While I was reading the book of Collector (very nice encyclopedia ), I was wondering about this contract : An american forward american option, that's mean an option in which after a time t you choose you will receive an at the money american call. How would you price that in the BS world ?

In BS world, and with usual assumptions, the price of American call is equal to price of European call and can be priced by BS formula. for the contract you mentioned, all you have to do is a backward dynamic programming algorithm deciding at each monte carlo step or PDE node whether it would maximize the price of contract by exercising the option at that point or by exercising it in the future. The exercise value is obtained directly by BS formula with ATM strike price there. If you know the exercise price, the rest of the work is standard LS method for monte carlo. Same thing holds for PDE methods for American options.It would be more interesting to find out the price of American option in which contract on delivery is an ATM put option.

QuoteOriginally posted by: AminIt would be more interesting to find out the price of American option in which contract on delivery is an ATM put option.Totally right ! I had my mind elsewhere when I typed that.The interesting stuff is the double optionnality of exercising (one with a critical time to obtain an option with a critical stock price for exercising). You should have a PDE with a double inequality constrain. A double intricated free boundary problem .

Last edited by frenchX on February 22nd, 2011, 11:00 pm, edited 1 time in total.

Yes it could also be possible. The problem is that for a backward PDE, we have to specify a strike beforehand and then we start from a final time back to present time. But one dimensional PDE is extremely cheap. You can set, for example, several strikes chosen with decent spacing and solve a PDE for each of them. We will have to store the solution of each PDE over all of the given domain. Since we have discrete strikes, it could be possible to find decent enough solution at intermediate strikes in between these discrete strikes by interpolation. This could easily give us exercise price of American ATM option at all future points. Once we have this, we will have to solve another PDE in which exercise price of ATM American option will be substituted from interpolation as I described earlier and we will follow a ususal backward dynamic procedure to find the value of the contract. The steps are as1. Find the solution of American put over all the solution domain with ATM strike price. As I mentioned, this can be achieved by solving several 1D pdes with discrete strikes and later interpolating over strikes. This should be decent enough to make the method work.2. Solve another PDE with exercise price of ATM American put coming from interpolation in step one. Rest is usual dynamic programming as for simple American options.

Last edited by Amin on February 22nd, 2011, 11:00 pm, edited 1 time in total.

And, if I may add, you can do this in one single backward sweep.

I never did this before but just thought right away how it could be done. It will be interesting how you would do it in one sweep. Yes you may try some analytic approximation for ATM American put and then it can certainly be done in one sweep. I would welcome your ideas. You have to keep in mind that option delivered at exercise is forward ATM at delivery.

Last edited by Amin on February 22nd, 2011, 11:00 pm, edited 1 time in total.

I think it's possible to make that in two PDEs (really not sure though).But it would be with a big dimensionnality. For the American put, you will have a PDE in 4D P(S,t,U,Tc) where Tc is the time when the forward option is triggered and U=S(Tc) is the strike price. You will have the american put constrain P>(U-S)+ and the final condition (U-S)+ (put payoff). Tc is important because if the forward option is triggered at Tc then the american option as only T-Tc as time to maturity.For the forward exotic option it's even more complicated because you will add a dimension which is P(U,T-Tc,U,Tc) (the ATM money american put with strike U and maturity T-Tc). The problem is that the constrain F>(P-strike_function)+ is very hard to estimate. The strike function is my problem at the moment. The main problem is because we have backward equations I think. EDIT: for me also the interesting fact is when the fixed maturity T is given for the total exotic contract which means that the classical american put will have a maturity T-Tc. I think it forces a constrain on the choice of Tc to avoid time decay on the underlying american put.

Last edited by frenchX on February 22nd, 2011, 11:00 pm, edited 1 time in total.

Given the time decay and the fact that there is no price scale other than the stock price, I will guess thesolution is the trivial one: exercise immediately. p.s. Proof.At time-0, with time T to expiration, and stock price S0, the critical exercise price must be Sc = f S0, withf dimensionless. So f = f(mu T, sig^2 T). Suppose S0=100 and f <> 1. For example, say f = 1.12Then Sc = 1.12 x 100 = 112. But if, instead, S0=112, then Sc = f 112 = 1.12 x 112 > 112. So we havenonsense unless (i) f=1, or (ii) f = 0 or infinity. But case (ii) means never exercise, which is clearlywrong since you would end up with 0. So the solution is f=1, which means "exercise immediately".Then, the value of the derivative is the same as the value of the ATM option you receive from exercising. To make the problem non-trivial, you need another price scale K.

Last edited by Alan on February 23rd, 2011, 11:00 pm, edited 1 time in total.

Very nice point Alan !So you can add the feature that you have to pay K if you want to exercise the start feature.So here you will have the same equation than I posted for the american vanilla put and you will have the constrain F>(P-K)+ for the forward start.Here you will have the famous double boundary feature F>(P-K)+ and P>(U-S)+ The PDEs for F and P are intricated through the common variables U and Tc because the P for the equation for F is not a free variable is the P(U,T-Tc,U,Tc) for the P equation.

Amin, my observation was quite trivial, it doesn't improve your solution in either running time or accuracy. Just in memory and probably a few lines of code. You can march backward calculating your several ATM american options, and do the interpolation and calculate the compound american option at the same time. On the other hand, I think that if the options you enter in are European, it could be possible to do this without interpolation. Not 100% sure, I'll think about it.

For me the main problem is that sure the american vanilla you will obtain will be at the money but with a time to maturity depending on when you exercised your exotic forward start. So the problem is not the simple BS equation for the american vanilla put, by quickly thinking (very raw though and maybe wrong), you may have a term like dP/dt-dP/dTc instead of simply dP/dt in your Black Scholes equation for the American Put. @Costeanu: I would like to see how you can calculate backward the ATM american put and the compound option in one step. I don't understand how it could be done. Could you explain a bit more please?

Last edited by frenchX on February 23rd, 2011, 11:00 pm, edited 1 time in total.

So after thinking a bit,for your vanilla american put,you have the invariant tau=T-Tc, your origin of time is when the compound option is exercized. dP/dtau+rS*dP/dS+(sigma*S)^2/2*d²P/dS²-rP=0 classical BS equation with P(S,T)=(U-S)+ and the constrain P(S,tau)>=(U-S)+ where U is the strike defined at S(Tc) id est the value of the asset when you exercise the american forward start and we are solving backward for all (U,S) to the instant tau=0 which means t=Tc.for the forward start exotic option, you will havedF/dt+rS*dF/dS+(sigma*S)^2/2*d²F/dS²-rF=0 with F(S,Tc)=(P-K)+ and the constrain F(S,t)>=(P-K)+ The dimension for P is 3 : S, tau, U while for F is 5 cause you add the dimension t and P itself.A very interesting but complicated problem !

QuoteOriginally posted by: CosteanuAmin, my observation was quite trivial, it doesn't improve your solution in either running time or accuracy. Just in memory and probably a few lines of code. You can march backward calculating your several ATM american options, and do the interpolation and calculate the compound american option at the same time. On the other hand, I think that if the options you enter in are European, it could be possible to do this without interpolation. Not 100% sure, I'll think about it.Costeanu, thanks for explaining your comments. I am glad I learnt it and if I ever have to code such a problem, I will keep it in mind.

I think I understood a bit your point (Amin & Costeanu).If I'm right it's a double loop for T and the exercising date Tcfor each Tc 1) backward PDE for ATM american put from T to Tc 2) backward PDE for the forward start from Tc to 0It's a very good idea (if I understood well) ! Thanks a lot guys

Maybe there could be some "approximative closed form formula" I don't guarantee the accuracy. All you need as Amin pointed out is a good approximation for the value of an ATM American put P(Strike,Time to maturity).At Tc, the expectation of the stock price (and thus the strike for the ATM american put) is S(Tc)=S(0)*exp[(r+sigma^2/2)*Tc]When need to find Tc which maximizes the value P(S(Tc),T-Tc). That would be the optimal exercising time for the exotic forward start. Then the forward start is like an EUROPEAN CALL with maturity Tc and Payoff= [P(S,T-Tc)-K]+This is a raw approximation which considers that you can only exercise the exotic forward OPTIMALLY. Does this approximation make sense or am I totally wrong ?Any comments are more than welcome

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