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Let us propose bivariate Black-Sholes Model. Assume, we have an arbitrage-free complete market. $r_{f}$ is risk-free rate. Under real-world measure $P$: $dS_{1} (t)=S_{1} (t) [\mu_{1}dt+\sigma_{1}... 1answer 489 views ### Radon-Nikodym derivative and risk natural measure I need help with my understanding of changing probability measure. Im not a mathematician so I hope for answers that are not too technical. As shown in this Wikipedia article http://en.wikipedia.org/... 1answer 457 views ### Girsanov Theorem and Quadratic Variation Girsanov theorem seems to have many different forms. I've got a problem matching the form in wiki to the one in Shreve's book, due to the difficulty of quadratic variation calculation. Below is the ... 2answers 307 views ### Uniqueness of equivalent martingale measure in Black Scholes-Model Let's consider standard Black-Scholes model with price process$S_t$satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where$B_t$is standard Brownian Motion for probability$\mathbb{P}$. I ... 1answer 602 views ### How to compute the Radon-Nikodym derivative? Suppose$B(t)$is a standard Brownian motion, and$B_{1}(t)$is given by$dB_{1}(t)=\mu dt+dB(t)$. Suppose$P$is the Wiener measure induced by$B(t)$on the$C[0,\infty)$, and$P_{1}$is the Law ... 1answer 320 views ### Use of Girsanov's theorem in bond pricing Assume that we want to calculate the time$t=0$price of a bond:$B(0,T) = E_P[\exp(-\int_0^T r_s ds)]$, where$r$is the interest rate following the SDE$dr_t=k(\theta-r_t)dt+\sigma dB_t=b(r_t)dt+\...
Let's take a GBM under $P$: $dS=\mu dt+\sigma dW_{t}^{P}$ and then under $Q$ $dS=r dt+\sigma dW_{t}^{Q}$, where $dW_{t}^{Q} = dW_{t}^{P} + (\mu - r)/\sigma dt$ Now, let's say that I have ...