We consider an stock price S following a normal model: dSt=σdWt
We can write this as dStSt=σStdWt
Hence we can see that S follows a "log-normal" diffusion with a local volatility function c(S)=σS which is downward sloping.
My question is: can we deduce that the log normal smile implied by this model will be downward sloping as well ? That is to say, if we have a local volatility function which is decreasing as a function of S, will the lognormal implied vol be decreasing as a function of the strike ?
Thanks !
Answer
Since ST=S0+σWT, C:=E((ST−K)+)=E((S0+σWT−K)+)=∫∞K−S0σ√T(S0+σ√Tx−K)1√2πe−x22dx=(S0−K)Φ(S0−Kσ√T)+σ√T√2πe−(S0−K)22σ2T, where Φ is the cumulative distribution function of a standard normal random variable. Then, dCdK=−Φ(S0−Kσ√T)<0. On the other hand, let σI(K) be the log-normal implied volatility, that is, C=C(K,σI(K)). Then dCdK=∂C∂K+∂C∂σI∂σI∂K. Here, ∂C∂K=−Φ(d2), where d2=lnS0K−12σ2ITσI√T. Since limK→∞S0−KlnS0K=∞, we can expect that, for K sufficiently large, d2>S0−Kσ√T. That is, ∂C∂σI∂σI∂K=Φ(d2)−Φ(S0−Kσ√T)>0. Then, ∂σI∂K>0, and the implied volatility is a increasing function of such strike levels. In conclusion, the implied volatility does not have to be a decreasing function of the strike.
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