# Backtesting One-Factor HJM model with selling European Receiver Swaption

I am attempting back test the performance of a model - namely the Musiela equation used to model instantaneous forward rates with constant time to maturity: $$r(t,x)=r(0,x)+\int_0^t\left(\frac{\partial}{\partial x}r(u,x)+\sigma(u,u+x)\int_0^x\sigma(u,u+s)ds\right)du+\int_0^t\sigma(u,u+x)dW_{\beta}(u)$$ Define instantaneous forward rates $$r(t,x)$$ with constant time to maturity $$x$$ by $$r(t,x)=f(t,t+x)$$

The continously compounded yield with constant time to maturity $$x$$ is $$y(t,x)=\frac1x\int_0^xr(t,s)ds$$

For practical purposes I have assumed: $$\begin{eqnarray*} \text{Var}[y(t+\Delta t,x)-y(t,x)] &\approx& \left(\frac1x\frac{\nu}a(1-e^{-ax})\right)^2\Delta t \end{eqnarray*}$$ Where a is the mean reversion parameter and $$\nu$$ is the volatility level.

Now in python I have a dataframe that contains daily CMT yields(converted to continuously compounded) for several maturities for 2018-2022. I have calculated the value of $$a$$ and $$\nu$$ and stored them in the named variables 'a' and 'nu' respectively.

I am wondering how I would calculate the price/payoff of selling a European receiver (sold at the start of Jan 2022) swaption under the one-factor Gauss/Markov one factor HJM model with parameters $$a$$ and $$\nu$$ defined via the Musiela equation. I am assuming the swaption expires in 6 months, payments are quarterly and the swap ends five years after the swaption expiry.

If I was to hedge this swaption with rebalancing on every day for which there is data, using the natural hedge instruments of the constituent zero coupon bond options, how would I find the profit/loss when this option matures?

As I am backtesting the model's performance I would only be using data avaliable to me at the time of selling (i.e. start of Jan 2022).

I am at a loss as to how to calculate the fixed rate of the swaption and also how to price the payoff and the swaption.

Any advice or tips is much appreciated. I am not looking for a fully fledged solution (though that would obviously be appreciated lol) but even a general framework as to how to approach this task would be useful.

• Couple of comments: CMT refers to Treasury yields not swap rates. Secondly, it sounds like you plan to backtest your model by comparing delta hedging pnl with original price. But that’s not going to tell you whether the model was correct. It just will show that realized and implied vol were different by some amount.
– dm63
Sep 28 at 10:40
• I understand CMT refers to yields - but for an at-the-money swaption am i correct in assuming you could construct yield curve and price the swaption based on the fixed rate being equal to the discounted expected cash flows of the floating rate at the time? I am assuming you would construct the yield curve using the Musiela equation. Also yeah I'm just looking to measure the performance of the model. Sep 28 at 13:53
• For example the 10yr Treasury is currently 4.60 but the 10yr swap is 4.30. Different curves.
– dm63
Sep 29 at 3:44