Long option positions have negative theta and short option positions have positive theta. Stated another way, time decay works against long option positions and benefits short option positions. But there is an exception. Long positions in deep in-the-money (ITM) European puts carry positive theta. While such options may not form part of your typical trading repertoire, it is interesting to understand why these puts are different. This week, we discuss how asset price behaviour explains why deep ITM European puts have positive theta.

Time benefits

Suppose you hold a deep ITM European put; and the underlying remains at the current levels till the expiry of the option. The put will carry positive theta as it approaches expiry. This means the option will gain in value with each passing day. Why?

Asset prices can go up more than they can come down. This is because asset prices have (theoretically) unlimited upside, but cannot go below zero (captured by lognormal distribution). While the upside and downside price movements may be limited for short time periods, prices typically carry an upward bias. 

Now, if you buy a put option and the underlying declines sharply, the option will become ITM. The deeper a put option is ITM, the greater your desire to exercise the option and capture the intrinsic value. Note that selling an option helps you capture both intrinsic value and time value whereas exercising an option helps you capture only the intrinsic value. 

Typically, deep ITM options are less preferred by traders, as such options require large capital outlay. Hence, such options have low liquidity. Therefore, exercising becomes a meaningful alternative to selling the put. But you cannot exercise a European put until expiry. And the more you wait to exercise the option, the more the risk that the underlying can move up, eroding the unrealised gains. Therefore, the faster a deep ITM European put approaches expiry, the better it is for the long position to capture the gains from intrinsic value. Hence, the increase in time value with each passing day or positive theta. This argument does not hold for a deep ITM American put, as you can exercise the option any time before expiry and capture the intrinsic value. 

Managing trades
If asset prices are upward biased, traders who go short on calls must price the risk of the underlying moving up continually
Optional reading

The lognormal distribution of asset prices is also helpful for explaining why deep ITM calls do not have positive theta. Note that the time value of an option consists of time to expiry and implied volatility. If asset prices are upward biased, traders who go short on calls must price the risk of the underlying moving up continually, lest a call becomes ITM and is exercised against them. This is true even if a trader shorts a deep ITM call, as the underlying can move up further and make the call more ITM. This risk is priced into the call through its time value. And this time value must decay as the option approaches expiry.  

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