Wilmott Forums - Options on Sums of Digital Payoffs

Spectator Global Risk Conference

Forum Navigation:

login

FORUMS > Technical Forum

Topic Title: Options on Sums of Digital Payoffs
Created On Thu Mar 06, 03 09:47 PM

Topic View:

View thread in raw text format

Moon
Member

Posts: 42
Joined: May 2002

Thu Mar 06, 03 09:47 PM

User is offline

Hi All,

I'd like to know if anyone has heard about the possibility of writing 'closed forms' pricing formulas for this kind of Equity Index Derivatives.

The pay-off of interest here is of the form :

Max(Floor, sum(over i) of indicator functions of events Ei)
where
Ei = { X(i+1) > X(i) } ; where X(i)=spot level at time i...

I don't know whether this has been addressed before (seemingly not, if I may rely on the Google search!) or if such valuation formulas are of interest to any people in the market (note: these products are usually OTC).

... of course we may use MC simulations, but does anyone know smthing simpler (i.e. an exact formula...)?

Thanks for sharing your views.

Moon
Member

Posts: 42
Joined: May 2002

Wed Mar 26, 03 10:18 AM

User is offline

Hello!?...Helllllloo !?
is anybody there?

As you can see, Paul, the idea of submitting the subject to the forum wasn't that great...

Drakkhen
Junior Member

Posts: 10
Joined: Mar 2003

Wed Mar 26, 03 09:03 PM

User is offline

Hello,

I would ask 2 questions :

- can you use Actuarial pricing technique => then use identical technique as in the weather derivatives market

- Isn't it something you could actually simply solve using a forward induction tree?

Of course MC simulations remains a very easy way.

kr
Senior Member

Posts: 1887
Joined: Sep 2002

Wed Mar 26, 03 09:22 PM

User is offline

when you put it this way it looks like a CDO tranche

you could make one with synthetic subleverage with the following recipe:

1) borrow a bunch of money
2) buy a bunch of stocks
3) sell a bunch of calls on said stocks, with strike at 20-30% of spot
4) hope to god that not too many of the stocks in your portfolio end up below the strike and demolish all those premiums you grabbed
5) if they do, give your lenders a call ASAP to tell them that you liquidated the collateral, but somehow it ain't covering what you borrowed

this is 'CDOs for kiddies' in that you can get some real risk on a reasonably short horizon, and you might even be bold enough to look good and hard about the portfolio risk in this kind of arrangement

-------------------------
Fools!
I'm Nonius!!!!

Drakkhen
Junior Member

Posts: 10
Joined: Mar 2003

Wed Mar 26, 03 10:05 PM

User is offline

Quote
Originally posted by: kr
when you put it this way it looks like a CDO tranche

Please would you mind to say what you mean by CDO? What you described looks like a perfect hedge for a ladder option or a roll down option. But I believe that if the underlying remain around a certain value then the payoff will be very high but your hedge technic would have failed.

What I personnally read from his initial post is something that really looks like Critical Event Day in the weather deriv market. The way it works is identical.
An option on a sum of digital indices. The way the weather market price it is by using actuarial or simulation techniques.

BUT because it is an equity linked derivtaive I would do it using trees or simulations. There are of course other ways to price it using imperfect hedges but we would then use ART technics which may not be desirable here.

I quite believe that it is a piece of cake to price it using a very fast forward trees.

kr
Senior Member

Posts: 1887
Joined: Sep 2002

Wed Mar 26, 03 10:30 PM

User is offline

CDO= collateralized debt obligation

You buy a bunch of less-than perfect bonds, prioritize the payments so that the first set goes to one party, the next goes to another party, and somebody else gets the leftovers. So, the average portfolio risk gets split into a less risky piece and a more risky piece, plus a highly uncertain leftover piece.

Practical situation - the bonds are junk bonds that pay a big fat coupon. You borrow 60% of the value from the first party, and pay a tiny coupon to them because they get first dibs on payments. You borrow 30% from a second party, who gets second dibs, so you will also give them a slightly bigger coupon. The last 10% is an equity investment without any guarantee at all, that receives residuals after all the coupons are paid and all the debt is paid.

Obviously the equity is risky, and the first-dibs piece is pretty safe. The guys in the middle are rather exposed, though. If the market tanks, all the shares fall together and they will get sliced to ribbons. If performance is random, then they will probably come out ok. The bonds are rather like digitals, in that most of the time you will get all your coupons plus par back, and in the cases where you don't, you're not getting 90% or 95% - you're getting 20% or 30%. The expectation is that most of the bonds will in fact survive, so the digitals are largely in-the-money in this formulation.

As for pricing, the main thing is to have a good correlation matrix. This contract has a nice MTM when everybody lines up and heads for DOW 36,000 or whatever. If you model the portfolio as a elliptical diffusion, then you're just deciding whether individual components have crossed into digital = 1 or in digital = 0. Closed-form means that you'd need to evaluate rectangular integrals of the normal distribution, and you'd be adding a whole bunch of these together. There are some analytical tricks, but I don't think anybody is using the approximation tools for integrating the normal over convex polytopes to get the answer (and adding up 2^n of them, where n=# of stocks). Instead, you try to rig MC so that you can get a lot of paths that end up in both outcomes for different stocks, or even jointly... and then just keep in mind that you're extremely dependent on the quality of the corr matrix in many cases. Specific parameter settings will change the character of this problem.

-------------------------
Fools!
I'm Nonius!!!!

Edited: Wed Mar 26, 03 at 10:38 PM by kr

spacemonkey
Senior Member

Posts: 340
Joined: Aug 2002

Thu Mar 27, 03 03:26 AM

User is offline

I may have totally misunderstood... however, I think that, depending on your assumptions about the behaviour of X(t), it is possible to calculate a closed form option price.

If X(t) is described by a time-homogenous diffusion (such as the standard black-scholes log-normal diffusion), then P(X(t+1)>X(t)) is independent of t.

Denote Y=\sum_{i=1}^N Ei, and the floor K the payoff is now max(K,Y) or alternatively K+(Y-K)^+

The option price (Im assuming the interest rate is zero here but I dont think it matters) is K+E[(Y-K)^+]

To calculate the expected value you need to know the distribution of Y, which can be written since it is a sum of bernoulli random variables (1 with a probability p=P(X(t+1)>X(t)), 0 with a probability 1-p) and is therefore binomially distributed,

P(Y=a)=(N,a) p^a(1-p)^(N-a) where (N,a) denotes the binomial coefficient.

So E[(Y-K)^+]=\sum_{i=int(K)}^N (N,i)p^i(1-p)^(N-i)*(i-K) int(K) is the first integer larger than K
and the option price is,

K+\sum_{i=int(K)}^N [ (N,i) p^i(1-p)^(N-i)*(i-K)]

which you can maybe simplify...

p will probably need to be written as an integral (maybe an error function) and is assumed to be a risk-neutral probability.

You would also need to modify this if you are between sampling dates.

Is this any use?

lazy
Member

Posts: 45
Joined: Aug 2002

Thu Mar 27, 03 04:06 PM

User is offline

Hi Moon,
Your description sounds like an exotic. The floor aspect is not used in that way because you can insert it by this way:
sums over indicator functions : XX% (1-Indicator) +YY% Indicator.

XX% and YY% are low and high retuns respectively.
In case your digital doesn't work you still have the low return by the relation (1 - Indicator) so if Indicator= 0 you receive XX%
The sums over all XX% will represent your floor.
Seen like this, the pricing is easier no

)))))) A little bit of finance is always a kind of help before brutal force.
I hope it helps

kr
Senior Member

Posts: 1887
Joined: Sep 2002

Thu Mar 27, 03 04:26 PM

User is offline

gang -

just take the two-asset case. You have an oval-shaped diffusion in the plane. The plane is divided into four quadrants which identify the terminal states of the two digitals. The range of the sum of digitals is {0,1,2} depending on which quadrant you end up in. The interesting contract in here would be Max(1, sum of digitals) - you always get 1, but if both assets perform well you get an extra kicker. To figure out whether you get that kicker or not, you need to know what the risk-neutral probability of ending up in the 'good' quadrant is. That will boil down to calculating the integral of the normal distribution over a (partially infinite) rectangular region. There may be some correlation between the two assets which could be accounted for by a linear change of variables, so that the region will come out to be an infinite wedge. This is the ultimate problem you face - calculating this integral. For low dimension, there are some tricks, but in high dimensions it's not too easy. As a first approximation, your result will tell you whether the expected mean of the distribution is inside the wedge region or not - this is an early indicator of the challenge. To a second approximation, you need the area of a sphere centered at the mean, which overlaps the wedge region. I can't imagine that this is easy to do, particularly if the mean is not the vertex of the wedge.

-------------------------
Fools!
I'm Nonius!!!!

lazy
Member

Posts: 45
Joined: Aug 2002

Thu Mar 27, 03 05:46 PM

User is offline

Hi kr,
It still looks like brutal force to me.
The method I suggested allows to retrieve the max and the floor. You just have to carrefully choose XX% to find back the floor you wanted.

kr
Senior Member

Posts: 1887
Joined: Sep 2002

Thu Mar 27, 03 08:04 PM

User is offline

I don't think it's brute force - it's a qualitative analysis of the boundary conditions on this kind of problem

you cannot 1_{X > a and Y > b} out of 1_{X > a} and 1_{Y > b}... the boundary condition depends on the joint information

it's the same reason you need the joint normal distribution function - you can't build it out of individual normals

and for the same reason I don't think you can build this dude's derivative out of your idea

-------------------------
Fools!
I'm Nonius!!!!

keris
Member

Posts: 43
Joined: Jan 2003

Fri Mar 28, 03 08:41 AM

User is offline

Quote
Max(Floor, sum(over i) of indicator functions of events Ei
where Ei = { X(i+1) > X(i) } ; where X(i)=spot level at time i...

what about thinking of this derivative as Put (maturity is up to n days) + (Sum of 1 day digital options up to n days)? What do u guys think?

lazy
Member

Posts: 45
Joined: Aug 2002

Fri Mar 28, 03 05:50 PM

User is offline

kr,
Sorry but I still don't see the need for 1_{X > a and Y > b} out of 1_{X > a} and 1_{Y > b}...
If my view is correct what we need is
Ei = { X(i+1) > X(i) } ; where X(i)=spot level at time i...
As to say {X(i+1)/ X(i) >1} It is of the form exp( ...+ ...W i+1 -Wi)>1
Everything is easy as we have carefully choosen a lognormal process admiting still an exact integrodifferential expression for its Euler form. Of course would we be under stochastic volatility driven process or under other assumptions it wouldn't be that easy.
Am I wrong?

kr
Senior Member

Posts: 1887
Joined: Sep 2002

Fri Mar 28, 03 10:17 PM

User is offline

it's the boundary conditions that make the closed form expression unpleasant

-------------------------
Fools!
I'm Nonius!!!!

Moon
Member

Posts: 42
Joined: May 2002

Fri Mar 28, 03 11:51 PM

User is offline

Thanks all for sharing your views...
Actually, I think that spacemonkey has got an appropriate approach (or at least, quite similar to mine) : indeed by doing some probability calculations I came up with a closed form for the problem.
Moreover, I checked out the results by using MC simulations and it seems to be working.

Anyway, if anyone is interested in checking the proof for the formula (who knows...) I can next time attach my paper (pdf; and hopefully this week end

Moon
Member

Posts: 42
Joined: May 2002

Fri Mar 28, 03 11:53 PM

User is offline

Quote
Originally posted by: lazy
Hi Moon,
Your description sounds like an exotic. The floor aspect is not used in that way because you can insert it by this way:
sums over indicator functions : XX% (1-Indicator) +YY% Indicator.

XX% and YY% are low and high retuns respectively.
In case your digital doesn't work you still have the low return by the relation (1 - Indicator) so if Indicator= 0 you receive XX%
The sums over all XX% will represent your floor.
Seen like this, the pricing is easier no)))))) A little bit of finance is always a kind of help before brutal force.
I hope it helps

Hi Lazy, am sorry but I don't think I got this right...(I mean, the relation to the payoff I specified)
could you elaborate more, please?

Thanks.

lazy
Member

Posts: 45
Joined: Aug 2002

Mon Mar 31, 03 02:07 PM

User is offline

Quote
Max(Floor, sum(over i) of indicator functions of events Ei)

This description of the product is not complete. What you really want is a sum of indicateur functions of events * by something, the real payoff as S ti+1 -Sti (ratchet style payoff) a coupon YY%....
So what we want is to suppress the Max and the Floor otherwise we would need a monte carlo approach. At least we want to suppress the Floor aspect. So we try to re-compose your product under another form. This form will of course integrate the floor aspect.
How
Well we try to find a linear form that will garantee us with the floor. In general the Floor in a financial product is a protection against the worst case, no. What is your worst case in your product. Well When all indicator functions are equal to 0, that seems pretty bad non? So why wouldn't use this case to find back our floor. Floor= XX% (1-Indicator). Using the probability of having all the indicator functions set to zero you find back the XX necessary for your equation.
After it is more a problem of formalism: do us the one in the ref of El Karaoui Rochet; change of numeraire. It helps.

View thread in raw text format

FORUMS > Technical Forum

Forum Navigation:

© All material, including contents and design, copyright Wilmott Electronic Media Limited - FuseTalk 4.01 © 1999-2010 FuseTalk Inc.