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 (*unverifiable claim due to missing citation!).
Stocks, portfolios and markets 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
d log X i ( t ) = γ i ( t ) d t + ∑ ν = 1 d ξ i ν ( t ) d W ν ( t ) {\displaystyle d\log X_{i}(t)=\gamma _{i}(t)\,dt+\sum _{\nu =1}^{d}\xi _{i\nu }(t)\,dW_{\nu }(t)}
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 d\geq n} , and the processes γ i {\displaystyle \gamma _{i}} and ξ i ν {\displaystyle \xi _{i\nu }} 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 _{\nu =1}^{d}\xi _{i\nu }(t)\xi _{j\nu }(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\rightarrow \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
α i ( t ) = γ i ( t ) + σ i i ( t ) 2 {\displaystyle \alpha _{i}(t)=\gamma _{i}(t)+{\frac {\sigma _{ii}(t)}{2}}}
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 measurable processes; the quantity π i ( t ) {\displaystyle \pi _{i}(t)} represents the proportion of total wealth invested in the i {\displaystyle i} -th stock at time 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
d log Z π ( t ) = ∑ i = 1 n π i ( t ) d log X i ( t ) + γ π ∗ ( t ) d t {\displaystyle d\log Z_{\pi }(t)=\sum _{i=1}^{n}\pi _{i}(t)\,d\log X_{i}(t)+\gamma _{\pi }^{*}(t)\,dt}
where the process γ π ∗ {\displaystyle \gamma _{\pi }^{*}} is called the excess growth rate process and is given by
γ π ∗ ( t ) := 1 2 ∑ i = 1 n π i ( t ) σ i i ( t ) − 1 2 ∑ i , j = 1 n π i ( t ) π j ( t ) σ i j ( t ) {\displaystyle \gamma _{\pi }^{*}(t):={\frac {1}{2}}\sum _{i=1}^{n}\pi _{i}(t)\sigma _{ii}(t)-{\frac {1}{2}}\sum _{i,j=1}^{n}\pi _{i}(t)\pi _{j}(t)\sigma _{ij}(t)}
This expression is non-negative for a portfolio with non-negative weights π i ( t ) {\displaystyle \pi _{i}(t)} and has been used in quadratic optimization of stock portfolios, a special case of which is optimization with respect to the logarithmic utility function. The market weight processes,
μ i ( t ) := X i ( t ) X 1 ( t ) + ⋯ + X n ( t ) {\displaystyle \mu _{i}(t):={\frac {X_{i}(t)}{X_{1}(t)+\cdots +X_{n}(t)}}}
where i = 1 , … , n {\displaystyle i=1,\dots ,n} define the market portfolio μ {\displaystyle \mu } . With the initial condition Z μ ( 0 ) = X ( 0 ) , {\displaystyle Z_{\mu }(0)=X(0),} the associated value process will satisfy Z μ ( t ) = X ( t ) {\displaystyle Z_{\mu }(t)=X(t)} for all t . {\displaystyle t.}
A number of conditions can be imposed on a market, sometimes to model actual markets and sometimes to emphasize certain types of hypothetical market behavior. Some commonly invoked conditions are:
A market is nondegenerate if the eigenvalues of the covariance matrix ( σ i j ( t ) ) 1 ≤ i , j ≤ n {\displaystyle (\sigma _{ij}(t))_{1\leq i,j\leq n}} are bounded away from zero. It has bounded variance if the eigenvalues are bounded. A market is coherent if lim t → ∞ t − 1 log ( μ i ( t ) ) = 0 {\displaystyle \operatorname {lim} _{t\rightarrow \infty }t^{-1}\log(\mu _{i}(t))=0} for all i = 1 , … , n . {\displaystyle i=1,\dots ,n.}
A market is diverse on [ 0 , T ] {\displaystyle [0,T]} if there exists ε > 0 {\displaystyle \varepsilon >0} such that μ max ( t ) ≤ 1 − ε {\displaystyle \mu _{\max }(t)\leq 1-\varepsilon } for t ∈ [ 0 , T ] . {\displaystyle t\in [0,T].}
A market is weakly diverse on [ 0 , T ] {\displaystyle [0,T]} if there exists ε > 0 {\displaystyle \varepsilon >0} such that
1 T ∫ 0 T μ max ( t ) d t ≤ 1 − ε {\displaystyle {\frac {1}{T}}\int _{0}^{T}\mu _{\max }(t)\,dt\leq 1-\varepsilon }
Diversity and weak diversity are rather weak conditions, and markets are generally far more diverse than would be tested by these extremes. A measure of market diversity is market entropy, defined by
S ( μ ( t ) ) = − ∑ i = 1 n μ i ( t ) log ( μ i ( t ) ) . {\displaystyle S(\mu (t))=-\sum _{i=1}^{n}\mu _{i}(t)\log(\mu _{i}(t)).}
Stochastic stability
We consider the vector process ( μ ( 1 ) ( t ) , … , μ ( n ) ( t ) ) , {\displaystyle (\mu _{(1)}(t),\dots ,\mu _{(n)}(t)),} with 0 ≤ t < ∞ {\displaystyle 0\leq t<\infty } of ranked market weights
max 1 ≤ i ≤ n μ i ( t ) =: μ ( 1 ) ( t ) ≥ μ ( 2 ) ( t ) ≥ ⋯ μ ( n ) ( t ) := min 1 ≤ i ≤ n μ i ( t ) {\displaystyle \max _{1\leq i\leq n}\mu _{i}(t)=:\mu _{(1)}(t)\geq \mu _{(2)}(t)\geq \cdots \mu _{(n)}(t):=\min _{1\leq i\leq n}\mu _{i}(t)}
where ties are resolved “lexicographically”, always in favor of the lowest index. The log-gaps
G ( k , k + 1 ) ( t ) := log ( μ ( k ) ( t ) / μ ( k + 1 ) ( t ) ) , {\displaystyle G^{(k,k+1)}(t):=\log(\mu _{(k)}(t)/\mu _{(k+1)}(t)),}
where 0 ≤ t < ∞ {\displaystyle 0\leq t<\infty } and k = 1 , … , n − 1 {\displaystyle k=1,\dots ,n-1} are continuous, non-negative semimartingales; we denote by Λ ( k , k + 1 ) ( t ) = L G ( k , k + 1 ) ( t ; 0 ) {\displaystyle \Lambda ^{(k,k+1)}(t)=L^{G^{(k,k+1)}}(t;0)} their local times at the origin. These quantities measure the amount of turnover between ranks k {\displaystyle k} and k + 1 {\displaystyle k+1} during the time-interval [ 0 , t ] {\displaystyle [0,t]} . A market is called stochastically stable, if ( μ ( 1 ) ( t ) , ⋯ , μ ( n ) ( t ) ) {\displaystyle (\mu _{(1)}(t),\cdots ,\mu _{(n)}(t))} converges in distribution as t → ∞ {\displaystyle t\rightarrow \infty } to a random vector ( M ( 1 ) , ⋯ , M ( n ) ) {\displaystyle (M_{(1)},\cdots ,M_{(n)})} with values in the Weyl chamber
{ ( x 1 , … , x n ) ∣ x 1 > x 2 > ⋯ > x n and ∑ i = 1 n x i = 1 } {\displaystyle \{(x_{1},\dots ,x_{n})\mid x_{1}>x_{2}>\dots >x_{n}{\text{ and }}\sum _{i=1}^{n}x_{i}=1\}} of the unit simplex, and if the strong law of large numbers
lim t → ∞ Λ ( k , k + 1 ) ( t ) t = λ ( k , k + 1 ) > 0 {\displaystyle \lim _{t\rightarrow \infty }{\frac {\Lambda ^{(k,k+1)}(t)}{t}}=\lambda ^{(k,k+1)}>0}
holds for suitable real constants λ ( 1 , 2 ) , … , λ ( n − 1 , n ) . {\displaystyle \lambda ^{(1,2)},\dots ,\lambda ^{(n-1,n)}.}
Arbitrage and the numeraire property Given any two investment strategies π , ρ {\displaystyle \pi ,\rho } and a real number T > 0 {\displaystyle T>0} , we say that π {\displaystyle \pi } is arbitrage relative to ρ {\displaystyle \rho } over the time-horizon [ 0 , T ] {\displaystyle [0,T]} , if P ( Z π ( T ) ≥ Z ρ ( T ) ) ≥ 1 {\displaystyle \mathbb {P} (Z_{\pi }(T)\geq Z_{\rho }(T))\geq 1} and P ( Z π ( T ) > Z ρ ( T ) ) > 0 {\displaystyle \mathbb {P} (Z_{\pi }(T)>Z_{\rho }(T))>0} both hold; this relative arbitrage is called “strong” if P ( Z π ( T ) > Z ρ ( T ) ) = 1. {\displaystyle \mathbb {P} (Z_{\pi }(T)>Z_{\rho }(T))=1.} When ρ {\displaystyle \rho } is κ ≡ 0 , {\displaystyle \kappa \equiv 0,} we recover the usual definition of arbitrage relative to cash. We say that a given strategy ν {\displaystyle \nu } has the numeraire property, if for any strategy π {\displaystyle \pi } the ratio Z π / Z ν {\displaystyle Z_{\pi }/Z_{\nu }} is a P {\displaystyle \mathbb {P} } −supermartingale. In such a case, the process 1 / Z ν {\displaystyle 1/Z_{\nu }} is called a “deflator” for the market. No arbitrage is possible, over any given time horizon, relative to a strategy ν {\displaystyle \nu } that has the numeraire property (either with respect to the underlying probability measure P {\displaystyle \mathbb {P} } , or with respect to any other probability measure which is equivalent to P {\displaystyle \mathbb {P} } ). A strategy ν {\displaystyle \nu } with the numeraire property maximizes the asymptotic growth rate from investment, in the sense that
lim sup T → ∞ 1 T log ( Z π ( T ) Z ν ( T ) ) ≤ 0 {\displaystyle \limsup _{T\rightarrow \infty }{\frac {1}{T}}\log \left({\frac {Z_{\pi }(T)}{Z_{\nu }(T)}}\right)\leq 0}
holds for any strategy π {\displaystyle \pi } ; it also maximizes the expected log-utility from investment, in the sense that for any strategy π {\displaystyle \pi } and real number T > 0 {\displaystyle T>0} we have
E [ log ( Z π ( T ) ] ≤ E [ log ( Z ν ( T ) ) ] . {\displaystyle \mathbb {E} [\log(Z_{\pi }(T)]\leq \mathbb {E} [\log(Z_{\nu }(T))].}
If the vector α ( t ) = ( α 1 ( t ) , ⋯ , α n ( t ) ) ′ {\displaystyle \alpha (t)=(\alpha _{1}(t),\cdots ,\alpha _{n}(t))'} of instantaneous rates of return, and the matrix σ ( t ) = ( σ ( t ) ) 1 ≤ i , j ≤ n {\displaystyle \sigma (t)=(\sigma (t))_{1\leq i,j\leq n}} of instantaneous covariances, are known, then the strategy
ν ( t ) = arg max p ∈ R n ( p ′ α ( t ) − 1 2 p ′ α ( t ) p ) for all 0 ≤ t < ∞ {\displaystyle \nu (t)=\arg \max _{p\in \mathbb {R} ^{n}}(p'\alpha (t)-{\tfrac {1}{2}}p'\alpha (t)p)\qquad {\text{ for all }}0\leq t<\infty }
has the numeraire property whenever the indicated maximum is attained. The study of the numeraire portfolio links SPT to the so-called Benchmark approach to Mathematical Finance, which takes such a numeraire portfolio as given and provides a way to price contingent claims, without any further assumptions. A probability measure Q {\displaystyle \mathbb {Q} } is called equivalent martingale measure (EMM) on a given time-horizon [ 0 , T ] {\displaystyle [0,T]} , if it has the same null sets as P {\displaystyle \mathbb {P} } on F T {\displaystyle {\mathcal {F}}_{T}} , and if the processes X 1 ( t ) , … , X n ( t ) {\displaystyle X_{1}(t),\dots ,X_{n}(t)} with 0 ≤ t ≤ T {\displaystyle 0\leq t\leq T} are all Q {\displaystyle \mathbb {Q} } −martingales. Assuming that such an EMM exists, arbitrage is not possible on [ 0 , T ] {\displaystyle [0,T]} relative to either cash κ {\displaystyle \kappa } or to the market portfolio μ {\displaystyle \mu } (or more generally, relative to any strategy ρ {\displaystyle \rho } whose wealth process Z ρ {\displaystyle Z_{\rho }} is a martingale under some EMM). Conversely, if π , ρ {\displaystyle \pi ,\rho } are portfolios and one of them is arbitrage relative to the other on [ 0 , T ] {\displaystyle [0,T]} then no EMM can exist on this horizon.
Functionally-generated portfolios Suppose we are given a smooth function G : U → ( 0 , ∞ ) {\displaystyle G:U\rightarrow (0,\infty )} on some neighborhood
U {\displaystyle U} of the unit simplex in R n {\displaystyle \mathbb {R} ^{n}} . We call
π i G ( t ) := μ i ( t ) ( D i log ( G ( μ ( t ) ) ) + 1 − ∑ j = 1 n μ j ( t ) D j log ( G ( μ ( t ) ) ) ) for 1 ≤ i ≤ n {\displaystyle \pi _{i}^{\mathbb {G} }(t):=\mu _{i}(t)\left(D_{i}\log(\mathbb {G} (\mu (t)))+1-\sum _{j=1}^{n}\mu _{j}(t)D_{j}\log(\mathbb {G} (\mu (t)))\right)\qquad {\text{ for }}1\leq i\leq n}
the portfolio generated by the function G {\displaystyle \mathbb {G} } . It can be shown that all the weights of this portfolio are non-negative, if its generating function G {\displaystyle \mathbb {G} } is concave. Under mild conditions, the relative performance of this functionally-generated portfolio π G {\displaystyle \pi _{\mathbb {G} }} with respect to the market portfolio μ {\displaystyle \mu } , is given by the F-G decomposition
log ( Z π G ( T ) Z μ ( T ) ) = log ( G ( μ ( T ) ) G ( μ ( 0 ) ) ) + ∫ 0 T g ( t ) d t {\displaystyle \log \left({\frac {Z_{\pi ^{\mathbb {G} }}(T)}{Z_{\mu }(T)}}\right)=\log \left({\frac {\mathbb {G} (\mu (T))}{\mathbb {G} (\mu (0))}}\right)+\int _{0}^{T}g(t)\,dt}
which involves no stochastic integrals. Here the expression
g ( t ) := − 1 2 G ( μ ( t ) ) ∑ i = 1 n ∑ j = 1 n D i j 2 G ( μ ( t ) ) μ i ( t ) μ j ( t ) τ i j μ ( t ) {\displaystyle g(t):={\frac {-1}{2\mathbb {G} (\mu (t))}}\sum _{i=1}^{n}\sum _{j=1}^{n}D_{ij}^{2}\mathbb {G} (\mu (t))\mu _{i}(t)\mu _{j}(t)\tau _{ij}^{\mu }(t)}
is called the drift process of the portfolio (and it is a non-negative quantity if the generating function G {\displaystyle \mathbb {G} } is concave); and the quantities
τ i j μ (