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The first equation expresses the option price as a discounted expected value of the payoff contingent on an asset price $S \geqslant 0$. Without loss of generality, we assume that the probability density function has support in $[0,\infty)$, and rewrite as \begin{align} P_{t,T}(K) &=e^{-r(T-t)} \int_{-\infty}^{\infty}\left(K-S\right)^+ q_T^S(S)\,dS \\... 3 For a very nice reference on this matter, I recommend Pykhtin and Zhu’s Guide to Modelling Counterparty Credit Exposure, a short paper that thoroughly defines these concepts. Expected Exposure EE(t) (also known as Expected Positive Exposure) for a trade with value V(t) is given by:EE(t)=\mathbb{E}[\max(0,V(t))]$$It is effectively “what you could ... 3 To lighten notation, we assume a constant accrual factor \tau, a swap rate S_n(T) which fixes at T and pays at T_p (e.g. T_p-T=\text{3 months}) and a simple CMS payoff of the form:$$\Phi(S_n(T))=(S_n(T)-K) fixed at time $T=T_m$. We are interested in pricing under a measure for which the underlying risk factor of interest (i.e. the swap rate) is ...