What are some common models for one-sided returns?

Question

One typically models the log returns of a portfolio of equities by some unimodal, symmetric (or nearly symmetric) distribution with parameters like the mean and standard deviation estimated by empirical estimators. For example, the most basic model would be to assume the returns are normal (yes, I know the tails are not fat enough), then fit the mean and volatility. To get fancier, one might try to fit returns to a fat-tailed distribution like a Tukey g-and-h, or a J-transformation, Lambert W x Gaussian, whatever.

What are the commonly used approaches, I am wondering, for modeling the returns of portfolios of one-sided instruments. For example, a portfolio consisting entirely of long vanilla put and call options. When considered from purchase to expiration, the maximum loss is bounded, but the upside is theoretically unbounded (if you have call options). Fitting to an unbounded and symmetric distribution like a normal seems unsatisfactory. What are the most commonly used models for this? I like simple models as much as fancy ones.

Answer 1

These returns are almost always modeled by finding some fundamental two-sided variable and modeling that. For options, we would model their prices as derivatives -- we would take the log-returns of underlying prices as the fundamental variable, possibly with other models for what would happen to volatilities and the like, and compute the consequences for option prices.

For a vanilla call with expiration $T$, imagine that we assume the volatilities and stock prices follow exponential brownian motion (Ornstein Uhlenbeck would be the more common assumption for vols) up to some intermediate time $\tau$ for which we want to know this call's price distribution.

The distribution of stock prices, based on a gaussian random variable $z \sim N(0,1)$ and ignoring variation on volatility before $\tau$, is

$$ S_\tau \sim S_0 \exp\left(\left(r- \sigma^2/2\right) \tau + z\sigma\sqrt{\tau} \right) $$

and similarly for volatilities with drift $\mu$, variability $\nu$ and another gaussian random variable $w \sim N(0,1)$

$$ \sigma_\tau \sim \sigma_0 \exp\left(\left(\mu- \nu^2/2\right) \tau + w\nu \sqrt{\tau} \right) $$

Define these distributions as $S(z)$ and $\sigma(w)$.

Now given the Black-Scholes formula $BS(S, \sigma)$ we can then construct the distribution of option price at tine $\tau$ from $z$ and $w$ as

$$ V_\tau \sim BS(S(z), \sigma(w)) $$

As you point out, this distribution for $V_\tau$ is very one-sided.

A more interesting one-sided example than options might be treasury bonds, whose value is bounded above. In this case, one typically simulates interest rate curves, but of course it is counterintuitive for them to dip (much) below zero. However, we can select distributions for the interest rates based on processes that, like Black-Scholes for stocks, don't go negative.

In this case, the bond prices are twice removed from the underlying two-sided variables.