# 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

• When you use the Girsanov theorem, what are the parameters you use to adjust the drift? Is there an error on these? – will Oct 30 '17 at 17:17
• I'm not using Girsanov. After the change of measure I just use the forward measure again. Of course is not correct but I would like to understand how far I am from the real value if I "ignore" the change of measure. – clarkmaio Oct 30 '17 at 17:19
• Ignoring the change of measure is the same as saying the correlation is zero... – will Oct 30 '17 at 17:20
• Could you explain this? I'm not following – clarkmaio Oct 30 '17 at 17:22
• Have a look in this question, where the effective forward, $F' = F \exp (\hat{c}) = F \exp \left(\hat{ \sigma}_{DOM} \cdot \rho \cdot \hat{\sigma}_{FOR} \cdot T\right)$. This is a change of measure from the domestic to foreign measure, and is very similar to what you're asking. If the correlation is zero though, there is no adjustment. – will Oct 30 '17 at 17:25