mj


Total Posts: 1045 
Joined: Jun 2004 


I have worked out to use a power of the stock price as numeraire. I think it's neat.
http://ssrn.com/abstract=2735944
it has applications to both analytic formulas and numerical methods.

More mathematical finance has been published.




Nonius

Founding Member Nonius Unbound

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Joined: Mar 2004 


hey that's kinda cool. I had always thought numeraires had to be traded securities but I guess the main thing you need is the value can't be zero (maybe it's enough it's not zero a.s.?) and probably has to keep the same sign? 
Chiral is Tyler Durden 


mj


Total Posts: 1045 
Joined: Jun 2004 


the trick is to take the value of a derivative that pays a power of the stock price, rather than the power of the stock price itself.

More mathematical finance has been published.





Hmm, but you could apply such trick to any function of the stock price? What's stopping me from taking the value of a derivative which pays a hyperbolic cosine of the stock price? 



Nonius

Founding Member Nonius Unbound

Total Posts: 12657 
Joined: Mar 2004 


I don't think anything would stop you from doing so, although I'm not sure how useful it would be as the hyperbolic cosine, on first glance doesn't seem like it would have a bunch of nice properties in respect of stochastic calc calcs, transforms etc. Seems like his idea is that you can use another function as the numeraire by creating a fictitious derivative whose payout is the function; then he zeros in on powers because there's some utility there for some other standard options + the calculations aren't wildly complex. 
Chiral is Tyler Durden 




OK, it pays the power of the stock price at some fixed time T. So analogous to the "terminal measure" in fixed income pricing (using a zerocoupon bond as a numeraire).
But then you could define a derivative which is an American option with zero strike and a payoff of S_t^alpha when exercised at time t. 



mj


Total Posts: 1045 
Joined: Jun 2004 


indeed you could take the price process of any positive payoff derivative as numeraire. The questions are can you compute with it and does it do anything useful?
would the American derivative you suggest be worth early exercising? In the BS model without dividends and \alpha>1 I doubt it.

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Why would it not be worth for alpha > 1 and worth exercising for alpha = 1? 



mj


Total Posts: 1045 
Joined: Jun 2004 


it's the convexity of the payoff.
You'd also have to decide what happened to the payoff over the period $[t,T]$ if it was exercised at time $t.$

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"it's the convexity of the payoff."
But its' convexity w/r to S_t. W/r to itself, S_t^alpha is not convex but linear. So why do we single out S_t?
"You'd also have to decide what happened to the payoff over the period $[t,T]$ if it was exercised at time $t.$"
What if I invest in a bank account? 



mj


Total Posts: 1045 
Joined: Jun 2004 


1. because it's $S_t$ that is the tradable.
2. Sure you could take it as numeraire but it would do anything useful and can you compute the process.

More mathematical finance has been published.




Lorenzo


Total Posts: 7 
Joined: Jan 2016 


Hi Mark,
Power numeraires are an attractive alternative to prices of hockeystick payoffs for modeling the dynamics of the smile as they make up a family of convex payoffs, just like vanillas for a fixed moneyness. Section 4.3 of my book is devoted to them  you may want to take a look. I consider the following issues:  expressing implied volatilities of power payoffs directly as a function of implied vols of vanillas.  deriving a dynamics for the variance curve associated to power numeraires for a given power.
The simplest power numeraire is obtained with  it's the logswap For other values of , figuring out a Markov representation of the associated variance curve is (much) harder.
Lorenzo 



mj


Total Posts: 1045 
Joined: Jun 2004 


thanks. I haven't got your book yet...
How do you get the log contract to work as a numeraire? It is not always positive.

More mathematical finance has been published.




Lorenzo


Total Posts: 7 
Joined: Jan 2016 


I'm not using them as numeraires, but as a family of convex payoffs whose implied vols  or variances curves  are easier to model than vanilla implied vols. 




Hi Mark,
you might want to have a look at this paper I wrote a longtime back, we use power numeraires to simplify the calculation of for example barrier options etc. It is a very useful thing for simplifying calculations as you also figured out.
Regards Jiri




