A pricing heuristic for risk applications based on Kelly Criterion

Problem Statement

Consider a systematic seller of insurance contracts. For a given insurance contract, is there a premium heuristic for the seller that on average leads to long term capital growth?


Proposed solution

Consider an LP who contributes an initial amount $C_0$ into a pool of capital and starts systematically selling insurance, achieving a certain random growth rate $g_i$ at each insurance expiration cycle.

Nth capital state $C_N$ can be expressed as:

$$ \begin{align*} C_N = C_0 \cdot (1 + g_1) \cdot (1+g_2)\cdot...\cdot(1+g_N) \tag{1.1} \end{align*} $$

Where $g_i$ are growth samples from growth rate function $G$:

$$ \begin{align*} G(u,p) = \text{Payoff}\Big(S, R, p(u)\Big) \cdot u \tag{1.2}

\end{align*} $$

Where $u$ is the utilization rate of capital $u\in[0,1]$ at each cycle, $p(u)$ is the premium charged as function of $u$, and $\text{Payoff}$ is a financial gain/loss function of the insurance contract at expiry:

$$ \begin{align*}

\text{Payoff}\Big(S,R,p(u)\Big)= \begin{cases} R-S+p(u) & \text{if $R<S$}\\ p(u) & \text{if $R \geq S$} \end{cases} \tag{1.3} \end{align*} $$

Where $S$ is the insurance strike price as percentage of current price , $R$ is the random return of the underlying asset being insured, and $p(u)$ is the premium charged by the LP as a function of utilization.

Nth capital (1.1) can be rewritten in exponential form as:

$$ \begin{align*} C_N = C_0 \cdot \exp\bigg( \sum_{i=0}^{i=N}\ln(1+g_i) \bigg) \tag{1.4}

\end{align*} $$

Per law of large numbers as N tends to infinity the sum of samples over N converges to the expected mean: