Hello Fellow QuantsIf someone could point me in the right direction it would be highly appreciated.I'm trying to come up with an Option Pricing Model with a closed form solution for underlying securities that have a cap and a floor. Unlike regular Black Scholes (which assumes returns are lognormally distributed, with a floor price of zero and a price cap of infinity), certain underlyings have a non zero floor and well defined cap price. I would like to build a pricing model which uses a modified distribution accounting for caps and floors and then returns a closed form solution.This is for energy markets.Thanks for the help!

You need to be specific. What are the underlyings? Are they tradeable? Presumably not, but -- if so -- what prevents an arbitrage when they hit their barriers? Post a chart which shows their behaviors.

Hi AlanThanks for the interest.Yes they are. It's a case where there is a floor at which you can buy the underlying (electricity or gas) in the spot market. So no one would sell a forward at below the spot floor because they would be selling below what they can buy in the spot market (arbitrage argument like you mentioned).In terms of option pricing it's the opposite of "fat tails" problem because the probability of the underlying going below the floor is zero. I would call it "thin tails". The normal distribution assumes a probability of the underlying trading below floor which is the problem.Any ideas?Sorry I don't have a chart at this stage.

but they are selling forward not spot.. unless you can arbitrage spot and forward markets there is no reason why forward price cannot be less than spot.

I don't see why someone would sell in the forward market below the spot floor price. I suppose there is nothing stopping them (maybe for a hedging reason where they can't sell in the spot market)Forward arguments aside, how would you price an option where the underlying is the spot market?

you would sell it forward if you can't sell spot... as I said if it is possible to arbitrage i.e. buy forward sell spot short then you should be indifferent.Are you now asking how to price an option where the underlying has a curtailed distribution ? I don't know how to do that.

Yes that's what I'm asking.Thanks for your insights into the forward market.

I can't see how an asset can have a "curtailed" distribution. Care to explain how that would come to pass ?

The underlying market is regulated not allowing an underlying to trade below or above a certain price. In a normal case a share price can't go below zero. Assume that the spot market for natural gas has a price floor and cap.

I don't know those markets, so can only make general comments. It might make sense to suppose you have a non-tradable (or at least non-storable) spot, underlying some futures that do trade.Let's call the spot process [$]X_t[$]; since it cannot be practically held I am guessing this allows [$]X_t[$] to move in a mean-reverting way between two barriers (f,c) for floor and cap without an arbitrage opp. (Is that correct -- it's the inability to store that prevents an arbitrage on a barrier hit?)Forward prices are given, under some market pricing process by [$]F_{t,T} = E_t[X_T][$]. Since the support of X is in (f,c), you will always find the forwards in the same interval.Now a good model will identify some good state variables.Putting aside the issue of good state variables, the nature of the barriers matters for your question. Does the spot sometimes reach one of the barriers? If so, what happens next?

it would seem that you have a couple of free options - a put at the lower barrier and a call at the upper. if the spot is volatile enough then these can be quite valuable. I am reminded of warrant valuation when there is dilution... I would imagine that valuation here is similarly iterative. If the spot is tradeable.

Yes the spot is tradable and storeable (natural gas). However it will never reach the floor and cap because market regulation prevents it. So any calls or puts at the barrier are worthless because markets will never trade through the strike.I think I may have confused the issue by talking about forwards. So lets just assume you are buying and selling options on spot with a non-normal distribution.Alan I'll think about your suggestion.

QuoteOriginally posted by: jonokrugerYes the spot is tradable and storeable (natural gas). However it will never reach the floor and cap because market regulation prevents it. So any calls or puts at the barrier are worthless because markets will never trade through the strike.I think I may have confused the issue by talking about forwards. So lets just assume you are buying and selling options on spot with a non-normal distribution.Alan I'll think about your suggestion.No they are not worthless... if you firmly believe that the caps and floors will hold then you have a limited loss strategy. you buy when its close to the floor and sell (or long) short when it gets close to the upper barrier. that to me are free options. the levels you buy and sell at will obviously be down to your risk appetite.

presumably it costs to store the natural gas. What about liquidity? If the price is at the cap, are participants forced to sell at that price or can they just say no?

Yes there would be a cost to store the natural gas.They can just say no if they don't want to sell at that level.