Let's say we are pricing a 11 year option on a $10 stock that pays quarterly dividend at $0.25. Let's say the interest rate r=0. In the risk-neutral world, this stock willbe expected to worth zero in 10 years. Let's say we want to value the long-dated option on this stock, how do we do this? Of course the firm hasn't gone bankrupt since the drift is caused by cost of carry. How do we deal with the boundary condition at S=0???

A timely question. This is the classic problem of discrete dividends.A new treatment of this old problem is given in an upcoming Wilmott mag. issue --probably the next issue -- article by Haug, Haug, and Lewis.The S=0 problem is the problem of what to do when thestock price is less than D = $0.25 instantaneously prior toan ex-date. Clearly, you have to modify "something".In the article, we handle this by assumg that the dollardividend actually paid is modfied to some d(S) if S < D on the ex-date.You could choose d(S) = 0, d(S) = S, or any function in between. If you choosed(S) < S, then the company never goes bankrupt under GBM. Whatever youchoose, you get the option price at any time by evaluating the discounted expectationof the future option value -- i.e. its value on a future date. The future date is either thenext ex-date or expiration, whichever comes first. This recursive solutionis pretty easily implemented for GBM, but the same principle works for any othermodel of the stock price evolution.For some more, see the thread "Stock dividends and volatility landscapes".regards,alan

Actually, I think this problem is somewhat different. You are making an extreme assumption: the stock is priced at the net present value of the known dividends. This is only consistent if the stock is known to be worthless at the end of ten years. Therefore you will have problems with any model that assigns volatility to the stock price.To get answers with a reasonable relation to any real stock you would have to allow uncertainty in the dividends or raise the price of the stock or the interest rate.

Thanks for the responses. About stock being NPV of future dividends, if dividend yield is greater than discount rate, then the stock will be worth infinite amount.Then I guess it makes more sense to discount at a risky rate taking into consideration default of S=0. Can we considerS=0 as default? But now let's cosider 2 firms with 2 different dividend policy: firm 1 issues cash dividends whereas firm 2 does share repurchase.In Black-Scholes (risk-neutral) world, firm 1 has finite probability of default where as firm 2 has zero probability of default. But we know that cash dividendsand share repurchases are same thing in tax-free world. So do we have a paradox???

Not sure I see the paradox. Suppose Firm 1 has 4 million shares ourstanding under yourprevious $0.25 dividend policy. Each quarter it distributes $1 million in dividends. Each quarter Firm 2 purchases $1 million market value of shares outstanding and retires them. The two firms, identical in everyway, (world with no frictions, as you said) should maintain the same total market capitalizations (= shares x dollars per share).Now suppose the total market capitalization of each firm falls below $1 million. If they each raise all the cashthey can by selling all their assets, then this scenario must mean that their "cash-on-hand" minus any debt oustanding (= equity) isless than $1 million (just prior to the distribution date). Faced with that, they will both have to modifiy theirstated policies. Their choices are (1) distribute/repurchase less than 100% of their current equity and keep the stock price above 0or (2) distribute/repurchase exactly 100% of the current equity and close up shop. (Caveats. I believe that the total market capitalization must equal the total equity = cash on hand minus debt at thispoint because, having liquidated all their assets, they have no earnings power (above the riskless rate r). Thismarket capitalization total will always be non-negative since shares are limited liability, although it may reach exactly zero. Given the firms balance sheet at this point (cash only, and some debt), I believe no rational lender would loan them more money to make their stated $1 million distribution. This means that the firms cannot take on more debt to maintain the original policy. Iam suggesting that all the caveats actually follow from the 'Black-Scholes frictionless market postulate and limited liability).So ... if both firms make the same policy modication decision at this point, it seems that parity will bepreserved. Unless I am missing something?.

Alan,Thank you for your post, it was very helpful. But one point I seem to be confused is that in the risk-neutral world where we price options, one firm can default while the other cannot. But can we even say that firm 1 can default in risk-neutral world? After all, the drift down by the dividendis caused by the cost of carrying a replicating portfolio, not because the firm actually is in danger of default.

Well, risk neutral evaluation is only valid if the two probability measures (actual and risk-neutral)have the same "null-sets" -- i.e. sets with zero probability. So S=0 must have either (i)strictly positive probabilityor (ii) strictly zero probability in both the 'real-world' and the 'risk-neutral world'. It can't have strictly positive probabilityin one world and zero in the other.

QuoteOriginally posted by: pleeAbout stock being NPV of future dividends, if dividend yield is greater than discount rate, then the stock will be worth infinite amount.No, if the dividend yield is greater than the discount rate and certain to continue forever, then the price of the stock must increase to correct that disequilibrium. But the stock price need not increase to infinity, just to the dividend payment divided by the discount rate. The infinite stock value only arises if the growth rate of dividends is greater than the appropriate risk-adjusted discount rate.There is no difference between Firm 1 and Firm 2. Neither one can default, because neither one has debt. Depending on your assumptions, either firm may find itself unable to raise the cash to fund a dividend or repurchase shares. But if one has the cash the other will.If we assume the failure to make the dividend or repurchase has some dramatic effect on the stock, there could be differences between the two securities from an investor's standpoint. With Firm 1, the investor receives the cash dividend until, at some random point, she gets her pro-rata share of the value of the firm after failure to pay dividends. With Firm 2, the investor receives nothing until that random point, but then gets a larger share of the post-failure firm value.There is no theoretical (and certainly no practical) reason to believe these two stocks would have the same value. Investors cannot convert one to the other costlessly, because you have assumed there is some dramatic firm-event based on the firm's failure.