The Golden Age of the Mathematical Finance
aa r X i v : . [ q -f i n . M F ] F e b The Golden Age of the Mathematical Finance
José Manuel Corcuera ∗ .February 15, 2021 Abstract
This paper is devoted, mainly, to show that the last quarter of the past century can be considered as thegolden age of the Mathematical Finance. In this period the collaboration of great economist and the bestgeneration of probabilists, most of them from the Strasbourg’s School led by Paul André Meyer, gaverise to the foundations of this discipline. They established the two fundamentals theorems of arbitragetheory, close formulas for options, the main modelling approaches and created the appropriate frameworkfor the posterior development.
Key words : Financial asset pricing; Options; Arbitrage; Complete markets; Semimartingale; Utilityindifference price; Fundamental theorems of asset pricing.
JEL-Classification
C61 · D43 · D44 · D53 · G11 · G12 · G14
MS-Classification 2020 : 60G35, 62M20, 91B50, 93E03
In the last quarter of the past century a new mathematical discipline emerged: the Mathematical Finance.It was the conjunction of great economists like F. Black, M. Scholes, R. Merton, at the United States, andmany others, with great mathematicians, most of them belonging to, or following, the famous
Séminairede Probabilités de Strasbourg under the leadership of Paul Andrè Meyer. The theory of stochastic integralswith respect to semimartingales developed in this Seminar was the mathematical basis to establish the mainresults of the new discipline. In this paper we are going to explain how arose and were proved the two maintheorems and how they give the framework to the Arbitrage Theory, the core of the Mathematical Finance.The number of results and new paths opened during this period was too huge to be described in one sin glepaper. Issues like local and stochastic volatility models, (Dupire et al., 1994), (Heston, 1993) the differentapproaches in credit risk, (Duffie & Singleton, 1999) the use of strict local martingales to describe bubbles,(Loewenstein & Willard, 2000) the Heath-Jarrow-Morton approach in interest rate models, (Heath, Jarrow,& Morton, 1992), the models under transaction costs,(Leland, 1985), and a long etc, are not treated here.However many excellent books on the matter, written before the end of the last century, already includethem, for instance, (Lamberton & Lapeyre, 1996), (Musiela & Rutkowski, 1997), (Björk, 1998), (Dana &Jeanblanc-Picqué, 1998) or (Shiryaev, 1999), among others.In the next section we explain the beginning, at the 70’s, and how this could happened. In the third sectionwe explain the two fundamental theorems and their proofs, showing this intertwining among economists andmathematicians. Finally we explain the arbitrage theory for pricing and some of the strategies to solve theproblem of incompleteness and the multiplicity of (non) arbitrage prices. ∗ Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, E-08007 Barcelona, Spain.
E-mail: [email protected] work of J. M. Corcuera is supported by the Spanish grant MTM2013-40782-P. The inception
Probably everything started at 1973 when Black and Scholes publish their famous paper, Black and Scholes(1973) where they give a formula for a call option based on an arbitrage argument, the Itô formula. Theyassume that the stock S t follows a geometric Brownian motion (a model proposed previously by Samuelson(1965) ), that is d S t = S t ( µ d t + σ d W t ) , S > . with B a Brownian motion, and that the value of the unit of money, say B t , evolves as d B t = B t r d t. for µ, σ, r constants. The way that Black and Scholes obtained the formula is the following. Suppose thatthe price, at time t , of the call option, where the payoff is ( S T − K ) + , is a smooth function of the form C t := f ( t, S t ) , and consider a portfolio with β t calls and α t stocks, the cost of this portfolio is β t C t + α t S t =: V t , and when it evolves, in a self-financed way its value changes as d V t = β t d C t + α t d S t , d V t = β t (cid:18) ∂ t f d t + ∂ x f d S t + 12 ∂ xx f S t σ d t (cid:19) + α t d S t , where we apply the Itô formula for stochastic differentials.Now if we take α t = − β t ∂ x f we have that the cost of this portfolio is d V t = β t (cid:18) ∂ t f + 12 ∂ xx f S t σ (cid:19) d t. It behaves like a bank account! remember that if we put V t in the bank account then d V t = V t r d t, then if wewant an equilibrium situation in such a way that it is not possible to do profit without risk, we must have β t (cid:18) ∂ t f + 12 ∂ xx f S t σ (cid:19) = rV t = r ( β t C t − β t ∂ x f S t ) = rβ t ( f − ∂ x f S t ) . So, the price of a call is the solution of the partial differential equation ∂ t f + rx∂ x f + 12 σ x ∂ xx f = rf, (1)with the boundary condition f ( T, x ) = ( x − K ) + . By doing a change of variable we obtain C t = S t Φ( d + ) − Ke − r ( T − t ) Φ( d − ) where Φ is the cdf of a standard normal distribution and d ± = log( S t K ) + ( r ± σ )( T − t ) σ √ ( T − t ) . If we apply the Feynman-Kac formula, it is easy to see that C t = E ∗ (cid:16) ( S T − K ) + e − r ( T − t ) (cid:12)(cid:12)(cid:12) S t (cid:17) where E ∗ is an expectation assuming that d S t = rS t d t + σS t d W ∗ t , W ∗ is a Brownian motion as well. In fact by the integration by parts formula, the boundary conditionand equation (1) ( S T − K ) + e − r ( T − t ) = f ( T, S T ) e − r ( T − t ) = f ( t, S t ) + Z Tt e − r ( u − t ) ∂ x f σS u d W ∗ u + Z Tt e − r ( u − t ) (cid:18) ∂ u f + rS u ∂ x f + 12 σ S u ∂ xx f (cid:19) d u − Z Tt e − r ( u − t ) rf d u = f ( t, S t ) + Z Tt e − r ( u − t ) ∂ x f σS u d W ∗ u . Now taking the expectation we obtain the result.However Samuelson and Merton in 1969, see also Merton (1973) where the author extends the analyticalmethod of Black and Scholes to price more complex options, wrote a paper where they give the followingformula for the price of a call option with strike Kf ( t, S t ) = e − r ( T − t ) Z ∞ StK ( zS t − K ) d Q T − t ( z ) (2)where Z ∞ z d Q T − t ( z ) = e r ( T − t ) . Then, in the special case when Q t is a log-normal distribution with log-variance equal to σ ( T − t ) we recoverthe Black-Scholes formula.Nevertheless we have to move back until 1900 to see the real birth of the mathematical finance. At thattime the French postgraduate student Louis Bachelier presented his thesis Théorie de la Speculation . Asit is said in Courtault et al. (2000) this thesis, supervised by Henri Poincaré, contains ideas of enormousvalue in both finance and probability. Bachelier was the first to consider the stochastic processes that todaywe call it Brownian or Wiener process. He use it to model the movements of stock prices. He considertwo constructions of the process, as a limit of (today called) a random walk and as a solution of a Fourierequation (today heat equation). Even he finds the distribution of the supremum of the Brownian motion.From a financial point of view he introduces a new idea to price an option. He uses the principle of themathematical expectation in games: a game is fair if the expectation of the profit is zero, but he considers theexpectation with respect to what today we call a risk-neutral probability . He establishes, as a fundamentalprinciple, that the mathematical expectation of the profit of a buyer or seller of a financial product has tobe zero. This principle allows him to calculate the price of a call option and he obtains, translated in theusual nowadays notation, see Bachelier (1900, p. 50), that C = Z ∞ S − K ( z + S − K ) d Q T ( z ) (3)where Q T ∼ N (0 , σ T ) . It is important to remark that for Bachelier the Gaussian distribution was theobjective one, the historical one. Even the same was true for the Black and Scholes model: they probablybelieved that the geometric Brownian motion was the real distribution of the stock and there were notdistinction between the real and the risk-neutral probability. At least is how Merton interpret the Black-Scholes formula. He says, in Merton 73, page 161: "...However, d Q [in 2] is a risk-adjusted ("util-prob")distribution, depending on both risk-preferences and aggregate supplies, while the distribution in (1) is the objective distribution of returns on the common stock". Moreover, note that (3) is an arithmetic version of(2), except for the discount factor e − r ( T − t ) . It seems that, at least in the French market, all payments wererelated to a single date. In this way the discount factor is not needed in (3), Bachelier was talking about forward values. The term Fundamental Theorems of Asset Pricing (FTAP) was coined by Phil Dybvig in 1987, to describework initiated by his thesis advisor, Steve Ross around 1978, see Ross (1978). The connection to martingaletheory is in Harrison and Kreps (1979) with an important extension to the continuous time setting in Harrisonand Pliska (1981). In fact there are two fundamental theorems as we will see.Let S = (cid:0) S t , S t ..., S dt (cid:1) t ∈ I be a non-negative d + 1 -dimensional semimartingale representing the price processof d + 1 securities, here I is either a discrete finite set I = { , , .., T } or a compact interval I = [0 , T ] . Thisprocess is assumed to be defined in a complete filtered probability space (Ω , F , F , P ) where F = ( F t ) t ∈ I is afiltration representing the flow of information and satisfying the usual conditions . We suppose that S t > is F t − -measurable, t = 1 , ..., T , that is S is predictable. We also assume that F is trivial (a.s.) and that F T = F . We define the discounted price as ˜ S t := S t S t , t ∈ I. Definition 1 A trading strategy is a predictable stochastic process φ = (( φ t , φ t , ..., φ dt )) ≤ t ≤ T in R d +1 , thatis φ it is F t − -measurable, for all ≤ t ≤ T . it indicates the number of units of security i in the portfolio at time t and that φ is predictable means thatthe positions in the portfolio at t is decided at t − , using the information available in F t − . Then we havealso the following definitions. Definition 2
The value of the portfolio associated with a trading strategy φ is given by V t ( φ ) = φ t · S t := d X i =0 φ it S it , t = 1 , ..., T, V ( φ ) = φ · S . and its discounted value ˜ V t ( φ ) := φ t · ˜ S t Definition 3
A trading strategy φ is said to be self-financing if V t ( φ ) = φ t +1 · S n , t = 1 , ..., T − . It is easy to see that the self-financing condition is equivalent to the equality ˜ V t ( φ ) = ˜ V ( φ ) + t X i =1 φ t · ∆ ˜ S t , t = 1 , ..., T. Definition 4 An admissible trading strategy φ is a self-financing strategy satisfying the following constraint V t ( φ ) ≥ a.s. for t = 0 , , ..., T Definition 5
An admissible trading strategy φ is an arbitrage opportunity if it satisfies V ( φ ) = 0 , a.s. V T ( φ ) ≥ , a.s. and P ( V T > > . In an analogous way we can say when a self-financing strategy is an arbitrage. In the discrete time settingwe have the following lemma
Lemma 1
The class of admissible trading strategies contains no arbitrage opportunities if and only if theclass of self-financing strategies contains no arbitrage opportunities.
Proof.
Let ϕ be a self-financing strategy that is an arbitrage. Define t = inf { u, ˜ V t ( ϕ ) ≥ a.s. for all u > t } , note that t ≤ T − since ˜ V T ( ϕ ) ≥ . Let A = n ˜ V t ( ϕ ) < o , define the predictable vector process θ , suchthat for all i = 1 , ..., d θ iu = (cid:26) u ≤ t A ϕ iu u > t Then, ˜ V u () = 0 , for all ≤ u ≤ t and for all u > t ˜ V u ( θ ) = u X v = t +1 A ϕ v · ∆ ˜ S v = A u X v =1 ϕ v · ∆ ˜ S v − t X v =1 ϕ v · ∆ ˜ S v ! = A (cid:16) ˜ V u ( ϕ ) − ˜ V t ( ϕ ) (cid:17) ≥ , so θ is admissible and ˜ V T ( θ ) = A (cid:16) ˜ V T ( ϕ ) − ˜ V t ( ϕ ) (cid:17) > in A, then θ is an admissible arbitrage.Notice that, according to this lemma, if we consider only the admissible strategies we are not reducingthe possibilities of arbitrage opportunities. Henceforth the term arbitrage opportunities refers to admissiblearbitrage opportunities. 5 efinition 6 Probabilities P and P ∗ defined on (Ω , F ) are said to be equivalent, we write P ∼ P ∗ , if theyhave the same null-sets. Now we can establish the first fundamental theorem of the asset pricing (FFTAP).
Theorem 1
There are not arbitrage opportunities if and only if there is a probability P ∗ ∼ P under whichthe process ˜ S is a martingale. Proof. (Sufficiency) Suppose that ˜ S is a martingale under some P ∗ ∼ P . Then, let φ be an arbitrageopportunity, so, since it is admissible, V t ( φ ) ≥ a.s. P ∗ (because the equivalence between P ∗ and P ) and E P ∗ ( V t ( φ )) is well defined, although possible infinite. Then we have E P ∗ (cid:16) ˜ V t ( φ ) (cid:17) = E P ∗ (cid:16) φ t · ˜ S t (cid:17) = E P ∗ (cid:16) E P ∗ (cid:16) φ t · ˜ S t (cid:12)(cid:12)(cid:12) F t − (cid:17)(cid:17) = E P ∗ (cid:16) φ t · E P ∗ (cid:16) ˜ S t (cid:12)(cid:12)(cid:12) F t − (cid:17)(cid:17) = E P ∗ (cid:16) φ t · E P ∗ (cid:16) ˜ S t (cid:12)(cid:12)(cid:12) F t − (cid:17)(cid:17) = E P ∗ (cid:16) φ t · ˜ S t − (cid:17) = E P ∗ (cid:16) φ t − · ˜ S t − (cid:17) = ... = E P ∗ (cid:16) ˜ V ( φ ) (cid:17) = 0 . where in the second line line we use the predictability of φ and the martingale property of ˜ S , and in thethird line the self-financing condition.(Necessity) Now, if the number of elements in Ω is finite we have simpler proofs than in the general case. Forthe finite case the standard proof is that of Harrison and Pliska (1981), based on the separation hyperplanetheorem in R k Ω k . If the sample space is not finite Morton (1988) and Dalang, Morton, and Willinger (1990)give a proof based in the following lemma for the case that d = 1 . Lemma 2
Let Y be a bounded d -dimensional random vector defined on some (Ω , F , P ) and assume that Y ∈ K for some compact set in R d then one of the following two conditions hold, . There exists α ∈ R d with α · Y ≥ a.s and P ( α · Y > > . . There exists a positive g ∈ C ( K ) (the space of real continuous functions on K ) with E ( g ( Y ) Y ) = 0 . For d > the idea is patching together the conditional martingale measures corresponding to each periodbut this involves subtle arguments with the Measurable Selection Theorem. For alternative proofs that tryto simplify the above one, see Schachermayer (1992), (Rogers, 1994) and Jacod and Shiryaev (1998).Before stating the second fundamental theorem of asset pricing (SFTAP) we need the following definitions. Definition 7
Let X be a claim, that is X ≥ , and X ∈ F T . We say that X is attainable, or replicable, if X is equal to the final value of and admissible strategy. Assume that the market model is free of arbitrage, then the set of risk-neutral probabilities is not empty. Ifwe choose one, say P ∗ , we have the following definition. Definition 8
We say that the market model is complete if every claim X ∈ F T , such that E P ∗ (cid:16) ˜ X (cid:17) < ∞ ,is attainable. Then the SFTAP reads as follows. 6 heorem 2
The market is complete if and only if the risk-neutral probability is unique
Proof. (Sufficiency) If the market is complete, the claims of the form S T A with A ∈ F T are attainable,therefore we can write A = ˜ V ( φ ) + T X t =1 φ t · ∆ ˜ S t where φ is admissible. Now, if we assume that there are two risk-neutral probabilities, say P ∗ and Q ∗ , wehave, since ˜ S is a martingale with respect to both probabilities P ∗ and Q ∗ , that E P ∗ ( A ) = E Q ∗ ( A ) , in such a way that P ∗ and Q ∗ are the same probability in F T .(Necessity) Assume first that Ω is finite. In the finite case. Let H be the subset of random variables of theform ˜ V ( φ ) + T X t =1 φ t · ∆ ˜ S t with φ predictable. H is a vector subspace of the vectorial space, say E , formed by all random variables.Moreover it is not a trivial subspace, in fact since the market is incomplete there will exist h such that hS n H (note that if h ≥ can be replicated by a non -admissible strategy then the market cannot beviable). Let P ∗ be a risk-neutral probability in E , we can define the scalar product h X, Y i = E P ∗ ( XY ) . Let X be an random variable orthogonal to H and set P ∗∗ ( ω ) = (cid:18) X ( ω )2 || X || ∞ (cid:19) P ∗ ( ω ) . Then we have an equivalent probability to P ∗ : P ∗∗ ( ω ) = (cid:18) X ( ω )2 || X || ∞ (cid:19) P ∗ ( ω ) > , X P ∗∗ ( ω ) = X P ∗ ( ω ) + E P ∗ ( X )2 || X || ∞ = X P ∗ ( ω ) = 1 , take A , with A ∈ F t − , then A ∆ ˜ S jt ∈ H for all j = 1 , .., d and X is orthogonal to H . E P ∗∗ (cid:16) A ∆ ˜ S jt (cid:17) = E P ∗ (cid:16) A ∆ ˜ S jt (cid:17) + E P ∗ (cid:16) X A ∆ ˜ S jt (cid:17) || X || ∞ = 0 in such a way that ˜ S is a P ∗∗ -martingale. The surprise for the general case is that if the risk-neutralprobability is unique then Ω is essencially finite for P !, in the sense that F T is purely atomic with respect to P with at most ( d + 1) N atoms, see Theorem 6 in Jacod and Shiryaev (1998). Now we have that I = [0 , T ] , S = (cid:0) S t , S t ..., S dt (cid:1) t ∈ I is a non-negative d + 1 -dimensional semimartingalerepresenting the price process of d + 1 securities. This process is assumed to be defined in a complete filteredprobability space (Ω , F , F , P ) where F = ( F t ) t ∈ I is a filtration satisfying the usual conditions . We supposethat S t > . We also assume that F is trivial (a.s.) and that F T = F . We define the discounted value as ˜ S t := S t S t , t ∈ I. efinition 9 A trading strategy is a predictable stochastic process φ = (( φ t , φ t , ..., φ dt )) ≤ t ≤ T in R d +1 , thatmeans that φ it is measurable with respect to the sigma-field generated by the càglàd adapted processes andthat is integrable with respect to S Then we have also the following definitions.
Definition 10
The discounted value of the portfolio associated with a trading strategy φ is given by ˜ V t ( φ ) = φ t · ˜ S t = d X i =0 φ it ˜ S it , Definition 11
A trading strategy φ is said to be self-financing if ˜ V t ( φ ) = φ · ˜ S + Z t φ t · d ˜ S t , t ∈ [0 , T ] . Definition 12 An admissible trading strategy φ is a self-financing strategy satisfying ˜ V t ( φ ) ≥ − a . for all t ∈ [0 , T ] for some a > . According to Delbaen Schachermayer (1994) we use the notation K = (Z T φ s · d ˜ S t , φ admissible ) C = K − L K = K ∩ L ∞ C = C ∩ L ∞ C the closure of C under L ∞ Definition 13
We say that the model satisfies the No Free Lunch with Vanishing Risk condition (NFLVR)if C ∩ L ∞ + = { } We have the FFTAP in continuous time
Definition 14
Let ˜ S be a locally bounded R d -valued semimartingale. There are not free lunches with van-ishing risk if and only if there is probability P ∗ ∼ P under which ˜ S is a local martingale. Proof. (Sufficiency) Let φ be an admissible strategy with intial value equal to zero and let P ∗ be aprobability such that ˜ S is a P ∗ -local martingale. Since ˜ V · ( φ ) = R · φ t · d ˜ S t is bounded below it is a localmartingale (see Ansel and Stricker (1994)) and in fact a supermartingale by Fatou’s lemma. Then we have E P ∗ (cid:16) ˜ V T ( φ ) (cid:17) ≤ and we obtain that E P ∗ ( h ) ≤ for every h in C and consequently for h in ¯ C by the Lebesgue theorem.Therefore ¯ C ∩ L ∞ + = { } . (Necessity) This is the difficult part, but essentially it consists in considering the weak* topology in L ∞ and to show that, under the NFLVR condition, C is closed with this topology (the proof of this is verytechnical and the reference is Delbaen and Schachermayer (1994)). Then we can apply the separation8heorem with this weak topology (e.g., 9.2 in Schaefer (1971)) and to show that there exist P ∗ ∼ P such that E P ∗ ( h ) ≤ for each h in C (see the details in Schachermayer (1994), 3.1). Now if we assume first that ˜ S isa bounded semimartingale we have that for each s < t, B ∈ F s and α ∈ R , α B (cid:16) ˜ S t − ˜ S s (cid:17) ∈ C . Therefore E P ∗ (cid:16) B (cid:16) ˜ S t − ˜ S s (cid:17)(cid:17) = 0 and P ∗ is a risk-neutral probability for ˜ S . If ˜ S is locally bounded then a localizationargument, and the result for the bounded case, allows us to obtain a locally martingale-measure P ∗ for ˜ S. Also we have a result for the case that ˜ S is not locally bounded. The result involves the concept of σ -martingale, that is a process that can be obtained as a stochastic integral of a positive integrand withrespect to a martingale, see Delbaen and Schachermayer (1999). Definition 15
Let ˜ S be a R d -valued semimartingale. There are not free lunches with vanishing risk if andonly if there is probability P ∗ ∼ P under which ˜ S is a σ -martingale. We have the analogous definitions for completeness in the the continuous time case, thought the meaning ofreplication now is in terms of stochastic integrals.
Definition 16
Let X be a claim, that is X ≥ , and X ∈ F T . We say that X is attainable, or replicable, if X is equal to the final value of and admissible strategy. Assume that the market model satisfies the NFLVR condition, then the set of risk-neutral probabilities isnot empty. If we choose one, say P ∗ , we have the following definition. Definition 17
We say that the market model is complete if every claim X ∈ F T , such that E P ∗ (cid:16) ˜ X (cid:17) < ∞ ,is attainable. Then the SFTAP for the continuous time setting reads the same way as in the discrete time case.
Theorem 3
The market is complete if and only if the risk-neutral probability is unique
Proof. (Sufficiency) The proof is exactly the same as in the discrete time case.(Necessity) Suppose that the risk neutral probability is unique. By Theorem 11.2 in Jacod (1979) ˜ S has therepresentation property with respect to P ∗ (any P ∗ -martingale can we written as an integral with respect to ˜ S )if and only P ∗ is an extremal measure in the set of all probability measures for which ˜ S is a local martingale.Suppose that P ∗ is not extremal, then there exist Q , Q ′ local martingale measures (not equivalent to P )such that P ∗ = λ Q +(1 − λ ) Q ′ for some λ ∈ (0 , . Also Q β := β Q +(1 − β ) Q ′ is a local martingale measurefor any β ∈ (0 , , and since Q << P ∗ and Q ′ << P ∗ we obtain that Q β ∼ P ∗ contradicting the uniquenessof P ∗ . Now by the representation property, if E P ∗ (cid:16) ˜ X (cid:17) < ∞ we can write E P ∗ (cid:16) ˜ X |F t (cid:17) = E P ∗ (cid:16) ˜ X (cid:17) + Z t φ s · d ˜ S s , t ∈ [0 , T ] and, in particular, with φ · ˜ S = E P ∗ (cid:16) ˜ X (cid:17) , ˜ X = φ · ˜ S + Z T φ t · d ˜ S t . Notice that the admissibility condition for φ is trivially satisfied since X ≥ . The arbitrage theory
From the the previous results we deduce that in a complete model we can price any payoff X such that E P ∗ (cid:16) ˜ X (cid:17) < ∞ and the price at time t is obviously the price of the replicating portfolio, V t ( φ ) , given by thestrategy φ such that ˜ V t ( φ ) is a P ∗ martingale. Then since ˜ X = ˜ V T ( φ ) we have that V t ( φ ) = S t E P ∗ (cid:16) ˜ X (cid:12)(cid:12)(cid:12) F t (cid:17) . (4)Notice that we can have other admissible strategies, say ϕ , replicating X but such that they are localmartingales and since they are bounded below they are supermartingales. Then ˜ V t ( φ ) = E P ∗ (cid:16) ˜ X (cid:12)(cid:12)(cid:12) F t (cid:17) = E P ∗ (cid:16) ˜ V T ( ϕ ) (cid:12)(cid:12)(cid:12) F t (cid:17) ≤ V t ( ϕ ) . Consequently they are more expensive and no one will pay for that.If the model is not complete but satisfying NFLVR, the formula (4) can be used for pricing if we take onerisk-neutral probability since, if we do that, we will have a price market model free of arbitrage. But whatis the good one? One way is to ask an additional property to the risk-neutral probability. An approach isto choose a risk neutral probability, say P ∗ , close to P , for instance if we take a strictly convex function wecan try to minimize E (cid:18) V (cid:18) d P ∗ d P (cid:19)(cid:19) . (5)For instance if V ( x ) = x log x we have the minimal entropy martingale measure, see Frittelli (2000), thatcoincides with the Esscher measure in the case we model the stock by a geometric Lévy model, see Chan(1999). Some risk-neutral measures, like the minimal martingale measure are also related to the minimizationof the cost to replicate perfectly the contingent claims, see Föllmer and Schweizer (1991). It can seen that, incertain cases, the minimal martingale measure minimizes (5) with V ( x ) = x , see Chan (1999) and Schweizer(1995).If we consider the (negative) Legendre transform of V we have U ( x ) = inf y ( V ( y ) + xy ) , for the examples above we obtain U ( x ) = − e − x if V ( x ) = x log x and U ( x ) = − x that can be inter-preted as utility functions. Then by the duality relationship we have that, under appropriate conditions see(Schachermayer, 2000), the optimal wealth, say W ∗ T , when we try to maximize the expected utility of thefinal wealth (by admissible strategies), satisfies U ′ ( W ∗ T ) E ( U ′ ( W ∗ T )) = d P ∗ d P where d P ∗ d P minimize (5). Then if we use this risk neutral to price derivatives what we obtain is the marginalutility indifference price proposed by Mark H.A. Davis, see . In fact, let define, for an initial wealth x , v ( x ) := sup E ( U ( W T,x )) assuming U : R + → R is strictly concave and strictly increasing C function, with U ′ ( ∞ ) = 0 and U ′ (0 + ) = ∞ . If we invest ε in a claim ξ with price p we can look for the value of p, say ˆ p, such that dd ε sup E (cid:18) U (cid:18) W T,x − ε + ε ˆ p ξ (cid:19)(cid:19) ε =0 = 0 , then, we have that E (cid:18) U (cid:18) W T,x − ε + εp ξ (cid:19)(cid:19) − E ( U ( W T,x − ε )) = ε E (cid:18) U ′ ( W T,x − ε ) ξp (cid:19) + o ( ε ) ˆ p = E (cid:16) U ′ (cid:0) W ∗ T,x (cid:1) ξ (cid:17) v ′ ( x ) , since v ′ ( x ) = E (cid:16) U ′ (cid:0) W ∗ T,x (cid:1)(cid:17) we obtain that ˆ p = E P ∗ ( ξ ) , with d P ∗ d P = U ′ ( W ∗ T,x ) E ( U ′ ( W ∗ T,x ) ) . References
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