Suggested by plessas. This should be an easy one! Explain it, when does it work, when not, assumptions...P

well to get this started ...Put-call parity is a strong arbitrage relation between Euro-stylecall prices C and put prices P with the same strike price K. It states that C - P = S - PV(Dividends) - K e^(-r T),where S is an underlying stock price, T is the option holding period, r is a continuously compoundedinterest rate and PV(Dividends) is the present value of thedividends received by the stock owner over the holding period. The only assumptions needed to derive this relation are that: (i) there is a constant interest rate environment for both borrowing and lending over the option holding period, and (ii) the dividends to be received are known and certain.To derive the relation, you construct riskless borrowing/lendingtrades with options. The riskless lending trade, termed a conversion,is to buy the stock, sell a call, and buy a put option with the samestrike. Regardless of the terminal stock price, you will receive atexpiration: K plus the future value of the dividends recieved. Soyour investment must grow at the rate e^(r T) and that yields therelation. Similarly if the riskless borrowing rate is also r, then thecorresponding (credit) trade is the reversal: short sell the stock, buya call, and sell the put option. A careful empirical study the put/call parity relation was performed byKamara and Miller: "Daily and Intradaily Tests of European Put-CallParity" (JFQA, 30, 1995, No. 4, 519-539). They found that, in practice,there are many small violations. But in investigating all violations in intradaytransaction data, they found that almost half of the "arbitrages" resultin a loss when execution delays are accounted for. They alsofound that the mean ex post profit in trying to exploit the violations wasnegative.The relation doesn't hold for American-style options.

Alan, you can drop the constant interest rate assumption by replacing the last term with the value of a zero-coupon bond paying K at the option maturity...should such a thing exist!P

Hi Paul,Ok, good modification (if the holding period is long and there are lots of large intermediate dividends, I guess a whole bunch of different forward/strip trades would be needed??).

QuoteOriginally posted by: AlanThe relation doesn't hold for American-style options.Why?

The relation doesn't hold for American-style options, whichallow an early exercise prior to expiration. For example, oneof the options legs in the conversion trade may disappearprior to expiration because of an exercise/assignment. Closingthe whole trade at this point produces a gain/loss that isunknown when the conversion is initiated. Not closing the trade leaves a risky position.

I'd add that put-call parity relies on the existence of either a liquidly tradeable underlying or forward contract.MJ

mj,good point worth adding. In fact, perhaps we could also add:Gross violations of put-call parity are sometimes seen inthe listed options market. These are almost always associatedwith corporate actions that have been announced, but will notbe completed, until sometime prior to the option's expiration. For example, gross violations may be associated with an outstanding partial tenderoffer for the stock, or other actions where the underlying securitymay change character, or be difficult to borrow.

QuoteOriginally posted by: AlanThe relation doesn't hold for American-style options, whichallow an early exercise prior to expiration. For example, oneof the options legs in the conversion trade may disappearprior to expiration because of an exercise/assignment. Closingthe whole trade at this point produces a gain/loss that isunknown when the conversion is initiated. Not closing the trade leaves a risky position.I don't share this view. Any early assignment will only speed up the arbitration game in the trader's favour... If one is careful in maintaining margin adequately below a margin call limit, there should be no problem.

Here we go again ... Apparently the failure of put-call parity for American optionsis controversial (see the thread on this in the General forum). My suggestion is for Paul to open a new FAQ or maybe two new FAQs entitled"Why does put-call parity fail (hold) for American-style options?"and you guys can battle it out. Otherwise, this one may never end

Or ...It can be proved under essentially the same weak assumptions thatthe above put-call parity relation does not hold for American-styleoptions. Here is a rough outline of the proof. Consider the case of a non-dividend payingstock and a strictly positive interest rate. Then, a similar arbitrage argument to the above shows that for aeuropean-style call option with price c, it must be true that c > max[0, S - K e^(-r T)].Since an American-style call C is always worth at least as much, C >= c > S - K.But if C > S - K, then the option will never be exercised, so C = c.That is, an American call has the same value as aEuro-style call when there are no dividends and early exercise for it isnever optimal. But, under the same circumstances, things are quite different for the put.An American-style put is worth strictly more thanits Euro-style counter-part. To prove it, assume otherwise. Then,since a perpetual Euro-style put is easily shown to be worth zero, a perpetual American-style put must be worth zero. But this is a nonsense conclusion since an American style put must not decrease in value as the time to expiration increases. The premise must havebeen wrong, so under a no-dividend assumption, the American-style put value P isstrictly greater the Euro-style value p. Hence P > p = C - S + K e^(-r T).In words, the American-style put value is *strictly larger* than the value given by the put-call parity relation. For more details see: R.C. Merton, "Theory of Rational Option Pricing",(Bell J. of Economics and Mgt. Science, 4, 1973, 141-183.)

QuoteOriginally posted by: AlanBut, under the same circumstances, things are quite different for the put.An American-style put is worth strictly more thanits Euro-style counter-part. To prove it, assume otherwise. Then,since a perpetual Euro-style put is easily shown to be worth zero, a perpetual American-style put must be worth zero. But this is a nonsense conclusion since an American style put must not decrease in value as the time to expiration increases.Interesting proof, but it is wrong, or at least has so many holes...1) If p->0 for t->oo, it's also the same for c: c->0 for t->oo (why? because you can't exercise the euro-call option with a strike at t=oo) 2) much more interesting, the method to prove P>p uses an hypothesis that is unreal (t.exp=oo), while the P-C parity is an arbitrage relation, and useable only for "real" cases (eg: you can't arbitrage a theoric irrealistic case). It's equivalent to consider a system where t=vector t = [0,t] + [1,0] for any t3) The P-C parity is an arbitrage relation, that means you must impose it, when you think you're able to impose it for a profit, but you can't expect that others will impose it for you when you need it. And if P>p you can impose it always for less risk than imposing the one for p /early exercise on the short side is only a favor to you/4) The P-C parity represents the convenience to enter in a synthetic portfolio, for real cases, and for real times (arbitrages times are of the order of months). In these cases the P-C parity stands especially for American options, while it is also valid for European Options (that are less good for arbitrages, given their long times to close). Eg: it's convenient to enter in the synthetic portfolio with american options instead of european.So, what are the results from the above?1) Is that the P-C parity always stands for american and european REAL options.2) It's always convenient to use american options to impose the P-C parity, why because the european options will follow or will create other arbitrage opportunities.

QuoteOriginally posted by: Alan It states that C - P = S - PV(Dividends) - K e^(-r T),I would also add that I'm used to look at P-C parity, in a way that I consider a lot more comprehensible for real time trading:Calling1) C(t) and P(t)= prices of Call and Put2) intrinsic value = Iv 3) premium= Pr(t)4) interest to get= R5) PV(div)=present value of dividends to pay ;-)it'easy to derive the P-C looking at values (where are /or should be/ values?)ok, it's easy to see that...........this is P-C for me...................C(t)=Iv(call) + Pr + RP(t)=Iv(put) + Pr + PV(dividends).........................................................

gerico, I suspect you have access to an option value calculator. Try this:enter no dividends for the call, a positive interest rate, and calculate values for both aEuro-style and American-style call with the same terms. Youget the same number, right?Now try the two puts. The American-style put value is greater thanthe Euro-style value, yes? If you agree, then you might thinkabout that. That's another proof for you, usiing a "real world" expiration.

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