On distributional robust probability functions and their computations

Transcription

1 On distributional robust probability functions and their computations Man Hong WONG a,, Shuzhong ZHANG b a Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong b Department of Industry and Systems Engineering, University of Minnesota, Church Street S.E. Minneapolis, MN USA Abstract Consider a random vector, and assume that a set of its moments information is known. Among all possible distributions obeying the given moments constraints, the envelope of the probability distribution functions is introduced in this paper as distributional robust probability function. We show that such a function is computable in the bi-variate case under some conditions. Connections to the existing results in the literature and its applications in risk management are discussed as well. Keywords: moment bounds, semidefinite programming SDP), robust optimization, worst-case scenario, nonlinear risk, risk management MSC: 6E5, 6P5, 9C. Introduction One of the tasks in risk management is to manage what to do in all scenarios, especially when worst comes to worst. With reference to a risk measure, a common way to describe worst-case is through distributional robustness, which refers to any distribution fitting some given moments, say m, m,, m n. Consequently, the formulation below is often known as the moment bound problem: GP ) sup x m,,m n ) E[ψx)] := sup s.t. E[ψx)] E[x x x] = m }{{} i, i =,, n. # of x=i A few words about our notations are in order here. For x R, denotes the usual scalar multiplication and m i R for all i, with m i := E[x i ]. For x R d, is the tensor multiplication or matrix Corresponding author addresses: mhwong@se.cuhk.edu.hk Man Hong WONG), zhangs@umn.edu Shuzhong ZHANG ) Preprint submitted to Elsevier August 3,

2 # of d=i {}}{ multiplication in case d = ) in the corresponding spaces and m i Rd d. For example, if x = [x ), x ) ] T R, then x x = xx T = x) ) x ) x ) R, which is in the same space x ) x ) x ) ) where m resides. We will use the lower case letters e.g. m) for scalars and vectors, capital letters e.g. M) for matrices, and fraktur lower case letters m for ambiguous implied) dimensions. Scarf 958) was the first to apply this worst-case analysis in inventory management, where he took ψx) = min{x, k} for some constant k and assumed the knowledge of the first two moments. Lo 987) and Grundy 99) applied the similar concept for option bounds. As a matter of fact, a sizeable amount of relevant literature can be found see e.g. Chen et al. ); Cox 99); Cox et al. 8); Han et al. 5); He et al. ); Jansen et al. 986); Liu and Li 9); De Schepper and Heijnen 7, ); De Vylder and Goovaerts 98, 983); De Vylder 98)). With the recent computational developments of moment bounds, applications have been introduced in different streams in financial engineering. For example, Bertsimas and Popescu 5) discussed the moment bounds using semidefinite programming and its relevance in probability theory; Popescu 7) considered the mean-covariance solutions for stochastic optimization; Chen et al. ) as well as Natarajan and Sim ) discussed the moment bounds in the context of robust portfolio selection; Wong and Zhang ) discussed the moment bounds in the context of nonlinear risk management; Lasserre et al. 6) discussed the pricing of a class of exotic options with moments and SDP relaxation. Regarding the theory underlying the computation of moment bounds, we refer the interested reader to Popescu 5) and Lasserre 8). In particular, when we choose ψx) = x E for some event E in the sample space Ω R d of x, GP ) is the worst-case probability, which can be regarded as an implicit function of the moments, given the event E: F d,n E) := sup P[x E] R d )x m,,m n) = sup E[ x E ] x m,,m n) ). ) As we shall see later, choosing E = {x R : x t}, F, E) is in fact a probability distribution itself. In general, however, this is not the case; although we always have F d,n, it may not satisfy the additivity of joint countable union, namely, for any countable sequence of pairwise disjoint event E, E,, we only have F d,n j E j F d,n E j ). In other words, only subadditivity is guaranteed and the equality holds only when the right hand side is j= attained by the same extremal distribution of x for all E j. The possibility of deriving an analytical form or devising a simple computational procedure for F d,n

3 remains open for a general n and d. Throughout this paper, we adhere our discussion to the case n =, unless specified otherwise. When d = and n =, there are nice distributional robust functions in analytical form. In our subsequent discussion we will revisit them while applying the Value-at-Risk in the context of portfolio selection. The formulation is in line with El Ghaoui et al. 3), who discuss the worst-case Value-at-Risk knowing the first two moments. To the best of our knowledge, even when d = there is no analytical form or method for exact computation. The closest approximation is due to Cox et al. 8), who use sum-of-squares SOS) polynomials to approximate F, E) for nonnegative random variables, where E = {x R : x t for some t R +}. Our key contribution is to provide the exact computational methods, in the form of semidefinite programming SDP), for F, ). The methodology is based on the characterization of copositive cones in R d+, where d 3, and some results in Luo et al. 4), which state that, given either i) x ) [, ] or ii) x ) R +, and x ) R m, the nonnegativity of a bi-quadratic function reduces to LMIs. The rest of this paper is organized as follows. In Section, we review F, and its connection to Valueat-Risk in the context of portfolio selection. In Section 3, we derive the LMIs for computing F,, where three events are taken into account as our base cases : E := {x R : x ) u ), x ) u ) }, E := {x R : l ) x u ), l ) x u ) } and E 3 := {x R : x ) u ), x u ) }. Model extensions are introduced in Section 4, followed by applications in Section 5, and finally we conclude the paper in Section 7.. Distributional robust function with a single random variable Take E = {x R : x t} and let µ and σ be the mean and variance respectively. F, E ) can be represented as a function of t Chen et al., ): F, E ) := F t) = σ µ t) +σ, t µ ;, t > µ. The above follows essentially from the Chebyshev-Cantelli inequality. ) It is also well-known that this worse-case probability is achieved by a two-point distribution of x. However, the story is completely different when F t) is regarded as a distribution function of some random variable ζ, since it now has a smooth and continuous distribution ), which allows us to compute its moments. It is interesting to note that the first two moments of ζ and x are no longer the same: Eζ) = µ π σ versus Ex) = µ ; and Eζ ) = versus Ex ) = µ + σ see Appendix). This infinite variance actually provides us with some insight about the huge fluctuation of ζ. In risk management, as extreme events are often associated with the Value-at-Risk VaR), let us apply 3

4 ζ with this risk measure and consider a portfolio selection problem. Suppose that θ R p is the vector of investment return from p assets with a mean m R p and second moment matrix M S p +. Let w R p be the portfolio weights and x = w T θ the portfolio return. Then Ex) = w T m and Ex ) = w T Mw. Applying F, E ) in the definition VaR, where we regard w T θ as the loss and choose t = α in E, we have VaR ϵ w T θ) := arg min{f, w T θ α) ϵ}, α where ϵ, ) is the level of confidence. The higher the α, the higher the risk. Therefore we would like to minimize the risk over the set of admissible portfolio W which typically incorporates the target of return, budget constraint and sometimes no short selling constraints) as follows: min α s.t. F, w T θ α) = F α) ϵ 3) w W, where ϵ is given. Recall that F α) = sup x wt m,w T Mw) Px α). Below we show that 3) is a convex optimization model and can be solved efficiently. Lemma. Problem 3) can be reformulated by second-order cone programming SOCP). Proof. The assertion follows from the observation that F α) ϵ ϵ)w T Mw ϵw T m + α) wt m + α SOCd + ). M w ϵ ϵ w T Mw w T m+α) +w T Mw ϵ Note that we have implicitly assumed w T m α. Otherwise F, w T θ α) = > ϵ, which contradicts the definition of VaR. Our result is in line with that in El Ghaoui et al. 3), who arrived at the same conclusion from a completely different angle. Let us introduce the worst-case probability of the event E := {x R : l x u} as: σ l µ ) +σ, µ < l; F, E ) =, l µ u; σ µ u) +σ, µ > u. 4) 4

5 3. Moment bound of joint probability In this section, we consider three base-case joint events: E := {x R : x ) u ), x ) u ) }, E := {x R : l ) x ) u ), l ) x ) u ) }, E 3 := {x R : x ) u ), l ) x ) u ) }. Our goal is to show the SDP formulations for F, E k ), k =,, 3. Let l := l ), l ) ) T and u := u ), u ) ) T and the mean µ R and covariance matrix Γ S + of x be given. Recall their primal form, P k ) sup P[x E k ] := sup x µ,γ) x P E[ Ek ] s.t. E[x] = µ E[xx T ] = Γ + µµ T. Since we can pick any feasible distribution from P for the optimal, the bound is known as distributional robust. Another remark is that P k ) is an infinite dimensional problem that is not trivial to solve. Therefore, we will look into their dual formulation: D ) inf z,z,z z + z T µ + Z Γ + µµ T ) s.t. z + z T x + Z xx T x u 5) z + z T x + Z xx T x R, D ) inf z,z,z z + z T µ + Z Γ + µµ T ) s.t. z + z T x + Z xx T l x u 6) z + z T x + Z xx T x R, D 3 ) inf z,z,z z + z T µ + Z Γ + µµ T ) s.t. z + z T x + Z xx T x ) u ), l ) x ) u ) 7) z + z T x + Z xx T x R. Obviously the second constraint in the above formulations are simply LMI. In the mean time, 5) is a copositive constraint in dimension 3 3, thus also an LMI. We will supplement the derivation for completeness. To show that 6) and 7) can be cast into LMIs as well, we base the results on Theorems and : 5

6 Theorem. c.f. Theorem 4.5 of Luo et al. 4)) Let px, y) := y T Cy + y T By)x + y T Ay)x be defined by p : R + R m R, where A, B, C are sub-matrices in Z L,m and L,m := C B S m : A, B, C S m B A. Then px, y) Z C B x R +, y R m B L,m : C A B B A E E T, E + E T for some E Theorem. c.f. Theorem 4.6 of Luo et al. 4)) Let px, y) := y T Cy + y T By)x + y T Ay)x be defined by p : [, ] R m R, where A, B, C are sub-matrices in Z L,m. Then px, y) Z C B x [, ], y R m B L,m : C A B E T B E A + E + E T, E + E T for some E. The key to apply Theorems and into the moment bounds is that we choose y =, ξ,, ξ m ) T, where ξ R, so that px, ξ) has a degree m in ξ. In other words, it is no longer bi-quadratic. Below we shall show the conversion can be done and how the theorems are invoked. 3.. Constraint 5) in LMIs Let x = u x. Then rewrite 5) with a few lines of algebra: z + z T u x) + u x) T Z u x) = z + z T u + u T Z u z T + u T Z ) x + x T Z x x R + z + z T u + u T Z u z T / u T Z z / Z u Z N S 3 +, where N R Here we use the fact that the copositive cone C m = S m + + R m m + for m 4. Hence D ) can be cast as an SDP: SDP ) inf z,z,z z + z T µ + Z Γ + µµ T ) s.t. z + z T u + u T Z u z T / u T Z z / Z u Z N ij), i, j =,, 3 z z T /. z / Z N, We remark here that D ) can be extended to compute F 3, E ). 6

7 3.. Constraint 6) in LMIs Let x ) = u ) l ) )η + l ) and x ) = l) +u ) +. Writing the expression in 6) componentwise and multiplying + ) on both sides, we have + ) z + z T x + Z xx T ) = + ) z + z ) x) + z ) x) + Z ) x ) ) + Z ) + Z ) )x ) x ) + Z ) x ) ) ) = z + z ) u) l ) )η + l ) ) + Z ) u ) l ) )η + l ) ) ) + + ξ 4 ) [ ) ] + z ) + Z ) + Z ) u ) l ) )η + l ) ) l ) + u ) ) + ) + Z ) l ) + u ) ) = c ξ) + b ξ)u ) l ) )η + a ξ)u ) l ) ) η, where c ξ) := = z + z ) l) + Z ) l ) ) ) [ )] + ) + z ) + l ) Z ) + Z ) l ) + u ) ) + ) +Z ) l ) + u ) ) T ξ C ξ, with := C Cl ), l ), u ) )) { Y S 3 Y ) = z + z ) l) + Z ) l ) ) ) + Y 3) + Y ) + Y 3) = z + z ) l) + Z ) l ) ) ) [ )] + z ) + l ) Z ) + Z ) l ) + u ) ) + Z ) l ) u ) ; Y 33) = z + z ) l) + Z ) l ) ) ) + } Y ) = Y ) = Y 3) = Y 3) =, [ )] z ) + l ) Z ) + Z ) l ) + Z ) l ) ) ; [ z ) + l ) Z ) + Z ) )] u ) + Z ) u ) ) ; and b ξ) := = z ) + Z ) T ξ B l )) + ) + ξ, ) Z ) + Z ) l ) + u ) ) + ) 7

8 with := B Bl ), l ), u ) )) { Y S 3 Y ) = z ) Y 3) + Y ) + Y 3) = Y 33) = z ) + Z ) l )) + + Z ) l )) + z ) Y ) = Y ) = Y 3) = Y 3) = Z ) + Z ) l )) + Z ) + Z ) }, + Z ) ) l ) ; Z ) + Z ) ) u ) ; ) l ) + u ) ); and a ξ) := Z ) + ) = ξ T A ξ, with A A) := {Y S 3 Y ) = Z ) ; Y 3) + Y ) + Y 3) = Z ) ; Y 33) = Z ) ; Y ) = Y ) = Y 3) = Y 3) = }. By Theorem, constraint 6) can be represented by LMIs: z + z T x + Z xx T x t c ξ) + b ξ)u ) l ) )η + a ξ)u ) l ) ) η η, ξ R C u ) l ) )B / E, u ) l ) )B / E T u ) l ) ) A + E + E T E + E T, where A A, B Bl ), l ), u ) ), C Cl ), l ), u ) ). Hence, D ) is equivalent to the following SDP: SDP ) inf z,z,z z + z T µ + Z Γ + µµ T ) C u ) l ) )B / E u ) l ) )B / E T u ) l ) ) A + E + E T E + E T s.t. A A, B Bl ), l ), u ) ), C Cl ), l ), u ) ) z z T /. z / Z, 8

9 3.3. Constraint 7) in LMIs Let x ) = u ) η and x ) = l) +u ) +. Writing the expression in 7) componentwise and multiplying + ) on both sides, we have + ) z + z T x + Z xx T ) = + ) z + z ) x) + z ) x) + Z ) x ) ) + Z ) + Z ) )x ) x ) + Z ) x ) ) ) = z + z ) u) η) + Z ) u ) η) ) + + ξ 4 ) [ ) ] + z ) + Z ) + Z ) u ) η) l ) + u ) ) + ) + Z ) l ) + u ) ) = c ξ) + b ξ)η + a ξ)η, where with c ξ) := = z + z ) u) + Z ) u ) ) ) + ) [ z ) + u ) Z ) + Z ) + ξ T C ξ, )] l ) + u ) ) + ) + Z ) l ) + u ) ) := and C Cu ), l ), u ) )) { Y S 3 Y ) = z + z ) u) + Z ) u ) ) ) + Y 3) + Y ) + Y 3) = z + z ) u) + Z ) u ) ) ) [ )] + z ) l ) + u ) ) + Z ) + u ) Z ) + Z ) z + z ) u) + Z ) Y 33) = u ) ) ) + } Y ) = Y ) = Y 3) = Y 3) =, b ξ) := z ) + Z ) T = ξ B l )) + ) ξ, [ )] z ) + u ) Z ) + Z ) l ) + Z ) l ) ) ; l ) u ) ; [ z ) + u ) Z ) + Z ) )] u ) + Z ) u ) ) ; ) Z ) + Z ) l ) + u ) ) + ) 9

10 with := B Bu ), l ), u ) )) { Y S 3 Y ) = z ) ) Z ) + Z ) + Z ) u )) l ) ; Y 3) + Y ) + Y 3) = z ) + Z ) u )) Z + Z ) l ) + u ) ); Y 33) = z ) + Z ) u )) ) + Z ) + Z ) u ) ; } Y ) = Y ) = Y 3) = Y 3) =, and a ξ) := Z ) + ) = ξ T A ξ, with A A) := {Y S 3 Y ) = Z ) ; Y 3) + Y ) + Y 3) = Z ) ; Y 33) = Z ) ; Y ) = Y ) = Y 3) = Y 3) = }. By Theorem, constraint 7) can be represented by LMIs: z + z T x + Z xx T < x ) < u ), < x ) < u ) c ξ) + b ξ)η + a ξ)η η, ξ R C B / E, E + E B / A E T T where A A, B B u ), l ), u ) ), C Cu ), l ), u ) ). Summarizing, the tractable formulation for the D 3 ) is SDP 3 ) inf z,z,z z + z T µ + Z Γ + µµ T ) C B / E, B / A E T E + E T s.t. A A, B Bu ), l ), u ) ), C Cu ), l ), u ) ), z z T / z / Z.

11 4. Several model extensions 4.. Moment bound of probability of union events For the tight bound of the union of two events, say sup x µ,σ) Pl ) x ) u ) or x ) u ) ), applying Theorems and to its dual is almost immediate: inf z + z T µ + Z Γ + µµ T ) z,z,z s.t. z + z T x + Z xx T l ) x ) u ) 8) z + z T x + Z xx T x ) u ) 9) z + z T x + Z xx T x R. ) We can see that Theorem is applied to 8) and Theorem to 9). 4.. Domain extensions We can extend to compute the corresponding bound of P k ), k =,, 3, for nonnegative random variables. The dual formulations are respectively D + ) inf z,z,z z + z T µ + Z Γ + µµ T ) s.t. z + z T x + Z xx T x u ) z + z T x + Z xx T x ), x ) R + D + ) inf z,z,z z + z T µ + Z Γ + µµ T ) s.t. z + z T x + Z xx T max{, l} x u ) z + z T x + Z xx T x ), x ) R + D 3+ ) inf z,z,z z + z T µ + Z Γ + µµ T ) s.t. z + z T x + Z xx T x ) u ), max{, l ) } x ) u ) 3) z + z T x + Z xx T x ), x ) R +. Approximations to D + ) with SOS polynomials are discussed in Cox et al. 8). Theorems can be applied to ), ) and 3) in the same way as that in the previous section. Their second constraint are now a copositive constraint of dimension 3 3, i.e., z z T / z / Z C 3,

12 and therefore an LMI. Then the SDP for D k+ ), k =,, 3, are SDP + ) inf z,z,z z + z T µ + Z Γ + µµ T ) s.t. z + z T u + u T Z u z T / u T Z z / Z u Z N ij), i, j =,, 3 z z T / N S+ 3 z / Z N ij), i, j =,, 3, N S 3 +, SDP + ) inf z,z,z z + z T µ + Z Γ + µµ T ) C u ) max{l ), })B / E u ) max{l ), })B / E T u ) max{l ), }) A + E + E T s.t. E + E T, A A, B Bmax{l ), }, max{l ), }, u ) ), C Cmax{l ), }, max{l ), }, u ) ) z z T / N S+ 3 z / Z N ij), i, j =,, 3, SDP 3+ ) inf z,z,z z + z T µ + Z Γ + µµ T ) C u ) B / E u ) B / E T u ) ) A + E + E T s.t. E + E T, A A, B B, max{l ), }, u ) ), C C, max{l ), }, u ) ) z z T / N S+ 3 z / Z N ij), i, j =,, 3. Meanwhile, it is worth noting that the results above imply that we can compute the joint probability bound for two random variables with a variety of support of x ) and x ), in any combination: I) x R; II) x R + hence any semi-infinite interval); III) x [a, b] for any constant a and b; IV) x j j= I j, where I j s can be intervals of the above forms.

13 4.3. Higher moments incorporated We have shown that Theorems and can be applied to compute the exact joint probability bound. As a matter of fact, the theorems provide us with the freedom of using either random variable s higher moments. For example, for the bound of Pl ) x ) u ), l ) x ) u ) ), if we are given the higher moments of x ), say λ 3,, λ n, in addition to µ and Γ, then the dual formulation is inf z,z,z,z 3,,z n s.t. z + z T µ + Z Γ + µµ T ) + n i=3 z iλ i z + z T x + Z xx T + n i=3 z ix ) ) i l x u z + z T x + Z xx T + n i=3 z ix ) ) i x ), x ) R. Every step in the previous setting can be applied and the dimension m in Theorems and now becomes m = n Applications of the moment bound 5.. Riemann integrable set approximation Given any bounded Riemann integrable subset R of a sample space Ω R and the first two moments µ and Γ) of an arbitrary probability measure of it, we can always approximate the distributional robust probability measure of R with finitely many rectangular partitions [x i, x i ] [y i, y i ]. In other words, there exists m N such that R m i= [x i, x i ] [y i, y i ] and sup x,y) µ,σ) P x, y) R) sup x,y) µ,σ) P x, y) ) m i= [x i, x i ] [y i, y i ] = inf z,z,z z + z T µ + Z Γ + µµ T ) z + z T x + Z xx T x [x i, x i ] [y s.t. i, y i ], i =,, m z + z T x + Z xx T x, y R. 5.. Worst-case simultaneous VaR VaR refers to the risk of a single asset or a whole portfolio. Given the international investment markets nowadays, dependence among them has raised a growing concern. Therefore, it is of a great interest to study the VaR of different portfolios simultaneously. Consider two investment markets i =, ). Suppose that θ i R p i be the vector of investment return from p i assets with a mean m i R p i, second moment matrix M i S p i +, and covariance matrix between the two markets C R p p. Let w i R p i be the portfolio weights and x i) = w T i θ i the portfolio return. If α i) is the VaR of the portfolio w T i θ i in the two markets, then we can compute the worst-case probability through SDP ), in which we take 3

14 µ = [ w T m, w T m ] T, Γ = wt M w w T C w and u = [α ), α ) ] T. In fact, SDP ) allows w T C w w T M w us to compute the probability exactly upon machine error) for at most three portfolios under SDP ). In general, when we let α = α ) =... = α d), we can define the worst-case simultaneous VaR WS-VaR) by WS-VaR ϵ w T θ,..., wd T θ d ) := arg min{f d, w T θ α,..., wd T θ d α) ϵ}, α Since F d, w T θ α,..., w T d θ d α) is monotone in α, WS-VaR can be obtained by the bisection method. 6. Illustrations of the distributional robust probability functions To get an impression how the distributional robust probability functions actually look like, we provide some computational results in this section. For simplicity, in our examples we assume the mean to be zero vector and the covariance matrix to be identity, although the technique works for any mean vector and covariance matrix. To start with, we plot F, which is readily computable. Figure shows F, x t). To illustrate F, l x u), we fix the interval width between l and u so that they depend on the same parameter. Figure and 3 represent F, t x t + ) and F, t x t + ) respectively...8 Function Value t Figure : Plot of F, x t) Corresponding plots of F, are illustrated in Figure 4 to 6. We see that the shape of F, is similar to that of F, in the sense that the latter looks like the cross-section of the former. 4

15 .. Function Value Function Value t 5 5 t Figure : Plot of F, t x t + ) Figure 3: Plot of F, t x t + ). Function Value t) 5 5 t ) Figure 4: Plot of F, x ) t ), x ) t ) ) For F, with random bivariate with different variance and non-zero covariance, plots with positive and negative covariance are illustrated in Figure 7 and 8 respectively. 7. Conclusion In this paper we introduce the notion of distributional robust probability function under various moment informational constraints. In particular, F, x R : x t) turns out to be a probability distribution itself while this may not be so in general. Our result can be applied to portfolio selection minimizing the VaR under the first two moments constraints, and the formulation reduces to a previous 5

16 .. Function Value Function Value ) t 5 5 ) t t) 5 t) Figure 5: Plot of F, t) x) t) +, t) x) Figure 6: Plot of F, t) x) t) +, t) x) t) t) + ) + ).. Function Value Function Value ) t 5 ) t 5 t) 5 5 ) t Figure 7: Plot of F, t) x) t) +, t) x) Figure 8: Plot of F, t) x) t) +, t) x) t) + ) with different variance and positive covariance t) + ) with different variance and negative covariance well-known result of El Ghaoui et al. 3). For two dimensional random variables, we propose an SDP approach to compute the distributional robust probability function for a fairly general collection of events. Under mild assumptions, we also propose the idea of worst-case simultaneous VaR and take into account the risk of two or three portfolios at the same time. Finally, we remark that the applications of these new computational tools in portfolio selection and risk management are abundant. References Bertsimas, D., Popescu, I., 5. Optimal inequalities in probability thoery: A convex optimization approach. SIAM Journal of Optimization 5,

17 Chen, L., He, S., Zhang, S.,. Tight bounds for some risk measures, with applications to robust portfolio selection. Operations Research 59, Cox, S.H., 99. Bounds on expected values of insurance payments and option prices. Transactions of the Society of Actuaries 43, 3 6. Cox, S.H., Lin, Y., Tian, R., Zuluaga, L.F., 8. Bounds for ruin probabilities and value at risk. Actuarial Research Clearing House Newsletters De Schepper, A., Heijnen, B., 7. Distribution-free option pricing. Insurance: Mathematics and Economics 4, De Schepper, A., Heijnen, B.,. How to estimate the value at risk under incomplete information. Journal of Computational and Applied Mathematics 33, 3 6. De Vylder, F., 98. Best upper bounds for integrals with respect to measures allowed to vary under conical and integral constraints. Insurance: Mathematics and Economics, 9 3. De Vylder, F., Goovaerts, M., 98. Analytical best upper bounds on stop-loss premiums. Insurance: Mathematics and Economics, 97. De Vylder, F., Goovaerts, M., 983. Maximization of the variance of a stop-loss reinsured risk. Insurance: Mathematics and Economics, El Ghaoui, L., Oks, M., Oustry, F., 3. Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Operations Research 5, Grundy, B., 99. Option prices and the underlying asset s return distribution. Journal of Finance 46, Han, D., Li, X., Sun, D., Sun, J., 5. Bounding option prices of multi-assets: A semidefinite programming approach. Pacific Journal of Optimization, He, S., Zhang, J., Zhang, S.,. Bounding probability of small deviation: a fourth moment approach. Mathematics of Operationa Research 35, 8 3. Jansen, K., Haezendonck, J., Goovaerts, M., 986. Upper bounds on stop-loss premiums in case of known moments up to the fourth order. Insurance: Mathematics and Economics 5, Lasserre, J.B., 8. A semidefinite programming approach to the generalized problem of moments. Mathematical Programming, Series b,

18 Lasserre, J.B., Preito-Rumeau, T., Zervos, M., 6. Pricing a class of exotic options via moments and sdp relaxations. Mathematical Finance 6, Liu, G., Li, W.V., 9. Moment bounds for truncated random variables. Statistics and Probability Letters 79, Lo, A., 987. Semiparametric upper bounds for option prices and expected payoff. Journal of Financial Economics 9, Luo, Z.Q., Sturm, J.F., Zhang, S., 4. Multivariate nonnegative quadratic mappings. SIAM Journal on optimization 4, 4 6. Natarajan, K., Sim, M.,. Tractable robust expected utility and risk models for portfolio optimization. Mathematical Finance, Popescu, I., 5. A semidefinite programming approach to optimal-moment bounds for convex classes of distributions. Mathematics of Operations Research 3, Popescu, I., 7. Robust mean-covariance solutions for stochastic optimization. Operations Research 55, 98. Scarf, H., 958. A min-max solution of an inventory problem, in: Arrow, K., Karlin, S., Scarf, H. Eds.), Studies in the Mathematical Theory of Inventory and Production. Stanford University Press. Wong, M.H., Zhang, S.,. Computing best bounds for nonlinear risk measures with partial information. Preprint. Appendix We can regard F t) in as a distribution of t with density function d F t) = ft) = dt σ µ t) [µ t) +σ ], µ > t, µ t. 8

19 Then Et) = µ = σ [ µ σ µ t)t [µ t) + σ ] dt = σ [ [ = σ σ [ = σ σ [ = σ σ = µ π σ π/ µ t) [µ t) + σ ] dt + σ 3 tan θ sec θ σ 4 sec 4 dθ + π/ π/ [ θ sin θdθ µ σ π/ µ π/ cos θ dθ) µ 4σ ] µµ t) [µ t) + σ ] dt ] σ µ tan θ sec θdθ σ 4 sec 4 dθ θ ] sin θ cos θdθ π/ ] sin θ µ π/ 4σ [ cos θ] π/ sin θdθ) ] ] and Et ) = µ σ µ t)t [µ t) + σ ] dt π/ = σ σ tan θµ σ tan θ) σ 4 sec 4 σ sec θdθ θ = = = π/ π/ π/ tan θµ cos θ σ sin θ) dθ µ sin θ cos θ + σ tan θ sin θ µσ sin θ)dθ π/ µ σ ) sin θ cos θ µσ sin θ)dθ + σ tan θd. We can check that in the last line, the first integral is finite while the second one is infinite. Hence, Et ) = +. 9