European Call Option with Random Maturity

ramessum · February 6th, 2006, 7:27 pm

How can we price the european call option where the maturity of the option (T) is a random variable independent of the filtration generated by the stock price? Is it look like modeling default as in the intensity based approaches? or it is given by a much more simple formula? Thanks.

PaperCut · February 7th, 2006, 2:24 am

ppauper · February 7th, 2006, 1:12 pm

You can write down the BS formula for a given maturity,call it C(S,t,T)and then pick a distribution for T and integrate\int_{T1}^{T2} C(S,t,T) p(T) dT

jrquant1 · February 7th, 2006, 1:20 pm

If it's an American option how can we do this perhaps in the context of a variable annuity.I would think one way to price such a contract is to pick a terminal point far enough away (for example if maturity follows Poisson than T>>1/lambda) and start working backwards weighted by the "survival" probability*value at time t+1 + "default" probability*intrinsic and apply the optimal exercise condition at each lattice point.

spursfan · February 8th, 2006, 9:27 am

have a look at leisen's paper on the random-time binomial modelhttp://ideas.repec.org/p/bon/bonsfb/399.html

sgelb · February 8th, 2006, 10:49 am

this has to be the silliest thread i have ever heard.. who would want to own that?

ppauper · February 8th, 2006, 1:30 pm

QuoteOriginally posted by: sgelbthis has to be the silliest thread i have ever heard.. who would want to own that?agreed, it's gambling rather than hedging

quantie · February 8th, 2006, 2:30 pm

QuoteOriginally posted by: ramessumHow can we price the european call option where the maturity of the option (T) is a random variable independent of the filtration generated by the stock price? Is it look like modeling default as in the intensity based approaches? or it is given by a much more simple formula? Thanks.Carr's paper on the american put could be usefulhere he gives a semi-explicit solution. The approximation consists of replacing the finite expiration by a Gamma random variable T(n) with parameters n and n/T.

Alan · February 8th, 2006, 5:09 pm

QuoteOriginally posted by: sgelbthis has to be the silliest thread i have ever heard.. who would want to own that?It's a mathematical method, essentially a Laplace transform. Sometimes, esp. with american options, you can solve that problem but not the fixed maturity one.Then, you invert the transform to get the real solution you were looking for.regards,

Collector · February 8th, 2006, 7:50 pm

>How can we price the european call option where the maturity of the option (T) is a random variable independent of the filtration generated by the >stock price? Is it look like modeling default as in the intensity based approaches? or it is given by a much more simple formula? Thanks.From valuation perspective this is basically "the same" as an option with stochastic volatility, at least when dealing with European options. Stochastic clocks are used to model stochastic volatility, so we can just as well use stochastic vol to model stochastic time.