# Analytical Asset Price Processes

## Abstract

In an arbitrage-free market, asset prices are characterised by the fundamental pricing equation (2.4) We have also seen that the work of Bick [14] and others has imposed further restrictions on asset price processes than just the absence of arbitrage possibilities Viable asset prices in their sense imply a path-independent pricing kernel While this class of asset prices is still large, it is hard to find analytical solutions for these processes unless the pricing kernel has constant elasticity In this chapter1 we derive analytical asset prices for alternative characterizations of the pricing kernel We begin with a new class of pricing kernels Its advantage is that although these pricing kernels generate analytical asset prices, they are relatively flexible and not restricted to constant elasticity Later, we will also derive analytical asset prices which are consistent with HARA-utility.

## Keywords

Asset Price Asset Return Sharpe Ratio Relative Risk Aversion Geometric Brownian Motion## Preview

Unable to display preview. Download preview PDF.

## Notes

- 1.This chapter is partly based on Lüders [130] and Lüders [131].Google Scholar
- 2.For a derivation see Appendix A.7.Google Scholar
- 4.Random numbers are generated with SAS.Google Scholar
- 5.All figures are given in Appendix B.Google Scholar
- 6.The option implied risk-neutral distribution is calculated from the implied volatilities of S&P 500 options provided by PMpublishing (http://www.pmpublishing.com/volatility/sp.html) Note that this shape of the distribution is typical The option implied risk-neutral distributions are usually more skewed and more leptokurtic than the lognormal distribution, see Subsect 4.1.2 and Jackwerth and Rubinstein (1996) and Rubinstein (1994).Google Scholar
- 7.To plot the graph, more than 2500 pricing rules (equation 6.9) are evaluated numerically.Google Scholar
- 8.For a related result in a one-period model see Camara [28] Camara [28] derives option prices when the distribution of asset prices is three-parameter lognormal.Google Scholar
- 9.For a derivation see Appendix A.9.Google Scholar
- 10.Note that we assume that every period is of length 1 However, the length of periods is arbitrary but setting the length of a period equal to 1 simplifies notation. We assume that
*T*≫ 1.Google Scholar