Can anybody give an example of an option with positive gamma and positive theta? I know that theoretically this is possible. But I simply cannot give an example. By the way, this is a question that I was asked on a phone interview.

Visualize a graph of a euro-style put price vs stock price, esp. near S=0.Now visualize it again closer to expiration.regards,

Alan's example is perfectly righ. This is how I like to imagine it (same as Alan I think?): assume deep in-the-money put, it is optimal to excersise it early, but because European style you can not excerzise before maturity, you have to wait until maturity to get full (non-discounted) instrinct value. Best you can do is to lock it in by close to 100% delta hedge. You can sell option in market, but buyer has same problem and will not pay more than discounted value. However for a big move in asset you can make more than instrinct value so option has positive gamma. If you plot European put value in P&L diagram you will see option value before expiration below exerzise value for deep-in-the money. For American style this will go away because you would excerzise option if value below instrinct.

This all very much depends on how you define theta. I always like to define it as d/dt assuming asset prices follow their forward curves and vols move to forward vols - and that theta doesn't include the costs of financing your portfolio. In that framework a zero coupon bond doesn't have theta, nor does a forward contract.Then under BS, theta really should just offset gamma, and you'll never get theta and gamma with opposite signs.But the world isn't really BS, and there are more sources of nonlinear value than just straight gamma. So in a stoch vol world, for example, you can construct a portfolio that's got positive theta and positive gamma so long as the nonlinear vol pnl terms are negative enough. For example by selling a strangle and buying some ATM options. But in practise it's quite hard to do because the gamma pnl is an order of magnitude bigger than the nonlinear vol terms.

consider a very steep yield curve and a short maturity floor with underlying rate, say 3m... current fixing is around for example 4% and u buy a 4.1% strike floor... if the curve is steep enough u will get positive carry and also positive gamma...