SERVING THE QUANTITATIVE FINANCE COMMUNITY

frenchX
Topic Author
Posts: 5911
Joined: March 29th, 2010, 6:54 pm

### Hypothetic strange contracts

In the quant "zoology" you have a lot of very exotic contracts which can be though at being weird and funny (despite they are usefull) : asian, lookback, mountain range note, russian option, timer option, etc...So it would be fun to see how you would price some kinds of "hypothetic" contracts (just to see the method and the corresponding PDE). So the first contract I propose to challenge you (and me because of course I will think on it too ) is what I called a "half maturity strike call option"this option is a european call on an asset with a maturity T but the strike is unknown at the beginning of the contract and it is given at the maturity by underlying close price at half the maturity. To make it simple we only consider one asset which follow a GBM.So fellow quantitative structurers how would you price such a contract ?

stilyo
Posts: 168
Joined: January 12th, 2009, 6:31 pm

### Hypothetic strange contracts

isn't that simply a "forward-starting" call option?as long as the strike is going to be proportional to the underlying (in your case, it is equal to the underlying), then you can use the fact that an option is linearly homogeneous with respect to the spot and the strike. at the time t when the strike is set, the call price will be:C(t)=C(S(t),K=a*S(t),T-t,r,q,vol)=S(t)*C(1,a,T-t,r,q,vol)=S(t)*X where in your case t=T/2, a=1. So at time t the option is worth X shares of stock, then today it must be worth X*S(0)*exp(-q*t), where X=C(1,a,T-t,r,q,vol)you don't really need GBM for this argument - an option should be linearly homogeneous with respect to the spot and strike for other underlying diffusion processes, otherwise our model will give us different prices in different currency denominations

frenchX
Topic Author
Posts: 5911
Joined: March 29th, 2010, 6:54 pm

I didn't know that this kind of option was already existing. Your analysis is very precise and right.In fact there are so many contracts that I think it's hard to invent new ones which are not already existing . So another one (which I hope does not exist). A kind of "pair contract". You have two assets S1 and S2 which forms a "pair", the contract gives 1$as long as |S1-S2|>K (a threshold for the breaking of the pair). How would you price that one ? stilyo Posts: 168 Joined: January 12th, 2009, 6:31 pm ### Hypothetic strange contracts for that I'll do a double integral (with appropriate boundaries) over the jointly lognormal pair, which will be simplified to a single integral over the Black-Scholes formula with respect to a lognormal density. we can price this numerically or one of the simplest closed-form approximations I've seen is assume the sum of lognormals is still lognormal and do moment matching. frenchX Topic Author Posts: 5911 Joined: March 29th, 2010, 6:54 pm ### Hypothetic strange contracts I didn't know what a "box option" is but after some research on the web I've found that.QuoteDual-option position that aims to profit from the discrepancies in the market prices of option contracts by using two pairs of puts and calls, all with the same expiration date but each pair having a different exercise price. Box options are considered immune to the fluctuations in the underlying asset's price.FX Box option overview from OANDA Last edited by frenchX on September 21st, 2010, 10:00 pm, edited 1 time in total. frenchX Topic Author Posts: 5911 Joined: March 29th, 2010, 6:54 pm ### Hypothetic strange contracts This OANDA version is very interesting. You draw a box on your price graph and you got money if the curve enters in the box It's true that I'm wondering how it's priced. frenchX Topic Author Posts: 5911 Joined: March 29th, 2010, 6:54 pm ### Hypothetic strange contracts Is it possible to model this by using the risk neutral probability measure with a Kolmogorov backward equation with the box as your final condition ? And then you take the expectation for the pricing with the no arbitrage argument.It's a 2$ idea.
Last edited by frenchX on September 21st, 2010, 10:00 pm, edited 1 time in total.

riccardo24
Posts: 93
Joined: August 25th, 2008, 8:20 pm

### Hypothetic strange contracts

wow!that's a cool stuff!Why the up and down barrier cannot be two partial barrier options? I'm wondering which model they use for the underlying. it's offered down to 5 minutes time!and if I use it before the NFP release?:-)I always saw kolgomorov equation with final condition at sime time T or fixed level. maybe in this caseit is not trivial to use it.I think the model should involve some high frequency model.

Manosgerms
Posts: 39
Joined: July 14th, 2002, 3:00 am

### Hypothetic strange contracts

Outrun I like the nice way they have described those options in the website but in general they have been around for some time and called window OT's or window DNT's basically putting a window time condition on a regular range bet. Most banks usually price them under mc engines and look at the regular DNT pricing for the limiting cases and calibration.

frenchX
Topic Author
Posts: 5911
Joined: March 29th, 2010, 6:54 pm

### Hypothetic strange contracts

A short brainteaser. Imagine that you have an option on two assets S1 and S2, which are two correlated GBM. You have the point x given by the coords (S1,S2). So at the time of the settlement x0 is equal so to (S1(0),S2(0)) and x will evolve in a 2D random walk in the plane given by the moving tip x(t). The contract expires when the x curve intersects itself and the payoff is abs(S1-S2). How would you price this ? Edit: Since the price is always a positive quantity, the 2D random walk will always be in the upper right quarter of the plane. I was thinking about using Stochastic Loewner Evolution for this kind of problem (even if it could be not the simplest thing). For some conditions your random walk will be self avoiding and then you would have NO EXERCISING
Last edited by frenchX on April 15th, 2011, 10:00 pm, edited 1 time in total.