If ${\frac {\partial^2 C} {\partial K ^2}}$ was zero, then the price-strike curve would just be a straight sloping-downwards line, and it would cost the same to buy either two call options at strike... Read More

It's really quite simple. It's just a matter of the fact that we can change measure on the stochastic volatility while not changing the fact that the stock is a martingale. Once we can do this, we ha... Read More

Try $$\mathbb{E}\frac{1}{T} \int0^T Vt dt = \frac{1}{T} \int0^T \mathbb{E} Vt dt$$ and use $$\frac{1}{dt}\mathbb{E} Vt = \kappa\theta - \kappa \mathbb{E}Vt + (\lambda0 + \lambda1\mathbb{E} Vt)\muV, $... Read More

Okay just to wind things down here, I think an important clarification is needed if readers might come and seek to a similar solution. The Geometric Brownian Motion (GBM) is a model of asset prices d... Read More

In fact, this is a confusion caused by a sloppy notation. The rigorous version of the setup should be $$A(K)\rightarrow \epsilon A(K).$$ Then we let $x:=\frac{f-K}\epsilon$. The rest is the usual sin... Read More

1) Gatheral expresses everything in forward terms: forward value of the spot and of the call. Consider an asset $A$. You need to hold $A$ at time $T$ but since you don't need it now you don't want t... Read More

Well if you think that this model represents reality more accurately than the Black-Scholes assumptions. A lot of people do indeed think so. But I wouldn't say you're "tweaking" Black-Scholes... you'... Read More

I would say that SABR is overkill for inflation options, because due to the scarcity of prices for inflation options, there isn't enough information to fit all the SABR parameters. It is probably bet... Read More

You don't want to use the SABR (or an extension) to price equity options or FX options. The lag of mean-reversion in the model's volatility dynamics leads to explosive behavior and to a implied distr... Read More

You might want to have a look at the stochvol vignette (http://cran.r-project.org/web/packages/stochvol/vignettes/article.pdf), where this process is described in detail in Algorithm 1. In particular... Read More