June 25th, 2015, 3:58 pm
QuoteOriginally posted by: acastaldoA stock is at 100, both you and daveangel buy an ATM 6-month put, but you buy a European put and daveangel an American put.The next day a terrible event happens and the stock plunges to 1 a share. Analysts predict that the price 6 months hence will be between 0.75 and 1.25 per share. Both put holders are overjoyed.But daveangel is in a better position, he can exercise his put now and get his cash immediately, while you have to wait 6 months. You have a piece of paper that will be worth about 99 in 6 months, so it is very much like a T-bill or a zero coupon bond. You can go to a discount house and they will give you the present value now 99*exp(-rt), but not the full amount. As time goes on the PV gradually increases to 99. So the put is gradually appreciating, just like a T-Bill or ZCB.Fantastic explanation! Just that the European holder is not as disadvantaged as it seems. He can try to sell the option on the market instead of waiting for maturity.I would also add we can think of this negative time value (positive theta) in 2 different angles.One is to think in terms of cost of carry. The higher the cost of carry, the more rational it is to dump the option and convert to stock.For calls, this occurs when dividends on the underlying stock becomes extremely attractive. Since call holders are not entitled to the dividend, the simple act of missing out on the dividend is a 'cost' to the 'carry'.Cost of carry becomes too high when dividends > interest rate. Interest rate is the money saved from not holding stock positions using borrowed cash.For puts, cost of carry is opposite, i.e. interest rate > dividend. If I long a put, and the interest rate is sufficiently high, it is more rational to sell the put (or exercise the put), and put the money in the bank. The simple act of missing out on the interest is that 'cost' to the 'carry'. Or we can think of it as a collateralised derivative. Having to pay prevailing rate on the collateral put up by counterparty which is higher than the embedded intrinsic value of the option--as dave has pointed out.The other effect is the limited liability property of stocks, i.e. they cannot go below 0! A drastic plunge from 100 to 1 may favour the put, but he has to realise the profits as soon as he can to reap any material benefits. There is very little upside (when stock goes to 0) and a whole lot of downside (when stock shoots back to 100). This again reflects the positive theta. This effect is much more dominant in puts than in calls. Therefore puts are much more likely to have positive theta than calls.You can extend this to digital options, where past the strike, there is nothing much else to gain, and a whole lot to lose in the event that stock moves below the strike again. This effect is equally strong in both digital calls and puts.Hope that helps!
Last edited by
karfey on June 25th, 2015, 10:00 pm, edited 1 time in total.