The Global Calibration paper outlines a method which is one approach to resolve inconsistencies between pricing and calculating risk measures.... Read More

The result you're looking for is $$ \left. \frac{d\Bbb{P}}{d\Bbb{Q}}\right\vert{\mathcal{F}t} = \left( \left. \frac{d\Bbb{Q}}{d\Bbb{P}}\right\vert{\mathcal{F}t} \right)^{-1} $$ This is a result from... Read More

You probably wonder whether $\mathbb{E}^\mathbb{P}[PT\mid\mathcal{F}t]= \mathbb{E}^\mathbb{Q}[PT\mid\mathcal{F}t]$. Note the $T$ as index, i.e. the future unknown payoff and not the current price $P_... Read More

If you assume that $$ Vt = V(t,St;\theta) = \Bbb{E}^\Bbb{Q} \left[ e^{-\intt^T r(s) ds} VT \mid \mathcal{F}t \right] $$ with $VT = h(ST;\theta), \Bbb{Q}-\text{a.s.}$ along with $$ dSt/St = r(t) dt +... Read More

Yes, you may as well take this as the definition of the risk-neutral probability $Q$. I will now try to give you some intuition for that kind of construction. Assume the risk-free interest rate $r$... Read More

Ad. 1. You are right: $Y{t}=e^{-rt}S{t}$ $dY{t}=d(e^{-rt}S{t})=-re^{-rt}S{t}dt+e^{-rt}dS{t}=(\mu-r)e^{-rt}S{t}dt+\sigma e^{-rt}S{t}dW{t}=(\mu-r)Y{t}dt+\sigma Y{t}dW{t}$ Now we have: $\hat{W}{t}=\frac... Read More

By definition $Q^{Ts}$ is risk neutral for the numeraire $P(t,Ts)$, and $Q^{Te}$ is risk neutral for the numeraire $P(t,Te)$, hence $$ \left(\frac{dQ^{Ts}}{dQ^{Te}}\right)t = \frac{P(t,Ts)}{P(t,Te)}... Read More

The risk-neutral probability density function $q(.)$ is indeed given by $$ q(S_T=s) = \frac{1}{P(0,T)} \frac{ \partial^2 C }{\partial K^2} (K=s,T) $$ where $P(0,T)$ figures the relevant discount fact... Read More

Just trying (I'm not sure I've well understood the question). I will assume the ususal risk neutral dynamic for $St$: $$ dSt = rSt dt + \sigma StdW_t $$ so that $\forall T>t$ we have: $$ ST = Ste^{\l... Read More

Yes. The risk neutral and the real path share the same volatility, so the difference is in the drift rate, where the risk-neutral path drifts with the risk-free rate r. You may want to check out Paul... Read More