Framework for Backtesting a One-Factor HJM Receiver Swaption Hedge
Summary
The document outlines a proposed backtest of a receiver swaption under a one-factor Hull–White or Gaussian Markov HJM framework. It describes forward rates in Musiela form, relates continuously compounded yields to average forward rates, and gives an approximation for yield-change variance using a mean-reversion parameter and volatility level estimated from historical constant-maturity Treasury yields. The proposed trade is a European receiver swaption sold at the start of January 2022, with six months to expiry, quarterly payments, and a five-year underlying swap.
The author asks how to determine the swap’s fixed rate, price the swaption, calculate its payoff, and measure profit or loss when hedging daily with constituent zero-coupon bond options. The intended backtest uses only information available at sale time. However, the document contains questions rather than a worked procedure: it supplies no calibration results, pricing derivation, hedge rebalancing rules, or performance evidence. It is best read as a statement of a modeling and implementation problem, not as a validated backtesting method.
Key ideas
- The proposed model represents instantaneous forward rates using the Musiela formulation.
- The yield-change variance approximation depends on mean reversion, volatility, and maturity.
- The proposed trade is a sold European receiver swaption hedged with zero-coupon bond options.
- A swaption backtest must establish the fixed swap rate, initial price, expiry payoff, and hedge profit or loss.
- The document poses these implementation questions but does not provide a worked solution or results.
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Full text
# Backtesting One-Factor HJM model with selling European Receiver Swaption
# 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.