A Relative-Entropy Bound on Log Growth from Derivative Payoffs

# Information-theoretic upper bound on portfolio log-growth


# Information-theoretic upper bound on portfolio log-growth












I recently formulated and successfully solved a simple model portfolio optimization problem. I'm not a mathematician, but the result seems to have close ties to information theory. I've tried searching for texts that cover this result, and while it has been proven in other contexts (see chapter 16.4 of Elements of Information Theory Second Edition by Cover & Thomas), I haven't found anything that frames the result the way I have and discusses the implications on portfolio theory. My primary question is: is this a well-known result, and where can I find more discussion about it?

The problem is as follows: Assume we have a market that consists of one riskless asset with risk-free rate $r$, and a risky asset $S$, as well as any possible derivative securities of $S$. We will model $S$ as a continuous variable, but over a discrete time step, with initial time $0$ and final time $T$. We assume there is a known real-world probability distribution function $P(S_T)$ of the value of $S$ at time $T$. We also assume that enough derivatives of $S$ exist in the market such that we can construct a portfolio with an arbitrary payoff function $V(S_T)$. Assuming the no-arbitrage condition holds for $V(S_T)$ and that the portfolio is self-financed (range $y>0$), what is the maximum expected geometric growth rate $G$ of the portfolio from time $0$ to $T$?

I solved the problem like this:

From the fundamental theorem of asset pricing, we know that there must exist some risk-neutral measure $\Bbb{Q}$ under which every portfolio yields the risk-free rate. We assume that both likelihood ratios $\frac{P(S_T)}{Q(S_T)}$ and $\frac{Q(S_T)}{P(S_T)}$ are well-behaved. Starting from the definition of $G$:

$\begin{equation}\begin{aligned} G&\stackrel{\text{def}}{=}\space\Bbb{E}^P[logV(S_T)]\\[8pt] &=\space\Bbb{E}^P\left[log\left(V(S_T)\frac{Q(S_T)}{P(S_T)}\frac{P(S_T)}{Q(S_T)}\right)\right]\\[5pt] &=\space\Bbb{E}^P\left[log\left(V(S_T)\frac{Q(S_T)}{P(S_T)}\right)\right]+\Bbb{E}^P\left[log\frac{P(S_T)}{Q(S_T)}\right]\\[5pt] &\stackrel{\text{def}}{=}\space\Bbb{E}^P\left[log\left(V(S_T)\frac{Q(S_T)}{P(S_T)}\right)\right]+D_{KL}(P\Vert{Q})\\[5pt] &\stackrel{\text{(a)}}{\leq}{log}\space\Bbb{E}^P\left[V(S_T)\frac{Q(S_T)}{P(S_T)}\right]+D_{KL}(P\Vert{Q})\\[8pt] &\leq\space{log}\space\Bbb{E}^Q\left[V(S_T)\right]+D_{KL}(P\Vert{Q})\\[5pt] &\stackrel{\text{(b)}}{\leq}\space{rT}+D_{KL}(P\Vert{Q}) \end{aligned}\end{equation}$

Where $D_{KL}(P\Vert{Q})$ is the Kullback–Leibler divergence, (a) follows from Jensen's inequality, and (b) follows from the fundamental theorem of asset pricing.

From that, it's not hard to find the payoff function of the optimal portfolio $V^*(S_T)=e^{rT}\frac{P(S_T)}{Q(S_T)}$ which when substituted into the definition of $G$ gives $G^*={rT}+D_{KL}(P\Vert{Q})$

This seems like an extremely useful result which shows that you can extract optimal log-growth from any risky asset without continuous rehedging (given a developed enough derivatives market), and that the geometric growth rate depends only on the relative entropy between the real-world and risk-neutral probability distributions, without invoking stochastic calculus or geometric Brownian motion. Could anyone point me to any discussion about this?