Dollars and Jens

Friday, April 30, 2004

Re: put-call parity

My brother, via email, points out that the facts of the article on put-call parity to which he linked the other day glossed over a couple of important things, specifically dividends and the cost of capital. For example, the article says:

Finance theory states that, holding the underlying stock and buying a put will deliver the same payoff as buying a call and investing the present value of the exercise price.This includes the cost of capital, but ignores the dividends.

Let's compare the strategies;

`S`is the present stock price,

_{0}`S`is the future stock price,

_{t}`D`is the future value of any dividend payments between now and the expiration of the options,

_{t}`X`is the exercise price,

`X`is the present value of

_{0}`X`(i.e., the amount you would have to pay now for a risk-free bond that pays you

`X`on the date the options expire),

`P`is the present price of a put, and

`C`is the present price of a call.

Strategy | present cost | future value if S_{t}>=X | future value if S_{t}<=X |
---|---|---|---|

Buy the stock and a put | S_{0}+P | S_{t}+D_{t} | X+D_{t} |

Buy a call, invest X_{0} | C+X_{0} | (S_{t}-X)+X, i.e., S_{t} | X |

Regardless of the future stock price, the "stock-plus-put" strategy ends up worth more by D

_{t}. If we define

`D`to be the

_{0}*present*value of the dividends, this stock-plus-put strategy should cost D

_{0}more than the call-plus-bond strategy. I.e.,

_{0}+P=C+X

_{0}+D

_{0}

_{0}-X

_{0}-D

_{0}

A simplified condition of this is where the strike price of the put and call is the same as the current underlying ETF price. In such a case,This is true only if -- and only if -- the dividend paid by DIA equals the risk-free rate (in DIA's case, I wouldn't be surprised if they were close). If you buy a share of DIA and a put, you get the dividends paid by DIA in between now and the maturity date of the option, so this strategy is more attractive when you consider the dividends. If you buy a call instead, you don't get the dividends, but you also don't have to put up much money -- you could invest the money in a treasury bond instead, and collect risk-free interest, so this makes the call option more attractive.

Put = Call at same strike and maturity

For example, as long as the Diamonds Trust trades at 104, the call option to buy it at a price of 104 anytime between now and the date of options expiration should cost the same as the put option to sell it at 104. A brief glance at an options price screen will confirm this. When on April 22, DIA traded 104, the June 04 put and call options both cost just over $2, the January 05 put and call both cost about $5, etc.

In terms of the formula (C-P=S

_{0}-X

_{0}-D

_{0}), C=P not when S

_{0}=X, but when

_{0}=X

_{0}+D

_{0}

As it happens, the bond ETF to which the author refers carries a dividend of 5.5%, which is higher than any reasonable risk-free rate between now and January 2006. Even if shorting were possible, the cost of an at-the-money put would probably exceed that of an at-the-money call. I don't care to look up the cost of a risk-free January 2006 zero-coupon bond, and I can't do a precise calculation without it. That said, I suspect the difference between the put and the call would work out to something less than $6, but certainly far more than $0.