Came across am option that is effectively a collar, where the call/ceiling option let's say is $1.5, straight up. The put/floor option lets say is $1.25, fine. However, if the underlying goes below the put option strike ($1.25), the actual intrinsic settlement uses a 3rd rate called the "conditional" strike which lets say is $1.35 (a value between the floor and ceiling) NOT the $1.25. Trying to price this is a meaningful but simple way. It does not have a knockout, barrier, corridor or other exotic type feature. Any thoughts on breaking this down to use a standard-type valuation for the time value?

By "standard-type valuation", I will first assume you mean under a GBM process. All you need is the joint density of (ST,MT): the terminal value and the minimum of the process over (0,T).

That density is known in closed form for drifting BM and hence GBM. (Googling shows it's in Mark Joshi's 'Concepts.." book, for example)

Then, the option value is simply the (discounted, double) integral of that density times the payoff function that you gave in your post. (Do the integral numerically and you're done -- if a GBM value is all you really wanted).

Of course, maybe you'll want to go on from there to a more market-consistent model. It would be nice to have the stand-alone ST density agree with the market's RN density which you can infer from Breeden-Litzenberger, for example. Can you infer a model-independent, market-consistent joint (ST,MT) density? Hard to say -- likely depends on what other types of options are liquid and well-quoted. You may need to calibrate various more complicated models to the vanilla (and other liquid barrier) options, and then do Monte Carlos to price the thing.

That density is known in closed form for drifting BM and hence GBM. (Googling shows it's in Mark Joshi's 'Concepts.." book, for example)

Then, the option value is simply the (discounted, double) integral of that density times the payoff function that you gave in your post. (Do the integral numerically and you're done -- if a GBM value is all you really wanted).

Of course, maybe you'll want to go on from there to a more market-consistent model. It would be nice to have the stand-alone ST density agree with the market's RN density which you can infer from Breeden-Litzenberger, for example. Can you infer a model-independent, market-consistent joint (ST,MT) density? Hard to say -- likely depends on what other types of options are liquid and well-quoted. You may need to calibrate various more complicated models to the vanilla (and other liquid barrier) options, and then do Monte Carlos to price the thing.

It sounds like this is a two leg option: a 1.5 strike call together with a 1.35 strike put with a knockin barrier at 1.25.

Good observation -- simpler than mine and a way to use standard formulas (for GBM).

Alan, this is helpful. Yes, using GBM. So can you elaborate on how I would adjust variables as it relates to N(d1) and N(d2) that you mentioned using a joint density, when we have a put option that has the 2 prices above? To start I'm going to take some type of average and then go from there. Also assuming S(t) will be 1.25 since the moneyness is based on that price. But in reality, intrinsically, the strike is 1.35.

Yes, or like travisf says, it operates like a simultaneous knock-in/knock-out. I was thinking 2 options with a 1.35 strike with a knock-in AND knock-out at 1.25 .

No, I believe travisf is correct. For the put(s) it is entirely (under GBM) a knock-in with a knock-in barrier at 1.25 and strike of 1.35. Under GBM, which is a continuous process (and assuming continuous monitoring), it's impossible for the separate strike of 1.25 to be effective without triggering the knock-in event.

So, just go look up standard formulas for that one.

So, just go look up standard formulas for that one.

Last edited by Alan on February 14th, 2018, 5:19 pm, edited 1 time in total.

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