Tải bản đầy đủ - 0 (trang)
A.5 Calculus, Fourier Transformations and Partial Differential Equations

# A.5 Calculus, Fourier Transformations and Partial Differential Equations

Tải bản đầy đủ - 0trang

348

A Mathematics

∂h(x, y)/∂x. This means that we cut out a thin slice of the hills along the

x-direction and look at the derivative of the elevation on this slice (which

is just a function in one variable). Of course we can generalize this idea to

arbitrarily many dimensions. An example for a vector composed of partial

derivatives is the gradient of a function h : (x1 , . . . , xn ) → h(x1 , . . . , xn ) ∈ R

which is deﬁned by

∇h(x) := ⎜

∂h(x)

∂x1

..

.

∂h(x)

∂xn

⎟,

where x = (x1 , . . . , xn ) and sometimes Dh is written instead of ∇h.

In the following we need to extend the real numbers R to complex numbers

C that can be written as a linear combination of a real and an imaginary

number, i.e., z = x + iy ∈ C with x, y ∈ R and i2 = −1. For complex numbers

particularly the Euler formula holds:

eiφ = cos φ + i sin φ.

We can now deﬁne a Fourier transformation: it maps a function f : R → R

to its Fourier transform F f : R → C and is deﬁned by

1

(F f )(ξ) := √

+∞

e−iξt f (t) dt.

−∞

The Fourier transformation describes in a certain way the frequency distribution of f . The most natural application is in acoustics where f describes the

oscillation (e.g., of the air) and its Fourier transform corresponds to the distribution of frequencies, i.e., its sound. A sine function would lead to a Dirac

distribution as Fourier transform: in other words there is only one frequency,

the sound is a pure tone. Such a tone would sound very harsh and artiﬁcial,

since natural tones (like the sound of a piano) are composed of many diﬀerent tones, i.e., they correspond to a weighted sum of sine functions. Their

Fourier transform is therefore a weighted sum of Dirac distributions. Another

example is a normal distribution: their Fourier transform is again a normal

distribution.

The Fourier transformation has a couple of interesting properties. We collect the most important in the following lemma:

Lemma A.12 (Properties of the Fourier transformation). Let F be the

Fourier transformation, then:

(i) F is a linear map.

(ii) The inverse of the Fourier transformation is given by

1

(F −1 f )(x) = √

+∞

eixt fˆ(t) dt.

−∞

A.5 Calculus, Fourier Transformations and Partial Diﬀerential Equations

349

(iii) The Fourier transform of a derivative is a polynomial, more precisely:

F

∂n

f (x) (ξ) = (iξ)n (Ff )(ξ).

∂xn

(iv) It is possible to deﬁne an appropriate space (of functions or distributions)

such that F is a bijective map on this space.

Property (iii) can be summarized by saying that the Fourier transformation turns derivatives into a product – although this has nothing to do with

the marketing of options... The property is often the main reason to use the

Fourier transformation and could even be used as an alternative deﬁnition

of diﬀerentiation. While this is certainly a complicated way for reaching this

goal, it enables us to generalize the deﬁnition of derivatives: if we take the

Fourier transform of a multiplication of f with, e.g., i|ξ|1/2 , we get something

like a “half” derivative. If that sounds esoteric to you, then be ensured that it

is not: very much to the contrary it is surprisingly useful. The concept leads

to the deﬁnition of pseudo diﬀerential operators and they are needed, e.g.,

in solving certain asset pricing problems when the underlying process is of

a more complicated form. We mention this point and generally use Fourier

transformations when we discuss L´evy processes in Section 8.8.

A partial diﬀerential equation (short: PDE ) is an equation that contains

diﬀerent partial derivatives of an unknown function and is used to determine

this function. In contrast, an ordinary diﬀerential equation (short: ODE ) only

involves one kind of derivative (e.g., only derivatives with respect to t). As an

example for a simple PDE we consider the heat equation 4 which is needed to

solve the Black-Scholes equation (see Section 8):

∂ 2 u(x, t)

∂u(x, t)

=

,

∂t

∂x2

where t ∈ [0, T ] and x ∈ R. (In the original physics model for heat transport,

t is the time and x the space variable, whereas u(x, t) is the temperature at

time t and position x.) The above equation therefore means that the partial

derivative of u with respect to t equals its second partial derivative with

respect to x.

Typically, PDEs can only be solved uniquely when we have additional

conditions. In the case above this would be an initial condition (specifying u

at t = 0) plus some boundary conditions (e.g., specifying the behavior of u

for x → ±∞).

PDEs are a central modeling tool in all scientiﬁc disciplines that rely on

sophisticated mathematical models (like physics, chemistry, biology, engineering – and some areas in ﬁnance). Therefore their analytical and numerical

4

This PDE was originally used to describe the transport of heat in a material.

There are, however, various other applications for this equation, therefore it is

sometimes also called, e.g., diﬀusion equation.

350

A Mathematics

investigation is very important. But how can we solve such a PDE? First,

we need to stress that there is no general method that works for all kinds of

PDEs. Very much to the contrary, speciﬁc methods need to be developed for

diﬀerent situations, and there is a whole research area in mathematics dedicated to this. In the case of the simple linear heat equation given above, things

look better, of course: there are in fact several methods that can be applied.

In the following we sketch one particularly simple method (the separation of

variables). Other methods that could be used are the Fourier transformation5 ,

variational methods or the ﬁnite element method. The latter is the standard

way for numerical computations and works for a large class of PDEs. We refer

the reader to [Eva98] and [RR04] for in-depth introductions to PDEs.

The key idea of the separation of variables is to look not for all possible

solutions, but only for solutions of a special form, namely u(x, t) = a(x) · b(t).

Once we have found such a solution, we only need to prove uniqueness, and

we know that the solution we have found is not any solution, but the only

solution. In fact, the uniqueness proof will be omitted here, but can be found

in most mathematical textbooks on PDEs.

Using the ansatz u(x, t) = a(x) · b(t) we can rewrite the PDE as follows:

a(x)b (t) = a (x)b(t).

Sorting terms (and assuming that non of them vanishes, which is another

point that would have to be justiﬁed later) we obtain

a (x)

b (t)

=

.

b(t)

a(x)

The central observation is now the following: the left side only depends on t

and the right side only on x. Since both sides agree for all x and t, both terms

have to be constant. Let us call this constant −λ, then we get two ordinary

diﬀerential equations:

b (t) = −λb(t),

a (x) = −λa(x).

The ﬁrst of these equations can be solved by an exponential function:

b(t) = b(0)e−λt ,

the second can be solved by a combination of sine and cosine functions, e.g.

5

Here we can exploit that the Fourier transform of a derivative is a simple multiplication. Thus after taking the Fourier transform of a PDE some of the derivatives

become multiplications (compare Lemma A.12). The result is typically an ODE

that can be solved much easier, either analytically or numerically. Finally the

solution needs to be Fourier transformed again to return to the original formulation.

A.6 General Axioms for Expected Utility Theory

351

a(x) = sin(x/ λ).

To determine the precise form of a and b we need to take into account the

initial and boundary conditions of the heat equation: we superimpose solutions

for a such that a(0) corresponds to the initial condition . For instance, if

the boundary condition is given by u(0, t) = u(1, t) = 0, then any ak (x) =

sin(kx/π) for k ∈ N satisﬁes the boundary condition, since sin(kπ) = 0 for

all k ∈ N. Denote uk (x) = ak (x) · bk (x), where bk (x) = b(0)e−kt/π . Then a

weighted sum of the uk can be constructed to ﬁt the initial condition. This

sum still solves the heat equation and the boundary and initial condition, since

the heat equation is linear, i.e., weighted sums of solutions are also solutions.

A.6 General Axioms for Expected Utility Theory

There are many diﬀerent ways to obtain a general characterization of Expected

Utility Theory for arbitrary probability measures. In the following we sketch

a rather new approach by Chatterjee and Krishna [CK]. The details of this

derivation can be found in their article.

Let Z be a compact metric space. Let P(Z) be the set of all probability

measures on Z.

The Independence Axiom can essentially be stated as in the ﬁnite case:

Axiom A.13 (Independence). Let p, q, r ∈ P(Z). Let p

then λp + (1 − λr) λq + (1 − λ)r.

q and λ ∈ (0, 1],

To state the Continuity Axiom we need to generalize the notion of continuity via the concept of open sets (compare App. A.3): we say that a function

f : X → Y is continuous if, for all open sets U ⊂ Y , the set f −1 (U ) is open.

Open sets in P(Z) can be deﬁned via weak- convergence (compare

Sec. 2.4.5): ﬁrst, deﬁne closed sets as sets which contain the limit of any

converging sequence (compare App. A.3). Second, deﬁne open sets as complement to these closed sets. We can do more and construct even a metric

d that measures the distance between two probability measures and reﬂects

the same convergence. Such a metric is given by the so-called “Wasserstein

metric”, compare, e.g., [AGS05].

The Continuity Axiom then becomes:

Axiom A.14 (Continuity). The sets {q ∈ P(Z) | q

q} are open.

p} and {q ∈ P(Z) | p

Theorem A.15. Let Z be a compact metric space (e.g. a bounded and closed

interval in R). Let

be a preference relation, i.e., a complete and transitive

relation on P(Z), satisfying the Continuity and Independence Axioms, then

can be represented by a von Neumann-Morgenstern Expected Utility function

u : Z → R.

352

A Mathematics

To prove Thm. A.15, Chatterjee and Krishna use intermediate steps: they

prove that the Independence Axiom together with the Continuity Axiom

implies a new axiom, the Translation Invariance Axiom. Together with the

Continuity Axiom, this new axiom implies the existence of a EUT function,

representing . Translation Invariance can be stated as follows:

Axiom A.16 (Translation Invariance). Let r be a signed measure on Z with

average r(Z) = 0, in other words, let r be the diﬀerence of two probability

measures on Z. Let p, q ∈ P(Z). Assume moreover that p + r, q + r ∈ P(Z).

Then p q implies p + r q + r.

The intuition behind the translation invariance is that adding a signed

measure r to a lottery does not change the preference relation. This means

that making certain outcomes more likely, others less likely in the same way

for p and q, does not change the original preference between p and q. This

is morally the same as the Independence Axiom and mathematically at least

close enough to show the equivalence of both axioms (under the condition of

continuity) relatively easy.

How can we now use the Independence Axiom to construct an EUT function?

First, one can prove that the indiﬀerence sets under the preference relation

are “thin”. This means: for any q ∈ P(Z) and any ε > 0 there are p, r ∈ P(Z)

which are “close” to q, i.e., d(p, q) < ε and d(q, r) < ε, and that p q r.

Second, one can show that, for any p ∈ P(Z), the contour sets {q ∈

P(Z) | q p}, {q ∈ P(Z) | q ∼ p} and {q ∈ P(Z) | q ≺ p} are all convex.

P(Z) is a convex subset of a vector space. We can pick a measure o ∈ P(Z)

and some δz ∈ P(Z) with δz

o. Let us choose moreover some q ∈ P(Z),

q = δz , q = o. The structure of the indiﬀerence sets derived above allows

us to ﬁnd a continuous aﬃne functional f : P(Z) → R such that f (o) = 0,

f (δz ) = 1 and such that the indiﬀerence set of q is a contour set of f , i.e., for

all p ∈ P(Z) with q p we have f (q) > f (p).6

Using the translation invariance, one can show that f reﬂects the preferences on all of P(Z). With a translation, we can also assume that f is not

only aﬃne, but linear.

In the ﬁnal step, we deﬁne u : Z → R by u(z) := f (δz ). We have to show

that this deﬁnition is correct, i.e., that

U (p) :=

u(z) dp(z) = f (p).

Z

This is easy to see for measures with ﬁnite support: let p =

by linearity of f ,

6

n

i=1

pi δzi , then

More precisely, we ﬁrst restrict ourselves to a ﬁnite dimensional subset, such

that the existence of the aﬃne functional f can be deduced from the Separating

Hyperplane Theorem (see App. A.1). Later one can show that f is independent

of the choice of this ﬁnite dimensional subset.

A.6 General Axioms for Expected Utility Theory

n

u(z) dp(z) =

Z

n

pi u(zi ) =

i=1

n

pi f (zi ) = f

i=1

353

p i δz i

= f (p).

i=1

We can approximate any measure p ∈ P(Z) by measures with ﬁnite support.

Since f is continuous, this proves that U (p) = f (p) for all p ∈ P(Z) and thus

u is an expected utility function representing the preference relation .

B

Solutions to Tests and Exercises

“Teachers open the door. You enter it by yourself.”

Chinese proverb

The tests are meant to provide an immediate feed back when studying by

yourself, hence we give solutions to all questions. Although some of the questions are tricky and require some thinking about the context of the chapter,

the student should be able of answering most questions correctly after working

through a chapter. If this is not the case, we would recommend to the reader,

to study the chapter a little bit more in detail. A good result, however, can

only ensure that the basic concepts have been understood and memorized.

The exercises then serve as a way to apply and train the ideas and methods

of the chapter. We only give solutions to some of the exercises. This may be

inconvenient for the self-learning student, but it allows to use some of the

exercises for homework assignments.

Solutions to Tests

Chapter 2

Exercise: 1 2 3 4 5 6 7 8 9 10 11 12 13 14

××

×

×

×

××

×

××

×

××

×

× ×

×

× ×

× × × ×

×

×

×

×

356

B Solutions to Tests and Exercises

Chapter 3

Exercise: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

×

×

×××

×

× ×

×

×

×

×

×

×

× ×

×

×

×

Chapter 4

Exercise: 1 2 3 4 5 6 7 8 9 10

××

××

×

××

×

× ×

××

×××

×

×

Chapter 5

Exercise: 1 2 3 4 5 6

×

×

×

××××

××

×

Solutions to Exercises

Solutions to the exercises are provided on the web page to this book. See

http:\\www.financial-economics.de

or the publisher’s web page for details.

References

AB03.

Abd00.

Abe89.

Abe90.

AGS05.

Ake70.

Arr74.

Aum77.

Bac00.

Ban81.

BC97.

BE82.

BE92.

Ber38.

C.D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with

Applications to Economics, Mathematical surveys and monographs, vol.

105, American Mathematical Society, 2003.

M. Abdeallaoui, Parameter-Free Elicitation of Utilities and Probability

Weighting Functions, Management Science 46 (2000), 1497–1512.

A. Abel, Asset Prices under Heterogeneous Beliefs: Implications for the

, Asset Prices under Habit Formation and Catching up with the

Joneses, The American Economic Review 80 (1990), 38–42.

L. Ambrosio, N. Gigli, and G. Savar´e, Gradient ﬂows in metric spaces

and in the space of probability measures, Birkhă

auser Verlag, Basel, 2005.

G.A. Akerlof, The Market for ”Lemons“: Quality Uncertainty and the

Market Mechanisms, The Quarterly Journal of Economics 84 (1970),

no. 3, 488–500.

K.J. Arrow, The Use of Unbounded Utility Functions in ExpectedUtility Maximization: Response, The Quarterly Journal Of Economics

88 (1974), no. 1, 136–138.

R.J. Aumann, The St. Petersburg Paradox: A Discussion of Some Recent Comments, Journal of Economic Theory 14 (1977), no. 2, 443–445.

L. Bachelier, Th´eorie de la Sp´eculation, Annales Scientiﬁques de l’Ecole

Normale Superieure 3 (1900), no. 17, 21–86.

R.W. Banz, The Relationship between Return and Market Value of Common Stocks, Journal of Financial Economics 9 (1981), no. 1, 3–18.

M.H. Birnbaum and A. Chavez, Tests of Theories of Decision Making:

Violations of Branch Independence and Distribution Independence, Organizational Behavior and Human Decision Processes 71 (1997), no. 2,

161–194.

L.E. Blume and D. Easley, Learning to Be Rational, Journal of Economic Theory 26 (1982), no. 2, 340–351.

L. Blume and D. Easley, Evolution and Market Behavior, Journal of

Economic Theory 58 (1992), no. 1, 9–40.

D. Bernoulli, Specimen Theoriae de Mensura Sortis, Commentarii

Academiae Scientiarum Imperialis Petropolitanae (Proceedings of the

royal academy of science, St. Petersburg) (1738).

358

References

Ber98.

Bew80.

BH97.

BH98.

BHS01.

BHW98.

Bir05.

BL02.

Bla05.

BN97.

BN98.

BP00.

Bre79.

BS73.

BSV98.

BT95.

BX00.

CH94.

CIR85.

CK.

CKL96.

J. Bertoin, L´evy Processes, Cambridge University Press, 1998.

T. Bewley, The optimum quantity of money, Models of Monetary Economics (1980), 169–210.

W.A. Brock and C.H. Hommes, A Rational Route to Randomness,

Econometrica: Journal of the Econometric Society (1997), 1059–1095.

, Heterogeneous Beliefs and Routes to Chaos, in a Simple Asset

Pricing Model, Journal of Economic Dynamics and Control 22 (1998),

1235–1274.

N. Barberis, M. Huang, and T. Santos, Prospect Theory and Asset

Prices, The Quarterly Journal of Economics 116 (2001), no. 1, 1–53.

S. Bikhchandani, D. Hirshleifer, and I. Welch, Learning from the Behavior of Others: Conformity, Fads, and Informational Cascades, The

Journal of Economic Perspectives 12 (1998), no. 3, 151–170.

M.H. Birnbaum, A Comparison of Five Models that Predict Violations

of First-Order Stochastic Dominance in Risky Decision Making, Journal

of Risk and Uncertainty (2005).

S. Boyarchenko and Levendorskiˇi, Barrier Options and Touch-and-Out

Options under Regular Levy Processes of Exponential Type, The Annals

of Applied Probability 12 (2002), no. 4, 1261–1298.

P.R. Blavatskyy, Back to the St. Petersburg Paradox?, Management Science 51 (2005), no. 4, 677–678.

O.E. Barndorﬀ-Nielsen, Normal Inverse Gaussian Distributions and

Stochastic Volatility Modelling, Scandinavian Journal of Statistics 24

(1997), no. 1, 1–13.

M.H. Birnbaum and J.B. Navarrete, Testing Descriptive Utility Theories: Violations of Stochastic Dominance and Cumulative Independence,

Journal of Risk and Uncertainty 17 (1998), 49–78.

H. Bleichrodt and J.L. Pinto, A Parameter-Free Elicitation of the Probability Weighting Function in Medical Decision Analysis, Management

science 46 (2000), 1485–1496.

D.T. Breeden, An Intertemporal Asset Pricing Model with Stochastic

Consumption and Investment Opportunities.

F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, The Journal of Political Economy 81 (1973), no. 3, 637–654.

N. Barberis, A. Shleifer, and R. Vishny, A Model of Investor Sentiment,

Journal of Financial Economics 49 (1998), no. 3, 307–343.

S. Benartzi and R.H. Thaler, Myopic Loss Aversion and the Equity

Premium Puzzle, The Quarterly Journal of Economics 110 (1995), no. 1,

73–92.

M. Brennan and Y. Xia, Stochastic Interest Rates and the Bond-Stock

Mix, European Finance Review 4 (2000), 197–210.

C. Camerer and T.H. Ho, Violations of the Betweenness Axiom and

Nonlinearity in Probability, Journal of Risk and Uncertainty 8 (1994),

167–196.

J.C. Cox, J.E. Ingersoll, and S.A. Ross, A Theory of the Term Structure

of Interest Rates, Econometrica 53 (1985), 385–407.

K. Chatterjee and R.V. Krishina, A Geometric Approach to Continuous

Expected Utility, Economic Letters In press.

L. Chan, J. Karceski, and J. Lakonishok, Momentum Strategies, Journal

of Finance 51 (1996), 1681–1711.

### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

A.5 Calculus, Fourier Transformations and Partial Differential Equations

Tải bản đầy đủ ngay(0 tr)

×