0
$\begingroup$

I'm trying to code in C++ Rebonato's formula for swaption volatilities

$$ v_{\alpha,\beta}^2=\frac{1}{T_\alpha} \sum_{i,j=\alpha+1}^{\beta} \frac{w_i(0)w_j(0)F_i(0)F_j(0)}{S_{\alpha\beta}^2(0)}\rho_{i,j}\int_{0}^{T_\alpha}\sigma_i(t)\sigma_j(t)dt$$

where the forward rate volatility is a piecewise-constant function $\sigma_i(t)=\sigma_{i,\beta(t)}=\phi_i\psi_{i-(\beta(t)-1)}$. This means that the integral term "reduces to a summation of products of volatility parameters", Brigo&Mercurio pag 316.

Assume the swap's floating leg is a 1-Year Libor rate, resetting every 360 days,while the fixed leg is set to make the IRS fair, i.e. $S_{\alpha\beta}$(t), also resetting every 360 days.

I've come up with the following code, where vol_m[i][h+1] represents the quantity $\sigma_{i,h+1}$ and corr_m[i][j] is the correlation matrix $\rho_{ij}$.

    Real sum=0;
    Real sum1=0;
    Real time_unit = 360;
    for (int i=0;i<=beta-1;i++){
        for (int j=0;j<=beta-1;j++){
            for (int h=0;h<=alpha;h++){
                sum1=sum1+time_unit*vol_m[i][h+1]*vol_m[j][h+1];
            }
            sum=sum+weight[i]*weight[j]*r[i]*r[j]*(corr_m[i][j]))/pow(swaprate,2)*sum1;
        }
    }

However, this code is not working properly. Can anyone help me improve it, please?

EDIT: The Real keyword you see in the code is just a QuantLib typedef for double

EDIT: In this article Brigo&Morini present how Rebonato's formula reads using the same piecewise-constant volality $\sigma_i(t)=\sigma_{i,\beta(t)}$ as above,

$$\int_{0}^{T_{\alpha}} \sigma_i(t)\sigma_j(t) dt = \sum_{h=0}^{\alpha}(T_h-T_{h-1})\sigma_{i,h+1}\sigma_{j,h+1}$$

Improve this question
$\endgroup$
4
  • $\begingroup$ Can you explain more? I've hard time aligning your code with the formula. $\endgroup$
    – Bob Jansen ♦
    Mar 19, 2021 at 13:19
  • $\begingroup$ @BobJansen sure, I wanna code a function that given weights, forward rate, swap rate, correlation and volatility computes the ATM swaption volatility, via Rebonato's formula above. The point is I really don't understand how to translate such formula in C++ terms and what I've written so far doesn't seem to be working. $\endgroup$
    – Matteo Campagnoli
    Mar 19, 2021 at 15:00
  • $\begingroup$ Shouldn't you divide time_unit (which seems to be number of days) by 365? $\endgroup$
    – ir7
    Mar 19, 2021 at 16:34
  • $\begingroup$ @ir7 I think it depends on your day couting assumption. This code is actually part of a bigger program written using QuantLib, throught which I've always used the convetion Actual360. For this reason I felt it was natural to use 360 as time_unit $\endgroup$
    – Matteo Campagnoli
    Mar 20, 2021 at 20:35

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.