Aren't options priced asymmetrically for Inverse and Leveraged ETFs, for the same underlying index?

Question

I ask merely about pairs of Bear (Inverse) and Bull (Leveraged) ETFs for the same underlying index. The Bear ETF's price shall be bounded below by 0. But the bull ETF's price is unbounded.

But doesn't this difference — in boundedness — cause an dissymmetry in pricing options on Bear and Bull ETFs? On one hand, they shall be priced the same. Why? Any difference shall be arbitraged away, because you can buy the pricier options on one direction, and sell the cheaper options on the other direction.

On the other hand, the return on the Bear ETF is bounded below by 0. As the bear ETF's price nears 0, its put option shall near its absolute maximum (strike price — option premium). But the bull ETF's price — and its option price — are UNbounded above!

Example With YINN (Daily FTSE China Bull 3X Shares) And YANG (Daily FTSE China Bear 3X Shares).

If the underlying FTSE China 50 Index rallies, then YINN can skyrocket unbounded, so can call options on YINN. But YANG must minimize at 0, so must put options on YANG.

The asymmetry is that YINN call option's price is unbounded. But YANG put option's price shall be bounded and maximized, when YANG's price = 0.

Conversely, if the underlying FTSE China 50 Index crashes, then YINN shall minimize at 0, so can put options on YINN. But YANG can skyrocket unbounded, so can call options on YANG!

The asymmetry is that YANG call option's price is unbounded. But YINN put option's price shall be bounded and maximized, when YINN's price = 0.

Answer 1

First, note that ordinary unleveraged ETFs and stocks also have this behavior -- "bounded below, unbounded above" -- and their options obey put-call parity (by an arbitrage argument). In particular, an at-the-money (ATM) call and put with the same expiration are priced about the same. Essentially, the unbounded upside of long assets is not a free lunch, because the probability of larger and larger gains is lower and lower.

For leveraged ETFs, or indeed any tradable asset, the options must likewise obey put-call parity (by the same arbitrage argument). To understand this in more detail: The frequent (usually daily) ETF rebalancing leads to two opposing effects that cancel out in option prices.

1. In a directional move, the ETF buys on the way up or sells on the way down, increasing returns and maintaining the behavior "bounded below, unbounded above".
2. In a volatile period, the ETF buys high and sells low, decreasing returns -- the leveraged time decay effect.

Together, this again reflects no free lunch: While the maximum loss is bounded, the probability of ultimately achieving (close to) this maximum loss is substantial -- highly leveraged ETFs often head toward zero in the long run.

At a technical level, the Black-Scholes option pricing model predicts (see Equation 5) that an option on a leveraged ETF is valued like an option on the ordinary underlying ETF or index except that the implied volatility is multiplied by the leverage ratio. Thus, as expected, the options on leveraged ETFs obey put-call parity as well, and are symmetric between long and short ETFs.

This explains why the price is the same for ATM calls on $1,000 worth of YINN and ATM puts on $1,000 worth of YANG (and also the same as calls on YANG or puts on YINN).