Between the mathematics Wednesday and the physics Friday, I have an easy part today. This is a link worthy of your attention: Eric Weinstein: What Math and Physics Can Do for New Economic Thinking - In my title I have replaced economy by finance.
When Variance is Negative
Michael has been writing on correlation matrices in his recent posts. Is there also a relevance in option pricing?
Obviously, there are multi-asset options, e.g., equity basket options, on the market. If we assume that each of the underlyings follows an individual geometric Brownian motion (Black-Scholes world) with correlated Wiener processes, then the value of the option satisfies the multi-dimensional Black-Scholes equation
When the variance covariance matrix is positive definite, then this equation is a backwards heat equation. With appropriate end conditions (the payoff of the multi-asset option), it obtains a uniques solution which depends continuously on the data.
If the space dimension is not too large (not larger than 4 oder 5), it can be solved numerically by finite elements or by convolution of the payoff with the Green kernel of the PDE (a multi-dimensional lognormal density).
Alternatively, Monte Carlo techniques could be applied.
What happens if, for some reason, the variance-covariance matrix turns out to be not positive definite meaning that at least one of the eigenvalues is not positive. For the sake of simplicity of arguing, let this eigenvalue be negative.
Finite elements
In this case, the finite element method tries to solve an equation which is a backwards equation with end condition for the directions corresponding to the positive eigenvalues and a forwards equation with end condition for the directions corresponding to the negative eigenvalues. This is severely ill-posed as we pointed out in our Traunsee example.
Green's kernel
Naively applying the multidimensional normal distribution when one of the eigenvalues is negative leads to a density which is not a probability density as it does not integrate to one. (But I like the 3d plot.)
This is not exactly the bell shape one would expect.
Monte Carlo
Monte Carlo simulation typically takes uncorrelated random numbers and transforms them to the correlated ones by applying the (roughly speaking) Cholesky square root ot he variance covariance matrix. This is the good news: If an eigenvalue is negative, then the Cholesky decomposition will terminate with an error.
Practical relevance
But is this of practical relevance? We have learnt that a variance covariance matrix is at least positve semidefinite.
Coming next Wednesday: Negative eigenvalues in practical finance.
Obviously, there are multi-asset options, e.g., equity basket options, on the market. If we assume that each of the underlyings follows an individual geometric Brownian motion (Black-Scholes world) with correlated Wiener processes, then the value of the option satisfies the multi-dimensional Black-Scholes equation
If the space dimension is not too large (not larger than 4 oder 5), it can be solved numerically by finite elements or by convolution of the payoff with the Green kernel of the PDE (a multi-dimensional lognormal density).
Alternatively, Monte Carlo techniques could be applied.
What happens if, for some reason, the variance-covariance matrix turns out to be not positive definite meaning that at least one of the eigenvalues is not positive. For the sake of simplicity of arguing, let this eigenvalue be negative.
Finite elements
In this case, the finite element method tries to solve an equation which is a backwards equation with end condition for the directions corresponding to the positive eigenvalues and a forwards equation with end condition for the directions corresponding to the negative eigenvalues. This is severely ill-posed as we pointed out in our Traunsee example.
Green's kernel
Naively applying the multidimensional normal distribution when one of the eigenvalues is negative leads to a density which is not a probability density as it does not integrate to one. (But I like the 3d plot.)
Monte Carlo
Monte Carlo simulation typically takes uncorrelated random numbers and transforms them to the correlated ones by applying the (roughly speaking) Cholesky square root ot he variance covariance matrix. This is the good news: If an eigenvalue is negative, then the Cholesky decomposition will terminate with an error.
Practical relevance
But is this of practical relevance? We have learnt that a variance covariance matrix is at least positve semidefinite.
Coming next Wednesday: Negative eigenvalues in practical finance.
UnRisk FACTORY 5 Released
30-Oct-13 - UnRisk takes UnRisk FACTORY and UnRisk FACTORY Capital Manager version 5 to financial institutions for advanced, individualized investment and risk management processes.
Language Is Too Clumsy an Instrument?
said Luitzen Egbertus Jan Brouwer, Dutch mathematician and philosopher, founder of the mathematical philosophy of intuitionism. This weekend I reread Dietmar Dath's Höhenrausch, Mathematics of the 20th Century in 20 brains - a collection of short stories and fictional portraits of Cantor, Hilbert, Poincaré, Brouwer, Noether, Ramanujan, Gödel, Dirac, Turing, Kolmogorow, von Neumann, Dieudonné, Grothendieck, Chaitin, Thom, Robinson, Mandelbrot, Witten, Wolfram … I am afraid, this book appears to be available in German only.
Portfolio Optimization and Correlation
As frequent readers of our blog may have noticed 'Physics Friday' has more or less become a 'Physics Weekend' post - hope despite this some of you still enjoy them.
Here Cij is the correlation matrix and in order to measure and optimize risk in a portfolio it is essential to obtain a reliable estimate for the correlation matrix.In general this is difficult as the number of assets in the portfolio N may not be significantly larger than the number of days T in the time series (4 years of data give 1000 entries in the time series and the typical size of a portfolio is several hundred assets). The order of entries to estimate for the correlation matrix is N2/2. An accurate estimation of the correlation matrix would require q=N/T to be significantly smaller than 1.
An optimal (Markowitz) portfolio using this empirical correlation matrix would have
as the corresponding risk (the risk of the portfolio over the period used to construct it). We call this the in sample risk. Now assume there is a "true" correlation matrix C which is perfectly known resulting in a risk
We now use this perfect correlation matrix to draw past and future x then we can construct a portfolio which risk is given by
In our last post we discussed some of the questions which may arise when working with random matrices. But I still owe you an explanation way random matrix theory may affect your life in quant finance. Today I try to answer parts of your question and we will make a short tour in portfolio optimization.
Suppose the task is to build a portfolio of N assets, then the daily variance of the portfolio return is given by
Here Cij is the correlation matrix and in order to measure and optimize risk in a portfolio it is essential to obtain a reliable estimate for the correlation matrix.In general this is difficult as the number of assets in the portfolio N may not be significantly larger than the number of days T in the time series (4 years of data give 1000 entries in the time series and the typical size of a portfolio is several hundred assets). The order of entries to estimate for the correlation matrix is N2/2. An accurate estimation of the correlation matrix would require q=N/T to be significantly smaller than 1.
An optimal (Markowitz) portfolio using this empirical correlation matrix would have
The gain of the portfolio is
with gi the predicted gains of a single asset.
If r is the daily stock return at time t the empirical variance of each stock is
and the empirical correlation matrix is obtained as
as the corresponding risk (the risk of the portfolio over the period used to construct it). We call this the in sample risk. Now assume there is a "true" correlation matrix C which is perfectly known resulting in a risk
We now use this perfect correlation matrix to draw past and future x then we can construct a portfolio which risk is given by
The subscript out refers to the fact, that the risk is constructed using E but observed on the next period (Remember that we can draw future samples of x!).
So we have three possible estimates and it remains to understand their biases. One can use convexity arguments for the inverse of positive definite matrices to show that the out-of-sample risk of an optimized portfolio is larger (and in practice, this can be much larger) than the in-sample risk, which itself is an underestimate of the true minimal risk. This is a general situation: using past returns to optimize a strategy always leads to over-optimistic results because the optimization adapts to the particular realization of the noise, and is unstable in time [Potters et al, Financial Applications of Random Matrix Theory: Old Laces and New Pieces].
Only in the limit q going to 0 these quantities will coincide, since in this case the measurement noise disappears.
The question is how to "clean" the empirical correlation matrix to avoid (f possible) such biases in the estimation of future risk. And here RMT enters the game - how read again next "Physics Weekend" post.
Apple Rolled Out New Things You May Buy?
First up: we make UnRisk cross platform and platform agnostic. But we also recognize that there are events, where virtually incremental improvements are giving new directions.
Black vs. Bachelier revisited
Last week we realized that one of the strengths of the Black76 model - interest rates must not become negative - is one of its weaknesses at the same time - interest rates must not become negative and that the Bachelier model may be a possibility to handle interest rates that are close to zero.
Black: dF= sigma F dW
Bachelier: dF= sigma W
Note that the Bachelier volatility is an absolute volatility not depending on the actual level of the underlying, whereas the Black volatility is mulitplied by the value of the underlying.
Black76
For the practitioner, the Black76 model is widely used to valuate vanilla interest rate options like caps, floors or swaption. The formula to be applied is
 |
| Black option formulae for call (C) and put (P) options |
Here, F is the forward rate (e.g. of the floating rate to be capped), K is the strike level (the cap rate), sigma is the annualized Black volatility and T is the time, when the floating rate is set. If sigma tends to infinity, then the call value converges to exp(-rT) F, independent of the strike level.
 |
| Cap value for at the money caplet as a function of sigma. (F = K = r = 0.5 % = 50 bp, T= 1) |
Bachelier
The same can be done for the Bachelier model.
 |
| Bachelier option formulae for call (C) and put (P) options. |
Here, if sigma tends to infinity, then d1 becomes zero, and the call value (and also the put value) grows unboundedly.
 |
| Cap value using the Bachlier model. Instrument data as above. |
Can the Black and the Bachelier volatilities be translated into one another?
For very small sigma (sigma = 0), the Black and the Bachelier option values coincide. For growing sigma, the Bachlier value grows unboundedly, whereas the Black value goes into saturation. This means that for every Black volatility, there is a corresponding Bachelier volatility delivering the same price for the specific vanilla instrument; this implied Bachelier volatility depends on the specific instrument. The following figure shows the mapping for various at the money caplets.
The other way round, i.e. obtaining the implied Black volatility from the Bachelier value, is more complicated, because the existence of a solution is not guaranteed.
Note that if the translation is possible at all, moderate Bachelier volatilities of 2 percent are translated into Black volatilities of 800 percent and more for small at the money rates.
Again, we have found a classical ill-posed problem. The solution need not exist and if it does exist, small perturbations in the data (Bachelier vola, termsheet data) may lead to arbitrarily large perturbations in the solution (the Black volatility).
Why We Did Not Install A Service Desk
To answer this question I am lucky - I just need to link to An ABSERD incident - a service desk satire the latest post of "Eight to Late". It is pointed but describes the core problem. How to make services effective and efficient.
Agenda 2014 - Package and Disseminate Know How
Having packed 15 years of experiences into the transformation of mathematical schemes from complex technical systems into finance, Andreas and Michael have written A Workout in Computational Finance - recently published by Wiley.
All Random with this Matrix
On our way from billiards to quantitative finance we stopped at quantum chaos in last "Physics Friday" blog post. Without going to deep into details we mentioned the term random matrix - a term some of the readers may not be too familiar with - so perhaps, as they have a direct application in quant finance, we will go a bit more into detail today.
As the name implies, a random matrix is a matrix with random entries. For such a matrix several question can be asked:
Next week we will see how random matrices are connected to quant finance.
- What is the limiting shape of the histogram of the distribution of eigenvalues when the size of the matrix becomes large?
- How does the distribution of spacings between eigenvalues look like and what is the limiting shape if the size of the matrix becomes large?
- Is there universality of these shapes with respect to small changes in the distribution of the matrix entries?
- Are the eigenvectors localized or are there no dominant components?
Different approaches to answer these questions exist - a few of them are mentioned in the following:
- Method of traces (combinatorial)
- Stieltjes transform method
- Orthogonal polynomials
- Stochastic differential equations
The figure shows a histogram of eigenvalues from a random real symmetric matrix (N=3000). Each matrix element is a random number and these numbers are Gaussian distributed with a mean of zero and a standard deviation of one (except the diagonal elements which have a standard deviation of square root of 2). The distribution thins out toward the edges as it should as the theoretical density of eigenvalues is known as Wigner's semicircle law (because the curve takes the shape of a semicircle centered at zero with radius Sqrt[2N],where N is the dimension of the random matrix (and hence the number of eigenvalues).
Next week we will see how random matrices are connected to quant finance.
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