In continuation of some previous posts on residual estimation risk (RER), we establish an upper bound for RER when the risk measure is VaR for any arbitrary error distribution , where the error distribution is defined as the difference between an actual loss distribution and a loss estimator (see [1] for more details). Asymmetric Error Distribution For an arbitrary…

# Financial Mathematics

# Sensitivities for Make-whole Callable Bonds

A callable bond is a bond that provides the issuer with the right to exchange the bond for its call value in cash. There are a few ways one can value a callable bond by extending the expression of a vanilla bond: Yield-to-x: Calculate the yield-to-maturity, yield(s)-to-call, and/or yield-to-worst and take the lowest of this set. Revalue the bond with…

# Lévy Processes For Finance: An Introduction In R

Prior to "opting for the herd" and leaving academia to work in the private sector, I began scribbling the details of what my PhD thesis might look like. The topic I was interested in writing my thesis on (at the time) was Lévy processes and in particular their applications to derivatives pricing. I began coding some of the more well-known…

# A Fast Fourier Transform Method for Mellin-type Option Pricing

Following from the previous post on Mellin-type option pricing, analytical pricing formulas and Greeks are obtained for European and American basket put options using Mellin transforms. In the manuscript below, we assume assets are driven by geometric Brownian motion which exhibit correlation and pay a continuous dividend rate. A novel approach to numerical Mellin inversion is achieved via the fast…

# Multi-Asset Option Pricing with Exponential Lévy Processes and the Mellin Transform

I'm a big fan of the Mellin transform for solving PDEs. As an application of this, I've provided the 'broad strokes' for solving multi-asset options driven by exponential Lévy processes in this brief manuscript: http://arxiv.org/abs/1309.3035 The extended version of this paper will be ready at the end of 2015.

# Using Fourier Transforms To Solve Option Prices

When the pdf of a distribution is not known analytically, it's common to compute by taking the inverse Fourier transform of its characteristic function. The same idea applies to financial options. For simplicitly, let's consider the discounted expectation formula of a European option . for log prices and time to expiry . In integral form this is where is the transition…

# Lévy Processes for Modelling Asset Returns

Over the past few decades analysis has shown that market data is inconsistent with some of the underlying assumptions of the Black-Scholes model (e.g. normality of asset log-returns). For example, if asset price changes remain large while time periods shrink, one cannot assume that prices are continuous. Under this observation, Cox and Ross assume prices follow a pure jump process…