6

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 ...


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