Saturday, March 5, 2016

options - Gamma is always positive on both put and call



I recently met the claim that for standard put and calls the gamma of the options are always positive. Is this a general result?


I am hoping not to assume any model, especially not Black-Scholes.



Answer



I'll use a European call option as an example, I think you can easily generalize it for a put option.


Given underlying S(t)=St, maturity T, strike K and risk-free rate r, the price of a call option as time t under the rik-neutral measure Q is


C(t,St)=EQ[er(Tt)max(STK,0)]=EQ[er(Tt)(STK)1STK]=EQ[er(Tt)ST1STK]EQ[Ker(Tt)1STK]=er(Tt)EQ[ST1STK]Ker(Tt)EQ[1STK]


where 1STK is a function values 1 when STK and 0 otherwise. For the first expectation, we can change the probability measure to make it more manageable. Call P the new measure; the Radon-Nikodym derivative between the P and Q is


dQ=StSTer(Tt)dP


Therefore you get


C(t,St)=er(Tt)EQ[ST1STK]Ker(Tt)EQ[1STK]=er(Tt)EP[ST1STKStSTer(Tt)]Ker(Tt)EQ[1STK]=er(Tt)EP[1STKSter(Tt)]Ker(Tt)EQ[1STK]=StEP[1STK]Ker(Tt)EQ[1STK]



Expanding the expectations as integrals you get:


C(t,St)=StKfP(ST)dSTKer(Tt)KfQ(ST)dST=StP1Ker(Tt)P2


where P1,P2 highlight that the integrals are probabilities.


Now the Greeks:


Δ=CSt=KfP(ST)dST=P1Γ=2CSt=ΔSt=fP(St)fP(St)St


The derivative in Γ is the key. I don't think you can prove Γ to be positive for any probability density (i.e. any model).


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