exponential utility 证明
基于限价订单簿的最优高频策略及其实证检验
基于限价订单簿的最优高频策略及其实证检验作者：刘冰等来源：《价值工程》2015年第15期摘要：本文针对日内高频交易，引入保留价格具体分析限价单到达率，设计限价单报价的最优高频交易策略，与一般高频交易策略结果进行对比分析，并提供了基于该策略的实证检验分析。
理论部分首先对证券价格路径进行了介绍，并通过指数效用函数引出保留价格的基本公式；之后通过对市场微观结构的研究，分析得出限价订单理论上的成交概率；最后通过建立并解出汉密尔顿-雅各比-贝尔曼方程获得最优高频策略，进行计算机模拟并将其与一般高频策略结果进行对比分析，证实了该模型在理论拟合中的优势与策略的稳定性。
文中实证分析部分，选取A股市场上的股票，测量股票波动率，并利用理论部分得出的结论，通过Wind终端实时接收数据，检验策略在模拟交易情况下的稳定性与有效性。
Abstract： In recent years， with the growth of electronic exchanges， anyone is willing to submit limit orders in the system， it can effectively play the role of a market maker. In this project，we used reservation price to control the inventory risk arising from uncertainty in the asset’s value. The pricing strategies of dealers have been studied extensively in the micro structure literature. In the first part， we will introduce the basic theory on the microstructure including the exponential utility function and some theory on the limit orders. One of the key steps in the analysis is to use the dynamic programming principle to show that this problem solves the Hamilton–Jacobi–Bellman equation. Based on the model， we explore a simple trading strategy and present an approximate solution and a real trading of the performance of our strategy’s profit and loss through the Wind database.关键词：高频交易；微型做市商策略；限价单策略Key words： high-frequency trading；mini-market-maker strategy；limit order strategy中图分类号：S731 文献标识码：A 文章编号：1006-4311（2015）15-0193-041 研究现状随着交易制度的不断完善，高频交易已然成为交易人员、研究学者与监管部门关注的焦点。
Copeland金融理论与公司政策习题答案03
Chapter 3The Theory of Choice: UtilityTheory Given Uncertainty1. The minimum set of conditions includes(a) The five axioms of cardinal utility• complete ordering and comparability • transitivity • strong independence • measurability • ranking(b) Individuals have positive marginal utility of wealth (greed).(c) The total utility of wealth increases at a decreasing rate (risk aversion); i.e., E[U(W)] < U[E(W)]. (d) The probability density function must be a normal (or two parameter) distribution.2. As shown in Figure3.6, a risk lover has positive marginal utility of wealth, MU(W) > 0, whichincreases with increasing wealth, dMU(W)/dW > 0. In order to know the shape of a risk-lover’s indifference curve, we need to know the marginal rate of substitution between return and risk. To do so, look at equation 3.19: U (E Z)Zf(Z)dZ dE d U (E Z)f(Z)dZ ′−+σ=σ′+σ∫∫(3.19) The denominator must be positive because marginal utility, U’ (E + σZ), is positive and because the frequency, f(Z), of any level of wealth is positive. In order to see that the integral in the numerator is positive, look at Figure S3.1 on the following page.The marginal utility of positive returns, +Z, is always higher than the marginal utility of equally likely (i.e., the same f(Z)) negative returns, −Z. Therefore, when all equally likely returns are multiplied by their marginal utilities, matched, and summed, the result must be positive. Since the integral in the numerator is preceded by a minus sign, the entire numerator is negative and the marginal rate of substitution between risk and return for a risk lover is negative. This leads to indifference curves like those shown in Figure S3.2.14 Copeland/Shastri/Weston • Financial Theory and Corporate Policy,Fourth EditionFigure S3.1Total utility of normally distributed returns for a risk loverFigure S3.2 Indifference curves of a risk lover3. (a)ln W 8.4967825E[U(W)].5ln(4,000).5ln(6,000).5(8.29405).5(8.699515)8.4967825e We $4,898.98W=+=+====Therefore, the individual would be indifferent between the gamble and $4,898.98 for sure. Thisamounts to a risk premium of $101.02. Therefore, he would not buy insurance for $125.(b) The second gamble, given his first loss, is $4,000 plus or minus $1,000. Its expected utility is=+=+====ln W 8.26178E[U(W)].5ln(3,000).5ln(5,000).5(8.006368).5(8.517193)8.26178e e $3,872.98WNow the individual would be willing to pay up to $127.02 for insurance. Since insurance costsonly $125, he will buy it.Chapter 3 The Theory of Choice: Utility Theory Given Uncertainty 154. Because $1,000 is a large change in wealth relative to $10,000, we can use the concept of risk aversionin the large (Markowitz). The expected utility of the gamble isE(U(9,000,11,000; .5)).5U(9,000).5U(11,000).5ln9,000.5ln11,000.5(9.10498).5(9.30565)4.55249 4.6528259.205315=+=+=+=+=The level of wealth which has the same utility isln W =9.205315W =e 9.205315=$9,949.87Therefore, the individual would be willing to pay up to$10,000 − 9,949.87 = $50.13 in order to avoid the risk involved in a fifty-fifty chance of winning or losing $1,000.If current wealth is $1,000,000, the expected utility of the gamble isE(U(999,000, 1,001,000; .5)).5ln 999,000.5ln1,001,000.5(13.81451).5(13.81651)13.81551=+=+=The level of wealth with the same utility is ln W =13.81551W =e 13.81551=$999,999.47Therefore, the individual would be willing to pay $1,000,000.00 − 999,999.47 = $0.53 to avoid the gamble.5. (a) The utility function is graphed in Figure S3.3.U(W)=−e −aW16 Copeland/Shastri/Weston • Financial Theory and Corporate Policy,Fourth EditionFigure S3.3 Negative exponential utility functionThe graph above assumes a = 1. For any other value of a > 0, the utility function will be amonotonic transformation of the above curve.(b) Marginal utility is the first derivative with respect to W.aW dU(W)U (W)(a)e 0dW−′==−−> Therefore, marginal utility is positive. This can also be seen in Figure S3.3 because the slope of a line tangent to the utility function is always positive, regardless of the level of wealth. Risk aversion is the rate of change in marginal utility.aW 2aW dMU(W)U (W)a(a)e a e 0dW−−′′==−=−< Therefore, the utility function is concave and it exhibits risk aversion.(c) Absolute risk aversion, as defined by Pratt-Arrow, is2aWaW U (W)ARA U (W)a e ARA a ae −−′′=−′−=−=Therefore, the function does not exhibit decreasing absolute risk aversion. Instead it has constant absolute risk aversion.(d) Relative risk aversion is equal toU (W)RRA W(ARA)WU (W)Wa′′==−′=Therefore, in this case relative risk aversion is not constant. It increases with wealth.Chapter 3 The Theory of Choice: Utility Theory Given Uncertainty 176. Friedman and Savage [1948] show that it is possible to explain both gambling and insurance if anindividual has a utility function such as that shown in Figure S3.4. The individual is risk averse todecreases in wealth because his utility function is concave below his current wealth. Therefore, he will be willing to buy insurance against losses. At the same time he will be willing to buy a lottery ticket which offers him a (small) probability of enormous gains in wealth because his utility function isconvex above his current wealth.Figure S3.4 Gambling and insurance7. We are given thatA >B >C >D Also, we know thatU(A) + U(D) = U(B) + U(C) Transposing, we have U(A) − U(B) = U(C) − U(D) (3.1) Assuming the individual is risk-averse, then 22U U 0 and 0W W∂∂><∂∂ (3.2) Therefore, from (1) and (2) we know that −−<−−U(A)U(B)U(C)U(D)A B C D(3.3) Using equation (3.1), equation (3.3) becomes11A B C DA B C DA D C B1111A D C B 22221111U (A)(D)U (C)(B)2222<−−−>−+>++>+ +>+In general, risk averse individuals will experience decreasing utility as the variance of outcomes increases, but the utility of (1/2)B + (1/2)C is the utility of an expected outcome, an average.18 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition8. First, we have to compute the expected utility of the individual’s risk.i i E(U(W))p U(W ).1U(1).1U(50,000).8U(100,000).1(0).1(10.81978).8(11.51293)10.292322==++=++=∑ Next, what level of wealth would make him indifferent to the risk?10.292322ln W 10.292322W e W 29,505=== The maximum insurance premium isRisk premium = E (W) – certainty equivalent$85,000.1$29,505$55,495.1=−= 9. The utility function is U(W)=−W −1Therefore, the level of wealth corresponding to any utility isW = –(U(W))–1Therefore, the certainty equivalent wealth for a gamble of ±1,000 is W.−−−=−−++−−111W [.5((W 1,000)).5((W 1,000))]The point of indifference will occur where your current level of wealth, W, minus the certainty equivalent level of wealth for the gamble is just equal to the cost of the insurance, $500. Thus, we have the condition−= −−= −− + +−−−= − =−−+= −−+==2222W W 5001W 50011.5.5W 1,000W 1,0001W 500W W 1,000,000W 1,000,000W 500W W W 1,000,000500WW 2,000Chapter 3 The Theory of Choice: Utility Theory Given Uncertainty 19Therefore, if your current level of wealth is $2,000, you will be indifferent. Below that level of wealth you will pay for the insurance while for higher levels of wealth you will not.10. Table S3.1 shows the payoffs, expected payoffs, and utility of payoffs for n consecutive heads.Table S3.1Number of Consecutive Heads = N Probability = (1/2)n +1 Payoff = 2NE(Payoff) U(Payoff)E U(Payoff) 0 1/2 1 $.50 ln 1 = .000 .0001 1/42 .50 ln 2 = .693 .1732 1/8 4 .50 ln 4 = 1.386 .1733 1/16 8 .50 ln 8 = 2.079 .130N (1/2)N +1 2N .50 ln 2N = N ln 2 N ln 22+=0 The gamble has a .5 probability of ending after the first coin flip (i.e., no heads), a (.5)2probability of ending after the second flip (one head and one tail), and so on. The expected payoff of the gamble is the sum of the expected payoffs (column four), which is infinite. However, no one has ever paid an infinite amount to accept the gamble. The reason is that people are usually risk averse. Consequently, they would be willing to pay an amount whose utility is equal to the expected utility of the gamble. The expected utility of the gamble isN i 1i12i 0Ni 1122i 0N 12ii 0E(U)()ln 2E(U)()i ln 2i E(U) ln 22+======∑∑∑ Proof that i i 0i 22∞==∑follows: First, note that the infinite series can be partitioned as follows: ∞∞∞∞====+−−==+∑∑∑∑i i i i i 0i 0i 0i 0i 1i 11i 12222 Evaluating the first of the two terms in the above expression, we have∞==++++⋅⋅⋅∑1124i i 011182 =+=−1/21211/220 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth Edition Evaluating the second term, we havei i 0i 11234104816322∞=−=−++++++⋅⋅⋅∑ The above series can be expanded as−=−++++⋅⋅⋅=+++⋅⋅⋅=++⋅⋅⋅=+⋅⋅⋅=1 111111 48163221111 816324111 1632811 3216Therefore, we havei i 0i i 0i 111111248162i 11102∞=∞=−=−+++++⋅⋅⋅−=−+=∑∑ Adding the two terms, we have the desired proof thati i i i 0i 0i 0i 1i 1202222∞∞∞===−=+=+=∑∑∑ Consequently, we have=====∑∑NN i i i 0i 0i i E(U) 1/2ln2 ln2, since 222 If the expected utility of wealth is ln2, the corresponding level of wealth isln2U(W)ln2e W $2===Therefore, an individual with a logarithmic utility function will pay $2 for the gamble.11. (a) First calculate AVL from the insurer’s viewpoint, since the insurer sets the premiums.AVL 1 ($30,000 insurance)=0(.98)+5,000(.01)+10,000(.005)+30,000(.005)=$250AVL 2 ($40,000 insurance)=0(.98)+5,000(.01)+10,000(.005)+40,000(.005)=$300AVL 3 ($50,000 insurance)=0(.98)+5,000(.01)+10,000(.005)+50,000(.005)=$350Chapter 3 The Theory of Choice: Utility Theory Given Uncertainty 21We can now calculate the premium for each amount of coverage:Amount of Insurance Premium$30,000 30 + 250 = $280$40,000 27 + 300 = $327$50,000 24 + 350 = $374Next, calculate the insuree’s ending wealth and utility of wealth in all contingencies (states). Assume he earns 7 percent on savings and that premiums are paid at the beginning of the year. The utility of each ending wealth can be found from the utility function U(W) = ln W. (See Table S3.2a.)Finally, find the expected utility of wealth for each amount of insurance,i i i E(U(W))P U(W )=∑and choose the amount of insurance which yields the highest expected utility.Table S3.2a Contingency Values Of Wealth And Utility of Wealth (Savings = $20,000) End-of-Period Wealth (in $10,000’s) Utility ofWealthU(W) = ln WWith no insuranceNo loss (P = .98) 5 + 2(1.07) = 7.141.9657 $5,000 loss (P = .01) 5 +2.14 − .5 = 6.641.8931 $10,000 loss (P = .005) 5 +2.14 − 1.0 = 6.141.8148 $50,000 loss (P = .005) 5 +2.14 − 5.0 = 2.140.7608 With $30,000 insuranceNo loss (P = .995) 5 + 2.14 − .0280(1.07) ≅ 7.111.9615 $20,000 loss (P = .005) 5 +2.14 − .03 − 2 ≅ 5.111.6312 With $40,000 insuranceNo loss (P = .995) 5 + 2.14 − .0327(1.07) ≅ 7.1051.9608 $10,000 loss (P = .005) 5 +2.14 − .035 − 1.0 ≅ 6.1051.8091 With $50,000 insuranceNo loss (P = 1.0) 5 + 2.14 − .0374(1.07) ≅ 7.10 1.9601 With no Insurance: E(U(W)) = 1.9657(.98) + 1.8931(.01) + 1.8148(.005)+ 0.7608(.005)= 1.9582With $30,000 insurance: E(U(W)) = 1.9615(.995) + 1.6312(.005)= 1.9598With $40,000 insurance: E(U(W)) = 1.9608(.995) + 1.8091(.005)= 1.9600With $50,000 insurance: E(U(W)) = 1.9601Therefore, the optimal insurance for Mr. Casadesus is $50,000, given his utility function.22 Copeland/Shastri/Weston • Financial Theory and Corporate Policy, Fourth EditionTable S3.2b Contingency Values of Wealth and Utility of Wealth(Savings = $320,000)End-of-Period Wealth (in $10,000’s)Utility ofWealthU(W) = ln W (Wealth in $100,000’s)With no insuranceNo loss (P = .98) 5 + 32.00(1.07) = 39.24 1.3671$5,000 loss (P = .01) 5 + 34.24 − .5 = 38.74 1.3543$10,000 loss (P = .005) 5 + 34.24 − 1.0 = 38.24 1.3413$50,000 loss (P = .005) 5 + 34.24 − 5.0 = 34.24 1.2308 With $30,000 insuranceNo loss (P = .995) 5 + 34.24 − .028(1.07) ≅ 39.21 1.3663$20,000 loss (P = .005) 5 + 34.24 − .03 − 2 ≅ 37.21 1.3140 With $40,000 insuranceNo loss (P = .995) 5 + 34.24 − .0327(1.07) ≅ 39.205 1.3662$10,000 loss (P = .005) 5 + 34.24 − .035 − 1.0 ≅ 38.205 1.3404 With $50,000 insuranceNo loss (P = 1.0) 5 + 34.24 − .0374(1.07) ≅ 39.20 1.3661(b) Follow the same procedure as in part a), only with $320,000 in savings instead of $20,000. (SeeTable S3.2b above for these calculations.)With no Insurance: E(U(W)) = .98(1.3671) + .01(1.3543)+ .005(1.3413) + .005(1.2308)= 1.366162With $30,000 insurance: E(U(W)) = .995(1.3663) + .005 (1.3140)= 1.366038With $40,000 insurance: E(U(W)) = .995(1.3662) + .005(1.3404)= 1.366071With $50,000 insurance: E(U(W)) = (1)1.366092 = 1.366092The optimal amount of insurance in this case is no insurance at all. Although the numbers are close with logarithmic utility, the analysis illustrates that a relatively wealthy individual may choose no insurance, while a less wealthy individual may choose maximum coverage.(c) The end-of-period wealth for all contingencies has been calculated in part a), so we can calculatethe expected utilities for each amount of insurance directly.With no insurance:E (U(W)) = .98(U(71.4)) + .01(U(66.4)) + .005(U(61.4)) + .005U(21.4)= –.98(200/71.4) – .01(200/66.4) – .005(200/61.4) – .005(200/21.4)= –2.745 – .030 – .016 – .047= –2.838With $30,000 insurance:E(U(W)) = .995(U(71.1)) + .005(U(51.1))= – .995 (200/71.1) – .005(200/51.1)= –2.799 – .020= –2.819With $40,000 insurance:E(U(W)) = .995(U(71.05) + .005(U(61.05))= – .995(200/71.05) – .005(200/61.05)= –2.8008 – .0164= –2.8172With $50,000 insurance:E(U(W)) = (1)( −200/71) = −2.8169Hence, with this utility function, Mr. Casadesus would renew his policy for $50,000.Properties of this utility function, U(W) = −200,000W −1:=>′=−<′−==>∂=−<∂==>∂=∂-2W -3W -1W W -2MU 200,000W 0 nonsatiationMU 400,000W 0 risk aversionMU ARA 2W 0MU ARA 2W 0 decreasing absolute risk aversion WRRA W(ARA)20RRA 0 Wconstant relative risk aversion Since the individual has decreasing absolute risk aversion, as his savings account is increased he prefers to bear greater and greater amounts of risk. Eventually, once his wealth is large enough, he would prefer not to take out any insurance. To see this, make his savings account = $400,000.12. Because returns are normally distributed, the mean and variance are the only relevant parameters.Case 1(a) Second order dominance—B dominates A because it has lower variance and the same mean. (b) First order dominance—There is no dominance because the cumulative probability functionscross.Case 2(a) Second order dominance—A dominates B because it has a higher mean while they both have thesame variance.(b) First order dominance—A dominates B because its cumulative probability is less than that of B. Itlies to the right of B.Case 3(a) Second order dominance—There is no dominance because although A has a lower variance it alsohas a lower mean.(b) First order dominance—Given normal distributions, it is not possible for B to dominate Aaccording to the first order criterion. Figure S3.5 shows an example.Figure S3.5 First order dominance not possible13. (a)Prob X X pi XiXi− E(X) pi(Xi− E(X))2.1 −10 −1.0 −16.4 .1(268.96)= 26.896 .4 5 2.0 −1.4 .4(1.96) = .784 .3 10 3.0 3.6 .3(12.96) = 3.888 .2 12 2.4 5.6 .2(31.36) = 6.272 E(X)= 6.4 var (X) = 37.840Prob Y Y pi YiYi− E(Y) pi(Yi− E(Y))2.2 2 .4 −3.7 .2(13.69) = 2.738.531.5 −2.7 .5(7.29) =3.645.2 4 .8 −1.7 .2(2.89) = .578.1 303.0 24.3 .1(590.49) = 59.049E(Y) = 5.7 var(Y) = 66.010 X is clearly preferred by any risk averse individual whose utility function is based on mean and variance, because X has a higher mean and a lower variance than Y, as shown in Figure S3.6. (b) Second order stochastic dominance may be tested as shown in Table S3.3 on the following page.Because Σ(F − G) is not less than (or greater than) zero for all outcomes, there is no second order dominance.Table S3.3Outcome Prob(X) Prob(Y) Σ Px = F Σ Py= G F − G Σ (F − G)−10 .1 0 .1 0 .1 .1−9 0 0 .1 0 .1 .2−8 0 0 .1 0 .1 .3−7 0 0 .1 0 .1 .4−6 0 0 .1 0 .1 .5−5 0 0 .1 0 .1 .6−4 0 0 .1 0 .1 .7−3 0 0 .1 0 .1 .8−2 0 0 .1 0 .1 .9−1 0 0 .1 0 .1 1.00 0 0 .1 0 .1 1.11 0 0 .1 0 .1 1.22 0 .2 .1 .2 −.1 1.13 0 .5 .1 .7 −.6 .54 0 .2 .1 .9 −.8 −.35 .4 0 .5 .9 −.4 −.76 0 0 .5 .9 −.4 –1.17 0 0 .5 .9 −.4 −1.58 0 0 .5 .9 −.4 –1.99 0 0 .5 .9 −.4 –2.310 .3 0 .8 .9 −.1 –2.411 0 0 .8 .9 −.1 –2.512 .2 0 1.0 .9 .1 –2.413 0 0 1.0 .9 .1 –2.314 0 0 1.0 .9 .1 –2.215 0 0 1.0 .9 .1 −2.116 0 0 1.0 .9 .1 –1.917 0 0 1.0 .9 .1 –1.818 0 0 1.0 .9 .1 –1.719 0 0 1.0 .9 .1 –1.620 0 0 1.0 .9 .1 –1.521 0 0 1.0 .9 .1 –1.422 0 0 1.0 .9 .1 –1.323 0 0 1.0 .9 .1 –1.224 0 0 1.0 .9 .1 –1.125 0 0 1.0 .9 .1 –1.026 0 0 1.0 .9 .1 –.927 0 0 1.0 .9 .1 –.828 0 0 1.0 .9 .1 –.729 0 0 1.0 .9 .1 –.630 0 .1 1.0 1.0 0 –.61.0 1.0Because Σ (F − G) is not less than (or greater than) zero for all outcomes, there is no second order dominance.Figure S3.6 Asset X is preferred by mean-variance risk averters14. (a) Table S3.4 shows the calculations.Table S3.4p i Co. A Co. B p i A i p i [A − E(A)]2 p i B i p i [B − E(B)]2.1 0 −.50 0 .144 −.05 .4000 .2 .50 −.25 .10 .098 −.05 .6125 .4 1.00 1.50 .40 .016 .60 0.2 2.00 3.00 .40 .128 .60 .4500.1 3.00 4.00 .30 .324 .40 .62501.20 .710 1.502.0875=σ==σ=A B E(A) 1.20, .84E(B) 1.50, 1.44(b) Figure S3.7 shows that a risk averse investor with indifference curves like #1 will prefer A, whilea less risk averse investor (#2) will prefer B, which has higher return and higher variance.Figure S3.7 Risk-return tradeoffs (c) The second order dominance criterion is calculated in Table S3.5 on the following page.15. (a) False. Compare the normally distributed variables in Figure S3.8 below. Using second orderstochastic dominance, A dominates B because they have the same mean, but A has lower variance. But there is no first order stochastic dominance because they have the same mean and hence thecumulative probability distributions cross.Figure S3.8 First order stochastic dominance does not obtain (b) False. Consider the following counterexample.Table S3.5 (Problem 3.14) Second Order Stochastic DominanceReturn Prob(A) Prob(B) F(A) G(B) F − G Σ (F − G) −.50 0 .1 0 .1−.1 −.1−.25 0 .2 0 .3−.3 −.40 .1 0 .1 .3−.2 −.6.25 0 0 .1 .3−.2 −.8.50 .2 0 .3 .3 0 −.8.75 0 0 .3 .3 0 −.81.00 .4 0 .7 .3 .4 −.41.25 0 0 .7 .3 .4 01.50 0 .4 .7 .7 0 01.75 0 0 .7 .7 0 02.00 .2 0 .9 .7 .2 .22.25 0 0 .9 .7 .2 .42.50 0 0 .9 .7 .2 .62.75 0 0 .9 .7 .2 .83.00 .1 .2 1.0 .9 .1 .9 3.25 0 0 1.0 .9 .1 1.0 3.50 0 0 1.0 .9 .1 1.13.75 0 0 1.0 .9 .1 1.24.00 0 .1 1.0 1.0 0 1.21.0 1.0Because Σ (F − G) is not always the same sign for every return, there is no second order stochastic dominance in this case.Payoff Prob (A) Prob (B) F (A) G (B) G (B) − F(A)$1 0 .3 0 .3 .3$2 .5 .1 .5 .4 −.1$3 .5 .31.0 .7 −.3$4 0 .31.0 1.0 01.0 1.0E(A) = $2.50, var(A) = $.25 squaredE(B) = $2.60, var(B) = $1.44 squaredThe cumulative probability distributions cross, and there is no first order dominance.(c) False. A risk neutral investor has a linear utility function; hence he will always choose the set ofreturns which has the highest mean.(d) True. Utility functions which have positive marginal utility and risk aversion are concave. Secondorder stochastic dominance is equivalent to maximizing expected utility for risk averse investors.16. From the point of view of shareholders, their payoffs areProject 1 Project 2Probability Payoff Probability Payoff.2 0 .4 0.6 0 .2 0.2 0 .4 2,000Using either first order or second order stochastic dominance, Project 2 clearly dominatesProject 1.If there were not limited liability, shareholder payoffs would be the following:Project 1 Project 2Probability Payoff Probability Payoff.2 −4000 .4 −8000.6 −3000 .2 −3000.2 −2000 .4 2,000In this case shareholders would be obligated to make debt payments from their personal wealthwhen corporate funds are inadequate, and project 2 is no longer stochastically dominant.17. (a) The first widow is assumed to maximize expected utility, but her tastes for risk are not clear.Hence, first order stochastic dominance is the appropriate selection criterion.E(A) = 6.2 E(D) = 6.2E(B) = 6.0 E(E) = 6.2E(C) = 6.0 E(F) = 6.1One property of FSD is that E(X) > E(Y) if X is to dominate Y. Therefore, the only trusts which might be inferior by FSD are B, C, and F. The second property of FSD is a cumulative probability F(X) that never crosses but is at least sometimes to the right of G(Y). As Figure S3.9 shows, A >C andD > F, so the feasible set of trusts for investment is A, B, D, E.Figure S3.9 First order stochastic dominance(b) The second widow is clearly risk averse, so second order stochastic dominance is the appropriateselection criterion. Since C and F are eliminated by FSD, they are also inferior by SSD. The pairwise comparisons of the remaining four funds, Σ(F(X) − G(Y)) are presented in Table S3.6 on the following page and graphed in Figure S3.10. If the sum of cumulative differences crosses the horizontal axis, as in the comparison of B and D, there is no second order stochastic dominance. By SSD, E > A, E > B, and E > D, so the optimal investment is E.Table S3.6 Second Order Stochastic DominanceRet. P(A)* P(B) P(D) P(E) SSD**(BA)SSD(DA)SSD(EA)SSD(DB)SSD(EB)SSD(ED)−2 −.1 −.1−1 0 .1 .2 0 .2 .2 0 0 −.2 −.20 0 .2 .2 0 .4 .4 0 0 −.4 −.41 0 .3 .2 0 .7 .6 0 −.1 −.7 −.62 0 .3 .4 0 1.0 1.0 0 0 −1.0 −1.03 0 .4 .4 0 1.4 1.4 0 0 −1.4 −1.44 0 .5 .4 0 1.9 1.8 0 −.1 −1.9 −1.85 .4 .5 .4 .4 2.0 1.8 0 −.2 −2.0 −1.86 .6 .5 .5 .4 1.9 1.7 −.2 −.2 −2.1 −1.97 .8 .5 .6 1.0 1.6 1.5 0 −.1 −1.6 −1.58 1.0 .6 .6 1.0 1.2 1.1 0 −.1 −1.2 −1.19 1.0 .6 .7 1.0 .8 .8 0 0 −.8 −.810 1.0 .7 .8 1.0 .5 .6 0 .1 −.5 −.611 1.0 .8 .8 1.0 .3 .4 0 .1 −.3 −.412 1.0 .9 .8 1.0 .2 .2 0 0 −.2 −.213 1.0 1.0 .8 1.0 .2 0 0 −.2 −.2 014 1.0 1.0 1.0 1.0 .2 0 0 −.2 −.2 0A >B A > D A < E no2ndorderdominance B < ED< E** SSD calculated according to Σ (F(X) − G(Y)) where F(X) = cumulative probability of X and G(Y) = cumulative probability of Y.18. (a) Mean-variance ranking may not be appropriate because we do not know that the trust returns havea two-parameter distribution (e.g., normal).To dominate Y, X must have higher or equal mean and lower variance than Y, or higher mean and lower or equal variance. Means and variances of the six portfolios are shown in Table S3.7. Bymean-variance criteria, E > A, B, C, D, F and A > B, C, D, F. The next in rank cannot bedetermined. D has the highest mean of the four remaining trusts, but also the highest variance. The only other unambiguous dominance is C > B.Figure S3.10 Second order stochastic dominanceTable S3.7E(X) var(X)B 6.0 26.80C 6.0 2.00D 6.2 28.36E 6.2 0.96F 6.1 26.89(b) Mean-variance ranking and SSD both select trust E as optimal. However, the rankings ofsuboptimal portfolios are not consistent across the two selection procedures.Optimal Dominance R elationshipsFSD A, B, D, E A > C, D > FSSD E A > B, A > DM-V E A > B, C, D, F; C > B。
关于函数一致连续性的研究
II
关于函数一致连续性的研究
目录
第一章 绪 论 ....................................................... 1 1.1 选题背景........................................................ 1 1.2 研究意义........................................................ 1
第四章 常见函数的一致连续性问题 .................................... 18 4.1 基本初等函数的一致连续性....................................... 18
4.1.1 幂函数.......................................................... 18 4.1.2 指数函数 ........................................................ 19 4.1.3 对数函数 ........................................................ 20 4.1.4 三角函数 ........................................................ 21 4.1.5 反三角函数 ...................................................... 22
3.8 二元函数的一致连续性问题....................................... 16
Lecture 3 _ Utility Maximization in Portfolio Choices
The decision variable is α , Setting we obtain
giving = 2p − 1.
5
Lesson Learnt
If we require α ≥ 0, then the existence of the above solution implicitly requires p ≥ 0.5. If p < 0.5, the value for α for optimal growth is given by α =0
2
General formula
Let Xk represent the capital after the kth trial, then Xk = Rk*Xk−1 where Rk is the random return variable. All Rk’s have identical probability distribution and they are mutually independent. (Optimal trading strategy alpha is constant over time) The capital at the end of n trials is Xn = (Rn)*(Rn−1 )· · (R2)*(R1)*X0: taking logarithm on both side, we have: or
7
Preference – pairwise comparison
Consider the set of alternatives B, how to determine which element in the choice set B that is preferred?
再保险-投资的M-V及M-VaR最优策略
再保险-投资的M-V及M-VaR最优策略王海燕;彭大衡【摘要】考虑保险公司再保险-投资问题在均值-方差(M-V)模型和均值-在险价值(M-VaR)模型下的最优常数再调整策略.在保险公司盈余过程服从扩散过程的假设及多风险资产的Black-Scholes市场条件下,分别得到均值-方差模型和均值-在险价值模型下保险公司再保险-投资问题的最优常数再调整策略及共有效前沿,并就两种模型下的结果进行了比较.【期刊名称】《经济数学》【年(卷),期】2011(028)003【总页数】6页(P71-76)【关键词】再保险-投资;均值-方差模型;均值-在险价值模型;常数再调整策略【作者】王海燕;彭大衡【作者单位】广东商学院数学与计算科学学院;广东商学院金融学院,广州 510320【正文语种】中文【中图分类】F830.9投资是保险公司获取利润的主要手段之一,再保险则主要用来控制保险公司的风险暴露.近年来,在风险模型中综合考虑再保险与投资策略成为一个研究的热点,最大化再保险-投资策略下盈余的期望效用和最小化再保险-投资策略下的破产概率是两种主要的模型选择.这两种模型的建立都依赖于对保险公司盈余过程的定量刻画.从已有文献来看,主要有两种刻画保险公司盈余过程的数学模型,一种是Cramer-Lundberg模型,也称为经典的风险过程;另一种是扩散模型,是对Cram er-L undberg风险过程的一种近似,在描述大型保险公司的盈余过程时,由于单个索赔额相对于盈余总量很小,用扩散模型近似Cramer-Lundberg风险过程被认为是可行的.B row ne[1]利用带飘移的布朗运动刻画保险公司的盈余过程,研究了使终端盈余在指数效用函数下的期望值最大化和使公司破产概率最小化的最优投资组合策略;利用与文献[1]相同的公司盈余过程的假设,Promislow和Young[2]研究了使公司破产概率最小的比例再保险-投资策略;Taksar和Markussen[3]也在盈余过程的扩散模型下得到了破产概率最小化的最优再保险-投资策略;Cao和Wan[4]在风险资产不允许卖空的条件下通过求解HJB方程得到指数效用函数和幂效用函数情形下的最大化期望效用的再保险-投资策略;Luo、Taksar和Tsoi[5]对Black-Scho les市场中风险资产具有不同投资约束时的情形进行了研究,等等.文献[1-5]只是在单一风险资产的Black-Scho les市场环境中开展研究.Bai和Guo[6]则在多风险资产的Black-Scho les市场环境中对具有卖空约束条件下的最优再保险-投资策略进行了研究,得到了指数效用函数期望值最大化和破产概率最小化的最优策略结果;Zhang、Zhang和Yu[7]在多风险资产市场环境中考虑具有交易成本时使终端盈余效用的期望最大化问题,得到了最优再保险-投资策略及最优值函数的显式解,同时,文[7]在建模中,增加了保险公司对盈余的条件VaR值的风险控制.其他盈余过程下,包括Schmidli[8]用Cramer-Lundberg模型、Yang和Zhang[9]用带跳的扩散模型、Irgens和Paulsen[10]在Cramer-L undberg模型中加入扩散扰动等,都有相应的研究结果.希望破产概率尽量小或终端盈余期望效用尽量大,都是保险公司进行再保险与投资的偏好结构的反映.破产概率最小化模型把公司安全性放在突出位置,这符合保险公司经营的客观需求,但模型对再保险-投资最优策略下收益并不直接反映,不利于在风险管理的同时对收益的考量;终端盈余期望效用最大化模型由于效用本身的抽象属性使模型更具理论价值而缺乏实际的可操作性.另外,现有文献对再保险-投资最优策略研究结果表明:最优策略都是关于时间变量连续变动的.这样的结果虽然具有理论上的一般性,但却给实际交易造成了困难,连续变动的交易及调整是不现实的.解决这一困难的办法是利用所谓的“常数再调整策略”进行近似处理.将再保险-投资策略设定为与时间无关的常量,试图建立关于再保险-投资的某一“共同基金”,通过每个决策初始时刻对共同基金作出一个倍数的调整来实现整个时间段上策略选择的近似最优.这种想法最初由Emmer等在文[11]中提出,之后,对风险度量指标为EaR(Earning-at-Risk,在险收益),或在具有投资机会约束等情形,文[12-15]在多风险资产的Black-Scholes市场环境中都得到了相应的最优常数再调整投资策略,但再保险风险转移机制在其中未予以考虑.本文考虑保险公司再保险-投资问题在均值-方差模型和均值-VaR模型下的最优常数再调整策略.在保险公司盈余过程服从扩散过程的假设及多风险资产的Black-Scho les市场条件下,分别得到均值-方差模型和均值-VaR模型下保险公司再保险-投资问题的最优常数再调整策略及其有效前沿,并就两种模型下的结果予以比较. 设保险公司遭遇的索赔额由带飘移的布朗运动刻画(参考文献[2]):其中,a,b为正的常数.保费增长速度为c=(1+ θ)a,θ(>0)为安全附加系数.基于(1),保险公司的盈余过程R1(t)满足:考虑保险公司在进行投资组合的同时,通过再保险进行风险管理.记q(t)表示时刻t 的再保险分出比例,η(>θ)表示再保险的安全附加系数.基于(2),保险公司通过再保险安排后的盈余过程R2(t)满足:假设X(0)=x>0表示保险公司的初始盈余.称α=(q(t),π(t))为再保险-投资策略.再保险-投资策略α被称为是可行的,如果全体可行的再保险-投资策略构成的集合记为αS.考虑再保险-投资的常数再调整投资策略,假设q(t)=q,πi(t)=πi,保险公司动态决策的特征只表现在每个计划期初对最优投资策略的一个常数倍的调整.从式(4)可以解得:其中Nβ是标准正态随机变量的β下侧分位数水平.把E[X(t)]-Cβ称作再保险-投资策略下盈余的在险价值(VaR),则有常数再调整投资策略下,再保险-投资最优策略的均值-方差模型如下:对某一给定的时刻T(>0),其中,是事先给定的某盈余水平.常数再调整投资策略下,再保险-投资最优策略的均值-在险价值(M-VaR)模型如下:对某一给定的时刻T(>0),首先,对任给ε>0,在椭球面||ασ||=ε上,目标函数为f(ε)=(2r)-1 e2rT-1ε2.其次,在ε可能的范围内,寻求使f(ε)达到最小值的最小的ε.由于约束条件左边满足时,式(9)取等号.因此,为了让模型(7a)中约束条件取到等号的ε＊值最小以实现目标函数值取到最小,当且仅当(变换后的)再保险-投资策略取α=αε＊.于是,上述分析结果可概括为:结论1 均值-方差(M-V)模型(7)在常数再调整投资策略下的最优再保险-投资策略由式(12)和(13)给出,再保险-投资策略的有效前沿由式(14)给出.注1 因为假设保险公司盈余的期望水平不小于x e rT,所以从式(14)得知Var[X(T)]的取值范围应该满足:保险公司的风险承受能力如果达不到这一下限要求,则最好的策略是把初始资金全部投资于无风险资产.把模型(8)变换为如下等价的模型(8a):回复到初始的变量表示,最优再保险-投资策略为:在最优策略下,目标函数值(即盈余的期望值)为对应的盈余V aR值为C.因此,最优常数再调整策略下,再保险-投资策略的有效前沿(V aR,上述分析结果可概括为:结论2 均值-在险价值(M-V aR)模型(8)在常数再调整投资策略下的最优再保险-投资策略由式(16)和(17)给出,再保险-投资策略的有效前沿由式(18)给出.注2 保险公司把初始资金全部投资于无风险资产即可得到确定的盈余x e rT,因此,从式(18)得知VaR应该满足:保险公司的风险承受能力若达不到这一下限要求,则最好的策略是把初始资金全部投资于无风险资产.分别是以标准差和在险价值作为风险度量时的风险价格.两模型的不同之处在于有效前沿的斜率不一定相同,即风险价格不一定相同,这是因为决策者的偏好、尤其是对待风险的态度不一样造成的.3)从注1和注2看出,无论在M-V模型还是在M-VaR模型下,保险公司风险承受能力都存在下限约束,表现为v1,v2>0.但同时应该看到,决策期限T越大,风险承受能力的下限约束越强,而T越小时,风险承受能力的下限约束越弱,特别地,若T→0,则v1,v2→0.因此,不同风险承受能力的保险公司可以选择不同的决策期限进而作出相应的再保险-投资组合的最优常数再调整策略,风险承受能力强的保险公司在决策期限的选择方面具有相对优势.4)索赔过程的随机驱动因素与风险资产的随机驱动因素相互独立,导致风险资产组合不能对冲索赔风险,但通过再保险可以进行风险转移,因此,把再保险作为策略的一部分,保险公司对来自索赔过程的风险可以根据自身的风险承受能力自由选择自留比例直至全部分出.有效前沿式(14)和式(18)与这种直观认识正好吻合.若没有再保险机制的安排,则有效前沿不一定表现为射线,例如,可以参看Xie、Li和Wang[16]的式(33).假设保险公司初始盈余X(0)=x=1亿元.T= 1,无风险利率水平r=0.05,索赔过程参数a=2,b =2,原保险安全附加系数θ=0.1,再保险安全附加系数η=0.15,置信水平β=0.05,从而Nβ=-1.65.设市场有一种无风险资产,三种风险资产,从而n=3,且假设三种风险资产的预期收益率为(μ1,μ2,μ3)’= (0.15,0.20,0.25)’,波动性矩阵为1)若保险公司希望期末盈余的期望值在不低于K=1.5亿元时最小化再保险-投资策略的方差,则根据式(12)和(13)计算得到:q=0.607,π= (0.503,0.587,0.291),投资于无风险资产的金额数为:1×(1-q)-(0.503+0.587+0.291)= -0.988(亿元).即保险公司通过分出60.7%的原保险业务,做空无风险资产获得0.988亿元,做多三种风险资产,金额分别为0.503亿元、0.587亿元和0.291亿元,可以实现期末盈余的期望值不低于K =1.5亿元时的盈余的方差最小,最小方差值为0.011 8平方亿元.2)若保险公司希望期末盈余的VaR值不超过0.6亿元时最大化盈余的期望值,则根据式(16)和(17)计算得到:q=0.795,π=(0.524,0.622, 0.303),投资于无风险资产的金额数为:1×(1-q) -(0.524+0.622+0.303)=-1.244(亿元).最大化盈余的期望值为1.8286亿元.为了便于保险公司对风险管理与收益考量的实际决策,采用与已有文献不同的风险/收益型再保险-投资模型,分别得到了均值-方差(M-V)模型和均值-在险价值(M-VaR)模型下的最优常数再调整策略及其有效前沿,并通过对两种模型下的结果比较发现:最优策略都表现为“共同基金”(¯μ-r1n+1)′(¯σ¯σ′)-1/‖¯σ-1(¯μ-r1n+1)‖的某个倍数,在每个决策期的初始时刻,决策者只要决定投资于该共同基金的一个最优倍数.由于考虑了再保险风险转移机制,本文所得“共同基金”与文献[12-15]所得共同基金虽然形式类似,但本质上是不一样的.M-V和M-VaR两模型下的再保险-投资策略有效前沿分别表现为从点(vi,x e rT)(i=1,2)出发向右上方延伸的射线,射线的斜率正是各自的风险价格.因为把再保险作为策略的一部分,保险公司对来自索赔过程的风险可以根据自身的风险承受能力自由选择自留比例直至全部分出.【相关文献】[1] S BROWNE.Optimal 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Insurance:Mathematics and Economics,2008,42:968-975.[7] X ZHANG,K ZHANG,X YU.Op timal p roportional reinsurance and investmentw ith transaction costs,I:M aximizing the terminal wealth[J].Insurance:Mathematics and Economics, 2009,44:473-478.[8] H SCHM IDL I.On minimizing the ruin p robability by investment andreinsurance[J].The Annals of App lied Probability, 2002,12(3):890-907.[9] H YANG,L ZHANG.Op timal investment for insurer with jump-diffusion risk process[J].Insurance:M athem atics and Economics,2005,37:615-634.[10]C IRGENS,J PAULSEN.Optimal control of risk exposure, reinsurance and investments for insurance portfolios[J].Insurance:Mathematics and Economics,2004,35:21-51.[11]S EMMER,C KLUPPELBERG,R KORN.Op timal portfolios with bounded capital atrisk[J].Math Finance,2001,11 (4):365-384.[12]李仲飞,汪寿阳.EaR风险度量与动态投资决策[J].数量经济技术经济研究,2003,20(1):45-51.[13]彭大衡.长期投资组合的连续时间模型[J].湖南大学学报:自然科学版,2004,31(1):103-107.[14]彭大衡,姚元端.带机会约束的动态投资决策模型研究[J].中国管理科学,2005,13(1):9-13.[15]王海燕,彭大衡.动态投资组合决策中机会收益与在险收益的权衡[J].经济数学,2008,25(1):28-35.[16]S X IE,Z L I,S WANG.Continuous-time po rtfolio selection w ith liability:M ean-va riance model and stochastic LQ approach[J].Insurance:Methematics and Economics,2008,42:943-953.。
UtilityFunctions...
Utility Functions, Risk Aversion Coefficients and TransformationsLecture VII. An examination of the Arrow-Pratt Coefficients for particular functions.A. Quadratic Utility Function: To specify the appropriate shape of the utility function, the quadratic function becomes()()()222U w aw bw U w a bw U w b=−′=−′′=−Arrow-Pratt absolute risk aversion coefficient:()()()()()()()()22222221202A A A b b R w R w a bw a bw d R w f x b d b dw dx f x a bw f x −=−⇒=−− ′=>⇐=− − Arrow-Pratt relative risk aversion coefficient()()()()22222222022R R b b R w w a a bw b w d R w b b b dw a bw a bw == − − =+>−−B. Power Utility Function:()()()111rrr w U w rU w w U w rw −−−−=−′=′′=−Arrow-Pratt absolute risk aversion coefficient:()()120r A r A rw r R w w w d R w r dw w−−− −=−= =−< Arrow-Pratt relative risk aversion coefficient:()()10r R r R rw w R w r w d R w dw−−− =−= = Constant relative risk aversion.C. Negative Exponential Utility Function:()()()()()()2exp exp exp U w w U w w U w w =−−ρ′=ρ−ρ′′=ρ−ρArrow-Pratt absolute risk aversion coefficient()()()()2exp exp 0A A w R w w d R w dw−ρ−ρ=−=ρ ρ−ρ =Constant absolute risk aversion. Arrow-Pratt relative risk aversion coefficient()()()()2exp exp 0R R w R w w w w d R w dw−ρ−ρ=−=ρ ρ−ρ =ρ>D. HARA–Hyperbolic Absolute Risk Aversion: ()()()()()112221,0111111111aw U w b b aw a U w b aw a b aw a U w a b aw a b γγ−γ−γ−γ− −γ=+> γ−γ′=−γ+ −γ−γ=+ −γ ′′=γ−+ −γ−γ=−+ −γArrow-Pratt absolute risk aversion coefficient()()()()()2211111111111A aw a b R w aw a b a a b aw aw b a a aw b aw b γ−γ− −+ −γ=− + −γ==−γ ++ −γ−γ−γ−γ==+−γ+−γ−γ1. As1γ→it becomes risk neutral. 2. 2γ= is a quadratic.3. γ→−∞ and 1b = is the negative exponential.4. 0b = and 1γ< is the power utility function. II. Interpretations and Transformations of Scale for the Pratt-Arrow Absolute RiskAversion coefficient: Implications for Generalized Stochastic DominanceA. To this point, we have discussed technical manifestations of risk aversion suchas where the risk aversion coefficient comes from and how the utility ofincome is derived. However, I want to start turning to the question: How dowe apply the concept of risk aversion?B. Several procedures exist for integrating risk into the decision making processsuch as direct application of expected utility, mathematical programmingusing the expected value-variance approximation, or the use of stochasticdominance. All of these approaches, however, require some notion of therelative size of risk aversion.1. Risk aversion directly uses a risk aversion coefficient to parameterizethe negative exponential or power utility functions.2. Mathematical programming uses the concept of the tradeoff betweenvariance and expected income.3. Stochastic dominance uses measures of risk aversion to bound theutility function.C. The current study gives some guidance on using previously published riskaversion coefficients. Specifically, the article looks at the effect of locationand scale on the risk aversion coefficient1. As a starting place, we develop an interpretation of the Pratt-Arrowcoefficient in terms of marginal utility()r x u x u x d u x d x u x d d x u x d u x u x d x()''()'()'()'()ln '()'()'()=−=− =−=−This algebraic manipulation develops the absolute risk aversion coefficient as the percent change in marginal utility at any level of income.a. Therefore, r is associated with a unit of change in outcomespace. If the risk aversion coefficient was elicited in outcomesof dollars, then the risk aversion coefficient is .0001/$.b. This result indicates that the decision-maker’s marginal utilityis falling at a rate of .01% per dollar change in income.2. This association between the risk aversion and the level of income then raises the question of the change in outcome scale.a. For example, what if the original utility function was elicitedon a per acre basis, and you want to use the results for a wholefarm exercise?b. Theorem 1: Let r (x )=u ’’(x )/u ’(x ). Define a transformation ofscale on x such that w =x /c , where c is a constant. Thenr (w )=cr (x ). (1) The proof lies in the change in variables. Given ()U x U wc d U d w d U d x d x d wcU x d U d wd d w d U d w d d w cU x c U x ()()'()'()''()==== ==222 (2) In other words, if the scale of the outcome changes byc , the scale of the risk aversion coefficient must bechanged by the same amount.c. Theorem 2: If v =x + c , where c is a constant, then r (v )=r (x ).Therefore, the magnitude of the risk aversion coefficient isunaffected by the use of incremental rather absolute returns.D. Example: Suppose that a study of U.S. farmers gives a risk aversion coefficient of r =.0001/$ (U.S.) Application to the Australian farmers whose dollar is worth .667 of the U.S. dollar is r =.0000667/$ Australian.。
Lecture00_绪论
• 以上特点非常适合于进行模型基础上的分析
电子表格建模
• 例1：一个房贷模型
基于固定利率及等额 分期付款方式，计算 贷款的每期付款额。
= 总价 – 首付金 = PMT(年利率/12,贷款期,-总贷款额) = 月贷款额×贷款期 – 总贷款额
贷款期 t 1 贷款期 t 月还贷额 (1 年利率 /12)
Arguments correspond to Adverbs
附：The Hierarchy of objects
Excel的所有对象，或者由其他对象组
成，或者是其他对象的一部分，或者二 者兼是。
例如，Workbook对象包含Worksheet对象， 而Worksheet对象包含Cell对象
Excel对象模型表示了对象之间的包含
关系。
附：The Hierarchy of objects
Excel 对象模型
进入VBA窗口， 选择 “帮助”菜 单，点击 “Microsoft Excel Visual Basic参考” 中的“Microsoft Excel对象模型 ”。
入了统一的应用程序自动化语言—— Visual Basic For Application (VBA)，并提供了 VBA 的 IDE 环境。 • VBA 将 Excel 作为开发平台来开发应用程序，
可应用 Excel 的所有现有功能，从而降低编程
难度，加快开发速度。
管理决策应用程序
• 应用程序主要由以下 4 个部分组成
Analogy:
“Things”:
Object correspond to Noun
Attributes of object:
Properties correspond to Adjectives
金融衍生品外文翻译
3.
在上一节中，我们提出了一个代表性定理对于那些添加剂的独立价格措施R.V.的。得到的价格表示可以被视为一个（混）Esscher变换概率预期下测量。在本节中，我们将介绍一个密切相关的概率测度变换和公理化的价格措施由它引起的。
对于一个给定的r.v。X,我们定义扩展实值函数 :
我们介绍了实值函数 定义为：
（10）
此后，函数 被称为Esscher-Girsanov 该R.V.X的价格。参数h 0和v> 0。鉴于V，存在X之间的唯一对应关系其Esscher-Girsanov价格在某种意义上说，X = Y在分布成立当且仅当
（11）
为了验证这种说法,有
（12）
上式可以被视为一个拉普拉斯变换,所以在函数 和 之间有一一对应且一致的关系。其中，函数 对于给出的h的导数为：
我们考虑一个齐次时间的主要资产的过程， ，其定义为一个随机微分方程的形式：
（17）
其中， 表示一个标准布朗运动。我们指出在一般的情况下S不需要是正数，因为它代表一个任意的主要资产。然而，如果一个已在考虑应用程序需要积极主要资产的过程，μ（·）和β（·）的附加条件就可以实施。
接着，我们考虑一个债券价格的过程 ，定义SDE为
由此不难验证为具有正常的CDF期望μ和方差 ，其Esscher-Girsanov变换与期望μ+ HV正常的CDF和方差σ2。在特别是，当v= ，我们平凡发现，对Esscher-Girsanov 变换是一个普通的Esscher变换。因此，对于一个正常的
CDF的Esscher-Girsanov变换，就像普通Esscher变换，改变了平均同时保留方差。请注意，对于平均值的变化，该值均值是无关紧要的。在下文中，我们令v是严格正和暂时固定和h的域限制到h 0。
ansys界面菜单翻译
Abbr --> 缩写Abbreviation --> 缩写词About --> 关于absolut --> 绝对Active --> 当前add --> 增加add/edit/delete --> 增加/编辑/删除Additional Out --> 附加输出adius --> 心Adjacent --> 相邻Adv --> 高级Advection --> 对流Algorithm --> 算法align --> 定位Align WP with --> 工作区排列按ALPX --> 热膨胀系数Also 副词再Ambient Condit'n --> 环境条件amplitude --> 振幅Analysis --> 分析Angle --> 角度Angles --> 角度Angular --> 角度Animate --> 动画Animation --> 动画Anno --> 注释Anno/Graph --> 注释/图Annotation --> 注释文字Annulus --> 环面ANSYS Multiphysics Utility Menu--> ANSYS 综合物理场有限元分析菜单Any --> 任意apply --> 应用Arbitrary --> 任意arccosine --> 反余弦Archive --> 合并Arcs --> 圆弧线arcsine --> 反正弦area --> 面Area Fillet --> 面圆角Area Mesh --> 已划分的面Areas --> 面Array --> 数组arrow --> 箭头Assembly --> 部件At Coincid Nd --> 在两节点间Attch 动词接触Attr --> 特征Attrib --> 属性Attributes --> 属性Auto --> 自动Automatic Fit --> 自适应Axes --> 坐标轴Axis --> 坐标轴Axi-Symmetric --> 轴对称back up --> 恢复Background --> 背景Banded --> 条状Based --> 基础BC --> 边界Beam --> 梁behavior --> 特性Bellows --> [密封]波纹管Bias --> 偏置Biot Savart --> 毕奥-萨瓦河Bitmap --> BMP图片Block --> 块Body --> 体Booleans --> 布尔操作box --> 框Branch --> 分支brick orient --> 划分块(方向)Builder --> 生成器Built-up --> 合成Buoyancy Terms --> 浮力项By Circumscr Rad --> 外切正多边形By End KPs --> 始点、终点By End Points --> 直径圆By End Pts --> 底圆直径By Inscribed Rad --> 内接?正多边形By Picking --> 鼠标选取By Side Length --> 通过边长确定多边形By Vertices --> 通过顶点确定多边形calc --> 运算Calcs --> 计算Capacitor --> 电容Capped/Q-Slice --> 切面透明度设置Capping --> 盖Capture --> 打印Cartesian --> 笛卡儿坐标系Case --> 情况CE Node Selected --> 约束节点选择cent 中心Center --> 中心centr 中心ceqn --> 约束CFD --> 计算流体力学(CFD) Change 动词更换Check --> 检查Checking --> 检查Checks --> 检查Circle --> 圆Circuit --> 电路circumscr --> 外接圆Clr Size --> 清除尺寸CMS --> 组件模式综合Cnst --> 常数Cntl --> 控制Cntrls --> 控制Coincident --> 重合Collapse --> 折叠收起Color --> 颜色Colors --> 颜色Common --> 普通Comp --> 组件complex variable --> 复数变量Component --> 组件Components --> 组件Compress --> 精减Concats --> 未划分Concentrate --> 集中concrete --> 混凝土Cond --> 导体Conditions --> 条件cone --> 圆锥Configuration --> 配置Connectivity --> 连通性Connt --> 连通区域consistent --> 固定Const --> 常数Constant Amplitude --> 恒幅Constants --> 常数Constr --> 约束Constraint --> 约束Constraints --> 约束constreqn --> 约束方程Contact --> 接触Contour --> 等值线Contour Plot --> 等值云图Contours --> 等值线contraction --> 收缩因子Control --> 控制Controls --> 控制CONVERGENCE INDICATOR --> 收敛精度CONVERGENCE VALUE --> 收敛值Convert ALPx -->热膨胀系数转换Coor --> 坐标系Coord --> 坐标Coord Sys --> 坐标系coordinate --> 坐标Coordinates --> 坐标Coords --> 坐标corner --> 对角Corners --> 对角cornr --> 对角correl field --> 相关性区域correlation --> 相关性count --> 总数Couple --> 耦合Coupled --> 耦合Coupling --> 耦合CP Node Selected --> 耦合节点选择Create 动词新建creep --> 蠕变criteria --> 准则cross product --> 向量积cross-sectional --> 截面CS --> 坐标系csys --> 坐标系ctr --> 中点ctrl --> 控制ctrls --> 控制Cupl --> 耦合Curr --> 电流curvature --> 圆弧Curvature Ctr --> 曲率中心Curve --> 曲线custom --> 定制Cyc --> 循环Cyclic Expansion -->循环扩展设置Cyclic Model --> 周向模型Cyclic Sector --> 扇型周向阵列cylinder --> 圆柱Cylindrical --> 柱坐标系Damper --> 阻尼[减震]器damping --> 阻尼系数Data --> 数据Data Tables --> 数据表格Database --> 数据库DB --> DB definitns --> 特征定义Deformed --> 已变形Degen --> 退化Degeneracy --> 退化Del --> 删除Del Concats --> 删除连接Delete --> 删除dependent --> 相关derivative --> 导数Design Opt --> 优化设计Device --> 设备differentiate --> 微分Digitize --> 数字化dimensions --> 尺寸Diode --> 二极管Directory --> 目录discipline --> 练习Displacement --> 变形Display --> 显示distances --> 距离Divide --> 划分Divs --> 位置DOF --> 自由度dofs --> 自由度dot product --> 点积Dupl --> 复制edge --> 边缘Edit --> 编辑Elbow --> 弯管[肘管]ElecMech --> 电磁ElecStruc --> 静电-结构electr --> 电磁Electric --> 电气类electromag --> 电磁electromagnetic --> 电磁Electromechanic --> 电-机械elem --> 单元Elem Birth/Death --> 单元生/死Element --> 单元Elements --> 单元Elems --> 单元Elm --> 单元EMT CDISP --> 电磁陷阱CDISP Enable 形容词允许ENDS --> 端energy --> 能量ENKE --> 湍动能量Entities --> 实体Entity --> 实体EPPL COMP --> 塑性应变分量EPTO COMP --> 总应变eq --> 方程Eqn --> 方程Eqns --> 方程equation --> 方程式Erase --> 删除Est. --> 估算Everything --> 所有EX --> 弹性模量EX exclude --> 排除Execute --> 执行Execution --> 执行Expansion --> 扩展Expend All --> 展开全部Exponential --> 幂数[指数] exponentiate --> 幂指数Export --> 模型输出Ext Opts --> 拉伸设置Extend Line --> 延伸线extra --> 附加extreme --> 极值Extrude --> 拉伸EY --> 弹性模量EY EZ --> 弹性模量EZ face --> 面Facets --> 表面粗糙fact --> 因子factor --> 系数factr --> 因子failure --> 破坏Fast Sol'n --> 快速求解Fatigue --> 疲劳FD --> 失效挠度field --> 区域Fill --> 填充Fill between KPs -->关键点间填入Fill between Nds --> 节点间填充fillet --> 倒角Fit --> 适当视图Flange --> 法兰Flip --> 翻转Floating Point --> 浮点FLOTRAN --> 流体FLOTRAN Set Up -->流体运行设置Flow --> 流量Fluid --> 流体Flux --> 通量Fnc_/EXI --> 退出Fnc_/GRAPHICS --> 图形界面Focus Point --> 焦点force --> 力Format --> 格式Fourier --> 傅立叶级数Free --> 自由Freq --> 频率From Full --> 完全Full Circle --> 完整圆Func --> 函数function --> 函数Functions --> 函数Gap --> 间隙Gen --> 一般General --> 通用General Options --> 通用设置General Postproc-->通用后处理器Generator --> 生成器Genl --> 普通Geom --> 单元Geometry --> 几何形状Get --> 获取Global --> 全局Globals --> 全局Glue --> 粘合gradient --> 梯度Graph --> 图Graphics --> 图形Graphs --> 图Gravity --> 引力(重力)Grid --> 网格GUI --> 图形用户界面GXY --> 剪切模量GXY GXZ --> 剪切模量GXZ GYZ --> 剪切模量GYZ hard --> 硬Hard Points --> 硬点Hard PT --> 硬点hardening --> 强化hex --> 六面体Hexagon --> 六边形Hexagonal --> 六棱柱hidden --> 隐藏higher-order --> 高阶Hill --> 希尔h-method --> 网格细分法hollow --> 空心Hollow Cylinder --> 空心圆柱体Hollow Sphere --> 空心球体hp-method --> 混合并行法I-J --> I-J imaginary --> 虚部Immediate --> 即时Import --> 模型输入Improve --> 改进independent --> 非相关Individual --> 单个Indp Curr Src --> 感应电流源Indp Vltg Src --> 感应电压源Inductor --> 电感Inertia --> 惯性Inertia Relief Summ --> 惯量概要Inf Acoustic --> 无穷声学单元init --> 初始化Init Condit'n --> 初始条件Initial --> 初始inquire --> 查询inscribed --> 内切圆Installation --> 安装int --> 强度integral --> 积分integrat --> 积分integrate --> 积分interactive --> 交互式Interface --> 接触面intermed --> 中间interpolate --> 插入Intersect --> 相交invert --> 切换is done --> 完成Isometric --> 等轴侧视图Isosurfaces --> 常值表面isotropic --> 各向同性Item --> 项目Items --> 项目Iteration --> 叠代Jobname --> 文件名Joint --> 连接Joints --> 连接KABS --> KABS Keypoint -->关键点Keypoints --> 关键点kinematic --> 随动KP --> 关键点KP between KPs -->关键点间设置kps --> 关键点Labeling --> 标志Layer --> 层Layered --> 分层Layers --> 层Layout --> 布局Lay-up --> 层布置Ld --> 载荷Legal Notices --> 法律声明Legend --> 图例Lib --> 库文件Library --> 材料库文件Licensing --> 许可Light Source --> 光源设置line --> 线Line Fillet --> 圆角Line Mesh --> 已划分的线Line w/Ratio --> 线上/比例Linear --> 线性Linearized --> 线形化Lines --> 线List --> 列出List Results --> 列表结果Ln' s --> 段Load --> 加载Load Step --> 载荷步Loads --> 载荷Loc --> 坐标值Local --> 局部Locate --> 定位Location --> 位置Locations --> 位置Locs --> 位置Log File --> 命令流记录文件lower-order --> 低阶LSDYNA --> LSDYNA(动力分析) LS-DYNA --> 显示动力分析Macro --> 宏命令Magnification --> 放大倍数management --> 管理Manager --> 管理器manual --> 手动ManualSize --> 手动尺寸Map --> 图Mapped --> 映射Mass --> 导体Mass Type --> 聚合量类型Master --> 主mat --> 材料Mat Num --> 材料编号Material --> 材料Materials --> 材料matl --> 材料Matls --> 材料maximum --> 最大Mechanical --> 机械类member --> 构件memory --> 内存MenuCtrls --> 菜单控制Merge --> 合并mesh --> 网格Mesher --> 网格Meshing --> 网格划分MeshTool --> 网格工具Message --> 消息Metafile --> 图元文件Meth --> 方法MIR --> 修正惯性松弛Miter --> 斜接[管]Mod --> 更改Mode --> 模式Model --> 模型Modeling --> 建模Models --> 模型Modify --> 修改Modle --> 模型Module --> 模块moment --> 力矩More --> 更多multi --> 多multi-field --> 多物理场耦合Multilegend --> 多图multilinear --> 多线性Multiple Species --> 多倍样式multiplied --> 乘Multi-Plot --> 多窗口绘图Multi-Plots --> 多图表Multi-Window --> 多窗口Mutual Ind --> 互感Name --> 名称Named --> 已指定natural log --> 自然对数nd --> 节点nds --> 节点NL Generalized -->非线形普通梁截面No Expansion --> 不扩展Nodal --> 节点Node --> 节点Nonlin --> 非线性Nonlinear --> 非线性Non-uniform --> 不均匀norm --> 法向Normal --> 法向Normals --> 没Num --> 编号NUMB --> NUMB Number -->编号Numbered --> 编号Numbering --> 编号Numbers --> 编号NUXY --> 泊松比Oblique --> 等角轴侧视图Octagon --> 八边形Octagonal --> 八棱柱offset --> 偏移Offset WP by Increments --> 指针增量偏移Offset WP to --> 指针偏移到Operate --> 操作Operations --> 运算OPT --> 优化Options --> 设置Optn --> 设置opts --> 设置Ord --> 指令Order --> 顺序Orders --> 指令Orient Normals --> 确定最外层法向Origin --> 原点Orthotropic --> 正交各向异性Other --> 其他Out Derived --> 输出派生outp --> 输出Output --> 输出Over Results --> 整个过程结果Over Time --> 规定时间内全过程Overlaid --> 覆盖Overlap --> 重叠Pair --> 偶Pairwise --> 新生成的Pan --> 移动pan-zoom-rotate --> 移动-缩放-旋转par --> 参数名parall --> 平行Parameters --> 参数Parms --> 参数Part IDs --> 部分ID号Partial --> 部分Partial Cylinder --> 部分圆柱体Particle Flow --> 粒子流迹Partition --> 分割Parts --> 局部Path --> 路径PDS --> 概率设计系统Pentagon --> 五边形Pentagonal --> 五棱柱Percent Error --> 误差率Periodic/Cyclic Symmetry--> 周期/循环阵列Perspective --> 透视phase --> 相位pick --> 选取Picked --> 已选取Piecewise --> 分段Piezoelectric --> 压电元件Pipe --> 管Pipe Run --> 管操作Pipe Tee --> T型管Piping --> 管Plane --> 平面Plane Strn --> 平面应变plasticity --> 塑性plot --> 绘图plotctrls --> 绘图控制Plots --> 绘图P-method --> 高次单元法Pointer --> 指针poisson --> 泊松Polygon --> 多边形POST1 --> 通用后处理器POST26 --> 时间历程后处理器postpro --> 后处理器postproc --> 后处理器potential --> 势POWRGRPH --> 激活窗体preferences --> 参数选项Pre-integrated --> 前集成处理PREP7 --> 前处理器preprocessor --> 前处理器PRES --> 压力Pre-tens Elements --> 删除单元后合并节点pretension --> 主张Pretensn --> 自划分prism --> 棱柱Pro --> Pro Prob --> 概率profiles --> 档案资料Prop --> 属性Properties --> 属性Props --> 属性PRXY --> 泊松比PRXY PRXZ --> 泊松比PRXZ PRYZ --> 泊松比PRYZ PT --> 点Pts --> 点Pulse --> 脉冲Q-Slice --> 切面Quad --> 积分Quadratic --> 二次qualities --> 质量query --> 查询QUIT --> 退出R --> 圆rad --> 半径radiation --> 辐射矩阵radius --> 半径Raise --> 升起random --> 随机range of variable --> 变量范围rate --> 率Rate of Change for ModelMainpulation --> 模型缩放变化率设定Reaction --> 反作用Read --> 读取Read Input from --> 读取命令流文件Real Constante --> 实常数RealConst --> 实常数Rectangle --> 矩形Redirect --> 重定向Reducer --> 接头ref --> 判定Refine --> 细化Reflect --> 阵列reflection --> 镜像Region --> 区域Regions --> 区域Relax/Stab/Cap --> 松弛/稳定/容量Relaxation --> 松弛release --> 版本Remesh --> 重划网格remove --> 删除rename --> 重命名Reorder --> 重置Replay Animation --> 重新播放动画Replot --> 重新绘图Report --> 报告Report --> 报告Res/Quad --> 结果/积分Reselect --> 分解Reset --> 取消Residual --> 余量Resistor --> 电阻response --> 响应Restart --> 重启动Restart/Clear --> 重启动/清除Restart/Iteration --> 重启动/迭代Restart/Load step --> 重启动/载荷步Restart/Set --> 重启动/设置Restart/Time --> 重启动/时间片Restore --> 恢复Result --> 结果Results --> 结果RESUM --> 恢复RESUM_DB --> 恢复_DB resume --> 恢复Reverse --> 相反Reverse Video --> 反色图像Rigid --> 刚性ROM --> 存储器Rotary --> 扭转Rotate --> 旋转Rotating --> 旋转rotational --> 旋转RUNSTAR --> 估计分析模块SAT --> SAT SAVE --> 保存SAVE_DB --> 保存_DBScalar --> 变量scale --> 比例scale factor --> 比例因子Scale Icon --> 图符尺度Scaling --> 比例Screen --> 屏幕se --> 超级单元secn --> 截面号sect --> 截面Sect Mesh --> 自定义网格Section --> 截面Sections --> 截面Sector --> 部分Segment --> 分段Segment Memory --> 分段保存segmented --> 分段Segments --> 分段Sel --> 选择sele --> 选择Select --> 选择Selected --> 已选择Selection --> 选择septagon --> 七边形septagonal --> 七边形的Set --> 设置Set Grid --> 设置栅格Set Up --> 设置Sets --> 设置Settings --> 设置Shaded --> 阴影Shape --> 形状Shell --> 壳Show --> 显示sided --> 边sine --> 正弦Singularity --> 奇异点sint --> 应力强度Sinusoidal --> 正弦Size --> 尺寸skinning --> 2线Slide Film --> 滑动薄膜Smart --> 精确SmartSize --> 智能尺寸Solid --> 实体Solid Circle --> 定圆心圆Solid Cylinder --> 定圆心圆柱体Solid Sphere --> 定圆心球体Solu --> 求解SOLUTION --> 求解器Solver --> 求解Sort --> 排序source --> 源Specification --> 约定Specifications --> 明细单Specified --> 指定Specified --> 指定Specified Loc --> 指定局部坐标spectrm --> 响应谱Spectrum --> 频谱Sphere --> 球体Spherical --> 球坐标系spline --> 样条Splines --> 样条曲线SpotWeld --> 点焊[缝、接点] Spring --> 弹簧Spring Support --> 弹性支撑Spring-Gap Supp -->弹性间隙支撑Src Waveform --> 屏幕波形Standed --> 标准Start --> 开始Start New --> 新建Start Num --> 初始编号Start Number --> 初始编号state --> 状态stats --> 状态Status --> 状态step --> 步store --> 存贮stress --> 应力Stresses --> 应力strn --> 应变Strnd Coil --> 线圈struct --> 结构structural --> 结构Style --> 样式submodeling --> 子模型Subtract --> 减去Summary --> 概要superelem --> 超单元superelement --> 超单元Superelements --> 超单元surf --> 表面Surface --> 面Surfaces --> 表面Sweep --> 扫描switch --> 转换Symbols --> 符号Symmetry Expansion --> 模型对称性扩展-镜像复制扫描Sys --> 系统Table --> 表tan --> 相切tangent --> 相切Taper --> 锥形Target --> 目标tech --> 技术TEMP --> 温度Temp Variatio --> 临时变量Temps --> 温度Tet --> 四面体Tets --> 测试Textured --> 纹理Texturing --> 材质thermal --> 热Thickness --> 壳厚度thickness func --> 函数定义变化的厚度Through --> 通过thru --> 通过Time Integration --> 时间积分Time Stepping --> 时间步设定Time-harmonic --> 时间-谐波timehist --> 时间历程TimeHist Postproc --> 时间历程后处理器Title --> 标题Toggle --> 扭转Tolerance --> 误差Toolbar --> 工具栏Topics --> 主题topological --> 拓扑torus --> 环行圆柱Trace --> 痕迹Trans --> 传递Transducer --> 传感器Transducers --> 传感器Transfer --> 移动Transient --> 暂态Translucency --> 半透视设置Traveling Wave --> 传导波Triangle --> 三角形Triangular --> 三棱柱ttribs --> 属性Turbulence --> 湍流Tutorials --> 指南Type --> 类型Types --> 类型Uniform --> 均布Units --> 单位Unload --> 卸载unpick --> 排除Unselect --> 不选择Update --> 更新user --> 用户User Numbered --> 自定义编号User Specified Expansion --> 自定义扩展模式utility --> 应用分析value --> 值Valve --> 阀Variables --> 变量Vector --> 矢量vectors --> 矢量Vector-Scalar --> 矢量-变量VFRC --> 体积含量View --> 视图Viewing --> 视图visco --> 粘Vltg --> 电压VOF --> 流体Volm --> 体Volms --> 体Volu --> 体volume --> 体Volumes --> 立体Volumes Brick Orient --> 沿Z向立方体Volus --> 体VS --> 电压源VX --> 速度X方向VY --> 速度Y方向VZ --> 速度Z方向w/Same --> w/相同节点Warning/Error --> 警告/错误warp --> 翘曲Wavefront --> 波前win --> 窗口Window --> 窗口Wire --> 导线wish --> 希望with --> 通过Working --> 工作Working Plane --> 工作平面WorkPlane --> 工作平面WP --> 工作平面WP Status --> 工作区指针状态Write DB log file --> 写入日志WrkPlane --> 工作面Zener --> 齐纳。
Heston模型下保险公司与再保险公司的博弈
Heston模型下保险公司与再保险公司的博弈∗王愫新;荣喜民;赵慧【摘要】This paper considers an optimal investment problem for both insurer and reinsurer. The insurer is allowed to purchase proportional reinsurance and both the insurer and reinsurer are allowed to invest in a risk-free asset and a risky asset whose price process satisfies the Heston’s stochastic volatility model. Firstly, we establish the objective function in the sense of maximizing the exponential utility of both the insurer and reinsurer on terminal wealth;Secondly, by solving the Hamilton-Jacobi-Bellman system, the closed-form expressions for the optimal reinsurance and investment strategies and the optimal value function are obtained;Finally, some numerical illustrations and sensitivity analysis for the proposed theoretical results are provided.%本文同时考虑保险公司和再保险公司的最优投资问题。
定量药理学名词中英对照表
English中文absolute prediction error(s) (APE)绝对预测误差absorption, distribution, metabolism, elimination (ADME)吸收、分布、代谢、消除active transport主动转运adaptive design自适应性设计additive error加和性误差adherence依从性administration给药affinity亲和力agonist激动剂allometric scaling异速生长antagonist拮抗剂area under curve (AUC)曲线下面积assumptions假设auto-induction自诱导backward elimination逆向剔除法base model基础模型baseline基线below the limit of quantification (BLQ)低于定量下限between-subject variability (BSV)个体间变异bias偏差biliary clearance胆汁清除率bioavailability生物利用度bioequivalence生物等效性biomarker生物标志物biopharmaceutics classification system (BCS)生物药剂学分类系统blood血body mass index (BMI)体质指数body surface area (BSA)体表面积bolus推注bootstrap自举法bottom-up appraoch自下而上的模式capacity-limited metabolism能力限制型代谢categorical data分类数据catenary compartment model链式模型causality因果chi-square test卡方检验clearance清除率Clinical trial simulation临床试验模拟clinical utility index临床效用指数Cmax峰浓度coefficient of variation (CV)变异系数Compartmental analysis房室模型分析competitive inhibition竞争性抑制compliance依从性concomitant medication effect联合用药效应condition number条件数conditional probability条件概率conditional weighted residuals (CWRES)条件加权残差confidence interval置信区间constitutive model本构模型continuous data连续数据convergence收敛correlation相关correlation coefficient相关系数correlation matrix相关矩阵count data计数数据covariacne matrix协方差矩阵covariance协方差covariate evaluation协变量评价covariate model协变量模型creatinine clearance肌酐清除率cross-over design交叉设计data analysis plan数据分析计划dataset assembly/construction数据集建立dataset specification file数据库规范文件degrees of freedom自由度dependent variable (DV)因变量determinant行列式deterministic identifiability确定性可识别性deterministic simulation确定性模拟diagonal matrix对角矩阵dichotomous二分类direct-effect model直接效应模型discrete离散disease progression疾病进程disease-modifying effect疾病缓解效应dose dependence剂量依赖性dose-normalized concentrations剂量归一化浓度double-blind双盲drug accumulation药物蓄积drug-drug interaction药物-药物相互作用duration of infusion输注持续时间efficacy功效eigenvalues特征值empirical Bayesian estimates (EBEs)经验贝叶斯估计endogenous内源性enterhepatic circulation肝肠循环estimate估计值estimation求参exogenous外源性exploratory data analysis (EDA)探索性数据分析exponential指数型external validation外部验证extrapolation外推extravascular administration血管外给药fasted禁食fed进食first in human (FIH) trial首次人体试验first-order absorption一级吸收first-order conditional estimation method (FOCE)一阶条件估计法first-order method (FO)一阶评估法first-pass effect首过效应Fisher information matrix Fisher信息矩阵fixed effect固定效应flip-flop翻转forward selection前向选择fraction of unbound (fu)游离分数full agonist完全激动剂gastric emptying胃排空generic products仿制药genetic polymorphism遗传多态性genome-wide association study (GWAS)全基因组关联研究genotype基因型global minimum全局最小值global sensitivity analysis全局敏感性分析glomerular filtration rate (GFR)肾小球滤过率goodness of fit拟合优度gradient梯度half maximal inhibitory concentration (IC50)半数抑制浓度half-life半衰期hepatic clearance肝清除率hierarchical层级homeostasis稳态homoscedasticity方差齐性hysteresis滞后identity matrix单位矩阵ill-conditioned matrix病态矩阵immunogenicity免疫原性in silico经由电脑模拟in situ原位in vitro体外in vivo体内independent variable自变量indirect response model间接反应模型individual parameter estimates个体参数估计individual prediction (IPRED)个体预测值individual residuals (IRES)个体残差individual weighted residuals (IWRES)个体加权残差infusion输注initial estimate起始参数估计inter-individual variability (IIV)个体间变异internal validation内部验证inter-occasion variability场合间变异interpolation插值intestinal absorption肠道吸收intra-individual variability个体内变异intramuscular administration (i.m.)肌肉注射intravenous administration (i.v.)静脉给药intrinsic clearance内在清除率inverse agonist反向激动剂inverse of matrix逆矩阵isobologram等效线图Jacobian matrix雅可比矩阵lag time滞后时间large-scale systems model大型系统模型lean body weight瘦体重level 1 random effect (L1)一级随机效应level 2 random effect (L2)二级随机效应ligand-receptor binding配体-受体结合likelihood ratio test似然比检验linear models线性模型linear pharmacokinetics线性药物动力学local minimum局部最小值local sensitivity analysis局部敏感性分析locally weighted scatterplot smoothing (LOWESS)局部加权散点平滑法logistic regression Logistic回归logit transform Logit变换log-normal distribution对数正态分布log-transformation对数变换maintenance dose维持剂量marginal probability边际概率mean均值mean absolute prediction error percent (MAPE)平均绝对预测误差百分比mean prediction error (MPE)平均预测误差mean residence time (MRT)平均滞留时间mean squared error (MSE)均方误差mechanism-based inhibition基于机制的抑制median中位数Michaelis-Menten constant米氏常数Michaelis-Menten kinetics米氏动力学missing dependent variable (MDV)缺失应变量mixed effect混合效应mixture models混合模型model diagnostic plots模型诊断图model evaluation模型评价model misspecification模型错配model specification file (MSF)模型规范文件model validation模型验证Model-based drug development基于模型的药物研发moment矩Monte Carlo simulation蒙特卡洛模拟multivariate linear regression多元线性回归negative feedback负反馈nested嵌套Non-compartmental analysis非房室模型分析noncompetitive inhibition非竞争性抑制nonlinear mixed effect models (NONMEM)非线性混合效应模型nonlinear pharmacokinetics非线性药物动力学normal distribution正态分布normalized prediction distribution errors (NPDE)归一化预测分布误差numerical predictive check (NPC)数值预测性能检查objective function value (OFV)目标函数值observation观测occupancy占有occupational model受体占有模型one-/two-compartment model一/二室模型onset of effect起效operational model操作模型optimal sampling最优采样Optimal study design优化试验设计oral口服ordered data有序数据outlier离群值parallel design 平行设计partial agonist部分激动剂peak concentration峰浓度perfusion灌注permeability渗透性Pharmacodynamics药效动力学Pharmacogenomics药物基因组学Pharmacokinetics 药物动力学Pharmacometrics定量药理学phase I reaction第一相反应phase II reaction第二相反应phenotype表型Physiologically based pharmacokinetics (PBPK)生理药物动力学piecewise linear models分段线性模型placebo安慰剂plasma血浆Poisson distribution泊松分布Poisson regression泊松回归population pharmacokinetics群体药物动力学positive feedback正反馈post hoc事后posterior distribution后验分布posterior predictive check (PPC)后验预测性能检查posterior probability后验概率potency效价强度power function幂函数precision精密度pre-clinical study临床前研究prediction (PRED)群体预测值prediction error (PE)预测误差prior distribution先验分布prodrug前药proof of concept study概念验证研究proportional error比例型误差Q-Q plot分位图quality assurance (QA)质量保证quality control (QC)质量控制random effect随机效应randomisation 随机化rate constant速率常数rate-limiting step限速步骤reference group对照组relative bioavailability相对生物利用度relative standard deviation (RSD)相对标准偏差relative standard error (RSE)相对标准误renal clearance肾清除率reparameterization重新参数化repeat dose重复剂量resampling重采样residual (RES)残差residual unexplained variability (RUV)残留不明原因的变异·rich sampling密集采样robust鲁棒性root mean square error (RMSE)均方根误差rounding errors舍入误差saturable可饱和的semi-logarithmic plot半对数图shirinkage收缩signalling transduction信号转导simulation模拟single dose单剂量singular奇异sparse sampling稀疏采样standard error (SE)标准误steady state (SS)稳态stochastic simulation随机模拟stratification分层structural identifiability结构可识别性subcutaneous administration (s.c.)皮下注射superposition叠加surrogate endpoint替代终点survival analysis生存分析symptomatic effect对症疗效synergism协同作用Systems pharmacology系统药理学target-mediated drug disposition靶点介导的药物处置therapeutic drug monitoring (TDM)治疗药物监测therapeutic index治疗指数time after dose (TAD)给药后时间time varying时间变化time-to-event analysis事件史分析tissue组织titration design滴定式设计tmax达峰时间tolerance耐受性top-down approach自上而下的模式total body weight总体重transit compartment model中转室模型transporter转运体transpose转置trough concentration谷浓度tubular reabsorption肾小管重吸收tubular secretion肾小管分泌turnover置换typical value paramters参数的群体典型值uncompetitive inhibition反竞争性抑制variance-covariance matrix方差协方差矩阵visual predictive check (VPC)可视化预测性能检查volume of distribution表观分布容积weighted residuals (WRES)加权残差well-stirred model充分搅拌模型within-subject variability个体内变异zero-order absorption零级吸收。
TREEPLAN使用详解
第一章安装Tree plan（以下称决策树）是在excel中画决策树的一个加载工具。
是由旧金山大学教授米歇尔R. 米德尔顿开发，并由杜克大学Fuqua商学院的詹姆斯E.史密斯教授改良使用。
一、安装方式决策树的所有功能都在一个名为TreePlan.xla的文件中，根据你的使用情况，共有以下3种安装方式。
第1种，偶然使用如果你只是偶然使用一次决策树，那么每次当你用的时候下载一次即可。
你也可以把TreePlan.xla文件放在一张软盘、电脑硬盘或网盘中。
直接双击“”加载第2种，选择性使用在这种场景下，你可以使用excel的加载项功能来安装决策树。
步骤如下：把TreePlan.xla保存在你电脑硬盘的某个地方。
如果你把TreePlan.xla文件保存在了excel 或office子目录文件夹里，请直接到第三步。
否则，打开excel——单击office按钮——excel 选项——加载项——转到——加载项对话框，单击浏览按钮，找到TreePlan.xla，单击确定。
在加载项对话框中，可以看到已经有TreePlan.xla选项，选中决策树前面的方框，单击确定。
如果你为了释放内存不再用决策树了，那么在加载项对话框中，去掉决策树前面的方框中的对勾。
当你要使用的时候，选择加载项，并选中决策树即可。
如果你要从加载项中移除决策树，直接在你保存TreePlan.xla文件的地方把它删除即可。
下次当你打开excel并使用加载项时，会出现一个“未找到加载的TreePlan.xla文件，是否从列表中删除？”的对话框，单价确定即可。
1)首先把加载宏放入如下安装文件件“D:\Program Files\Microsoft Office\Office15\Library‘’library 英[?la?br?r?] 美[la??br?r?] n. 图书馆，藏书室；文库C:\Program Files\Microsoft Office\Office15\Library2）加载：文件-选项-加载项-加载项-treeplan-转到第3种，经常使用如果你希望只要打开excel就能够使用决策树，那么就把TreePlan.xla文件保存在Excel XL Start文件夹里。
mean-varianceutility
Below is given annual work summary, do not need friends can download after editor deleted Welcome to visit againXXXX annual work summaryDear every leader, colleagues:Look back end of XXXX, XXXX years of work, have the joy of success in your work, have a collaboration with colleagues, working hard, also have disappointed when encountered difficulties and setbacks. Imperceptible in tense and orderly to be over a year, a year, under the loving care and guidance of the leadership of the company, under the support and help of colleagues, through their own efforts, various aspects have made certain progress, better to complete the job. For better work, sum up experience and lessons, will now work a brief summary.To continuously strengthen learning, improve their comprehensive quality. With good comprehensive quality is the precondition of completes the labor of duty and conditions. A year always put learning in the important position, trying to improve their comprehensive quality. Continuous learning professional skills, learn from surrounding colleagues with rich work experience, equip themselves with knowledge, the expanded aspect of knowledge, efforts to improve their comprehensive quality.The second Do best, strictly perform their responsibilities. Set up the company, to maximize the customer to the satisfaction of the company's products, do a good job in technical services and product promotion to the company. And collected on the properties of the products of the company, in order to make improvement in time, make the products better meet the using demand of the scene.Three to learn to be good at communication, coordinating assistance. On‐site technical service personnel should not only have strong professional technology, should also have good communication ability, a lot of a product due to improper operation to appear problem, but often not customers reflect the quality of no, so this time we need to find out the crux, and customer communication, standardized operation, to avoid customer's mistrust of the products and even the damage of the company's image. Some experiences in the past work, mentality is very important in the work, work to have passion, keep the smile of sunshine, can close the distance between people, easy to communicate with the customer. Do better in the daily work to communicate with customers and achieve customer satisfaction, excellent technical service every time, on behalf of the customer on our products much a understanding and trust.Fourth, we need to continue to learn professional knowledge, do practical grasp skilled operation. Over the past year, through continuous learning and fumble, studied the gas generation, collection and methods, gradually familiar with and master the company introduced the working principle, operation method of gas machine. With the help of the department leaders and colleagues, familiar with and master the launch of the division principle, debugging method of the control system, and to wuhan Chen Guchong garbage power plant of gas machine control system transformation, learn to debug, accumulated some experience. All in all, over the past year, did some work, have also made some achievements, but the results can only represent the past, there are some problems to work, can't meet the higher requirements. In the future work, I must develop the oneself advantage, lack of correct, foster strengths and circumvent weaknesses, for greater achievements. Looking forward to XXXX years of work, I'll be more efforts, constant progress in their jobs, make greater achievements. Every year I have progress, the growth of believe will get greater returns, I will my biggest contribution to the development of the company, believe in yourself do better next year!I wish you all work study progress in the year to come.Mean Variance UtilityIn this note I show how exponential utility function and normally distributed consumption give rise to a mean variance utility function where the agent’s expected utility is a linear function of his mean income and the variance of his income.The analysis is taken from p.154-155in T.Sargent,Macroeconomic Theory ,2nd.edition.Suppose that the utility function from consumption,C,is exponential and given byThis utility function is increasing and concavesince(1)Since the utility function is concave,it reflects risk aversion.Moreover note that theArrow-Pratt (2)index of absolute risk aversion is given byThis means that the larger λis,the more risk averse the agentis.(3)Next,suppose that C is distributed normally with mean,µ,and standard deviation,σ.Then the density of C is given by:Therefore,expected utility is givenby:(4)It is no useful to rewrite the exponent so as to group terms that depend on C and terms thatdo (5)not depend on C.To this end note thatSubstituting in EU(C),gives(6)Now,for all µ’,(7)because the left hand side is just the area under the density function over the entire supportwhen (8)the mean is µ’and the standard deviation is γ.Since this is so for any µ’including µ’=µ-λγ2,it follows thatHence,the objective of the agent is to maximize theexpression(9)That is,the agent is interested in maximizing his mean consumption minus thevariance (10)multiplied by a constant.As we saw before,the constant λmeasures the degree of risk aversion:the larger λis,the more risk averse the agent is.Hence the utility of the agent is increasing with the mean of his consumption and decreases with the variance.The rate of decrease with the variance is larger the more risk averse the agent is.。
随机波动与VaR下带实业的保险最优投资
随机波动与VaR下带实业的保险最优投资孙宗岐西安思源学院数学教研室陕西西安 710038摘要为了考虑一类带有实业项目投资的保险最优投资策略问题，假定保险公司盈余服从跳-扩散过程，在最小化保险公司破产概率准则下，使用动态规划原理建立了线性消费率下保险资金最优投资选择模型，通过求解HJB方程得到了最优投资决策和最小破产概率的解析式解，最后分析了线性消费、索赔强度、索赔额以及实业项目投资额对最小化破产概率和最优投资策略的影响。
关键词跳-扩散过程；实业项目投资；破产概率；线性消费率；投资策略；HJB方程；中图分类号：O211.63 文献标识码：A 文章编号：Optimal insurance approach with real investment under jump-diffusion processesSUN Zong-qiDepartment of Mathematics, Xi’an Siyuan University, Xi’an ，710038Abstract:Under the hypothesis that the insurance’s reserve price follows a jump-diffusion process,a class of optimal portfolio problem that combines a real investment is studied in the present paper.Based on the criterion of minimizing the insurance’s ruin probability, the optimal investment choice model was established using dynamic programming principle under the linear consumption rate. The optimal analytic solutions of the optimal investment approach and the minimizing ruin probability were obtained by solving the HJB equation. Finally, the relationship between the linear consumption rate、the claim strength 、the claim amount 、the project investment、and the Optimal Financial Approach was analyzed.Key words:jump-diffusion processes；real investment；ruin probability；linear consumption rate；investment approach；HJB equation1 引言2006年中国保监会发布了《保险资金间接投资基础设施项目试点管理办法》，标志着国家正式放行保险资金投资基础设施领域。
Ho-Lee利率模型下资产-负债管理的最优投资策略
Ho-Lee利率模型下资产-负债管理的最优投资策略常浩;荣喜民【摘要】Under the assumption that the risk-free interest rate dynamics follows the Ho-Lee interest rate model, we investigate the optimal investment strategy for portfolio selection with liability. The financial market is composed of two risky assets: a stock and a zero-coupon bond. The stock price and the bond price influenced by the interest rate volatility are all driven by the extended geometric Brow-nian motion. The liability process follows the extended Brownian motion with drift and is correlated with the interest rate and stock price. Under the utility maximization criterion, the maximum principle is used to obtain the HJB equation for the value function, and we study the optimal investment strategies under power utility and exponential utility, respectively. Finally, we obtain the closed-form solutions for the optimal investment strategies by applying the variable transformation approach.%假设无风险利率是服从Ho-Lee利率模型的随机过程,本文研究负债情形下组合投资的最优投资策略.我们考虑金融市场存在两种风险资产:一种股票和一种零息票债券,其中股票和零息票债券由于受到利率波动的影响而服从扩展的几何布朗运动,而负债服从扩展的带漂移的布朗运动,且和利率与股票价格存在相关性.文章应用最大值原理得到效用最大化下值函数的Hamilton-Jacobi-Bellman (HJB)方程,并研究幂效用和指数效用函数下的最优投资策略.利用变量变换方法得到了两种效用函数下最优投资策略的显示表达式.【期刊名称】《工程数学学报》【年(卷),期】2012(029)003【总页数】10页(P337-346)【关键词】Ho-Lee利率模型;资产-负债管理;最大值原理;HJB方程;幂效用;指数效用;最优投资策略【作者】常浩;荣喜民【作者单位】天津工业大学数学系,天津300387;天津大学管理学院,天津300072;天津大学理学院,天津300072【正文语种】中文【中图分类】F830.481 引言资产–负债管理是银行、基金和保险公司等金融机构进行风险管理的重要手段．近年来，关于资产–负债管理方面的研究已经取得了一些研究成果，其中负债情形下的投资组合选择理论已经成为银行、基金和保险公司等金融机构进行投资决策的重要依据．1990年，Sharpe和Tint[1]首次在均值–方差框架下对负债情形下的投资组合选择问题进行了研究．之后，Leippold和Trojani[2]研究了多周期情形下的资产–负债管理问题；Chiu和Li[3]在负债和资产价格服从相同的几何布朗运动的环境下，运用随机二次规划和嵌入方法研究了均值–方差意义下的资产–负债管理问题；Xie等[4]进一步对负债和资产价格服从不同的布朗运动条件下的资产–负债管理问题进行了研究；Papi与Sbaraglia[5]应用动态规划对限制性的资产–负债管理问题进行了研究．这些研究一定程度上丰富和发展了组合证券投资理论在资产–负债管理中的应用，但这些市场模型中利率都是常数或时间的确定函数，而这一假设并不符合投资者和投资机构的实际投资环境．本文将资产–负债管理问题中的常数利率推广到随机利率模型，并在此基础上研究幂效用与指数效用函数下的最优投资决策问题．近年来，关于随机利率模型下的组合证券投资理论也取得了一些研究成果，如文献[6,7]应用鞅方法对Vasicek利率模型和CIR利率模型下的组合证券投资问题进行了研究；而文献[8]则应用随机最优控制和HJB方程对Ho-Lee利率模型和Vasicek利率模型下的投资问题进行了研究；文献[9–11]研究了随机利率模型下养老基金投资的最优投资策略问题；文献[12]对随机利率和随机波动率模型下的投资–消费问题进行了研究．与此同时，国内对随机利率环境下组合证券投资的研究也取得了一些研究成果[13-16]．但这些文献仅仅研究了随机利率环境下的组合证券投资问题，并没有考虑负债．金融实务中，投资过程往往伴随着负债，负债的引入将使得连续时间情形下的最优投资组合选择模型更符合实际．本文将负债引入文献[8]的市场模型中，并将利率假定为服从Ho-Lee利率模型[17]的随机过程，负债和随机利率的引入使得市场模型更贴合实际，更具有操作性和针对性，更能为投资者和投资机构提供科学的理论依据．文章应用最大值原理得到效用最大意义下值函数的HJB方程，并对幂效用与指数效用函数下的最优投资策略进行研究．最后，通过求解相应的HJB方程得到两种效用函数下最优投资组合的解析表达式．2 理论框架文中E(·)表示随机变量的数学期望，[0,T]是固定的有限投资周期．假设(W1(t),W2(t))是定义在完备概率空间(Ω,F,P,{Ft}0≤t≤T)上的标准适应的二维独立布朗运动，其中{Ft}0≤t≤T是由(W1(t),W2(t))所产生的信息流．L2(Ω,F,P)是均方可积的随机变量构成的Hilbert空间．假设在[0,T]内金融市场中有三种可连续交易的资产，其中第一种资产是无风险资产，比如银行账户，其t时刻的价格记为S1(t)，且S1(t)满足下列微分方程其中r(t)>0表示t时刻银行账户的无风险利率．本文假设r(t)是服从Ho-Lee利率模型的随机过程，满足随机微分方程其中a(t)=˜α(t)+b˜η(t)，且˜α(t),˜η(t)是时间t的确定函数，b>0是常数．第二种资产是股票，其t时刻股票价格记为S2(t)，且S2(t)满足下列随机微分方程其中µ1(t)是股票的收益率，σ1(t)是由布朗运动W1(t)引起的波动率，σ2(t)是由利率波动对股票价格的影响所引起的波动率，且µ1(t),σ1(t),σ2(t)都是[0,T]上Borel-可测的有界确定函数．第三种资产是投资期长为T1>T的零息票债券，其t时刻价格记为P(t)，且P(t)满足[8]其中σ3(t)=−b(T1−t),η(t)为时间t的确定函数．假设投资者在时刻t=0有初始财富w>0与初始负债l(l∈R)，那么投资者在时刻t=0有净初始财富x0=w−l>0．记L(t)为t时刻投资者的累积负债，假设L(t)满足其中d1(t)表示股价波动所引起的负债的波动率，d2(t)表示利率波动对负债的影响所引起的波动率，且c(t),d1(t),d2(t)都是时间t的确定函数．假设t时刻投资者投资于股票和零息票债券的资金数额分别记为π1(t),π2(t),t∈[0,T]，记π(t)=(π1(t),π2(t))，X(t)表示t时刻投资者的净财富，则t 时刻投资于债券的资金数额为X(t)−π1(t)−π2(t)．在借贷利率相同，且不考虑卖空、交易成本和消费的情况下，那么交易策略π(t)下财富过程X(t)满足下列随机微分方程即可行投资组合策略π(t)所形成的集合记为Γ={π(t):0≤t≤T}．假设投资者总是希望其终端财富的期望效用值达到最大，即求解下列优化问题其中效用函数U(·)是定义在(−∞,+∞)上的严格凹的连续可微函数．由凹函数的性质可知存在唯一的最优投资策略π(t)=(π1(t),π2(t))使得投资者的终端财富的期望效用值最大．3 最优投资策略本节应用最大值原理得到值函数所满足的HJB方程，然后对幂效用函数与指数效用函数下的最优投资策略进行研究，通过求解相应的HJB方程，得到两种效用函数下最优投资组合的解析表达式．值函数H(t,r,x)可定义如下其边界条件为H(T,r,x)=U(x)．为方便起见，在下列方程中将各参数中的(t)省略不写．应用最大值原理得到值函数满足如下HJB方程其中Ht,Hr,Hrr,Hx,Hxx,Hrx分别表示值函数H(t,r,x)对t,x,r的各阶偏导数．式(7)左边取得最大值的必要条件为解关于π1,π2的方程组可得最优解为将上述最优值代入(7)可得值函数H(t,r,x)满足下列的非线性二阶偏微分方程其中最优解下终端财富方程满足下面假设投资者对风险的偏好分别满足幂效用和指数效用函数，并在此两种效用函数下分别求解方程(10)．3.1 幂效用函数幂效用函数是一种常系数相对风险厌恶效用函数，其函数表达式为幂效用函数下，为了求解方程(10)，我们假设值函数具有如下结构其边界条件满足f(T,r)=1,g(T,r)=0．其中f(t,r),g(t,r)是未知函数，我们的目标是寻找f(t,r)与g(t,r)的解析表达式使得(11)是方程(10)的解．值函数H(t,r,x)对t,x,r求各阶偏导数如下将上述各阶偏导数代入方程(10)可得于是可得f(t,r)与g(t,r)分别满足如下的非线性二阶偏微分方程方程(12)与(13)的求解如下面两个引理所示．引理1 假定非线性二阶偏微分方程(12)的解为f(t,r)=A(t)eB(t)r，其边界条件为A(T)=1,B(T)=0，则有证明将f(t,r)=A(t)eB(t)r代入(12)可得由A(t)和B(t)满足(14)，可得B′(t)+β=0．注意到边界条件B(T)=0，于是有B(t)=β(T−t)．而A(t)则满足如下常微分方程解方程可得A(t)的表达式．证毕引理2 设二阶偏微分方程(13)解的结构为g(t,r)=D(t)eE(t)r，其边界条件D(T)=0，则有证明将g(t,r)=D(t)eE(t)r代入偏微分方程(13)，经分离变量后可得上式右端是一常数，而左端是eE(t)r与t的函数的乘积，要想上式对任意的r成立，只能取E(t)=0．于是可得D(t)满足如下常微分方程解方程可得考虑到引理1与引理2的结论，则有将引理1与引理2的结论代入(8)与(9)，可得幂效用函数下的最优投资策略为在此最优投资策略下，财富过程的期望值满足解方程可得其中因此，幂效用函数下我们有如下结论．定理1 如果效用函数为则Ho-Lee利率模型下资产–负债管理问题(6)的最优投资策略由(15),(16)给出，且终端财富期望值为其中g(t,r)由引理2所确定．3.2 指数效用函数指数效用函数即是常系数绝对风险厌恶型效用函数，其函数表达式为指数效用函数下，为了求解方程(10)，我们假设值函数具有如下结构其边界条件为u(T,r)=1,v(T,r)=0,w(T,r)=0．对值函数求各阶偏导数如下将值函数的各阶偏导数代入方程(10)可得于是得到如下三个二阶偏微分方程方程(18)–(20)的求解如下面三个引理所示．引理3设非线性二阶偏微分方程(18)的解为u=e(t)+(t)r，其边界条件为(T)=0,˜B(T)=0，则有证明将u=e(t)+(t)r代入方程(18)可得则有′(t)+1=0，注意到边界条件(T)=0，则有(t)=T−t．于是得到(t)满足如下常微分方程解方程可得(t)的解析表达式．证毕引理4设偏微分方程(19)解的结构为v=(t)eE(t)r，边界条件(T)=0，则有证明将引理3得到的u=e(t)+B(t)r代入方程(19)可得下面解法同引理2．证毕引理5设方程(20)解的结构为，边界条件为则有证明将u=e(t)+(t)r与w(t,r)=(t)+(t)r代入方程(20)可得则有′(t)=0，注意到边界条件(T)=0，于是有(t)=0．所以(t)满足如下常微分方程解方程可得(t)的解析表达式．证毕由引理3至引理5可得将引理3至引理5的结论代入(8)和(9)可得指数效用函数下的最优投资策略为在此最优投资策略下，财富过程的期望值满足解方程可得其中因此，指数效用函数下我们有如下结论．定理2 如果效用函数为则Ho-Lee利率模型下资产–负债管理问题(6)的最优投资策略由(21),(22)所确定，且终端财富最优期望值为其中u(t,r),v(t,r)由引理3和引理4所确定．4 结论本文假设利率是服从Ho-Lee利率模型的随机过程，并对该利率模型下资产–负债管理问题的最优投资策略进行了研究．文章应用最大值原理得到了值函数的HJB 方程，通过求解HJB方程分别得到了幂效用函数与指数效用函数下最优投资策略的解析表达式．进一步的研究内容可考虑利率服从Cox-ingersoll-Ross利率模型且利率与风险资产存在一般的相关性，进一步研究负债情形下多种风险资产的最优投资策略问题．致谢：衷心感谢匿名审稿人提出的创造性修改意见！参考文献：[1]Sharpe W F,Tint L G.Liabilities—a new 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Gram-Charlier展开在生产和套期保值的应用
㊀第４１卷第５期２０２０年９月闽江学院学报ＪＯＵＲＮＡＬＯＦＭＩＮＪＩＡＮＧＵＮＩＶＥＲＳＩＴＹＶｏｌ.４１Ｎｏ.５Ｓｅｐｔ.２０２０ＤＯＩ:１０.１９７２４/ｊ.ｃｎｋｉ.ｊｍｊｕ.２０２０.０５.００３Ｇｒａｍ￣Ｃｈａｒｌｉｅｒ展开在生产和套期保值的应用陈玲菊１ꎬ２(１.闽江学院数学与数据科学学院ꎬ福建福州３５０１０８ꎻ２.生态与资源统计福建省高校重点实验室ꎬ福建福州３５０１０８)摘要:将一类指数型效用函数的密度函数进行Ｇｒａｍ￣Ｃｈａｒｌｉｅｒ(ＧＣ)展开ꎮ在此基础上ꎬ利用效用函数均值的一阶条件下得到了最佳产量只取决于成本和远期价格ꎮ同时发现ꎬ在价格的偏度和峰度满足正态条件下ꎬ当远期交货价格等于即期交货价格均值时不存在投机ꎮ关键词:效用函数ꎻＧｒａｍ￣Ｃｈａｒｌｉｅｒ展开式ꎻ最佳产量ꎻ远期交货价格中图分类号:Ｆ８３２㊀㊀㊀㊀㊀文献标识码:Ａ㊀㊀㊀㊀文章编号:１００９－７８２１(２０２０)０５－００１２－０５ＡｐｐｌｉｃａｔｉｏｎｏｆＧｒａｍ￣ＣｈａｒｌｉｅｒＥｘｐａｎｓｉｏｎｔｏＰｒｏｄｕｃｔｉｏｎａｎｄＨｅｄｇｉｎｇＤｅｃｉｓｉｏｎＣＨＥＮＬｉｎｇｊｕ１ꎬ２(１.ＣｏｌｌｅｇｅｏｆＭａｔｈｅｍａｔｉｃｓａｎｄＤａｔａＳｃｉｅｎｃｅꎬＭｉｎｊｉａｎｇＵｎｉｖｅｒｓｉｔｙꎬＦｕｚｈｏｕꎬＦｕｊｉａｎ３５０１０８ꎬＣｈｉｎａꎻ２.ＫｅｙＬａｂｏｒａｔｏｒｙｆｏｒＥｃｏｌｏｇｙａｎｄＲｅｓｏｕｒｃｅｓＳｔａｔｉｓｔｉｃｓｏｆＦｕｊｉａｎＰｒｏｖｉｎｃꎬＦｕｚｈｏｕꎬＦｕｊｉａｎ３５０１０８ꎬＣｈｉｎａ)Ａｂｓｔｒａｃｔ:ＩｎｔｈｉｓｐａｐｅｒꎬｗｅｐｒｏｐｏｓｅａＧｒａｍ￣Ｃｈａｒｌｉｅｒ(ＧＣ)ｅｘｐａｎｓｉｏｎｆｏｒｔｈｅｐｒｏｂａｂｉｌｉｔｙｄｅｎｓｉｔｙｆｕｎｃ￣ｔｉｏｎｏｆａｎｅｘｐｏｎｅｎｔｉａｌｕｔｉｌｉｔｙｆｕｎｃｔｉｏｎ.Ｔｈｅｏｐｔｉｍａｌｏｕｔｐｕｔｑｕａｎｔｉｔｙｉｓｄｅｒｉｖｅｄｕｎｄｅｒｆｉｒｓｔｏｒｄｅｒｃｏｎｄｉｔｉｏｎｓｏｆｔｈｅｅｘｐｅｃｔａｔｉｏｎｏｆｔｈｅｕｔｉｌｉｔｙｆｕｎｃｔｉｏｎꎬｗｈｉｃｈｄｅｐｅｎｄｓｏｎｔｈｅｃｏｓｔａｎｄｔｈｅｆｏｒｗａｒｄｐｒｉｃｅ.Ｍｅａｎｗｈｉｌｅꎬｔｈｅｒｅｅｘｉｓｔｓｎｏｓｐｅｃｕｌａｔｉｖｅｔｒａｄｉｎｇｉｆｔｈｅｆｏｒｗａｒｄｐｒｉｃｅｉｓｅｑｕａｌｔｏｔｈｅｅｘｐｅｃｔｅｄｓｐｏｔｐｒｉｃｅｗｉｔｈｔｈｅｓｋｅｗｎｅｓｓａｎｄｋｕｒｔｏｓｉｓｓａｔｉｓｆｙｉｎｇｎｏｒｍａｌｄｉｓｔｒｉｂｕｔｉｏｎ.Ｋｅｙｗｏｒｄｓ:ｕｔｉｌｉｔｙｆｕｎｃｔｉｏｎꎻＧｒａｍ￣Ｃｈａｒｌｉｅｒｅｘｐａｎｓｉｏｎꎻｏｐｔｉｍａｌｏｕｔｐｕｔꎻｆｏｒｗａｒｄｐｒｉｃｅ０㊀引言近年来ꎬＧｉｌｂｅｒｔ等应用一种负指数效用函数和３阶的泰勒展开近似式研究偏度对套期保值决策的影响[１]ꎮ由于泰勒展开在３阶截尾ꎬ所以就无法考虑更高阶的影响ꎬ如峰度ꎮ类似地ꎬＬｉｅｎ运用负指数效用函数分析偏度对最佳产量和套期保值决策的影响中使用的偏正态分布不是厚尾[２]ꎮ这些局限性启发了Ｌｉｅｎ和Ｗａｎｇ使用厚尾的偏学生分布ꎬ但是该分布的４个系数并不一一对应于股价的前４阶矩ꎬ这样就无法分离４阶矩各自在策略中的作用[３]ꎮ基于此ꎬ运用㊀㊀收稿日期:２０２０－０６－２９㊀㊀基金项目:福建省教育厅中青年教师教育科研项目(ＪＡＴ１６０３８３)㊀㊀通信作者:陈玲菊(１９７６ )ꎬ女ꎬ福建福安人ꎬ副教授ꎬ硕士ꎬ主要研究方向为应用统计学ꎮ第５期陈玲菊:Ｇｒａｍ￣Ｃｈａｒｌｉｅｒ展开在生产和套期保值的应用Ｇｒａｍ￣Ｃｈａｒｌｉｅｒ(ＧＣ)展开式作为半参数工具来放松通常的正态分布假设ꎮ由于ＧＣ分布的偏度和峰度对应其参数ꎬ所以具有正态分布所没有的弹性ꎮ因此ꎬ被广泛应用于金融数据分析ꎮ例如ꎬＬｅｏｎ用ＧＡＲＣＨ￣ＧＣ展开式模型研究股票和外汇市场ꎬ发现动态高阶矩模型具有更好的拟合预测能力[４]ꎮ王鹏和王建琼利用ＧＣ展开式研究中国股票市场的高阶矩波动特征ꎬ其结果表明能描述偏度和峰度时变特征的高阶矩波动模型有着更强的刻画现实的能力[５]ꎮ更多关于方差及高阶矩在套期保值和资产分配等金融决策问题中的应用ꎬ可参考文献[６－９]ꎮ基于以上的研究ꎬ本文尝试将ＧＣ展开式应用于生产和套期保值模型ꎮ由于ＧＣ分布的偏度和峰度对应不同的参数ꎬ所以可以将偏度和峰度在套期保值中各自的作用分离出来ꎬ因此具有比较大的弹性ꎮ文章的主要工作是在将效用函数进行ＧＣ展开的基础上ꎬ讨论最佳产量及最佳套期保值策略ꎮ由于本文是第一次将ＧＣ展开应用于研究偏度和峰度对套期保值的影响ꎬ所以是对套期保值研究方法的一个拓展的一个尝试ꎬ得到的结论也具一定的一般性ꎮ１㊀基本模型假定生产商在时刻０生产了Ｑ个单位的产品计划在时刻１出售ꎮ生产成本是个严格递增的凸函数Ｃ(Ｑ)ꎮ时刻１的价格是个随机变量ꎬ记为ｐ~ꎮ假定同时还签订了一份时刻１到期交付的远期合同ꎬＰ为时刻０合同的价格ꎮ进一步地ꎬ假设该产商可以按合同交付产品ꎬ则该生产商在时刻１获得的利润为π~＝ｐ~(Ｑ－Ｈ)＋ＰＨ－Ｃ(Ｑ)ꎬ(１)其中ꎬＨ是远期合同中出售的产品数量ꎮ参照文献[２－３]定义效用函数具有指数形式Ｕ(ω)＝－ｅｘｐ(－ｋω)ꎬ其中ꎬω为利润ꎬｋ>０为Ａｒｒｏｗ￣Ｐｒａｔｔ风险规避参数ꎮ在引入ＧＣ展开式之前ꎬ先记μπ㊁σπ㊁Ｓπ和Ｋπ分别为π~的均值㊁方差㊁偏度和峰度ꎮ将π~的密度函数按ＧＣ展开表示如下:ｆ(ｘ)＝１σπφｘ－μπσπæèçöø÷１＋Ｓπ６ψ３ｘ－μπσπæèçöø÷＋Ｋπ－３２４ψ４ｘ－μπσπæèçöø÷éëêêùûúúꎮ(２)其中ꎬψ３ｚ()＝ｚ３－３ｚꎬψ４ｚ()＝ｚ４－６ｚ２＋３ꎬφｚ()＝(２π)－１/２ｅ－ｚ２/２是标准正态分布的密度函数ꎮ通过该密度函数的ＧＣ展开式可以计算出效用函数的均值Ｅ[ｕ(π~)]ꎬ其显示表达式由以下定理１给出ꎮ定理１㊀效用函数的均值可以表示为Ｅ[Ｕ(π~)]＝－ｅｘｐ(－ｋμπ)ｅｘｐ[(１/２)ｋ２σ２π]Δ(σπꎬＳπꎬＫπ)ꎬ其中ꎬΔ(σπꎬＳπꎬＫπ)＝１＋Ｓπ６(－σπｋ)３＋Ｋπ－３２４σπｋ()４ꎮ证明㊀由(２)及指数效用函数可得Ｅ[Ｕ(π~)]＝－ʏξ(ｘꎬμπꎬσπꎬｋ)１＋Ｓπ６ψ３ｘ－μπσπæèçöø÷＋Ｋπ－３２４ψ４ｘ－μπσπæèçöø÷éëêêùûúúｄｘꎬ(３)其中ꎬξ(ｘꎬμπꎬσπꎬｋ)＝１σπφｘ－μπσπæèçöø÷ｅｘｐ(－ｋｘ)ꎮ显然ꎬ３１闽江学院学报第４１卷ξ(ｘꎬμπꎬσπꎬｋ)＝１２πσπｅｘｐ－(ｘ－μπ)２２σ２πæèçöø÷ｅｘｐ(－ｋｘ)＝１２πσπｅｘｐ－[ｘ－(μπ－σ２πｋ)]２２σ２πæèçöø÷ｅｘｐ(－ｋμπ＋(１/２)ｋ２σ２π)ꎮ㊀㊀记φｘꎬμπꎬσπꎬｋ()为Ｎ(μπ－σ２πｋꎬσ２π)的密度函数ꎬｃ＝ｅｘｐ(－ｋμπ＋(１/２)ｋ２σ２π)ꎮ则有ξ(ｘꎬμπꎬσπꎬｋ)＝ｃφｘꎬμπꎬσπꎬｋ()ꎮ(４)令ｘ∗＝ｘ－(μπ－σ２πｋ)σπꎬ则ｘ∗－σ２πｋ＝ｘ－μπσπꎮ因此成立ψ３ｘ－μπσπæèçöø÷＝ψ３(ｘ∗－σπｋ)＝ｘ∗３－３ｘ∗２σπｋ＋３ｘ∗σ２πｋ２－σ３πｋ３－３ｘ∗＋３σπｋꎻψ４ｘ－μπσπæèçöø÷＝ψ４(ｘ∗－σπｋ)＝ｘ∗４－４ｘ∗３σπｋ＋６ｘ∗３σ２πｋ２－４ｘ∗３σ３πｋ３＋σ４πｋ４－６ｘ∗２＋１２ｘ∗σπｋ－６σ２πｋ２＋３ꎮ㊀㊀进一步地ꎬ注意到φｘꎬμπꎬσπꎬｋ()为Ｎ(μπ－σ２πｋꎬσ２π)的密度函数ꎬ由正态密度函数的性质可得ʏｘ∗φｘꎬμπꎬσπꎬｋ()ｄｘ＝ʏｘ∗ｋ１２πｅ－ｘ∗２２ｄｘ∗＝０ꎬｋ＝２ｎ－１１ꎬｋ＝０ꎬ２３ꎬｋ＝４ìîíïïïïꎬｎ＝１ꎬ２ꎬ３ꎬ ꎮ因此ꎬ有ʏφｘꎬμπꎬσπꎬｋ()ｄｘ＝１ꎬ(５)ʏφｘꎬμπꎬσπꎬｋ()ψ３ｘ－μπσπæèçöø÷ｄｘ＝－σ３πｋ３ꎬ(６)ʏφｘꎬμπꎬσπꎬｋ()ψ４ｘ－μπσπæèçöø÷ｄｘ＝σ４πｋ４ꎮ(７)㊀㊀最后ꎬ由式(３)－式(７)可得Ｅ[Ｕ(π~)]＝－ｅｘｐ(－ｋμπ)ｅｘｐ[(１/２)ｋ２σ２π]１＋Ｓπ６(－σπｋ)３＋Ｋπ－３２４σπｋ()４æèçöø÷ꎮ㊀㊀证毕ꎮ在定理１给出的效用函数的ＧＣ展开式的基础上ꎬ讨论最佳产量与最佳套期保值策略ꎮ２㊀最佳产量注意到Ｅ[Ｕ(π~)]ɤ０ꎬ所以Δ(σπꎬＳπꎬＫπ)ȡ０ꎮ这意味着最佳的产量和套期保值决策在最大化Ｅ[Ｕ(π~)]时取得ꎮ定理２将证明最佳产量仅仅取决于成本函数Ｃ( )和远期价格Ｐꎮ定理２㊀存在最佳产量Ｑ∗ꎬ使得Ｃᶄ(Ｑ∗)＝Ｐꎮ因此ꎬ即期价格的偏度和峰度对最佳生产决策没有影响ꎮ证明㊀注意到μπ＝μｐ(Ｑ－Ｈ)＋ＰＨ－Ｃ(Ｑ)ꎬσπ＝σｐ｜Ｑ－Ｈ｜ꎬＳπ＝Ｓｐꎬ且４１第５期陈玲菊:Ｇｒａｍ￣Ｃｈａｒｌｉｅｒ展开在生产和套期保值的应用Ｋπ＝Ｋｐꎬ最大化Ｅ[Ｕ(π~)]相当于最大化以下的ｌｎＥ[Ｕ(π~)]:ｌｎＥ[Ｕ(π~)]＝ｋμπ－(１/２)ｋ２σ２π－ｌｎΔ(σπꎬＳπꎬＫπ)ꎮ(８)㊀㊀由于均值－方差套期保值者只考虑均值和方差而不考虑偏度和峰度ꎬ所以Ｓｐ＝０ꎬＫｐ＝３ꎬ因此有ｌｎΔ(σπꎬＳπꎬＫπ)＝０ꎮ对比均值－方差方法ꎬ式(８)中ＧＣ展开方法包含特殊项ｌｎΔ(σπꎬＳπꎬＫπ)ꎮ接着对Ｑ和Ｈ求偏导ꎬ可以得到其一阶条件如下:ｋ[μｐ－Ｃ'(Ｑ)－ｋσ２ｐ(Ｑ－Ｈ)]－∂Δ(σπꎬＳπꎬＫπ)/∂σπˑ∂σπ/∂ＱΔ(σπꎬＳπꎬＫπ)＝０ꎻ(９)ｋ[Ｐ－μｐ＋ｋσ２ｐ(Ｑ－Ｈ)]－∂Δ(σπꎬＳπꎬＫπ)/∂σπˑ∂σπ/∂ＨΔ(σπꎬＳπꎬＫπ)＝０ꎮ(１０)㊀㊀注意到∂σπ/∂Ｈ＝σｐｓｉｇｎ(Ｈ－Ｑ)＝－∂σπ/∂Ｑꎬ且∂Δ(σπꎬＳπꎬＫπ)/∂σπ＝－Ｓｐ２ｋ３σ２π＋Ｋｐ－３６ｋ４σ３π＝－Ｓｐ２ｋ３σ２ｐ(Ｑ－Ｈ)２＋Ｋｐ－３６ｋ４σ３ｐ｜Ｑ－Ｈ｜３ꎮ结合式(９)和式(１０)ꎬ可以得到最大值点Ｑ∗满足Ｃᶄ(Ｑ∗)＝Ｐꎮ这样ꎬ最佳产量Ｑ∗仅仅取决于成本函数Ｃ( )和远期交货价格Ｐꎮ由于Ｃ( )是严格凸函数ꎬ所以Ｑ∗是唯一的ꎮ３㊀最佳套期保值策略讨论最佳套期保值决策的一阶条件ꎮ首先定义ｙ＝Ｈ－Ｑꎮ当ｙ>０时为对冲过度ꎬ而ｙ<０时为对冲不足ꎮ式(１０)可以改写为Ｐ－μｐｋσ２ｐ－ｙ＝ｙＫｐ－３６ｋ２σ２ｐｙ２－Ｓｐ２ｋσｐ｜ｙ｜１－Ｓｐ６ｋ３σ３ｐ｜ｙ｜３＋Ｋｐ－３２４ｋ４σ４ｐｙ４ꎮ(１１)令ｘ＝ｋσｐｙꎬ则Ｐ－μｐｋσ２ｐ－ｙ＝Ｋｐ－３６ｘ３－Ｓｐ２ｘ｜ｘ｜ｋσｐ１－Ｓｐ６｜ｘ｜３＋Ｋｐ－３２４ｘ４æèçöø÷ꎮ(１２)令ｙ∗＝Ｈ∗－Ｑ∗是最佳的对冲过度(当ｙ∗>０)或对冲不足(当ｙ∗<０)量ꎮ如果Ｓｐ＝０ꎬＫｐ＝３ꎬ则有ｙ∗＝Ｐ－μｐｋσ２ｐꎮ㊀㊀此时ꎬｙ∗是期货市场的最佳远期头寸ꎬ即最佳投机交易量ꎮ进一步地ꎬ当Ｐ＝μｐ时ｙ∗＝０ꎬ且ｙ∗独立于Ｓｐ和Ｋｐꎬ这意味着当远期交货价格等于即期现货价格时ꎬ不存在投机ꎬ即最佳期货头寸等于商品产量ꎮ４㊀结论本文对一类指数型效用函数的均值进行正态分布的ＧＣ展开如定理１所示ꎬ然后利用定理１５１６１闽江学院学报第４１卷中效用函数均值的一阶条件得到了最佳产量只取决于成本Ｃ( )和远期价格Ｐ的结论ꎮ由于成本是严格的凸函数ꎬ所以最佳产量取值具有唯一性ꎮ这些结论在当未来的现货价格服从偏正态分布或偏ｔ分布时仍然成立[２－３]ꎮ同时ꎬ运用定理１中效用函数均值的一阶条件ꎬ文章给出了投机不存在的条件ꎬ即在Ｓｐ＝０ꎬＫｐ＝３条件下ꎬ当远期交货价格等于即期交货价格均值时ꎬ最佳期货头寸等于商品产量ꎮ此时投机交易量与峰度或者偏度无关ꎮ本文是ＧＣ展开式在生产与套期保值中应用的初步探讨ꎮ关于利用ＧＣ展开式研究偏度和峰度在套期保值策略中的应用是未来研究工作的重点ꎮ参考文献[１]ＧＩＢＥＲＴＳꎬＪＯＮＥＳＳＫꎬＭＯＲＲＩＳＧＨ.Ｔｈｅｉｍｐａｃｔｏｆｓｋｅｗｎｅｓｓｉｎｔｈｅｈｅｄｇｉｎｇｄｅｃｉｓｉｏｎ[Ｊ].ＪｏｕｒｎａｌｏｆＦｕｔｕｒｅｓＭａｒｋｅｔｓꎬ２００６(２６):５０３－５２０.[２]ＬＩＥＮＤ.Ｔｈｅｅｆｆｅｃｔｓｏｆｓｋｅｗｎｅｓｓｏｎｏｐｔｉｍａｌｐｒｏｄｕｃｔｉｏｎａｎｄｈｅｄｇｉｎｇｄｅｃｉｓｉｏｎｓ:ａｎａｐｐｌｉｃａｔｉｏｎｏｆｔｈｅｓｋｅｗ￣ｎｏｒｍａｌｄｉｓｔｒｉｂｕｔｉｏｎ[Ｊ].ＪｏｕｒｎａｌｏｆＦｕｔｕｒｅｓＭａｒｋｅｔｓꎬ２０１０ꎬ３０:２７８－２８９.[３]ＬＩＥＮＤꎬＷＡＮＧＹＱ.Ｅｆｆｅｃｔｓｏｆｓｋｅｗｎｅｓｓａｎｄｋｕｒｔｏｓｉｓｏｎｐｒｏｄｕｃｔｉｏｎａｎｄｈｅｄｇｉｎｇｄｅｃｉｓｉｏｎｓ:ａｓｋｅｗｅｄｔｄｉｓｔｒｉｂｕ￣ｔｉｏｎａｐｐｒｏａｃｈ[Ｊ].ＴｈｅＥｕｒｏｐｅａｎＪｏｕｒｎａｌｏｆＦｉｎａｎｃｅꎬ２０１５ꎬ２１:１１３２－１１４３.[４]ＬＥＯＮＡꎬＲＵＢＩＯＧꎬＳＥＲＮＡＧ.Ａｕｔｏｒｅｇｒｅｓｓｉｖｅｃｏｎｄｉｔｉｏｎａｌｖｏｌａｔｉｌｉｔｙꎬｓｋｅｗｎｅｓｓａｎｄｋｕｒｔｏｓｉｓ[Ｊ].ＴｈｅＱｕａｒｔｅｒｌｙＲｅｖｉｅｗｏｆＥｃｏｎｏｍｉｃｓａｎｄＦｉｎａｎｃｅꎬ２００５ꎬ４５(４):５９９－６１８.[５]王鹏ꎬ王建琼.中国股票市场的高阶波动特征研究[Ｊ].管理科学ꎬ２００８ꎬ２１(４):１１５－１２０.[６]ＨＡＲＶＥＹＣꎬＬＩＥＣＨＴＹＪꎬＬＩＥＣＨＴＹＭꎬｅｔａｌ.Ｐｏｒｔｆｏｌｉｏｓｅｌｅｃｔｉｏｎｗｉｔｈｈｉｇｈｅｒｍｏｍｅｎｔｓ[Ｊ].ＱｕａｎｔｉｔａｔｉｖｅＦｉｎａｎｃｅꎬ２０１０ꎬ１０:４６９－４８５.[７]ＬＡＩＪＹ.Ａｎｅｍｐｉｒｉｃａｌｓｔｕｄｙｏｆｔｈｅｉｍｐａｃｔｏｆｓｋｅｗｎｅｓｓａｎｄｋｕｒｔｏｓｉｓｏｎｈｅｄｇｉｎｇｄｅｃｉｓｉｏｎｓ[Ｊ].ＱｕａｎｔｉｔａｔｉｖｅＦｉｎａｎｃｅꎬ２０１２ꎬ１２:１８２７－１８３７.[８]ＡＤＣＯＣＫＣꎬＥＬＩＮＧＭꎬＬＯＰＥＲＤＯＮ.Ｓｋｅｗｅｄｄｉｓｔｒｉｂｕｔｉｏｎｉｎｆｉｎａｎｃｅａｎｄａｃｔｕａｒｉａｌｓｃｉｅｎｃｅ:ａｒｅｖｉｅｗ[Ｊ].ＴｈｅＥｕ￣ｒｏｐｅａｎＪｏｕｒｎａｌｏｆＦｉｎａｎｃｅꎬ２０１５ꎬ２１:１２５３－１２８１.[９]ＦＥＲＮＡＮＤＥＺ￣ＰＥＲＥＺＡꎬＦＲＩＪＮＳＢꎬＦＵＥＲＴＥＳＡＭꎬｅｔａｌ.Ｔｈｅｓｋｅｗｎｅｓｓｏｆｃｏｍｍｏｄｉｔｙｆｕｔｕｒｅｓｒｅｔｕｒｎｓ[Ｊ].Ｊｏｕｒ￣ｎａｌｏｆＢａｎｋｉｎｇａｎｄＦｉｎａｎｃｅꎬ２０１８ꎬ８６:１４３－１５８.[责任编辑:金㊀甦]。
延迟折扣的任务呈现方式、数学模型与测量指标
延迟折扣的任务呈现方式、数学模型与测量指标佟月华，韩颖（济南大学教育与心理科学学院，山东济南250022）【摘要】延迟折扣是指未来奖赏当前的主观价值随着时间的延长而减少的心理现象。
研究者通常采用虚拟奖金选择任务来探讨个体的延迟折扣，在呈现材料时使用算机编程法、卡片呈现法和问卷法，常用的数学模型包括单参数模型和双参数模型，使用的测量指标为延迟折扣率、曲线下的面积和ED50。
未来的研究应重点探讨双参数模型的预测力、测量指标的普适性及简易测量法的适用性问题。
【关键词】延迟折扣；虚拟奖金选择任务；数学模型；延迟折扣率（k ）；曲线下的面积（AUC ）中图分类号:R395.1文献标识码:A文章编号:1005-3611(2011)05-0585-04Task Presenting Modes ，Mathematical Models ，and Measures in Delay DiscountingTONG Yue-hua ，HAN YingSchool of Education and Psychology ，University of Jinan ，Jinan 250022，China【Abstract 】Delay discounting refers to the tendency to discount the subjective value of future reward as a function ofthe delay to receiving them.Researchers usually use hypothetical money choice task to ask subjects to make choices be -tween smaller-sooner rewards and larger-later rewards.Experiment task can be presented via the mode of computer,index card,and questionnaire.The mathematical models include one-parameter model and two-parameter model.Discounting rate (k),area-under-the-curve (AUC),and ED50are used to measure delay discounting.Future research should focus on the utility of two-parameter models,measurement indicators,and applicability of the one-shot delay discounting measure.【Key words 】Delay discounting ；Hypothetical money choice task ；Mathematical model ；Discounting rate (k)；Area-un -der-the-curve (AUC)【基金项目】山东省社会科学规划研究项目（09BJYJ03）；山东省高校人文社科研究计划项目（J10WH11）和山东省研究生教育创新计划项目（SDYC10020）资助延迟折扣（delay discounting ）是指未来奖赏的当前主观价值随着时间的延长而减少的心理现象，又称时间贴现（temporal discounting ）[1]。
exponential disutility function
exponential disutility function
"Exponential disutility function" 是一个经济学术语，通常用于描述一个函数，该函数表示的是某种物品或服务给消费者带来的不愉快程度（或称为“负效用”）与该物品的数量或消费量之间的关系。
具体来说，如果一个消费者在消费更多的某种物品时，他会感受到更多的不愉快（负效用）。
这种关系通常可以表示为一种指数形式或"exponential"形式的函数。
也就是说，随着消费量的增加，不愉快感以一个固定比率上升。
这个概念在经济学中用于理解消费者的偏好和需求，特别是在研究需求理论和市场行为时。
理解这种函数对于预测消费者行为、制定经济政策以及进行市场分析等都有重要意义。
双指数效用函数组合投资决策
双指数效用函数组合投资决策周庆健;吕思瑶;焦佳;赵建;魏连鑫;闫博【期刊名称】《大连理工大学学报》【年(卷),期】2011(051)005【摘要】Double exponential utility function is one kind of risk-averse utility function, being classic and comprehensively used by investors. Firstly, non-difference curve method in investment theory was used to calculate the maximum expected return for investors. Then, the optimal portfolio investment decision-making was derivated according to Markowitz＇s mean-variance model, and the corresponding investment proportion was given. The optimal portfolio investment decision-making problem with double exponential utility function was solved very well. At last, a numerical example was provided to illustrate the proposed method.%双指数效用函数是一类典型且被投资者广泛应用的风险厌恶型效用函数．首先应用投资学中的无差异曲线法理论求出具有该类型效用函数的投资者的最大期望收益，然后根据Markowitz的均值一方差模型理论推导出投资者的最优组合投资决策方案，给出了相应的组合投资比例，较好地解决了具有该类型效用函数的投资者的最优投资组合决策问题，最后给出实例对所得结果予以验证．【总页数】5页(P766-770)【作者】周庆健;吕思瑶;焦佳;赵建;魏连鑫;闫博【作者单位】大连理工大学系统工程研究所,辽宁大连116024;大连民族学院理学院,辽宁大连116600;大连民族学院理学院,辽宁大连116600;大连民族学院理学院,辽宁大连116600;同济大学经济与管理学院,上海200092;上海理工大学理学院,上海200093;大连海事大学交通运输管理学院,辽宁大连116026【正文语种】中文【中图分类】N945【相关文献】1.单时期证券市场中负指数效用函数的消费投资组合 [J], 肖翔;许伯生;李路2.基于指数效用函数的保险基金投资决策及保费确定 [J], 王后春;崔玉乐3.均值-方差效用函数在证券组合投资决策中的应用 [J], 万上海4.指数效用函数下的组合投资决策 [J], 刘树人;安雪梅5.指数效用函数下投资组合问题的有效集方法 [J], 刘树人因版权原因，仅展示原文概要，查看原文内容请购买。
risk aversion
% π = ⎢− var[ g ] 2 ⎣ u '( w) ⎥ ⎦ % var[ g ] = 0.5 × (20010 − 20000) 2 + 0.5 × (19990 − 20000) 2 = 100 u '( w) = 1/ w, u ''( w) = −1/ w2 u ''( w) / u '( w) = −1/ w = −1/ 20000
aversion 定义为： A( w) = − u ''( w) ≥0 u '( w)
因为 A(w)是与每单位绝对风险的风险溢价相联系的，因此也被称为绝对风险厌恶。 绝对风险厌恶不仅依赖于效用函数，它也依赖于财富水平 w。通常把绝对风险厌恶的倒 数称作 risk tolerance coefficient：
这样就可以得到参与者的相对风险厌恶，记作 R( w) ，定义为
1 ⎡ wu ''( w) ⎤
（7.4）
R ( w) ≡ −
wu ''( w) u '( w)
7-8
（7.5）
% 因此，如果参与者面临的风险是与他的财富 wg 成比例的，相应的风险溢价作为其
财富的一部分，是与他的相对风险厌恶以及风险相对于财富的大小成比例的。 例： u=ln(w)， w=$20000， G(10， ： ％， ％)： -10 50 50 50% will win 10, 50% will lose 10。 计算风险溢价。 Arrow－Pratt Measure
% % E[u ( w + g )] = u[ E ( w + g ) − π ] = u ( w − π ) = u (CE )