risk-neutral-measure


risk management - Consistency of economic scenarios in nested stochastics simulation

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


stochastic calculus - Girsanov Theorem, Radon-Nikodym Derivative backward

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


option pricing - Risk Neutral and Real World Valuations using Monte Carlo

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


stochastic calculus - Feynman-Kac converse

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


"The drift of stock price becomes the risk-free interest rate" under RNP

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


brownian motion - Calculating the stochastic integral of $\exp(-rt)S_t$

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


martingale - Change of measure between T-forward and T*-forward contract?

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


r - Obtaining risk-neutral probability from option prices

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


Estimation Risk-Neutral Variance of Returns

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


Is drift rate the same as interest rate in risk-neutral random walk when using Monte Carlo for option pricing?

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