options - pricing using dupire local volatility model

Assemble the data, consisting of a matrix of quoted option prices $\{C(Ti,Kj^i)\}{i=1}^{N}$ where $j=1,2,...,Mi$ together with the yield curve to determine $r$ . Interpolate and extrapolate these pr... Read More

black scholes - What are the main differences in Jump Volatility and Local Volatility

Jump volatility is a term sometimes used to describe randomly varying jump sizes in a model with asset value jumps. So strictly speaking it is merely a parameter in generic jump diffusion. Both local... 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

Euler Discretization to use with Monte Carlo simulation and Local Volatility Model

Use the risk free rate for pricing You use the risk free rate (using the risk neutral measure $\mathbb{Q}$) so that you can use the formula $$ V(t) = \underbrace{\exp(-r(T-t))}{\text{because we used... Read More

Price Down and In Barrier Option Using Local Vol and Monte Carlo

For the first question, you can just plug in t for T and S for K: $\sigma^2 \left(t, S \right)=\left. \sigma^2 \left(T,K\right) \right|_{T=t,K=S}$ For the Monte Carlo part, the barrier would apply to... Read More

calibration - Local volatility SVI parametrization

Gatheral and Jacquier discuss this issue in section 4 of the paper. Instead of using the raw parameterization of the SVI, they use the natural parameterization of the total implied variance: $$ w(k)... Read More

Three questions regarding local volatility implementation (based on the Andreasen, Huge article "Volatility interpolation")

Ok, I did some investigations, asked around and got some answers to most of my questions. Since it might be of general interest for other people I present my findings here. How to transfer market da... Read More

black scholes - Vega of exotic options

In the interest rate world, the vega of an exotic is usually defined by bumping all the relevant volatilities by a multiplicative factor , typically 1.01, 1.05 or 1.10. This could be done in at least... Read More

derivatives - Local Volatility implementation

The usual way is to fit a surface (e.g. smoothing splines) to the grid and to compute derivatives off the surface. Note however that the entire process tends to be more stable when applying the Dupir... Read More

Local Volatility with Monte Carlo Simulation

Let the risk-neutral dynamics under your LV model be given by $$ \frac{d St }{St } = \mut dt + \sigma(t,St) dWt $$ Let's drop the drift contribution (not relevant here) and apply Itô's lemma to obtai... Read More