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Since American style options allow early exercise, put-call parity will not hold for American options (unless they are held to expiration). In practice, there is also a difference between calls and puts for European options as well. The full description is here: What causes the call and put volatility surface to differ?


if only one person can make a choice, it strikes me as unlikely that it can reduce value. Ultimately, a choice means that the holder can choose between one of a number of portfolios on a given date. They will choose the one of maximal value. As long as the without choice portfolio was one of the ones they could have chosen, value can only go up.


Pricing via characteristic functions arises naturally in models that involve Levy processes. Therefore I can see how Black's formula for swaptions can be generalized for Levy dynamics: As in Black's model take the annuity as numeraire, and define the relevant measure $Q$ Black assumes that under this measure the swap rate is martingale GBM, that is to say ...


Let $\{F(t, T), 0 \leq t \leq T\}$ be the forward process that satisfies an SDE of the form \begin{align*} dF(t, T) = \sigma F(t, T) dW_t, \end{align*} where $\sigma$ is the constant volatility, $\{W_t, t>0\}$ is a standard Brownian motion. The payoff at time $T_1$, where $0 < T_1 \leq T$, of a vanilla European forward option is of the form ...

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