volatility of 1/S

rweinsh · July 20th, 2011, 5:18 pm

Guys!What is the volatility of an asset with a price of 1/S if the volatility of S is q?

bearish · July 20th, 2011, 5:33 pm

Try taking logs.

rweinsh · July 20th, 2011, 6:18 pm

Would you be so kind to explain how?

bearish · July 20th, 2011, 8:42 pm

Somewhat loosely (although exact for a Black-Scholes world): vol(ln(1/S))=vol(ln(1)-ln(s))=vol(0-ln(S))=vol(-ln(S))=vol(ln(S)). For asset price processes we almost always mean the volatility of the log price process when we talk about volatility in general.

Orbit · July 22nd, 2011, 12:57 pm

call the asset f. f=1/S. Let's assume S goes like dS = m*S*dt+q*S*dW (Geometric brownian motion)f_S=-1/S^2 and f_SS = 2/S^3df = f*(-m+q^2)*dt - q*f*dWSo the volatility of S was q, and the volatility of f is -qHTH

Orbit · July 22nd, 2011, 1:34 pm

actually I suppose you can simply say the the volatility of f is just q also.

list · July 22nd, 2011, 2:07 pm

Either methods should imply that S does not reach 0 with probability 1 during [ 0 , T ]. Taking this into account apply Ito formula to 1/S and you will get volatility.

Orbit · July 22nd, 2011, 2:10 pm

QuoteOriginally posted by: listEither methods should imply that S does not reach 0 with probability 1 during [ 0 , T ]. Taking this into account apply Ito formula to 1/S and you will get volatility.That's what I did, right

list · July 22nd, 2011, 2:31 pm

QuoteThat's what I did, rightYes, you did it.

HOOK · July 22nd, 2011, 4:53 pm

If S is lognormaly distributed, then 1/S will have the same volatility as S, i.e., q

Orbit · July 22nd, 2011, 7:53 pm

QuoteOriginally posted by: HOOKIf S is lognormaly distributed, then 1/S will have the same volatility as S, i.e., qsigh

rweinsh · July 23rd, 2011, 5:51 am

QuoteOriginally posted by: Orbitcall the asset f. f=1/S. Let's assume S goes like dS = m*S*dt+q*S*dW (Geometric brownian motion)f_S=-1/S^2 and f_SS = 2/S^3df = f*(-m+q^2)*dt - q*f*dWSo the volatility of S was q, and the volatility of f is -qHTHIt is still interesting that you got the negative volatilityRemember put-call supersymmetry?If you put a negative volatility(-q) in a call formula you get negative value of put with volatility qAre these things linked?

list · July 23rd, 2011, 10:31 am

SDEs with diffusion q ( t , S ) and - q ( t , S ) has the same distribution and we are study probabilities of the SDEs there is no sense to distinct them as to distinct w and - w. Nevertheless the latter two is an example when two processes have equal distributions but with probability 1 do not equal to each other.

spacemonkey · July 26th, 2011, 4:45 am

QuoteOriginally posted by: rweinsh...If you put a negative volatility(-q) in a call formula you get negative value of put with volatility q...No you don't.

rweinsh · July 26th, 2011, 6:06 am

QuoteOriginally posted by: spacemonkeyQuoteOriginally posted by: rweinsh...If you put a negative volatility(-q) in a call formula you get negative value of put with volatility q...No you don't.Option pricing formulas (Espen Haug),page 11 or try Excel