# how would you value this call option

to all the option gurus out there… I am trying to value this call option embedded in another security there are 2 features in the call option that I am not able to incorporate in my blck scholes model - the call can be exercised only if the stock price stays above \$50 for 30 days in a 45 day period. I have historical stock price data for 5 yrs. - the call will be bought now but can be execised only starting Jan 2008 and expires in dec 2010. I quickly did a google search but could not find much. have you come across such an instrument? any ideas on how to value this will be appreciated.

Either build a binomial tree or use monte carlo (which how I’d imagine people who trade these things value them…)

What question are you asking exactly? Are you looking for the risk neutral calculations (the historical stock price data doesn’t have much to do with this)? If so, a) Recognize that your option value = value of an American option expiring in 2010 - value of an American option expiring in 2008. Hence problem is just about valuing an American option with the exceedance conditions. b) Let P = # days stock price > 50 in previous 45 day period. Work out the approximate bivariate distribution of P(T) and S(T) by observing that P(T) | S(T) is approximately normal. c) Payoff of the option = Max (S(T) - K, 0) * I{P(T) > 30}. Proceed as usual with doing the integrals. Unless you are pretty good at this stuff, that’s going to take you awhile and there’s no way I’m doing it. Of course, you could simulate it or use a binomial tree but neither of those would be as chic. Edit: a) might not be exactly true because of the exceedance conditions. On the day that the option becomes active do the previous 45 days count or not?

Also, I bet this is some contingent convertible, yes? If so there might be some other things to think about.

Interesting. I was wondering how this might be done. Instinct was to value the option as an option, then compute the probability of the stock being exercisable because of the 30days out of 45 condition. Value = (value of straight option)*(prob of being exercisable). Joey picked up on the fact that a “start date” for exercisability means that you can compute the value of the option as (american option for 2010)-(american option for 2008). I hadn’t caught that. But since Jan 2008 has passed, you can still just value the straight option as an American option for 2010. But Black Scholes doesn’t give you the value of American options anyway (europeans only), so you may be stuck with Monte Carlo or binary trees. If you’re stuck with monte carlo, you might just take the past 5 year returns and then figure out the probability of the option exceeding the 30/45 conditions from there. You could simply count up the number of 45 day periods where this is true, divide by the number of days and get the probability in the last 5 years that you would have bought something that could potentially be exercised. Or you could use the statistical properties of the last 5 years (avg return, avg vol, maybe add skew and kurtosis), and figure out from there. I don’t think you’re likely to find an analytical solution, given that the first-order underlying is an american style option. So numerical solutions are where you’re at.

Yeah, I guess I switched to European there. It’s a pretty hard problem I think to do the risk neutral analytic thing. I think it’s even difficult to do the Monte Carlo thing because the optimal exercise problem is pretty tough. For high enough S(t) you don’t care about the exceedance condition because you are a long way from it and you are back in the American option = European option. For K sufficiently below 50, you’ve got a nasty dynamic programming problem you need to solve first (“e.g., I exercise in 2008 and collect my S- K = 20 because i have 30 out of 45 and blow away my time value or I hold onto my time value because it is likely that I will have another better stock price and 30 out of 45 exceedances”). I think this is a serious quant problem.

JoeyDVivre Wrote: ------------------------------------------------------- > Also, I bet this is some contingent convertible, > yes? If so there might be some other things to > think about. if it’s a convertible then the COCO structure does not impact the price. it wouldn’t happen to be the BofA convertible preferred?

parmesan Wrote: ------------------------------------------------------- > to all the option gurus out there… > I am trying to value this call option embedded in > another security > there are 2 features in the call option that I am > not able to incorporate in my blck scholes model > > - the call can be exercised only if the stock > price stays above \$50 for 30 days in a 45 day > period. I have historical stock price data for 5 > yrs. > - the call will be bought now but can be execised > only starting Jan 2008 and expires in dec 2010. > > I quickly did a google search but could not find > much. > have you come across such an instrument? any ideas > on how to value this will be appreciated. here’s a quick and dirty estimation of the ballpark option price. the only complication here is the barrier feature (knock-in at \$50 for 30 of 45 days). To estimate the probability of the option becoming active, run a GBM monte carlo on the stock price from option inception and count the number of times that the barrier condition is satisfied. Divide this indicator function over the total simulation, and you will get an approximation of the probability of the option value > 0, or P(A). Once the option is alive, it is then simply an American call. Assume that the stock is non-dividend paying, you can use BS and value it like an European call. Call this V(A). A rough value of the option at inception is then V(A)P(A).

bchadwick Wrote: ------------------------------------------------------- > American option for 2010. But Black Scholes > doesn’t give you the value of American options > anyway (europeans only), so you may be stuck with > Monte Carlo or binary trees. That’s true, except that it is never optimal to exercise an call on a non-dividend paying stock before expiry. In fact, unless the divident paid out is expected to be very substantial, the BS formula can be used to value the American call.

Typical propanol bullshit. Both posts are wildly wrong.

Oh Joey boy can actually bark. Gee… how I miss those earlier days in AF when there are actually people who actually know about stuff in the “trader” way, and chide at posers who pretend they know in the Joey way. Some things never change.

parmesan Wrote: ------------------------------------------------------- > to all the option gurus out there… > I am trying to value this call option embedded in > another security > there are 2 features in the call option that I am > not able to incorporate in my blck scholes model > > - the call can be exercised only if the stock > price stays above \$50 for 30 days in a 45 day > period. I have historical stock price data for 5 > yrs. > - the call will be bought now but can be execised > only starting Jan 2008 and expires in dec 2010. > > I quickly did a google search but could not find > much. > have you come across such an instrument? any ideas > on how to value this will be appreciated. I am a bit confused by your question. You are supposed to value the call but in your last para you said the call will be bought now (meaning it has a market value?). Then why do we need to compute the fair value of the call option. As part of my job, I do valuation of employees stock option which has the following features (vesting period of 3 years where holder cant exercise option, strike price = share price at grant date, upon vesting it has an exercisable period of upto 7 years). We use binomial bec it is american-style option (i.e exercisable after 3 years and justified by auditors but IMO I think it should be european-style bec we have determine an expected useful live for the option which is different from the actual option contract life i.e. 10 years based on 3 years vesting and bal 7 years exercisable period). Key variables and how we determine them (can obtain from Bloomberg): Risk free rate = zero coupon risk free govt bond according to the expected life of the option Expected life = based on the past historical average exercise pattern stock volatility = based on weekly historical share price volatility for the same period according to the expected life of the option Dividend yield = expected dividend yield based on forward dividend and current share price at option grant date From here you can work out the value of the call using software easily.