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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 \\...


5

In swaptions, there is the expiration of the swaption into an underlying swap. When the dealers provide the vol surface, in the first column, they typically put the expiry of the swaption from earliest to farthest. Along the top row, they put maturity of the underlying swap from shortest to farthest. So when the dealers describe the upper left having high ...


5

In a practical manner, here is how you get to the PDE of your option: Use Girsanov theorem to go from the real-world measure to the risk-neutral measure (basically subtract the market price of risk $\mathrm dW^Q_t = \mathrm d W^P_t - \frac{\mu -r}{\sigma} \mathrm dt$). This will change your SDE. Discounted option price $e ^{-rt} v(t, S_t)$ has to be a ...


4

To compute the price of an American option or a callable instrument in general, at each potential exercise date, one is required to compare its continuation value (discounted risk-neutral expectation of what the option would pay off if it was not exercised) to the relevant exercise value/early redemption price. By construction, lattice and finite difference ...


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 ...


1

Exactly what @Alex C said. It's the time homogeneous diffusion proprety. You can't state such an argument in models where volatility is no longer time homogeneous ( that's being time independant and depending only on the underlyings).


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