Stochastic portfolio theory

Stochastic portfolio theory (SPT) is a mathematical theory for analyzing stock market structure and portfolio behavior introduced by E. Robert Fernholz in 2002. It is descriptive as opposed to normative, and is consistent with the observed behavior of actual markets. Normative assumptions, which serve as a basis for earlier theories like modern portfolio theory (MPT) and the capital asset pricing model (CAPM), are absent from SPT. Stochastic portfolio theory (SPT) is a mathematical theory for analyzing stock market structure and portfolio behavior introduced by E. Robert Fernholz in 2002. It is descriptive as opposed to normative, and is consistent with the observed behavior of actual markets. Normative assumptions, which serve as a basis for earlier theories like modern portfolio theory (MPT) and the capital asset pricing model (CAPM), are absent from SPT. SPT uses continuous-time random processes (in particular, continuous semi-martingales) to represent the prices of individual securities. Processes with discontinuities, such as jumps, have also been incorporated into the theory. SPT considers stocks and stock markets, but its methods can be applied to other classes of assets as well. A stock is represented by its price process, usually in the logarithmic representation. In the case the market is a collection of stock-price processes X i , {displaystyle X_{i},} for i = 1 , … , n , {displaystyle i=1,dots ,n,} each defined by a continuous semimartingale where W := ( W 1 , … , W d ) {displaystyle W:=(W_{1},dots ,W_{d})} is an n {displaystyle n} -dimensional Brownian motion (Wiener) process with d ≥ n {displaystyle dgeq n} , and the processes γ i {displaystyle gamma _{i}} and ξ i ν {displaystyle xi _{i u }} are progressively measurable with respect to the Brownian filtration { F t } = { F t W } {displaystyle {{mathcal {F}}_{t}}={{mathcal {F}}_{t}^{W}}} . In this representation γ i ( t ) {displaystyle gamma _{i}(t)} is called the (compound) growth rate of X i , {displaystyle X_{i},} and the covariance between log ⁡ X i {displaystyle log X_{i}} and log ⁡ X j {displaystyle log X_{j}} is σ i j ( t ) = ∑ ν = 1 d ξ i ν ( t ) ξ j ν ( t ) . {displaystyle sigma _{ij}(t)=sum _{ u =1}^{d}xi _{i u }(t)xi _{j u }(t).} It is frequently assumed that, for all i , {displaystyle i,} the process ξ i , 1 2 ( t ) + ⋯ + ξ i d 2 ( t ) {displaystyle xi _{i,1}^{2}(t)+cdots +xi _{id}^{2}(t)} is positive, locally square-integrable, and does not grow too rapidly as t → ∞ . {displaystyle t ightarrow infty .} The logarithmic representation is equivalent to the classical arithmetic representation which uses the rate of return α i ( t ) , {displaystyle alpha _{i}(t),} however the growth rate can be a meaningful indicator of long-term performance of a financial asset, whereas the rate of return has an upward bias. The relation between the rate of return and the growth rate is The usual convention in SPT is to assume that each stock has a single share outstanding, so X i ( t ) {displaystyle X_{i}(t)} represents the total capitalization of the i {displaystyle i} -th stock at time t , {displaystyle t,} and X ( t ) = X 1 ( t ) + ⋯ + X n ( t ) {displaystyle X(t)=X_{1}(t)+cdots +X_{n}(t)} is the total capitalization of the market. Dividends can be included in this representation, but are omitted here for simplicity. An investment strategy π = ( π 1 , ⋯ , π n ) {displaystyle pi =(pi _{1},cdots ,pi _{n})} is a vector of bounded, progressively measurableprocesses; the quantity π i ( t ) {displaystyle pi _{i}(t)} represents the proportion of total wealth invested in the i {displaystyle i} -th stock attime t {displaystyle t} , and π 0 ( t ) := 1 − ∑ i = 1 n π i ( t ) {displaystyle pi _{0}(t):=1-sum _{i=1}^{n}pi _{i}(t)} is the proportion hoarded (invested in a money market with zero interest rate). Negative weights correspond to short positions. The cash strategy κ ≡ 0 ( κ 0 ≡ 1 ) {displaystyle kappa equiv 0(kappa _{0}equiv 1)} keeps all wealth in the money market. A strategy π {displaystyle pi } is called portfolio, if it is fully invested in the stock market, that is π 1 ( t ) + ⋯ + π n ( t ) = 1 {displaystyle pi _{1}(t)+cdots +pi _{n}(t)=1} holds, at all times. The value process Z π {displaystyle Z_{pi }} of a strategy π {displaystyle pi } is always positive and satisfies where the process γ π ∗ {displaystyle gamma _{pi }^{*}} is called the excess growth rate process and is given by