Random Time Forward Starting Options

TL;DR

This paper introduces Random Time Forward Starting (RTFS) options, a generalization where the strike-determination time is random, and provides arbitrage-free pricing methods, practical implementations, and credit risk adjustments.

Contribution

It extends forward-starting options to include random strike times, offering explicit pricing formulas under certain market assumptions.

Findings

Explicit arbitrage-free pricing formulas derived

Pricing methods applicable in classical market models

Credit risk adjustment for OTC RTFS options provided

Abstract

We introduce a natural generalization of the forward-starting options, first discussed by M. Rubinstein. The main feature of the contract presented here is that the strike-determination time is not fixed ex-ante, but allowed to be random, usually related to the occurrence of some event, either of financial nature or not. We will call these options {\bf Random Time Forward Starting (RTFS)}. We show that, under an appropriate "martingale preserving" hypothesis, we can exhibit arbitrage free prices, which can be explicitly computed in many classical market models, at least under independence between the random time and the assets' prices. Practical implementations of the pricing methodologies are also provided. Finally a credit value adjustment formula for these OTC options is computed for the unilateral counterparty credit risk.