firm with interacted subsidiaries.See Example2.1and2.2.Then we study the problem of optimal risk sharing for γ-tolerant g-risk measures. Consider two groups A and B,measuring their risks by a multidimensional g-risk measure p9with different risk tolerant coefficients-γA.γB∈R+,i.e.,their risks are measured by respectively. For each k=1,…, n, we define inf-convolutionTheorem2.11Let ρt,Tg be a multidimensional convex g-risk measure with g satisfying (H1)～(H3) and (2.9) and g(·,0,0)≤0. Then (?)γA,γB∈R+, ρgγA□ρgγB is(γA+γB)-tolerant g-risk measure, i.e., it is the unique solution of the following BSDE Moreover, is the optimal transfer for all k=1,…, n.We also consider the problem of risk contribution by multidimensional coherent g-risk measures. When the firm has n interacted subsidiaries (or a portfolio consists of different currencies), they have to use a multidimensional risk measure. If the sub-sidiaries have an income X=(X1,…, Xn)∈L2(FT;Rn), so how much risk X brings to those subsidiaries? In this case we have to consider the problem of risk contribution by multidimensional risk measures. Let ρg[·] be a multidimensional coherent g-risk measure. We define and the set of extreme measures for Y We have the following representation.Theorem2.12For X, Y∈L2(FT;Rn), for each k=1,…, n, we have(Ⅲ) In Chapter3we investigate financial markets with mean-volatility uncertainty. Models for stock markets and option markets with uncertain pri-or distribution are established by Peng’s G-stochastic calculus. The process of stock price is described by generalized geometric G-Brownian motion in which the mean uncertainty may move together with or regardless of the volatility uncertainty. On the hedging market, the upper price of an (exot-ic) option is derived following the Black-Scholes-Barenblatt equation. It is interesting that the corresponding Barenblatt equation does not depend on the risk preference of investors and the mean-uncertainty of underlying s-tocks. Hence under some appropriate sublinear expectation, neither the risk preference of investors nor the mean-uncertainty of underlying stocks pose effects on our super and subhedging strategies. We call this phenomenon risk-neutral&mean-certain pricing. We reconsider conditions of no-arbitrage for super and sub-hedging strategies in the framework of model uncertainty. Minimal superstrategy with no-arbitrage is obtained by tools of backward s-tochastic differential equations. In particular, the term of bounded variation (Kt) arising in the upper pricing formula is interpreted as "pricing error" for a superhedging strategy. We also show that the put-call parity relation still holds for superhedging and subhedging strategies. In Markovian setting, stochastic control representation for the minimal superstrategy is given. The term of bounded variation also has a more clear form and is interpreted as the max Profit&Loss of being short a delta-hedged option. Finally we show the price fluctuation could be controlled by volatility fluctuation multiplying the Gamma of the option.The main results of this chapter comes fromMean-volatility uncertainty, arbitrage ambiguity and continuous time asset pricing,1-46, Completed Manuscript.We consider a stock market with mean-uncertainty and volatility-uncertainty to-gether. We do not have confidence in which direction the expected rate μ of return and the volatility a will move or even their distribution in future but they are sure to change within [μ,μ] and [σ.σ]. let r be the riskless interest rate. If μ. varies in [μ,μ], thenμ—varies in [μ-r, μ-r]. Let (βt) be a bounded-variation G[μ-r,μ-r]-Brownian motion,(Bt) a zero-mean G[σ2,σ2-Brownian motion under a given sublinear expectation E. Then the price process of a stock is in form of dSt=St(rdt+dβt+dBt).(38)It is important to keep in mind that we do not assume a risk-neutral world in advance in model (38). We can estimate volatility uncertainty by real data of stock prices, see Section3.2.5.In Section3.3, we deduce PDEs for both state-dependent payoffs and discrete-path- dependent payoffs. For state-dependent payoffs, the corresponding HJB equation is If there are uncertainty for the riskless interest rate, i.e., r∈[r, r], then the superhedging PDE should beBy an analogous procedure as in Section3.3.1, the superhedging price for discrete-path-dependent payoffs over [tk-1,tk] should satisfy The sequence of PDEs Vk, k=1,..., N, is defined recursively in a backward manner. The terminal conditions are defined respectively by VN(T,X (N-1),x)＝(?)(X(N-1),x), Vk,(tk,x(k-1,x)=Vk+1(tk,x(k-1),x,x).(42) What is new is that, although we put risk preference and uncertainty into stock appre-ciation//, the BSB equation does not involve any variables that are affected by the risk preference of investors.In Section3.4, we consider the conditions of arbitrage under model uncertainty. Especially the no arbitrage argument for subhedging is different from Vorbrink [155].Definition3.14There is an arbitrage for a subhedging strategy (V.π,K) satisfying (3.32), if the value process (Vt) satisfies V0=0and VT≥0, q.s. andTheorem3.5The substrategy (V, π, K) is arbitrage-free.In fact, if (43) holds, then by the strict comparison theorem in Li [102]. we have Thus the substrategy (V，π, K) is arbitrage-free. The put-call parity relation for superhedging strategies still holds in an incomplete market.Theorem3.6Let ct and pt be the superhedging prices of a European call option and a European put option underlying the same stock St and sharing the same strike price L. ThenSection3.4.5is related to asset with strictly non-zero upper price and generalized geometric G-Brownian motion.Definition3.15A sublinear expectation E is said to be risk-neutral if under E, the discounted stock price DtSt (paying no dividend) is a symmetric G-martingale.Proposition3.5Let E be a risk-neutral sublinear expectation in a market model. Then the upper price of every discounted portfolio is a G-martingale (not necessarily symmetric) under E.Definition3.16A process (Vt) is called a geometric G-Brownian motion if it follows dVt=Vt (rtdt+atdKt+0tdBt)(44) where (Bt) is a G-Brownian motion,(Kt) is a right-continuous increasing adapted pro-cess, K0=0,(at)∈M1G and equivalentlyAn asset with strictly non-zero upper price is a security paying Vt at time T whose upper price Vt≠0, q.s. for each t∈[0, T].Theorem3.7The upper price of an asset is strictly non-zero if and only if the upper price is a generalized geometric G-Brownian motion with V0≠0,(-dKt) being a G-martingale with bounded variation.In Section3.5.1, by stochastic control methods, we have the followingTheorem3.8Let V (t,x) be the unique viscosity solution of (3.17). ThenIn Section3.5.3, we give an interpretation of r) and K in the decomposition of G-martingale in Markovian setting. ●η corresponds to Gamma Γ of the option;●Kt coincides with the maximal P&L: of being short a delta-hedged option. That is, by choosing appreciate managing volatility σ, we obtain a nonnegative P&L (or K) for a robust strategy.Then we come back to equality (3.26) in section3.4, which now has a clear meaning:●The minimal superstrategy satisfies:changes of values of the portfolio minus the instantaneous P&L, equals to the change of the managing price of the option. That is to say, we can withdraw money P&L(t,t+dt) along the way and end up with the terminal payoff.For option buyers, to guarantee a profit, he/she has to choose the "minimal volatil-ity" such that his/her P&L on (t,t+dt) will always be nonnegative.Cont [25] proposed to measure the impact of model uncertainty on the value of a contingent claim ξ by In Section3.5.4, we show that e-p(·) depends on closely the volatility uncertainty and gamma risk.Theorem3.9For all ξ=(?)(ST),(?) is a Lipschitz functional of path St, we have where Therefore the ask-bid spread depends on closely the fluctuations of volatility.(IV) In Chapter4we show that the solution of a backward stochastic differential equation under G-expectation provides a probabilistic interpreta-tion for the viscosity solution of a type of path-dependent Hamilton-Jacobi-Bellman equation. Particularly, a G-martingale can be considered as a non-linear path-dependent partial differential equation (PDE). We also show that certain class of path-dependent PDEs can be transformed into classical multi-ple state-dependent PDEs. As an application, the path-dependent uncertain volatility model can be described directly by path-dependent Black-Scholes-Barrenblett equations.The main results of this chapter comes fromXu Yuhong, Probabilistic solutions for a class of path-dependent HJB equations, Stochastic Analysis and Applications,31(3)(2013),440-459.Generally a solution of a state-dependent BSDE can be viewed as a path solution of the corresponding parabolic PDE. Shige Peng conjectured in his ICM paper [135] that a path dependent solution of a BSDE and/or a G-martingale can be considered as a nonlinear path-dependent PDE of parabolic and/or elliptic types. The present paper gives a partial answer:Theorem4.3The BSDE under G-expectation where provides a probabilistic interpretation for the viscosity solution of the following path-dependent Hamilton-Jacobi-Bellman (HJB) equation where L(ωt,α) is the second order elliptic partial differential operator parameterized by the control variable a∈Γ (?) Rd,Let (Y(s)) be the solution of BSDE (4.12). If all the coefficients are Lipschitz-continuous, then by Theorem4.3, we haveTheorem4.4U (t, J1, J2, x):=Y (t) is the unique viscosity solution of the following state-dependent PDE:Lastly we consider the path-dependent uncertain volatility model. We give the pric-ing formula and show that the path-dependent Black-Scholes PDE can be transformed into the path-dependent heat equation. See Section4.5.