金融衍生品外文翻譯--金融衍生品定價(jià)的精算風(fēng)險(xiǎn)措施

1、<p><b>　　中文3715字</b></p><p><b>　　外 文 翻 譯</b></p><p>　　Actuarial risk measures for financial derivative pricing</p><p>　　金融衍生品定價(jià)的精算風(fēng)險(xiǎn)措施</p><p&

2、gt;<b>　　翻譯</b></p><p><b>　　學(xué)院</b></p><p><b>　　目 錄</b></p><p><b>　　1. 引言1</b></p><p>　　2. 隨機(jī)排序和Esscher轉(zhuǎn)換2</p>&

3、lt;p>　　3. Esscher-Girsanov轉(zhuǎn)換4</p><p>　　4. 金融衍生品定價(jià)的Esscher-Girsanov 轉(zhuǎn)換6</p><p><b>　　1. 引言</b></p><p>　　無(wú)論是直接或間接由一個(gè)公理來(lái)描述，風(fēng)險(xiǎn)措施的精算定價(jià)通常都是合理的。金融衍生產(chǎn)品定價(jià)通常依賴(lài)于無(wú)套利原則。本文建立了一個(gè)新的

4、關(guān)系。</p><p>　　本文的關(guān)系基于歷史悠久的Esscher轉(zhuǎn)換。Esscher轉(zhuǎn)換是十分有用的保險(xiǎn)和金融產(chǎn)品的定價(jià)的工具。Buhlmann (1980)，在溢價(jià)原則的基礎(chǔ)上指出Esscher變換是在一個(gè)一般均衡派生模型中，決策者必須服從負(fù)指數(shù)效用函數(shù);Iwaki（2001）以多段設(shè)置延伸了該模型。 Gerber和Goovaerts（1981）建立了遞增法的溢價(jià)原則其涉及到了Esscher的混合變換。<

5、;/p><p>　　在金融環(huán)境，Gerber和Shiu（1994，1996）使用Esscher變換構(gòu)造等價(jià)鞅措施為了L´evy過(guò)程（帶獨(dú)立和固定增量）。受此啟發(fā)，Buhlmann（1996）更多在 一般條件下使用Esscher變換來(lái)構(gòu)造等價(jià)鞅測(cè)度類(lèi)半鞅。</p><p>　　在本文中，建立風(fēng)險(xiǎn)評(píng)估機(jī)制的方法是由一個(gè)公理化特性用來(lái)描述一個(gè)可以生成無(wú)套利金融衍生品價(jià)格近似值的價(jià)格機(jī)制。特

6、別地，本文提出一個(gè)價(jià)格的表示定理。價(jià)格表示衍生涉及概率測(cè)度變換，它是 密切相關(guān)的Esscher變換，我們稱(chēng)之為 Esscher-Girsanov變換。我們證明了在金融市場(chǎng)，其中，由主資產(chǎn)價(jià)格被表示關(guān)于布朗運(yùn)動(dòng)的隨機(jī)微分方程， 近似值無(wú)套利金融衍生品價(jià)格一致隨著價(jià)格的代表性得出。</p><p>　　建立表示定理的步驟可以制定如下:</p><p>　　1、有序Esscher-Girsano

7、v變換意味著有序價(jià)格。如果價(jià)格措施適用于正態(tài)分布 隨機(jī)變量，則這個(gè)公理是等價(jià)于“遵守二階隨機(jī)支配”。</p><p>　　2、價(jià)格措施是適當(dāng)?shù)貥?biāo)準(zhǔn)化，使得的C非隨機(jī)的單位價(jià)格等于C非隨機(jī)的單位。</p><p>　　3、Esscher-Girsanov轉(zhuǎn)換金額的可加性。如果價(jià)格措施適用于正態(tài)分布的隨機(jī)變量，這公理等價(jià)于“超可加性和單調(diào)可加性?xún)r(jià)格措施“，從而掌握多樣化的好處。</p&g

8、t;<p>　　4、連續(xù)性條件，對(duì)于建立定理是必要的數(shù)學(xué)證明。</p><p>　　這篇文章的要點(diǎn)如下：在第2章中，源于它和我們討論的公理化混合Esscher原則，我們考慮Esscher變換，我們研究了一些隨機(jī)序列關(guān)系。在第3章中，我們將介紹Esscher-Girsanov變換和公理化的價(jià)格措施由它引起的。第4章涉及的金融定價(jià)衍生工具由Esscher-Girsanov變換手段。</p>

9、<p>　　2. 隨機(jī)排序和Esscher轉(zhuǎn)換</p><p>　　我們修復(fù)一個(gè)概率空間，在本文中，除非另有說(shuō)明，一個(gè)隨機(jī)變量（RV）代表凈收入或利潤(rùn)在未來(lái)某個(gè)時(shí)間。自始至終，我們假設(shè)對(duì)于任何R.V.在概率空間中定義的，它的瞬間生成函數(shù)存在，對(duì)于任何的R.V.有X：。</p><p><b>　?。?）</b></p><p>　　

10、累積分布函數(shù)對(duì)應(yīng)于給定r.v.X,我們定義：</p><p><b>　?。?）</b></p><p>　　其Esscher變換含有參數(shù)h。Esscher（1932）使用變換中的（2）代替了原來(lái)的建議累積分布函數(shù)，以眾所周知的漸近近似適用于; 見(jiàn)格柏（1979）。其原因是，埃奇沃思近似的預(yù)期附近表現(xiàn)良好，但在執(zhí)行的結(jié)尾效果十分差。注意當(dāng)h=0時(shí)，原始微分出現(xiàn)，且和E

11、sscher轉(zhuǎn)換是等效分布的且它們有相同的空集。不難確認(rèn)一個(gè)正常的cdf期望μ和變量，　　其Esscher轉(zhuǎn)換是一種正常的cdf與期望μ+h和變量。</p><p>　　其次,對(duì)于一個(gè)給定的r.v.X,我們定義實(shí)值函數(shù)如下：</p><p><b>　　（3）</b></p><p>　　其中，　被稱(chēng)為Esscher溢價(jià)，其中h為參數(shù)。注意在參

12、數(shù)h中是非遞減的。這可以證明容易使用Holder不等式，并將于稍后使用;同時(shí)，觀(guān)察到在（3）中最后一個(gè)表達(dá)式的衍生物可以是解釋為方差。</p><p>　　在下文中，我們用函數(shù)表示一個(gè)風(fēng)險(xiǎn)測(cè)量或——因子X(jué)被解釋為凈收入或利潤(rùn)而分配一個(gè)實(shí)數(shù)的任何RV的價(jià)格措施或它的累積分布函數(shù)。</p><p><b>　　A1、若，則</b></p><p>

13、　　A2、，對(duì)于全部的c</p><p>　　A3、,其中，X,Y是獨(dú)立的</p><p>　　A4、若弱收斂于X,，則</p><p>　　在一般的環(huán)境下，公理A1可以成立。格柏（1981） 已經(jīng)指出，Esscher溢價(jià)不單調(diào)。如果X是Y的一階隨機(jī)導(dǎo)數(shù),它不成立。表示為,然后，。</p><p>　　因此,公理 A1不保證函數(shù)的單調(diào)性。&l

14、t;/p><p>　　Goovaerts (2004)取代公理A1的更多限制性遵守拉普拉斯變換順序公理保障功能的單調(diào)性。若，則從拉普拉斯變換順序來(lái)看X比Y更小。寫(xiě)作。在期望效用模型，拉普拉斯變換順序表示與決策者的偏好由下式給出負(fù)指數(shù)效用函數(shù)：</p><p><b>　?。?）</b></p><p>　　在這里，-h是Arrow–Pratt措施的

15、絕對(duì)風(fēng)險(xiǎn)厭惡。</p><p>　　在下面的章節(jié)中，正態(tài)分布隨機(jī)變量r.v.是主要介紹額。假設(shè)X和Y是正太分布。則條件是</p><p><b>　　（5）</b></p><p>　　當(dāng)于條件,為了驗(yàn)證上述公式，注意正太分布r.v.滿(mǎn)足：</p><p><b>　?。?）</b></p&g

16、t;<p>　　此外,不難驗(yàn)證,如果X和Y 正態(tài)分布，當(dāng)且僅當(dāng)條件（5）成立，。</p><p>　　更為普遍的是,如果X和Y正態(tài)分布，且，那么X是二階隨機(jī)以Y為主，所以Y是優(yōu)先于X的任何風(fēng)險(xiǎn)規(guī)避的期望效用決策制造者。特別的，當(dāng)且僅當(dāng)，r.v.是正態(tài)分布的且。</p><p>　　在經(jīng)濟(jì)學(xué)文獻(xiàn)中，公理A2有時(shí)被稱(chēng)為確定性等價(jià)條件。請(qǐng)注意，C扮演著兩個(gè)在公理A2角色：一個(gè)退化的

17、r.v.位于c上的左手側(cè)和在右手側(cè)的實(shí)數(shù)。公理A3指出價(jià)格相加為獨(dú)立隨機(jī)變量的的可取性。公理A4是在價(jià)格上衡量一個(gè)連續(xù)性條件 。我們給出如下引理：</p><p>　　引理2.1 價(jià)格測(cè)量滿(mǎn)足條件A1到A4，當(dāng)且僅當(dāng)存在非負(fù)函數(shù)H：，因此</p><p><b>　?。?）</b></p><p>　　注2.1 Gerber和Goovaerts

18、（1981）建立了一個(gè)的混合Esscher原則公理化刻畫(huà)。Goovaerts（2004）提出了公理化的混合指數(shù)原則。它是直接驗(yàn)證為常分布隨機(jī)變量的任何混合Esscher保費(fèi)是一個(gè)混合指數(shù)溢價(jià)，反之亦然。在一般情況下，它僅認(rèn)為任何混合指數(shù)溢價(jià)是一個(gè)混合Esscher溢價(jià)。</p><p>　　注2.2 可以將混合函數(shù)H（·）視為一個(gè)累積分布函數(shù)，支撐在（-，0]和可能的故障與轉(zhuǎn)折點(diǎn)在-。它可以作為一個(gè)先驗(yàn)

19、分布為Arrow–Pratt度量絕對(duì)風(fēng)險(xiǎn)厭惡。要知道為什么參數(shù)-H參與在Esscher變換可以被解讀為箭，普拉特Arrow–Pratt衡量絕對(duì)風(fēng)險(xiǎn)厭惡相應(yīng)的決定制造商與負(fù)指數(shù)效用函數(shù)。</p><p>　　注2.3 引理2.1中的價(jià)格測(cè)量可以表示為，其中期望計(jì)算使用微分。</p><p>　　3. Esscher-Girsanov轉(zhuǎn)換</p><p>　　在上一節(jié)中

20、，我們提出了一個(gè)代表性定理對(duì)于那些添加劑的獨(dú)立價(jià)格措施R.V.的。得到的價(jià)格表示可以被視為一個(gè)（混）Esscher變換概率預(yù)期下測(cè)量。在本節(jié)中，我們將介紹一個(gè)密切相關(guān)的概率測(cè)度變換和公理化的價(jià)格措施由它引起的。</p><p>　　對(duì)于一個(gè)給定的r.v。X,我們定義擴(kuò)展實(shí)值函數(shù):</p><p><b>　?。?）</b></p><p>　　

21、表示逆的分布函數(shù)的標(biāo)準(zhǔn)正態(tài)分布。眾所周知的，如果函數(shù)FX 是連續(xù)的，那么，R.V. 是正態(tài)分布的，且均值為0，方差為1，在本節(jié)的其余部分，除非另有說(shuō)明者外，我們限制RV與連續(xù)CDF。我們有如下定義：</p><p>　　定義3.1 (Esscher——Girsanov 轉(zhuǎn)換)。對(duì)于累積分布函數(shù)且由其微分d對(duì)應(yīng)于給定r.v.X,和一個(gè)給定的實(shí)數(shù)v,我們定義如下：</p><p><b&

22、gt;　?。?）</b></p><p>　　其中Esscher-Girsanov轉(zhuǎn)換參數(shù)為h、v(絕對(duì)的分別為風(fēng)險(xiǎn)規(guī)避和懲罰參數(shù))。Igor V Girsanov的名稱(chēng)被安裝在概率上述措施變換定義強(qiáng)調(diào)的密切中所用的Radon-Nikodym導(dǎo)數(shù)之間的相似性（9），并在Girsanov的使用的Radon-Nikodym導(dǎo)數(shù)定理;參見(jiàn)，Karatzas和Shreve（1988）。</p>

23、<p>　　在這個(gè)階段，h和v只有產(chǎn)品似乎相關(guān)。然而，這兩個(gè)參數(shù)下面將扮演兩個(gè)不同的角色。按照烏爾曼（1980）中，h可以被解釋為的絕對(duì)風(fēng)險(xiǎn)厭惡系數(shù)，而V可能把握總體市場(chǎng)風(fēng)險(xiǎn)。憑借CLT，在通常的的情況下，總體市場(chǎng)風(fēng)險(xiǎn)可以很好地近似由正常R.V.此外，當(dāng)只有正常個(gè)體風(fēng)險(xiǎn)被視為（如在第4節(jié)，至少無(wú)窮）總體市場(chǎng)風(fēng)險(xiǎn)是完全正常的。</p><p>　　由此不難驗(yàn)證為具有正常的CDF期望μ和方差，其Essche

24、r-Girsanov變換與期望μ+ HV正常的CDF和方差σ2。在特別是，當(dāng)v=，我們平凡發(fā)現(xiàn)，對(duì)Esscher-Girsanov 變換是一個(gè)普通的Esscher變換。因此，對(duì)于一個(gè)正常的 </p><p>　　CDF的Esscher-Girsanov變換，就像普通Esscher變換，改變了平均同時(shí)保留方差。請(qǐng)注意，對(duì)于平均值的變化，該值均值是無(wú)關(guān)緊要的。在下文中，我們令v是嚴(yán)格正和暫時(shí)固定和h的域限制到h0。&

25、lt;/p><p>　　我們介紹了實(shí)值函數(shù)定義為：</p><p><b>　?。?0）</b></p><p>　　此后，函數(shù)被稱(chēng)為Esscher-Girsanov 該R.V.X的價(jià)格。參數(shù)h0和v> 0。鑒于V，存在X之間的唯一對(duì)應(yīng)關(guān)系其Esscher-Girsanov價(jià)格在某種意義上說(shuō)，X = Y在分布成立當(dāng)且僅當(dāng)</p>

26、<p><b>　　（11）</b></p><p>　　為了驗(yàn)證這種說(shuō)法,有</p><p><b>　?。?2）</b></p><p>　　上式可以被視為一個(gè)拉普拉斯變換,所以在函數(shù)和之間有一一對(duì)應(yīng)且一致的關(guān)系。其中，函數(shù)對(duì)于給出的h的導(dǎo)數(shù)為：</p><p><b>　

27、?。?3）</b></p><p>　　其中括號(hào)中的表達(dá)式可以被視為X和的Esscher-Girsanov協(xié)方差且非負(fù)。正如有人指出，在Goovaerts（2004），價(jià)格衡量其特征引理2.1有一個(gè)對(duì)應(yīng)的分配一個(gè)實(shí)數(shù)的函數(shù)。類(lèi)似的，用表示分配一個(gè)實(shí)數(shù)的任何函數(shù)。我們令價(jià)格測(cè)量定義為：</p><p><b>　?。?4）</b></p>&l

28、t;p>　　為了能夠遵循前面的設(shè)定需滿(mǎn)足：</p><p><b>　　B1、若，則</b></p><p>　　B2、，對(duì)于全部的c</p><p><b>　　B3、</b></p><p>　　B4、如果弱收斂于，則</p><p>　　注意公理B2、B4和A2、

29、A4相似。而且，</p><p>　　我們注意到，對(duì)于正態(tài)分布隨機(jī)變量r.v.，公理B1是類(lèi)似公理A1的，而且引起的二階隨機(jī)占優(yōu)維護(hù)的權(quán)益。容易驗(yàn)證，如果X和Y是兩個(gè)正態(tài)分布隨機(jī)變量，則其線(xiàn)性相關(guān)系數(shù)為xy，則</p><p><b>　　（15）</b></p><p>　　因此，對(duì)于r.v.的正態(tài)分布，公理B3相當(dāng)于投資組合的條件是價(jià)格X

30、+ Y等于價(jià)格r.v.正太分布。</p><p>　　定義3.2 （離散時(shí)間Esscher-Girsanov轉(zhuǎn)換）。對(duì)于累積分布函數(shù)，其微分為，相似與連續(xù)的r.v.。對(duì)于一個(gè)給定的嚴(yán)格的正函數(shù)v(·),我們定義為：</p><p><b>　　（16）</b></p><p>　　其離散時(shí)間Esscher-Girsanov轉(zhuǎn)換參數(shù)為h

31、 和罰函數(shù)為v(·)。</p><p>　　離散時(shí)間Esscher-Girsanov轉(zhuǎn)換視為一個(gè)特定條件Esscher轉(zhuǎn)換的例子（見(jiàn)Buhlmann （1996））,盡管有一個(gè)微妙的區(qū)別在于,按照經(jīng)濟(jì)解釋和公理化,我們使用一個(gè)常數(shù)Arrow-Pratt絕對(duì)風(fēng)險(xiǎn)規(guī)避的措施。</p><p>　　4. 金融衍生品定價(jià)的Esscher-Girsanov 轉(zhuǎn)換</p>&l

32、t;p>　　在本節(jié)中,我們將顯示,在金融市場(chǎng)的主要資產(chǎn)是由一個(gè)隨機(jī)表示微分方程(SDE)對(duì)于布朗運(yùn)動(dòng), 價(jià)格機(jī)制基于Esscher-Girsanov轉(zhuǎn)換可以生成無(wú)套利近似值金融衍生品的價(jià)格。</p><p>　　我們考慮一個(gè)有限的時(shí)間范圍。信息流所代表的是正確的連續(xù)過(guò)濾，對(duì)于全部的都成立。因此，對(duì)于給定的r.v.x，我們用表示的條件期望。</p><p>　　我們考慮一個(gè)齊次時(shí)間的主

33、要資產(chǎn)的過(guò)程，，其定義為一個(gè)隨機(jī)微分方程的形式：</p><p><b>　?。?7）</b></p><p>　　其中，表示一個(gè)標(biāo)準(zhǔn)布朗運(yùn)動(dòng)。我們指出在一般的情況下S不需要是正數(shù)，因?yàn)樗硪粋€(gè)任意的主要資產(chǎn)。然而，如果一個(gè)已在考慮應(yīng)用程序需要積極主要資產(chǎn)的過(guò)程，μ（·）和β（·）的附加條件就可以實(shí)施。</p><p>　

34、　接著，我們考慮一個(gè)債券價(jià)格的過(guò)程，定義SDE為</p><p><b>　　（18）</b></p><p>　　其中，足夠光滑對(duì)于存在的積分</p><p><b>　?。?9）</b></p><p>　　雖然我們限制自己齊次時(shí)間初選資產(chǎn)的過(guò)程,一般擴(kuò)散過(guò)程是可行的。然而,最著名的擴(kuò)散過(guò)程已經(jīng)

35、包含在公式(17)。</p><p>　　定理4.1 假設(shè)S是一個(gè)交易的資產(chǎn)。如果且</p><p>　　然后，伴隨著近似無(wú)套利金融衍生品的價(jià)格在時(shí)間。</p><p>　　證明：在一個(gè)完整的金融自由市場(chǎng)，著名的偏微分方程能夠描述近似無(wú)套利金融衍生品價(jià)格，考慮偏微分方程：</p><p>　　在這種情況下，且。證明完畢。</p>

36、<p><b>　　MENU</b></p><p>　　1 Introduction9</p><p>　　2 Stochastic ordering and the Esscher transform11</p><p>　　3 The Esscher–Girsanov transform13</p><

37、;p>　　4 Financial derivative pricing by means of Esscher–Girsanov transforms15</p><p>　　Actuarial risk measures for financial derivative pricing</p><p>　　1 Introduction</p><p>　　

38、Risk measures for actuarial pricing are usually justified,either directly or indirectly, by means of an axiomatic characterization.Financial derivative pricing usually relies on principles of no arbitrage. Various attemp

39、ts to connect the two approaches are available in the literature; the interested reader is referred to Embrechts (2000) for a review. This paper establishes a new connection.</p><p>　　The connection is based

40、 on the time-honored Esscher transform. The Esscher transform has proven to be a valuable tool for the pricing of insurance and financial products.In Buhlmann (1980), a premium principle based on the Esscher transform is

41、 derived within a general equilibrium model in which decision makers have negative exponential utility functions; see Iwaki et al. (2001) for an extension of that model to a multi-period setting. Gerber and Goovaerts (1

42、981) established an axiomatic charact</p><p>　　In a financial environment, Gerber and Shiu (1994, 1996) use the Esscher transform to construct equivalent martingale measures for L´evy processes (with in

43、dependent and stationary increments). Inspired by this, Buhlmann et al. (1996) more generally use conditional Esscher transforms to construct equivalent martingale measures for classes of semi-martingales.</p><

44、;p>　　In this paper, the approach of establishing risk evaluation mechanisms by means of an axiomatic characterization is used to characterize a price mechanism that can generate approximate-arbitrage-free financial de

45、rivative prices. In particular, this paper presents a representation theorem for price measures that are superadditive and comonotonic additive for normally distributed random variables. The price representation derived

46、involves a probability measure transform that is closely related to </p><p>　　The axioms imposed to establish the representation theorem can be formulated as follows:</p><p>　　1. Ordered Esscher

47、–Girsanov transforms implies ordered prices. If the price measure is applied to normally distributed random variables, this axiom is equivalent to “respect for second-order stochastic dominance”.</p><p>　　2.

48、 The price measure is appropriately normalized such that the price of c non-random units is equal to c non-random units.</p><p>　　3. Additivity for sums of Esscher–Girsanov transforms. If the price measure i

49、s applied to normally distributed random variables, this axiom is equivalent to “superadditivity and comonotonic additivity of the price measure”, thus capturing the benefits of diversification.</p><p>　　4.

50、Continuity conditions, which are necessary for establishing the mathematical proofs.</p><p>　　The outline of this paper is as follows: in Section 2, we consider the Esscher transform, we study some stochasti

51、c order relations derived from it and we discuss the axiomatization of the mixed Esscher principle. In Section 3, we introduce the Esscher–Girsanov transform and axiomatize a price measure induced by it. Section 4 addres

52、ses the pricing of financial derivatives by means of Esscher–Girsanov transforms.</p><p>　　2 Stochastic ordering and the Esscher transform</p><p>　　We fix a probability space,In this paper, unle

53、ss otherwise stated, a random variable (r.v.) represents net income or profit at a future point in time. Throughout, we assume that for any r.v. defined on the probability space, its moment generating function exists, i.

54、e., for any r.v.X：.</p><p><b>　　(1)</b></p><p>　　For the cumulative distribution function (cdf),corresponding to a given r.v. X, we define by</p><p><b>　　(2)</b

55、></p><p>　　its Esscher transform with parameter h. Esscher (1932) suggested using the transform in (2) instead of the original cdf, to apply the well-known Edgeworth approximation to;see also Gerber (1979

56、). The reason was that the Edgeworth approximation performs well in the vicinity of the expectation,but performs worse in the tails. Notice that for h = 0, the riginal differential appears, and that and its Esscher trans

57、form are equivalent distributions in the sense that they have the same null sets. It i</p><p>　　Next, for a given r.v. X, we define the real-valued function as follows:</p><p><b>　　(3)&l

58、t;/b></p><p>　　The number is known as the Esscher premium with parameter h. Notice that is non-decreasing in h.This can be proved easily using the Holder inequality and will be used later; also,observe

59、that the derivative of the last expression in (3) can be interpreted as a variance.</p><p>　　In the following, we denote by the functional a risk measure or – since X is interpreted as net income or profit

60、–rather a price measure that assigns a real number to any r.v. Or its cdf.We introduce a set of axioms that must satisfy:</p><p>　　A1.If ,then </p><p>　　A2.,for all c.</p><p>　　A3.

61、,when X and Y are independent.</p><p>　　A4.If converges weakly to X, with ,</p><p><b>　　then .</b></p><p>　　In a general setting, axiom A1 can be criticized. Gerber (198

62、1)already pointed out that the Esscher premium is not monotonic,it does not hold that if X is first-order stochastically dominated by Y , denoted by ,then ,Hence, axiom A1 does not guarantee monotonicity of the functiona

63、l .</p><p>　　Goovaerts et al. (2004) replaced axiom A1 by the more restrictive axiom of respect for Laplace transform order, which does guarantee monotonicity of the functional .We say that X is smaller than

64、 Y in Laplace transform order if ,We write .In the expected utility model, the Laplace transform order represents preferences of decision makers with a negative exponential utility function given by</p><p>&

65、lt;b>　　(4)</b></p><p>　　Here, ?h is the Arrow–Pratt measure of absolute risk aversion.The interested reader is referred to Denuit (2001) for a comprehensive treatment of the Laplace transform order.

66、</p><p>　　In the following sections, normally distributed r.v.’s are of particular interest. Suppose that X and Y are normally distributed. Then the condition that</p><p><b>　　(5)</b>

67、;</p><p>　　is equivalent to the condition that both .To verify this statement, notice that for normally distributed r.v.’s</p><p><b>　　(6)</b></p><p>　　Furthermore, it i

68、s not difficult to verify that if X and Y are normally distributed, then if and only if condition (5) is satisfied .</p><p>　　More generally, it is well known that if X and Y are normally distributed with ,t

69、hen X is second-order stochastically dominated by Y and hence Y is preferred to X by any risk averse expected utility decision maker</p><p>　　and Rothschild and Stiglitz (1970) for the original work on secon

70、d-order stochastic dominance. In particular, notice that for normally distributed r.v.’s,if and only if .Hence, axiom A1 is appealing for the case of normally distributed r.v.’s.</p><p>　　In the economics li

71、terature, axiom A2 is sometimes referred to as the certainty equivalent condition. Notice that c plays two roles in axiom A2: a degenerate r.v. at c on the left-hand side and a real number on the right-hand side. The des

72、irability of price additivity for independent r.v.’s, as imposed by axiom A3, was already pointed out by Borch .</p><p>　　Axiom A4 is a continuity condition on the price measure .We state the following lemm

73、a:</p><p>　　Lemma 2.1. A price measure satisfies the set of axioms A1–A4 if and only if there exists some non-decreasing function H: such that</p><p><b>　　(7)</b></p><p&g

74、t;　　Remark 2.1. Gerber and Goovaerts (1981) established an axiomatic characterization of the mixed Esscher principle. Goovaerts et al. (2004) axiomatized the mixed exponential principle. It is straightforward to verify t

75、hat for normally distributed r.v.’s any mixed Esscher premium is a mixed exponential premium, and vice versa. In general, it only holds that any mixed exponential premium is a mixed Esscher premium.</p><p>　

76、　Remark 2.2. The mixture function H(·) can be regarded as a cdf, supported on （-，0]and possibly defective with a jump at -.It can serve as a prior distribution for the Arrow–Pratt measure of absolute risk aversion;

77、see in this respect Savage (1954). To see why the parameter ?h involved in the Esscher transform can be interpreted as the Arrow–Pratt measure of absolute risk aversion corresponding to a decision maker with a negative e

78、xponential utility function. </p><p>　　Remark 2.3. The price measure characterized in Lemma 2.1 can be expressed as where the expectation is calculated using the differential</p><p>　　3 The Ess

79、cher–Girsanov transform</p><p>　　In the previous section, we presented a representation theorem for price measures that are additive for independent r.v.’s. The price representation derived can be regarded a

80、s an expectation under a (mixed) Esscher transformed probability measure. In this section, we introduce a closely related probability measure transform and axiomatize a price measure induced by it.</p><p>　　

81、For a given r.v. X, we define the extended real-valued function as follows:</p><p><b>　　(9)</b></p><p>　　in which denotes the inverse distribution function of the standard normal d

82、istribution. It is well known that if FX is continuous, then the r.v.is normally distributed with mean 0 and variance 1. In the remainder of this section, unless otherwise stated, we restrict ourselves to r.v.’s with a c

83、ontinuous</p><p>　　cdf. We state the following definition:</p><p>　　Definition 3.1 (Esscher–Girsanov Transform). For the cdf FX (·) with differential dFX (·) corresponding to a given r

84、.v. X,and a given real number v, we define by</p><p><b>　　(9)</b></p><p>　　its Esscher–Girsanov transform with parameters h and v.The name of Igor V. Girsanov is attached to the prob

85、ability measure transform defined above to emphasize the close resemblance between the Radon–Nikodym derivative used in(9) and the Radon–Nikodym derivative used in Girsanov’s Theorem.</p><p>　　At this stage,

86、 only the product of h and v seems relevant. However, below the two parameters will play two distinct roles. In accordance with B¨uhlmann (1980), h could be interpreted as the coefficient of absolute risk aversion w

87、hile could capture aggregate market risk. By virtue of the CLT, in usual circumstances, aggregate market risk can be well approximated by a normal r.v. Moreover, when only normal individual risks are considered aggrega

88、te market risk is exactly normal.</p><p>　　It is not difficult to verify that for a normal cdf with expectation μ and variance , its Esscher–Girsanov transform is a normal cdf with expectation μ + h and vari

89、ance . In particular, if v =we trivially find that the Esscher–Girsanov transform is an ordinary Esscher transform. Hence, for a normal cdf, the Esscher–Girsanov transform, just like the ordinary Esscher transform, chang

90、es the mean while preserving the variance. Notice that for the change of the mean, the value of the mean is irreleva</p><p>　　We introduce the real-valued function defined by</p><p><b>　

91、　(10)</b></p><p>　　Henceforth, the number is called the Esscher–Girsanov price of the r.v. X, with parameters h0 and v > 0. Notice that given v, there exists a unique correspondence between X and i

92、ts Esscher–Girsanov price in the sense that X = Y in distribution if and only if</p><p><b>　　(11)</b></p><p>　　To verify this statement, notice that</p><p><b>　　(1

93、2)</b></p><p>　　which can be regarded as a Laplace transform, so that the oneto-one correspondence between and follows. The derivative of with respect to h is given by</p><p><b>　　

94、(13)</b></p><p>　　in which the expression between brackets can be regarded as the Esscher–Girsanov covariance of X and and is nonnegative.As was pointed out in Goovaerts et al. (2004), the price measu

95、re characterized in Lemma 2.1 has a counterpart that assigns a real number to the function .</p><p>　　Similarly, we denote by a functional that assigns a real number to any</p><p>　　function ,

96、we let the price measure be defined by</p><p><b>　　(14)</b></p><p>　　and state the following set of axioms that should satisfy:</p><p>　　B1.If ,then </p><p>

97、;　　B2.,for all c.</p><p><b>　　B3.</b></p><p>　　B4.If converges to for all,then</p><p>　　Notice that axioms B2 and B4 are similar to axiom A2 and A4. Notice furthermore

98、that</p><p>　　We note that for normally distributed r.v.’s, axiom B1 is similar to axiom A1 and gives rise to the appealing secondorder stochastic dominance preserving property for. One easily verifies that

99、if X and Y are two normally distributed r.v.’s with linear correlation coefficienxy , then</p><p><b>　　(15)</b></p><p>　　Hence, for normally distributed r.v.’s, axiom B3 is equivalen

100、t to the condition that the price of the portfolio X +Y is equal to the price of a normally distributed r.v.</p><p>　　Definition 3.2 (Discrete-Time Esscher–Girsanov Transform). For the cdf FXn (·) with

101、differential dFXn (·) corresponding to a given continuous r.v. Xn, and a given strictly positive function v(·), we define by</p><p><b>　　(16)</b></p><p>　　its discrete-time

102、 Esscher–Girsanov transform with parameter h and penalty function v(·).The discrete-time Esscher–Girsanov transform can be regarded as a particular example of a conditional Esscher transform, though there is a subtl

103、e difference being that, in accordance with the economic interpretation and axiomatization, we use a constant Arrow–Pratt measure of absolute risk aversion.</p><p>　　4 Financial derivative pricing by means o

104、f Esscher–Girsanov transforms</p><p>　　In this section, we will show that in a financial market in which the primary asset is represented by a stochastic differential equation (SDE) with respect to Brownian

105、motion, the price mechanism based on the Esscher–Girsanov transform can generate approximate-arbitrage-free financial derivative prices.</p><p>　　We consider a finite time horizon . The flow of information i

106、s represented by the completed and right continuous filtration with for all .Henceforth, for a given r.v.X, we denote by the conditional expectation of X given .</p><p>　　We consider a time-homogeneous prima