Let \begin{align} L(t; T, T + \Delta) = \frac{1}{\Delta} \left[ \frac{P(t,T)}{P(t, T+\Delta)} - 1 \right] \end{align} be the forward Libor rate at time $t$ for the period $[T, T+\Delta]$. Consider a... Read More

The Bachelier model which assumes that price follows a normal distribution is a correct approximation for the Black-Scholes one for short times t. When time is short it's fine to ignore drift becaus... Read More

To provide a straight forward answer: It is not a good model. It never was, it never will be. Until we all do not come up with a better model that provides better modeling accuracy while it is equal... Read More

There is a deeper relationship between the two risk-neutral measures. Take any event in the binomial model with a finite number of steps and calculate the risk-neutral probability of it. Take the sam... Read More

Except in highly unusual cases, financial PDEs lack analytic solutions. The mathematical tools used are Monte Carlo, plus the usual ones for solving PDEs on grids, almost always one of the following:... Read More

European option on future To price European Option on Future, you just need to replace $S_0$ with $Fe^{-rT}$ in your original BS formula or you can use risk neutral approach. Both will lead to same V... Read More

The choice of hedging strategy cannot affect the expected p/l, because hedging just consists of doing at-market purchases or sales of the underlying, each of which have zero expected value at the tim... Read More

first - a nice and short note for the calculation can be found here. second: what do they mean by cheaper? The pay-off is different - so what can we compare. The only meaning is that if the stock ha... Read More

Intuitively, in a (log)-space homogenous diffusion model $$ St \propto S0, \forall t \geq 0 $$ such that implied volatilities will only depend on the moneyness level and not on the absolute spot leve... Read More

You are onto something, it is inconsistent to be calculating vega with Black-Scholes considering it assumes that volatility is constant. Black-Scholes is not a good for modeling option prices/implied... Read More