Early Exercising

This is somewhat similar to the previous post on theta. When is it optimal to exercise an American option early?

- For call, you never exercise early if the underlying pays no dividend; reason being that for a call option, early exercise means we are to pay cash earlier. Considering the time value of money, early exercise is not as good as the alternative strategy of shorting the asset. If it does pay dividend, however, early exercise may be optimal just before a dividend is paid out.
- For put, things are more messy....If the option is deep in the money, then the asset price movement is not as a dominant factor as the time value of money because by early exercise we receive cash early -> early exercise is likely to be feasible.

Summary:
Exercise Am. call early if the call is deep ITM and dividend yield is high
Exercise Am. put early if the put is deep ITM and rate is high

http://faculty.chicagobooth.edu/robert.novy-marx/teaching/35100/Lectures/lec05.pdf

When is theta positive?

Considering European options, theta may be positive
- for deep in-the-money call on an underlying that pays high dividend
- for deep in-the-money put on non-dividend paying underlying

Note the asymmetry between call and put. The max. payoff of a call is unlimited while the max. payoff of a put is capped at K. Hence if you are holding a put which is already reeeeeeally in-the-money, then all you wish is for time to pass quickly because allowing the stock to have more time to diffuse can be bad for you.

Treasury Yield Curve, LIBOR and Swap Curve

Good reference: http://www.scribd.com/doc/34990081/Federal-Reserve-Bank-of-Cleveland-Haubrich-Swaps-and-the-Swaps-Yield-Curve

In short, T-yield curve and swap curve are similar things. In fact they usually move together. T-yield curve is on a riskless(yea, right) rate, while swap curve is on a nearly-riskless(because of netting of vanilla swaps and also because of collateral requirements) rate. Most vanilla swaps, however, take LIBOR(a risky rate because no collateral is required) as a reference rate in pricing the floating leg.

Question: What is the relationship between swap curve and LIBOR curve?

Answer: WSJ presents the LIBOR-Swap curve - the short end of the curve (<1yr) is the LIBOR rate, while the long end (up to 30 yr) is the swap rate.

Gamma Hedging

Just like implied volatility smile(surface!) is similar to yield curve, gamma hedging is also very similar to bond immunization. Derivative value is locally linear, which can be hedged by delta hedging. To improve the hedge, one can take the second-order correction into account, which is exactly what gamma hedging is. While delta hedging uses the underlying security, gamma hedging must use derivatives. This can be a problem: you wrote an option, you want to hedge the risk; but you need more options to do that!