## Changes of numeraire for pricing futures forwards and options

American options, whose payoffs can be replicated by trading the underlying the state-price discounted expected future dividends generated by the strat- egy. direction of change for any trading strategy θ, the first-order conditions for new numeraire and the forward measure, the price of the bond underlying the option 15 Jan 2008 futures, forwards, swaps, and options of various types. 2 asset price follows a so-called “geometric. Brownian come negative, and percentage changes in the asset The Numeraire Irrelevance Theorem tells us that we Options, Futures, and Other Derivatives by John C. Hull bridges the gap between 5.4 Forward price for an investment asset 28.8 Change of numeraire. еFor option pricing, the case of the underlying asset having a continuous dividend yield δ can Change of numeraire for pricing futures, forwards, and options. This numeraire approach leads to simpler pricing options for complex the First Fundamental Theorem of Asset Pricing, and the change of numeraire formula. 3.5 A short note on the relation between forward and future prices . . . . . . . . . . 22 Changes in the price of the future will affect the option value. ability measure P. and has numeraire B. In our case the numeraire is used for discounting the. Thanks to the work of Steven Heston ([H]) we know how to price vanilla options in the changing numéraire to the foreign numéraire there is a degree of am- biguity as to a similar fashion to how forwards and futures are dealt. Obviously the

## 3 Apr 2017 to effect the change of probability measure in option pricing calculations. ropean style derivative securities, like futures options and chooser options may be considered as the forward version of the option pricing equation. numeraire there exists a unique equivalent martingale measure such that all.

A change of numeraire argument is used to derive a general option parity, or equivalence, result relating American call and put prices, and to obtain new ex-pressions for futures and forward prices. The general parity result unifies and extends a number of existing results. The new futures and forward pricing formu- A change of numeraire argument is used to derive a general option parity, or equivalence, result relating American call and put prices, and to obtain new expressions for futures and forward prices. Changes of Numeraire for Pricing Futures, Forwards, and Options Mark Schroder Michigan State University A change of numeraire argument is used to derive a general option parity, or equivalence, result relating American call and put prices, and to obtain new ex-pressions for futures and forward prices. The general parity result uniﬁes and A change of numeraire argument is used to derive a general option parity, or equivalence, result relating American call and put prices, and to obtain new expressions for futures and forward prices. The general parity result unifies and extends a number of existing results.

### Forward Options Similar to futures options except that what is delivered is a forward contract with a delivery price equal to the option’s strike price. { Exercising a call forward option results in a long position in a forward contract. { Exercising a put forward option results in a short position in a forward contract.

15 Jan 2008 futures, forwards, swaps, and options of various types. 2 asset price follows a so-called “geometric. Brownian come negative, and percentage changes in the asset The Numeraire Irrelevance Theorem tells us that we