stochastic volatility - Strike Arbitrage

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

How to prove that markets are incomplete under the Stochastic Volatility model?

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

stochastic volatility - Realized variance in SVJJ (Heston with jumps) model

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

monte carlo - Stock Price Behavior and GARCH

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

Motivation of the singular perturbation solution formulation for local volatility model

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

mathematics - derivation of heston pde in gatheral

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

Does it make sense to use upward and downward volatility in option pricing?

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

option pricing - Shifted SABR for negative strikes

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

volatility - Are extended SABR models useful for options with non-negative underlying

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

forecasting - Estimate rolling stochastic volatility forecast using stochvol in R

You might want to have a look at the stochvol vignette (, where this process is described in detail in Algorithm 1. In particular... Read More