效用無差異價格於不完全市場下之應用 - 政大學術集成

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(1)國立政治大學應用數學系碩士學位論文. Utility Indifference Pricing in 政治大 Incomplete Markets 立. ‧ 國. 學. 效用無差異價格於不完全市場下之應用. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. 碩士班學生 : 胡介國撰指導教授 : 胡聯國博士中華民國九十九年六月.

(2) 謝辭能完成這篇論文，我十分感謝指導教授胡聯國老師給予的細心教導，老師在百忙之中總是不厭其煩地給我寶貴的意見與方向，才使這篇論文能充實嚴謹的順利完成。同時謝謝口試委員姜祖恕老師與劉明郎老師，給我許多珍貴的建議，使得論文更加完善。在念研究所的這些日子，感謝系上所有老師們的認真教學，特別感謝陳天進老師與蔡隆義老師，讓我學習嚴謹扎實的數學訓練。最後謝謝家人和同學朋友們的鼓勵與支持，讓我更有信心完成這篇論文，謝謝焱騰學長和天財學長在這段期的照顧，不論是課業上或是生活上幫助我很多事情，雖然相處的時間不長，但我會永遠想念這兩年來和大家一. 政治大. 起努力、一起相聚的美好回憶。. 立. ‧ 國. 學. 胡介國謹誌于. ‧. 國立政治大學應用數學所中華民國九十九年六月. n. er. io. sit. y. Nat. al. Ch. engchi. i. i n U. v.

(3) Abstract In incomplete markets, prices of a contingent claim can be obtained between the upper and lower hedging prices. In this thesis, we will use utility indifference pricing to find an initial payment for which the maximal expected utility of trading the claim is indifferent to the maximal expected utility of no trading. From the central duality result, we show that the gap between the seller’s and the buyer’s utility indifference prices is always smaller than the gap between the upper and lower hedging prices under the exponential utility function.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. ii. i n U. v.

(4) 中文摘要在不完全市場下，衍生性金融商品可利用上套利和下套利價格來訂出價格區間。我們運用效用無差異定價於此篇論文中，此定價方式為尋找一個初始交易價，會使在起始時交易商品和無交易商品於商品到期日之最大期望效用相等。利用主要的對偶結果，我們證明在指數效用函數下，效用無差異定價區間會比上套利和下套利定價區間小。. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. iii. i n U. v.

(5) Contents. 謝辭. i. Abstract. ii. 中文摘要. iii. Contents. iv. 立. 1. 學. ‧ 國. 1 Introduction. 政治大. 2 The Fundamental Financial Market Model. ‧. 3 Superreplication and Subreplication. 4. Nat. sit. y. 7. io. n. al. 11. er. 4 Utility Indifference Pricing. Ch. 5 Proof of the Central Duality Result. engchi. i n U. v. 15. 6 Conclusion. 18. References. 19. iv.

(6) 1. Introduction. Pricing contingent claims, also called derivative securities or options, are very important topics in finance. Every contingent claim can admit a unique arbitrage-free price in complete markets. On the other hand, most contingent claims only acknowledge intervals of arbitrage-free prices in incomplete markets. The main result of the thesis is to price smaller intervals of arbitrage-free prices for bounded European contingent claims in incomplete markets. In the fundamental financial market, we describe risky assets by process. 治政 dS(t) = S(t)[µ(t)dt + σ(t) · 大 dB(t)], 立. where B(t) is a Brownian motion. If there exists an equivalent local martingale. ‧ 國. 學. measure with respect to the normalized market, then the market has no arbitrage. We say that the contingent T -claim G is attainable if there exists an admissible. io. sit. y. ‧. Nat. portfolio π(t) and initial capital x ∈ R such that Z T π(t) · dS(t) a.s.. G(ω) = x + 0. n. al. er. In complete markets, every contingent T -claim can be attainable by some admissi-. i n U. v. ble portfolios at maturity time T . Moreover, the contingent claim price at initial. Ch. engchi. time is taking the expectation of a risky free asset discount of it’s maturity value with respect to the (local) martingale measure. We can accept the unique (local) martingale measure by the Girsanov’s theorem. In incomplete markets, there are infinite equivalent local martingale measures, so that it isn’t definite to take expectation. There have been some studies of contingent claim prices in an incomplete market. We compare superreplication, subreplication and utility indifference pricing in this thesis. Superreplication is defined as a minimal price of the initial capital such that the maturity capital is more than the contingent claim for an investment strategy. The price is also called the seller’s price or the upper hedging price of the contingent claim. Similarly, subreplication. 1.

(7) is to consider a maximal payment for this security such that the buyer’s maturity capital adding the contingent claim for an investment strategy is always to make money. The payment is also called the buyer’s price or the lower hedging price of the contingent claim. From the seller’s and buyer’s hedging prices, we can determine the price interval of the contingent claim. In general, the gap between the upper and lower hedging prices is too broad. Other famous studies are about the utility function. We study the idea of utility indifference pricing. The pricing formula was introduced by Hodges and Neuberger (1989). It is based on an initial payment for which the maximal expected utility. 政治大 Øksendal and Sulem (2009) introduced risk indifference pricing. It follows from a 立 convex risk measure. 學. ‧ 國. of trading the claim is indifferent to no trading. As utility indifference pricing,. ρ : F → R,. where F is the set of some random variables; see Föllmer and Schied (2002). There-. ‧. fore, we can get an initial payment for which the risk of trading the claim is indiffer-. y. Nat. ent to no trading. Moreover, Øksendal and Sulem (2009) proofed the price interval. io. sit. of risk indifference pricing belongs to the price interval of the upper and lower hedg-. n. al. er. ing prices in jump diffusion markets. The purpose of this thesis is to prove that the. i n U. v. price interval of utility indifference pricing also falls between the upper and lower. Ch. engchi. hedging prices for every bounded T -claim in the fundamental market. To verify the main result, we use the relative entropy which measures the difference between two probability measures in probability theory and information theory. Under an exponential utility function, the relative entropy can lead to a special duality, called the central duality result. The central duality result can help us to solve the utility indifference price. The framework of the thesis is organized as follows: in section 2, we consider the fundamental finical market model and describe definitions of superreplication, subreplication and utility indifference pricing. In section 3, we give some principles of local martingale and prove that the upper and lower hedging prices can be repre-. 2.

(8) sented by the maximal and minimal local martingales of bounded claims. In section 4, we introduce the relative entropy and verify the price interval of utility indifference pricing falls between the price interval of the upper and lower hedging prices, based on the central duality result. In section 5, we prove that the central duality result of the maximal expect utility by applying the relative entropy. Finally, we give the conclusions and further researches in section 6.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 3. i n U. v.

(9) 2. The Fundamental Financial Market Model. Consider a filtered probability space (Ω, F, {Ft }0≤t≤T , P ), where we define Ft = (m). Ft. to be the σ-algebra generated by the m-dimensional Brownian motion Bt (ω).. Let a financial market S(t) with n + 1 assets. The price processes of assets are Ft -adapted and have the forms: (1) a risk free asset with unit price, S0 (t) = 1 for every time 0 ≤ t ≤ T . (2) risky asset prices Si (t) for i = 1, ..., n where we take a risk free asset as the. 政治大 dS (t) = 立S (t)[ µ (t)dt + σ (t) · dB(t)]. unit of account are given by i. i. i. i. ‧ 國. 學. Si (0) = si .. for t ∈ [0, T ],. ‧. We consider µi (t) and σi (t) are Ft -adapted processes for i = 1, ..., n with ) Z T (X n m X n X |µi (s)| + |σij (s)|2 ds < ∞ a.s... Nat. j=1 i=1. sit. i=1. y. 0. er. io. A portfolio in the market {S(t)}t∈[0,T ] is an (n+1)-dimensional (t, ω)-measurable. n. al. i n C of a portfolio π(t) = (π (t), π (t),h ..., π (t)) is defined e n g c h i Uby. v. and Ft -adapted stochastic process. The value process with initial value x at time t 0. 1. Xx(π) (t, ω). n. = π(t) · S(t) =. n X. πi (t)Si (t). i=0. where Xx(π) (0) = x. Definition 2.1 (1) The portfolio π(t) is called self-financing if  " n #2  Z T X n m  X X | πi (s) µi (s)| + πi (s) σij (s) ds < ∞ a.s.  0  i=1. j=1. i=1. and Xx(π) (t). Z. t. π(s) · dS(s). =x+ 0. 4. for t ∈ [0, T ].. (2.1).

(10) (2) A portfolio π(t) which is self-financing is called admissible if the value process Xx(π) (t) is (t, ω) a.s. lower bounded. Note that we say a portfolio with (2.1) is S-integrable. Moreover, if the initial value x and π1 (t), ..., πn (t) are given such that (2.1) holds, then there always exists a π0 (t) such that π(t) is a self-financing portfolio. A bounded European contingent T claim G is a bounded FT -measurable random variable and taken a risk free asset as the unit of account. It represents the net payoff of a derivative security at maturity T . In order to study incomplete market, we introduce pricing formula below.. 治政 folios. Superreplication of a T -claim G is to consider大 a minimal price as the initial 立 capital x such that the maturity capital is always more than G for an investment. Superreplication and subreplication. Let P be the set of all admissible port-. ‧ 國. 學. strategy. The minimal price which the seller is willing to sell is. ‧. Pup (G) = inf{x| there exits π ∈ P such that Xx(π) (T ) ≥ G a.s. }. y. Nat. and also called the seller’s price or the upper hedging price of G. Similarly, subrepli-. sit. cation is to consider a maximal payment for this security G such that the buyer’s. n. al. er. io. maturity capital adding the contingent claim for an investment strategy is always. i n U. v. to make a profit. The maximal price which the buyer is willing to pay is. Ch. engchi. (π). Plow (G) = sup{x| there exits π ∈ P such that X−x (T ) + G ≥ 0 a.s. } and also called the buyer’s price or the lower hedging price of G. In the complete market, it follows from the Girsanov’s theorem there exists a martingale measure Q such that Plow (G) = EQ [G] = Pup (G) : see [10, Theorem 12.3.2]. Utility indifference pricing. Given a utility function U : R → R. Let x be the initial capital before the claim G is being traded.. 5.

(11) (1) If we give the price p1 for the guarantee to the seller, then the seller can use x + p1 as the initial value to invest and needs to pay the guarantee G at maturity time. The maximal expected utility for the seller is (π). Vs (G, x + p1 ) = sup E[U (Xx+p1 (T ) − G)]. π∈P. (2) There is no claim G to trade. The maximal utility expected utility at maturity time T is V (G, x) = sup E[U (Xx(π) (T ))]. π∈P. (3) If we give the payment p2 for the guarantee to the buyer, then the buyer use. 政治大 time. The maximal 立 expected utility for the buyer is (π). 學. ‧ 國. an initial fortune x − p2 to invest and can gain the guarantee G at maturity. Vb (G, x − p2 ) = sup E[U (Xx−p2 (T ) + G)]. π∈P. ‧. The seller’s and buyer’s utility indifference prices are to find initial prices p1 = pus. y. Nat. and p2 = pub with Vs (G, x + pus ) = V (G, x) and Vb (G, x − pub ) = V (G, x). Denote. n. al. U (x) = −e−γx ,. Ch. engchi. where γ > 0 is the risk aversion parameter.. 6. er. io. exponential utility. sit. Pseller (G) := pus and Pbuyer (G) := pub for the T -claim G. In this study, we given an. i n U. v.

(12) 3. Superreplication and Subreplication Z. t. π(s) · dB(s) exist and are martingales under the condition. Itô integrals 0. T. Z. |π(t)| dB(t) < ∞. 2. E 0 t. Z. π(s) · dB(s) exist under the weak condition. In fact, Itô integrals 0. T. Z. |π(t)|2 dt < ∞ a.s... 0. 政治大 tion. In general, we discuss 立local martingale rather than martingale in stochastic t. Z. π(s) · dB(s) are martingales under the weak condi-. However, we can’t pledge. 0. ‧ 國. 學. differential equations.. Definition 3.1 An Ft -adapted stochastic process M (t) ∈ Rn is called a local mar-. ‧. tingale with respect to {Ft } if there exists an increasing sequence of Ft -stopping. as. k→∞. io. and M (t ∧ τk ) is an Ft -martingale for all k.. n. al. Ch. engchi. We can get following properties:. er. τk → ∞ a.s.. sit. y. Nat. times τk such that. i n U. v. Property 3.2 Let a filtered probability space (Ω, F, {Ft }0≤t≤T , P ) and M (t) be a local martingale with respect to {Ft }. (1) If M (t) is lower bounded, then M (t) is a supermartingale. Z t (2) Let φ(t, ω) be a Ft -adapted process such that φ(s) · dM (s) exist. Then, 0. Z. t. φ(s) · dM (s). Z(t) := 0. is a local martingale for 0 ≤ t ≤ T .. 7.

(13) Proof. (1). Given t ≥ s. Suppose τk is an increasing sequence of Ft -stopping times such that M (t ∧ τk ) is an Ft -martingale for all k. From Fatou’s lemma, E[ lim inf M (t ∧ τk )] ≤ lim inf E[M (t ∧ τk )]. k→∞. (3.1). k→∞. A process M (t ∧ τk ) is an Ft -martingale and E[M (t ∧ τk )] = E[E[M (t ∧ τk )|F0 ] ] = E[M (0 ∧ τk )] = E[M (0)]. Since M (t ∧ τk ) exists as k → ∞ and (3.1), we get E[M (t)] = E[ lim M (t ∧ τk )] ≤ E[M (0)] k→∞. and M (t) is integrable. It follows from Fatou’s lemma for conditional expectations,. 治政 E[ lim inf M (t ∧ τ )|F ] ≤ lim inf E[M (t ∧ τ )|F 大] = lim inf M (s ∧ τ ) 立 and hence E[M (t)|F ] ≤ M (s) almost surely. k. k→∞. s. k. k→∞. s. k→∞. k. s. ‧ 國. 學. (2). Since M (t) is a local martingale with respect to {Ft }, there exists an. ‧. increasing sequence of stopping times τk0 such that definition 3.1 holds. Suppose. for k ∈ N.. io. sit. Nat. τk = inf{t ≥ 0| φ(t) ∈ / Uk } ∧ τk0. y. {Uk } is an increasing sequence of bounded open sets in Rn . Let. n. al. er. Since φ(t ∧ τk ) is bounded and M (t ∧ τk ) is a martingale for t ∈ [0, T ], we obtain Z t∧τk φ(s) · dM (s) Z(t ∧ τk ) :=. Ch. engchi 0. i n U. v. is a martingale for t ∈ [0, T ]. The proof is complete. A probability measure Q is called a local martingale measure with respect to P , if it is equivalent to P and the process Sˆ = (S1 (t), ..., Sn (t)) is a Q-local martingale. ˆ By M we denote the set of all equivalent local martingale measures for process S. In this section, we show the upper hedging price and lower hedging price can be represented by Pup (G) = sup EQ [G], Q∈M. Plow (G) = inf EQ [G]. Q∈M. 8.

(14) The following theorem 3.3 and property 3.4 are adapted from the paper of Kramkrov (1996).. Theorem 3.3 (Optional Decomposition, [8, Theorem 2.1]) Let F (t) ∈ R be a positive Ft -adapted process. Then F is a super-martingale for each measure Q ∈ M if and only if there exists an S-integrable predictable process H(t) ∈ Rn and an Ft -adapted increasing positive process C such that Z t ˆ − C(t), t ∈ [0, T ]. H(s) · dS(s) F (t) = F (0) + 0. 治政大left limit exists process with sup E f < ∞. There is a right continuous and 立. Proposition 3.4 ([8, Proposition 4.2]) Let f be a positive variable on (Ω, F, P ) Q. Q∈M. ‧ 國. t ≥ 0.. 學. F (t) = ess sup EQ [f |Ft ], Q∈M. The process F is a Q super-martingale whatever Q ∈ M.. ‧. We consult theorem 3.5 of Kunita (2004) and proof the main result in this. al. er. io. sit. y. Nat. section as follows:. n. Theorem 3.5 Let G be a bounded European contingent T -claim. We get. i n U. CP h(G) = sup E [G], engchi up. Q. v. Q∈M. Plow (G) = inf EQ [G], Q∈M. where M denotes the set of equivalent local martingale measures Q. Proof. Without loss of generality, we assume G ≥ 0. Set p = sup EQ [G] and Q∈M. Aup = {x| there exits π ∈ P such that Xx(π) (T ) ≥ G a.s. }. We first prove p ≤ Pup (G). Let x0 ∈ Aup , then there exists a π 0 ∈ P such that Z T (π 0 ) 0 Xx0 (T ) = x + π 0 (t) · dS(t) ≥ G a.s.. 0. 9.

(15) Given a Q ∈ M. Since G is lower bounded and Q P , we get Z T (π 0 ) 0 π 0 (t) · dS(t) ≥ G a.s. w.r.t. Q Xx0 (T ) = x +. (3.2). 0 (π 0 ). and it is a lower bounded local Q-martingale. By property 3.2, Xx0 (t) is a supermartingale with respect to Q and Z Z T 0 π (s) · dS(s)] = EQ [EQ [ EQ [. T. π 0 (s) · dS(s)| Ft ]] ≤ 0.. 0. 0. Taking the expectation of (3.2) with respect to Q we obtain x0 ≥ EQ [G].. Hence. 立. 政治大 p ≤ Pup (G).. (3.3). ‧ 國. 學. Next, we verify the reverse inequality. From property 3.4., ess sup EQ [G|Ft ] Q∈M. t. ˆ − C(t), t ∈ [0, T ], H(s) · dS(s). y. Nat. Z. ess sup EQ [G|Ft ] = EQ [G] + Q∈M. sit. that. ‧. is a super-martingale for Q ∈ M and lower bounded. It follows from theorem 3.3. 0. n. al. er. io. where H is an S-integrable process and C is a positive increasing process. There. i n U. v. exists a π ∈ P such that πi = Hi for i = 1, ..., n. We get Z T EQ [G] + π(t) · dS(t) ≥ ess sup EQ [G|FT ] = G. Ch. engchi. a.s.,. Q∈M. 0. and EQ [G] ∈ Aup . Therefore, Pup (G) ≤ EQ [G] ≤ p. From (3.3), we obtain Pup (G) = p. It is similar to verify Plow (G) = inf EQ [G]. Q∈M. . 10.

(16) 4. Utility Indifference Pricing. In this section, we introduce the relative entropy and the central duality theorem to show the main result. Denote Pa the set of local martingale measures for Sˆ absolutely continuous to P . The relative entropy H(Q|P ) is defined by   dQ dQ  E ln if Q P , dP dP H(Q|P ) =   +∞ otherwise. In probability theory and information theory, the quantity H(Q|P ) measures the. 政治大 with finite relative entropy, H(Q|P ) < ∞. 立. difference between two probability measures Q and P . Define Pf = Pf (P ) to be the set of Q ∈ Pa. 學. ‧ 國. assumption. Pf ∩ M 6= ∅.. We consider the. (4.1). ‧. There is a unique measure Q0 ∈ Pf ∩ M minimizing H(Q|P ) over all Q ∈ Pf and call Q0 the minimal P -entropy martingale measure: see [7, Property 3.1 and. sit. y. Nat. Property 3.2].. n. al. i n U. central duality result. Let G be a bounded T -claim.. Ch. engchi. er. io. The main result of this study is obtained from the following theorem called the. v. Theorem 4.1 Given λ ∈ R and consider the function u(x; λ) := −. 1 inf ln E[exp (−γ(Xx(π) (T ) + λG))]. γ π∈P. (4.2). Then, we can get 1 u(x; λ) = x + inf {λEQ [G] + H(Q|P )}. Q∈Pf γ. (4.3). Moreover, the infimum in (4.3) is attained for a unique Qλ ∈ Pf ∩ M whose RadonNikodym derivative is given by Z T dQλ λ = exp cλ − γ π · dS + λG , dP 0 11.

(17) Z. λ. where π is a self-financing portfolio and cλ is a constant. In particular, if is (t, ω) a.s. lower bounded, then π λ ∈ P attains the infimum in (4.2).. t. π λ ·dS. 0. Because ln x/γ is a continuous function on (0, ∞), we can formulate the central duality result to utility indifference pricing. The proof of theorem 4.1 is obtained in ˙ section 5. Moreover, we improve the following result of Ilhan, Jonsson and Sircar (2005) by theorem 4.1. Denote Θ to be the set of Ft -adapted, S-integrable Rn+1 -valued processes θ (θ). such that θ is self-financing and X0 (T ) are Q-martingales for all Q ∈ Pf .. 政治大. Theorem 4.2 ([6, Theorem 4.1], [1, Theorem 2.2] and [7, Theorem 2.1]) Given. 立. u0 (x; λ) := −. 1 inf ln E[exp (−γ(Xx(π) (T ) + λG))]. γ θ∈Θ. 學. ‧ 國. λ ∈ R and consider the function (4.4). Then, we can get. ‧. (4.5). y. Nat. 1 u0 (x; λ) = x + inf {λEQ [G] + H(Q|P )}. Q∈Pf γ. sit. Moreover, the infimum in (4.5) is attained for a unique Qλ ∈ Pf ∩ M whose Radon-. al. er. io. Nikodym derivative is given by. v. n. Z T dQλ λ = exp cλ − γ θ · dS + λG , dP 0. Ch. engchi. i n U. where θλ ∈ Θ attains the infimum in (4.4), and cλ is a constant. The seller’s and buyer’s utility indifference price of the bounded claim G is the solution pus and pub of the stochastic differential equations Vs (G, x + pus ) = V (G, x) = Vb (G, x − pub ).. (4.6). Put an exponential utility U (x) = −e−γx into (4.6), where γ > 0 is the risk aversion parameter. We get equations (π). − inf E[exp(−γ(Xx+pus (T ) − G))] = − inf E[exp(−γXx(π) (T ))], π∈P. − inf. π∈P. π∈P. (π) E[exp(−γ(Xx−pub (T ). + G))] = − inf. π∈P. 12. (4.7) E[exp(−γXx(π) (T ))]..

(18) Since ln x is a continuous function on (0, ∞), it follows from (4.2) and (4.7) that we rewrite (4.6) by u(x + pus ; −1) = u(x; 0) = u(x − pub ; 1).. (4.8). Hence, utility indifference pricing for bounded T -claim G are solutions pus and pub of (4.8) and we consider Pseller (G) = pus and Pbuyer (G) = pub . Lemma 4.3 Let G be a bounded T -claim. Then Pbuyer (G) ≤ Pseller (G).. 政治大 1. Proof. From the Theorem 4.1 and (4.8), we get. 立. ‧ 國. 學. Pseller (G) = sup {EQ [G] − (H(Q|P ) − H(Q0 |P ))} γ Q∈Pf 1 Pbuyer (G) = inf {EQ [G] + (H(Q|P ) − H(Q0 |P ))} Q∈Pf γ. Q ∈ Pf . Let a function ζ : Pf → R with. n. al. er. io. We obtain. 1 (H(Q|P ) − H(Q0 |P )) ≥ 0. γ. sit. Nat. ζ(Q) =. y. ‧. where Q0 is a unique measure in Pf ∩ M and minimize H(Q|P ) over all. Ch. engchi. i n U. v. Pseller (G) − Pbuyer (G) = sup {EQ [G] − ζ(Q)} − inf {EQ [G] + ζ(Q)} Q∈Pf. Q∈Pf. = sup {EQ [G] − ζ(Q)} + sup {−EQ [G] − ζ(Q)} Q∈Pf. Q∈Pf. ≥ sup {−2ζ(Q)} = 0. Q∈Pf. Using lemma 4.3 we can get the following inequality.. Theorem 4.4 Let G be a bounded T -claim. Then Plow (G) ≤ Pbuyer (G) ≤ Pseller (G) ≤ Pup (G).. 13.

(19) Proof. It suffices to verify Pseller (G) ≤ Pup (G). From theorem 4.1 and lemma 4.3, we get Pseller (G) =. sup {EQ [G] − ζ(Q)} Q∈Pf ∩M. ≤ sup EQ [G] = Pup (G). Q∈M. It is similar to show Plow (G) ≤ Pbuyer (G). . 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 14. i n U. v.

(20) 5. Proof of the Central Duality Result. We consult the idea of Delbaen, Grandits, Rheinländer, Samperi, Schweizer, and Stricker (2002) to the setting of theorem 4.1. Consider a bounded contingent T claim G and define a probability measure PG equivalent to P by dPG := CG eγG , dP. (5.1). where CG−1 := E[eγG ] ∈ (0, ∞). Since G is bounded, we can get for Q P dQ H(Q|P ) = EQ ln + ln CG + γG = H(Q|PG ) + ln CG + EQ [γG] dPG. (5.2). 政治大. and H(Q|P ) < ∞ implies H(Q|PG ) < ∞. In particular, denote the set. 立. Pf (PG ) := {Q ∈ Pa |H(Q|PG ) < ∞}. ‧ 國. 學. = {Q ∈ Pa |H(Q|P ) < ∞} = Pf (P ). ‧. and write Pf simply. Recall the assumption (4.1), there is a unique measure Q0G ∈. sit. y. Nat. Pf ∩ M minimizing H(Q|PG ) over all Q ∈ Pf and call Q0G the minimal PG -entropy. io. er. martingale measure.. al. v i n assumption (4.1), the density C function of the minimal h e n g c h i U entropy Q n. To verify the central duality result, we need one additional arrangement. Under 0. with respect to. P has the form. dQ0 (θ) = K exp (X0 (T )) = K exp dP. Z. T. θ · dS. (5.3). 0. for some constant K > 0 and some self-financing θ such that Z T (θ) X0 (T ) = θ · dS 0. is a Q0 -martingale: see [3, Corollary 2.1] and [4, Proposition 3.2 and the proof of Theorem 4.13]. Taking the natural logarithm and the expectation under Q0 , we could rewrite (5.3) by ln K = EQ0 [ ln. dQ0 ] = H(Q0 |P ). dP 15. (5.4).

(21) In the same way for Q0G , we get the density function Z T dQ0G (θG ) = KG exp (X0 (T )) = KG exp ( θG · dS) dPG 0. (5.5). for some constant KG > 0 and some self-financing θG such that Z T (θG ) θG · dS X0 (T ) = 0. is a Q0G -martingale. Let 1 θ∗ = − θG γ. (5.6). be a self-financing portfolio such that (θ∗ ). exp (−γX0. 政治大 (θG ). (T )) = exp (X0. 立. is exactly in L1 (PG ).. (T )) =. 1 dQ0G KG dPG. (5.7). ‧ 國. 學. Proof of theorem 4.1. Since we give λ ∈ R, λG is always bounded. Without loss of generality, we suppose λ = −1 and consider an initial value x ∈ R. Writing. ‧. u1 (x) and u2 (x) for equations (4.2) and (4.3) imply Z T 1 u1 (x) = − inf ln E[exp (−γ(x + π · dS − G))] γ π∈P 0 Z T 1 1 = x − inf ln EPG [ exp (− γπ · dS)] γ π∈P CG 0 Z T 1 ln CG − inf ln EPG [exp (− π · dS)] =x+ γ γ π∈P 0. n. er. io. sit. y. Nat. al. Ch. and. engchi. i n U. v. 1 u2 (x) = x + inf {−EQ [G] + H(Q|P )} Q∈Pf γ 1 = x + inf {−EQ [G] + (H(Q|PG ) + ln CG + E[γG])} Q∈Pf γ ln CG 1 =x+ + inf H(Q|PG ) γ γ Q∈Pf by (5.1) and (5.2). As argued above and further including the definition of Q0G , we can verify that Z T − inf ln EPG [exp (− π · ds)] = inf H(Q|PG ) = H(Q0G |PG ). π∈P. Q∈Pf. 0. 16.

(22) From (5.5), we can get H(Q0G |PG ) = EQ0G [ ln. Z. Q0G. For any π ∈ P, it follows from Z 0 is a QG -supermartingale. Since. dQ0G ] = ln KG . dPG T. π · dS. ∈ Pf ∩ M and property 3.2 that 0. T. θG · dS is a Q0G -martingale, (5.5) and Jensen’s. 0. inequality, we obtain Z T Z T Z T 1 ln EPG [exp (− π · dS)] = ln EQ0G [ θG · dS )] π · dS − exp (− KG 0 0 0 Z T Z T π · dS − θG · dS ] ≥ − ln KG + EQ0G [−. 政治大 ≥ − ln K . 0. 立. Therefore,. G. T. π · dS)] ≤ ln KG .. − inf ln EPG [exp (− π∈P. 學. Z. ‧ 國. 0. 0. ‧. Z EPG [exp (. T. θG · dS)] =. 0. 1 . KG. io. sit. Nat. portfolio θG such that. y. Next, we verify the reverse inequality. From (5.5), there exists a self-financing. n. al. er. Choose a self-financing portfolio πG such that for t ∈ [0, T ], ω ∈ Ft consider  Z t   θG (s, ω) · dS(s) > 0,  θG (t, ω) if 0 Z πG (t, ω) = t   0 if θG (s, ω) · dS(s) ≤ 0. Ch. engchi. i n U. v. 0. and hence Z. T. Z πG · dS)] ≤ ln EPG [exp (−. ln EPG [exp (− 0. T. θG · dS)] = − ln KG . 0. Since πG is lower bounded, we can get πG ∈ P and Z T − inf ln EPG [exp (− πG · dS)] ≥ ln KG . π∈P. 0. Choose π λ = θ∗ = −θG / γ, Qλ = Q0G and cλ = ln (KG CG ). Thus the proof is complete. . 17.

(23) 6. Conclusion. Summarizing the thesis, it follows from theorem 3.5 and theorem 4.1 that we transform superreplication, subreplication and the maximal expect utility into representations of equivalent local martingale measures with respect to given risky asset prices. Therefore, we can compare pricing formulas and determine the main result. Note that this study took a step in the exponential utility function and bounded claims in the fundamental financial market. If the contingent claim is unbounded, the proof of theorem 4.1 is not complete. But the central dual result of the max-. 政治大 Grandits, Rheinländer, Samperi, 立 Schweizer, and Stricker (2002).. imal expect exponential utility still holds under some assumptions; see Delbaen, On the other. hand, it is possible of course to consider other utility functions or risky asset pro-. ‧ 國. 學. cesses.. ‧. In additional, risk indifference pricing is established by convex risk measure. y. sit. Nat. ρ : F → R,. er. io. satisfying some axioms, where F is the set of FT measurable random variables.. al. n. v i n has the representation about aCfamily of probability measures Q P on F , for h e nL g hi U c which L is given. Therefore, further research is required to compare risk indifference Based on theorem 2.2 of Øksendal and Sulem (2009), the convex risk measure T. pricing with utility indifference pricing for some suitable L.. 18.

(24) References [1] Delbaen, F., P. Grandits, T. Rheinländer, D. Samperi, M. Schweizer, and C. Stricker (2002): Exponential hedging and entropic penalties, Math. Finance 12, 99-123. [2] Föllmer, H., and A. Schied (2002): Convex Measures of Risk and Trading Constraints, Finance Stochast. 6, 429-447. [3] Fritelli, M. (2002a): The minimal Entropy Martingale Measure and the. 政治大 [4] Grandits, P., and T. Rheinländer (1999): On the Minimal Entropy Martin立 Valuation Problem in Incomplete markets, Math. Finance 10, 39-52.. gale Measure, Preprint, Technical University of Berlin, to appear in Annals. ‧ 國. 學. of Probability.. ‧. [5] Hodges, S. D., and A. Neuberger (1989): Optimal replication of contingent claims under transaction costs, Rev. Future Markets 8, 222-239.. y. Nat. n. al. Ch. er. io. tive securities, Finance Stochast. 9, 585-595.. sit. ˙ [6] Ilhan, A., M. Jonsson,and R. Sircar (2005): Optimal investment with deriva-. i n U. v. [7] Kabanov, Y. M., and C. Stricker (2002): On the optimal portfolio for the. engchi. exponential utility maximization: remarks to the six-author paper, Math. Finance 12, 125-134. [8] Kramkrov, D. O. (1996): Optimal decomposition of supermartingales and hedging of contingent claims in incomplete security markets. Probab. Theory and Relat. Fields 105, 459-479. [9] Kunita, H. (2004): Representation of Martingales with Jumps and Application to Mathematical Finance, Advanced Studies in Pure Mathematics, Math. Soc. Japan, Tokyo, 41, 209-232.. 19.

(25) [10] Øksendal, B.: Stochastic Differential Equations: an introduction with applications, 6ed, Springer 2003. [11] Øksendal, B., and A. Sulem (2009): Risk indifference pricing in jump diffusion markets, Math. Finance 19, 619-637.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 20. i n U. v.

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