Consider the 2 following approaches to pricing a security:
Monte-carlo ($\mathbb{Q}$-measure)
$\begin{equation} C = \frac{1}{N} \sum_{i=1}^{n} e^{-rT} max(S_i(t) - K, 0) \end{equation}$
Monte-carlo ($\mathbb{P}$-measure)
$\begin{equation} C = \frac{1}{N} \sum_{i=1}^{n} D_i max(S_{i}(t) - K, 0) \end{equation}$
where $D_i = \left(\frac{q_i}{p_i}\right) \frac{1}{(1+r)^t}$ is the stochastic discount factor / deflator / pricing kernel under scenario $i$.
Furthermore, I assume that
\begin{align} ln(S_t/S_0) &\sim N(r - 0.5\sigma^2, \sigma\sqrt{T}) \mbox{ under the $\mathbb{Q}$ measure} \\ ln(S_t / S_0) &\sim N(\alpha - 0.5\sigma^2, \sigma\sqrt{T}) \mbox{ under the $\mathbb{P}$ measure} \end{align} I am wondering how to calculate $q_i$ and $p_i$. Then I would like to compare $C$ under the Black-Scholes, Monte-carlo $\mathbb{Q}$ measure and Monte-carlo $\mathbb{P}$ measure.
I expect that $C_{BS}(S=100, \ K=100, \ \sigma=0.25, \ r=0.03,\ T=1) = 11.35$ is close to $C_{MC, \mathbb{Q}} \ (N=10,000)$ and $C_{MC, \mathbb{P}} \ (N=10,000)$.
I use the standard Monte-Carlo technique - i.e. I simulate values of $S(1)$ by taking a random draw from a normal distribution and converting this to the appropriate lognormal distribution.
Di=(qi/pi) * 1/(1+r)^t
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