Mathematics and Economics

Archive for the ‘Mathematics’ Category

Ross Recovery

In Economics, Finance, Mathematics, Uncategorized on January 12, 2016 at 6:09 am

In this post, we discuss the interesting recent paper by Steve Ross,”The Recovery Theorem”, in which a method is proposed to disentangle the risk aversion component from the subjective probability measure from state prices. In particular, a method is proposed to back out the market’s forecast of returns (a distribution over returns) from option prices. Attached is the pdf summary detailing the results.

Ross_Recovery_Summary

 

Bubbles and Crashes: The Local Martingale Characterization of Asset Price Bubbles

In Bubbles and Crashes, Economics, Finance, Mathematics on July 9, 2013 at 2:57 pm

We resume our Bubbles and Crashes series with this post discussing the recent influential works of Robert Jarrow and Phillip Protter on the characterization of bubbles as Local Martingales: The Local Martingale Characterization of Bubbles pdf

246A: Complex Analysis, Notes 2 – Meromorphic Functions and Properties of Analytic Functions

In Mathematics on February 10, 2011 at 2:24 am

We continue with a discussion about meromorphic functions and the properties of analytic functions. Later notes will consider the Riemann mapping theorem, harmonic functions and the Dirichlet problem among other topics.

Definition 2.1 A function {f} on an open set {\Omega} is meromorphic if there exists a discrete set of points {S = \left\{z: z \in \Omega\right\}} such that {f} is holomorphic on {\Omega\setminus S} and has poles at each {z \in S}. Furthermore, {f} is meromorphic in the extended complex plane if {F(z) = f(1/z)} is either meromorphic or holomorphic at {0}. In this case we say that {f} has a pole or is holomorphic at infinity.

By collecting results from the previous section, we are immediately led to the following proposition regarding the Laurent expansions of complex valued functions.

Proposition 2.2 Let {S} be the discrete set of singularities of a complex function {f:\Omega \rightarrow \mathbb{C}} where {\Omega} is an open set in {\mathbb{C}}. For a fixed {z_{0} \in S}, suppose the Laurent expansion for {f} in an annulus about {z_{0}} is given by {\sum_{-\infty}^{\infty}a_{n}(z-z_{0})^{n}}. Then,

  1. The function {f} has a removable singularity at {z_{0}} if and only if {a_{n} = 0} for all {n < 0}.
  2. The function {f} has a pole at {z_{0}} if and only if there exists {N \in \mathbb{Z}} with {N < 0} such that {a_{n} = 0} for all {n < N}; that is, the Laurent expansion of {f} about {z_{0}} has only finitely many negative terms.
  3. The function {f} has an essential singularity at {z_{0}} if and only if the Laurent expansion of {f} about {z_{0}} has infinitely many negative terms.
  4. Furthermore, {f} is meromorphic on the extended complex plane if and only if there exists {N \in \mathbb{Z}^{+}} such that {a_{n} = 0} for {n > N}.


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246A: Complex Analysis, Notes 1 – Analytic functions, Cauchy’s formula and singularities.

In Mathematics on January 4, 2011 at 7:36 pm

The following series of posts comprises our introduction to complex analysis as taught by Professor Rowan Killip at the University of California, Los Angeles, during the Fall quarter of 2009. Where necessary, course notes have been supplemented with details written by the authors of this website using assistance from Complex Analysis by Elias Stein and Rami Shakarchi. The basic properties of complex numbers will be assumed allowing us to begin with the definition of a holomorphic (or complex-differentiable) function, the central notion in our study of complex analysis.

The basic properties of complex numbers will be assumed, allowing us to begin with the definition of a holomorphic (or complex-differentiable) function, the central notion in our study of complex analysis.

Definition 1.1 Suppose {\Omega \in \mathbb{C}} is an open set and {f:\Omega \rightarrow \mathbb{C}}. We say {f} is holomorphic (or complex-differentiable) at {z_0 \in \Omega} if there exists {f^{\prime}(z_0) \in \mathbb{C}: f(z) = f(z_0) + f^{\prime}(z_0)(z - z_{0}) + o(\left |z-z_0\right |).} We say {f} is holomorphic on {\Omega} if it has this property for all {z \in \Omega}.

We can rewrite this formula in terms of the real and imaginary parts of {f} to surmise the relationship between complex differentiability and real analytic differentiability. Let {z, z_{0} \in \mathbb{C}} with {z = x+iy} and {z_{0} = x_0 + iy_{0}}, {x, y, x_0, y_0 \in \mathbb{R}} and write {f(z) = u(x,y) + iv(x,y)} where {u,v: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}}. Then,

\displaystyle  \left[ \begin{array}{cc} u(x,y) \\ v(x,y) \end{array} \right] = \left[ \begin{array}{cc} u(x_0, y_0) \\ v(x_0, y_0) \end{array} \right] + \left[ \begin{array}{cc} {\rm Re}f^{\prime}(z_{0}) & -{\rm Im}f^{\prime}(z_{0}) \\ {\rm Im}f^{\prime}(z_{0}) & {\rm Re}f^{\prime}(z_{0}) \end{array} \right] \left[ \begin{array}{cc} x-x_{0} \\ y-y_{0} \end{array} \right] + o(\left |x-x_{0}\right | + \left |y-y_{0}\right |).

We first notice that this is stronger than the differentiability of the real map {(x, y) \mapsto (u(x,y), v(x,y))} in {\mathbb{R}^2 \rightarrow \mathbb{R}^2}. In the real, multivariable case, the derivative of this map is a linear operator, namely, the Jacobian, {J_{f}(x,y)}; in our equation above, the {2 \times 2} matrix on the right hand side is {J_{f}(x,y)}. Clearly, it is endowed with a distinct structure summarized in the following proposition.

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