I have to price an option with the following characteristics:Let X% = daily variation of S&P (between two consecutive days)The bank pays a fixed coupon daily each day X% > -10%The bank receives (-X% - 10%) if X% < -10%Does anybody know how to price this rare event?Regards,

Off-hand, the essence of the thing is to develop market-consistent pseudo-probabilities for the 'jump'. But the devil may be in thedetails. For example, the structure seems highly dependent onthe circuit breaker rules, and how it handles trading halts, etc.So, you probably need to post an -exact- term sheet.Who is the bank? regards,

i would suggest the myriad of 10,000 year floods in the last 10y are trying to tell you to stip trying to "model" a market operated by beings with emoton.... people are not robots... fear and greed....

While I agree that definitions are important, you can also price this theoretically. There is a reasonably liquid market with strikes 10% below the money and expiries more than a week away. You can put together a rolling portfolio of out-of-the-money puts that will cover your losses and generate fairly small P&L otherwise. It's not hard to estimate those P&L's historically or with a model.You won't get an exact price, because there's no perfect hedge. But you'll get a price such that your hedge will make money with whatever confidence you choose.

Another approach is to fit the tails of the return data to a power law distribution. This model assumes that each increasing increment of price decline has a multiplicative decrease in probability( P(x) = a*x^k or log(P(x)) = k*log(x) + log(a) ). The distribution is useful for self-organizing critical systems (e.g. avalanches and earthquakes) in which an interaction of stochastic effects, accumulated energy and positive feedback can create events of radically different scales.Alan's comment on circuitbreakers & trading halts is germane. Each time the markets crash, people try to impose policies to prevent it from happening again (thus reducing the probability of large declines and crimping the tail of the distribution). Yet one can argue that markets always seem to find a way around such policies. The bank's term sheets would tell you how it would have handled an event of the nature of, but more serious than, 9/11.Also note that curve-fitting to the power law will leave you extrapolating the probabilities of extreme events. Small variations in the best-fit parameters could mean very large variations in expected return from tail events. Your bank's risk methodologies will come into play because the estimated sigma on the extrapolated expected return from the tail will be very large.

I don't think the power-law approach, or any other curve-fitting, will lead to a useful price. You don't care about the theoretical probability of this happening, you care how much it will cost to hedge the contract. You have solid data, both historical and implied forward predictions, for the event; the problem is figuring the chance of it happening in one day versus over a week or a month. That's only happened once in history but, (a) when it did happen, it went all the way to -22% and (b) it happened without obvious news.You're going to hedge this thing with longer-term 10% out of the money options, and write some other longer-term but less out of the money options to make back some of the premium. There's good long-term data on those prices, so you can put together a pretty good hedge, and get a pretty good idea of your residual risk.

QuoteOriginally posted by: marchisiThe bank pays a fixed coupon daily each day X% > -10%The bank receives (-X% - 10%) if X% < -10%This is a very strange set of terms -- the bank agrees to pay a fixed coupon unless there's a force majeur event in the markets as defined by a 1-day drop of greater than 10%. It's almost like the bank owns a automatically rolling/repurchased next-day 10% OTM call option.I do think that Aaron is right about looking at traditional OTM options. Although I'm not sure why the bank would need to hedge this risk (the bank pays out money in the near 100%-probable scenario that 10% events never occur and the bank gains significant money when the rare event happens), options pricing does provide insight. I can see looking at traditional options because this might be how the bank's counterparty looks at this deal (they are the ones accepting the big risk in the event of a serious decline). Speaking of counterparties, I don't know how you price the chance that the counterparty fails to pay the bank because the counterparty is wiped out by this serious event (that's not a low conditional probability in my mind).The one area that seems challenging is in translating prices on longer-term (multi-week) options into a probability/price on a daily event. On the one hand, the chance of a 10% decline with a traditional OTM (which may have weeks of life) is much much higher than the chance of a 10% decline on any given day because a seres of small declines can accumulate to 10% without declining 10% on any one day. (So does one look at the 10%*f(Y) OTM option that has Y days remaining to get the appropriate price of the daily 10% OTM option?). On the other hand the price of a traditional OTM option would tend not to fully price the chance of a one-day 10% decline that then recovers prior to expiration (the bank's situation include automatic exercise of the OTM option for each and every 10% day-to-day decline regardless of recovery on a longer time frame). The relationship between traditional long-expiration options and the bank's daily call option will be a strong function of whether the market is trending or mean-reverting.

You hedge because (a) you don't want to be writing checks every day in hopes of a lottery-ticket payoff one day and (b) you want to be sure you've priced it correctly.I have in mind something like buying 25% at the money one-month puts, writing 50% 5% out-of-the-money and writing 75% 10% out-of-the-money puts. This is crude, of course, and the exact ratios will depend on the prices and expiry of the liquid contracts. There should be a small carrying profit to this position, which would not be affected much by upward movements or small downward movements. In a really big downward movement, where the time value of the options disappears, you will have to pay over your receivable from the contract to settle.In real life, you'd want to use more contracts, and you'd want something managable to rebalance (unless your option book was so big, these just merged in without noticeable effect).