Swaption Pricing and the Error from Using the Wrong Measure

# Pricing of Swaption by Proxy and Monte Carlo


# Pricing of Swaption by Proxy and Monte Carlo












here's the problem. Suppose you want to compute the price of a Call option on a Swap contract. Let $T$ and $T+S$ the times (in year fraction) where the Swap lives and suppose that the fluxes of the swap are every $6$ months in the times: $$ T=T_0< T_1< \dots < T_N = T+S $$ Let $S_{fwd}$ the swap forward rate: $$ S_{fwd} = S(t; T, T+S) = \frac{B(t, T) - B(t, T+S)}{\sum_{i=1}^M (T_i - T_{i-1})B(t, T_i)} = 2\frac{B(t, T) - B(t, T+S)}{\sum_{i=1}^M B(t, T_i)} $$ while the spot Swap rate is: $$ S_t(s) = \frac{1-B(t, t+s)}{\sum_{i=1}^M (T_i - T_{i-1})B(t, T_i)} $$ where $B(0, T)$ is the usual risk-free discount. The above formula is just the rate which let the swap be fair. ($S_t(s)$ is just $S_{fwd}$ with $t=T$)

Our Call option will use this $S_{fwd}$ as strike and it will exercise $2$ times a year so that the payoff will be:

$$ \sum_i^{2M}\mathbb{E}^{T_i} \left[ \frac{1}{2}B(T, T_i)(S_T(s) - S_{fwd})^+ | \mathcal{F}_T\right] = A(T)(S_T(s) - S{fwd})^+ $$

Now we can compute the expected value. In order to have a simpler term we can (I have to...for the exercise spirit this passage is mandatory) change of measure:

$$ B(0, T)\mathbb{E}^T \left[ A(T)(S_T(s) - S{fwd})^+ \right] = A(0) \mathbb{E}^A \left[(S_T(s) - S_{fwd} )^+ \right]. $$

Finally I come to the main question: At this point the exercise ask me to do some approximations and to estimate the errors. One of this approximations is to compute the integral in the forward measure $T$ (the measure we are using before the change of measure) even if we've changed of numeraire. It seems like a good idea since all the trajectories are available in that measure and, in this way, no Girsanov or Brownian motion corrections is needed.

So that we have the following approximation: $$ A(0) \mathbb{E}^A \left[(S_T(s) - S_{fwd} )^+ \right] = A(0) \mathbb{E}^T \left[(S_T(s) - S_{fwd} )^+ \right] + \epsilon $$

The problem is that I don't understand how to compute the error $\epsilon$ generated by a change of measure: in fact it is a "theoretical" technique and I cannot see how to "write down" the error (the estimate can be done both in the analytic way and in a numerical way: Montecarlo, finite differences...).

Thank you in advice. I know it could seem like confusing...and in fact it is form me a little bit.

Ciao! AM