hi, came across hard to borrow stocks. when you are long this stock- broker pays you interest, when you short it- you pay interest to broker. question- should option pricing models for such stocks be adjusted? what if riskfree rate<borrowing rate, so should one use negative rate in the pricing models?appreciate any comments

The general answer is yes, the theory changes, and I believe all thesevariations have been worked out, although I don't have references at myfinger tips. One effect of constraints is that the Black-Scholes price canturn from a unique arbitrage-free value to an arbitrage-free bound.For example, take a world in which the stock follows GBM but you can'tshort it (your first example). But you can own it in any amount.Then, you can still replicate a long call by dynamic trading in thestock, long only. If you see a call selling in the market for more thanthe regular BS price, you can sell it and capture the price difference exactly.But, if the market price is less than the BS price, you're stuck, so theBS price becomes merely an arbitrage-free upper bound.regards, p.s. -- even 'worse' is that the inability to short can mess up strong,model-independent relations like put/call parity. Every now and then therewill be some take-over deal in which you will see some gross put/callparity violations -- I vaguely remember such an effect with a palm computerspin-off, for example.

tanya - it's exactly as alan describes - just thought i'd add that the interest rate is unaffected in your calculation. you would shift the "dividend yield" upward by the stock borrow so that the forward goes down; calls get cheaper and puts get more expensive.the interest rate is also used for discounting, so you don't want to shift that.on the other hand, if your european model is all set up for fixed div amounts and there is no easy way to shift the yield, you can still add in stock borrow.what you do is shift the fixed dividend amount by whatever amount you need, such that you get the forward price consistent with stock borrow.that is, F = S * exp[ (r-y-b)*T ] where "b" is the borrow rate and y is the div yield.you get the forward implied by your div/borrow assumptions by using your call and put prices:F = (C - P) * exp(r*T) + Kfor a binomial tree you should adjust the divs so you get the right forward at every binomial step.even better is if you can alter the code to take into account stock borrow correctly, but that isn't always possible!

this isn't a technical comment on your assumptions, but a practical one--you assume a different interest rate on your debit/credit spread because you're long a hard to borrow...you are not likely receive interest as the holder of HTB stock paying negative rebates unless you're a unique client at wherever you're holding that stock...typically the broker who's lending it is the one who will pocket the differerence between your FF+ margin rate and whatever they are being paid to lend it

note that in the US, you borrow stock and short it with the same guy, and get "the rebate", which is equal to "rate - borrow".in europe you borrow the stock from one guy, pay him the borrow rate, and you might short the stock the next day through someone else.your economics are the same, just a difference of terminology. if trading in the US i would work out what the implied borrow rate would be, and use that in place of "b" in my analysis.but you shouldn't use "the rebate" (which could be negative) as your interest rate in a black scholes calc.you don't discount the option payoff at the rebate, right?we are probably saying the same thing in different ways.

you all gave me important comments. I need time to think on all of this, thanks a lot! .quick question: so market makers who quote options on such stocks- they may not have stock positions and not geting any interest for being long or paying being short, but do they typically adjust these options pricing to borrowing rates as well? it looks clear for me that if I am long stock and getting extra interest for that, I would pirce my calls cheaper and puts higher, but what if I don't have stk position.

If stock borrow worked in a way that you were guaranteed you could lend your longs out and get a sensible credit, then the answer is yes, you would adjust the prices of both puts and calls by the stock borrow rate, as per my description.In reality what tends to happen is that the equity derivatives books get long of stock, the stock loan departments lends it out and keeps the money, claiming it as its PNL! In this case the option market maker won't be getting paid for lending out the stock (even though the market maker's company gets paid) so will not price call options more cheaply.However he would be charged the full amount of stock borrow if he borrowed stock, so he would price puts higher.Anecdote: at one bank i worked in, the CB department was paying a ton to borrow stock i was long of in my equity derivs book. The stock loan people snatched my stock and lent it to the CB desk and claimed this as PNL. So we started doing equity swaps between the equity derivs and the CB desk. The stock loan people indignantly squawked about us doing their job and demanded both desks pay them money because of business they were losing!! (We didn't pay up).

erstwhile: what if I just enter into options models (r-b) instead of (y+b)? my issue that currently I don't have special input for "b" rate into models and want to find out way how do w/o changing inputs much and w/o adding new inputs.so, one way as you higlighted would be enter (y+b) instead of usual (y), however as you correctlt mentioned, divs are not always used as yeild, but are entered as stream for equities. what if instead of adjusting each amount in this stream, I just enter (r-b) instead of (r) , leaving dividends inputs as they are. I guess divs will be adjusted by (r-b) as well further. Does this make sense as well?so, if this can make sense - sometimes (r-b) may require entering negative value if b>r , correct?

tanya - if you use (r-b) for the rate, you will get the right forward, but you will discount at the wrong rate.but what you can do is correct for this!let's assume your r and b are continuous rates.in that case you want to "undiscount" at (r-b) so you multiply by exp[ (r-b) * T].then you want to discount at the correct rate, so you multiply by exp(-r*T)so overall you are multiplying by the factor exp(-b*T).in summary, yes you can do this, as long as you multiply the overall result by exp(-b*T).-----------------------------------------------note that i woundn't do this with american options, as you don't know "T".but there is a good "old-school trick" that you might try if need be.the ratio of vega/gamma is vega/gamma = vol* S^2 * Tso T = vega/(gamma*vol* S^2)check it out for a european option first - depending on your definition of gamma you may have other factors of spot or 100 in there.now calculate the right hand side of that equation for an american optionyou will find that if there is no early exercise you get T=maturity. if it is an american call option that is highly likely to be exercised just before the dividend, you will find that T = dividend date!this would be a better value to use in the exp(-b*T) correction factor, which by the way will flow through to all your greeks...i "discovered" this T trick, and then used it to adjust the interest rate hedges of my american options back in the early 1990s before we had modern-type models that used the entire yield curve in the binomial tree.i called this T thing "fugit" (i.e., tempus fugit) to make fun of "vega" which is not a greek letter but is the name of a star...i also later heard that someone else had a time-related greek they called "fugit" as well - not sure hat it was for...

Hi All, A related question about index options. Should the borrow costs of the constituent components of the index have any affect on index options as well, given the trader should be using futures instead, and most exchanges would allow long and short futures. I do see index traders put in the borrow cost, maybe this is for any additional futures transaction cost of shorting the futures?

of course you should. if you are trading index arb you would ? Also, you may be short Dec options and hedging delta with jun futures. you have to account for the jun/sep sep/dec rolls as well otherwise you will be arbed by the market. if there are persistently high borrow costs and you dont account for them calls will be "cheap" and puts expensive so you will also be buying forwards and shorting futures and probably losing money on the roll.

To reply to 2fingers, it really depends on the construction of the index, but I would think not. In the no-arb world, the hard to borrow friction should be priced into the future.

QuoteOriginally posted by: MikeNNTo reply to 2fingers, it really depends on the construction of the index, but I would think not. In the no-arb world, the hard to borrow friction should be priced into the future.Are you saying that you should not adjust the forward price to account for borrowing costs ?

QuoteOriginally posted by: daveangelQuoteOriginally posted by: MikeNNTo reply to 2fingers, it really depends on the construction of the index, but I would think not. In the no-arb world, the hard to borrow friction should be priced into the future.Are you saying that you should not adjust the forward price to account for borrowing costs ?Assuming that he is he is continuing the OP's question he is asking if you should adjust if a stock's borrow changes (perhaps it became hard to borrow). Assuming you're using the index's future as the underlying, there should be no adjustment needed. Perhaps I misunderstood his question.Secondly futures have no cost of carry. However, there is a cost of carry to options.Hopefully that's better stated.

QuoteOriginally posted by: MikeNNQuoteOriginally posted by: daveangelQuoteOriginally posted by: MikeNNTo reply to 2fingers, it really depends on the construction of the index, but I would think not. In the no-arb world, the hard to borrow friction should be priced into the future.Are you saying that you should not adjust the forward price to account for borrowing costs ?Assuming that he is he is continuing the OP's question he is asking if you should adjust if a stock's borrow changes (perhaps it became hard to borrow). Assuming you're using the index's future as the underlying, there should be no adjustment needed. Perhaps I misunderstood his question.Secondly futures have no cost of carry. However, there is a cost of carry to options.Hopefully that's better stated.what if your option is cash settled and expires in Dec 2011 and you are hedging delta with Mar11 futures and there was a persistent borrow cost of 1% across the board and the roll always trade below fair value ?