Calibrating Correlated Vasicek Models for Risk-Free and Risky Rates
Summary
The document describes a pricing problem involving a derivative exposed to both a near risk-free short rate and a credit-risky short rate. The proposed setup models the rates with correlated Vasicek processes. For the near risk-free rate, the author can calibrate to traded caps, floors, and swaptions, while the risky rate lacks comparable option prices and would instead be calibrated approximately from its current term structure.
The central challenge is using a common numeraire and incorporating the external deterministic discount curve into calibration of the risky-rate model. The author also considers modeling the credit spread directly, but notes that this leaves the risk-neutral calibration problem unresolved. The document poses these issues without offering a solution or empirical results, so it serves as a question about model consistency and calibration inputs rather than a validated method.
Key ideas
- The proposed framework uses correlated Vasicek models for near risk-free and risky short rates.
- The near risk-free model can be calibrated to available interest-rate options.
- The risky-rate model lacks comparable option prices and may require approximate calibration from its term structure.
- Using an external deterministic curve for discounting raises questions about calibrating the second rate model consistently.
- Modeling the credit spread directly does not by itself resolve the risk-neutral calibration challenge.
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Full text
# Model risk-free and risky short rate with vasicek model # Model risk-free and risky short rate with vasicek model I aim to price a derivative product which depends on both, an (almost) risk-free interest rate and a risky interest rate (the latter is essentially the interest rate for companies with a specific rating). My idea: model both short rates via (correlated) Vasicek models. For the first model (almost risk-free interest rate), this is straightforward: I can find a risk-neutral calibration of the model using market prices for caps / floors / swaptions, which are available. However, for the second model, there are no caps / floors / swaptions are available on the market for this underlying interest rate. Hence, I have to use e current interest rates (risky) for different maturities for a risk-neutral calibration (approximatively). My main issue: different numeraires. In order to have a common numeraire for both models, my (pragmatic) approach would be to use an external risk-free (deterministic) curve to discount payoffs in both models. For the first model, I can use this curve for discounting payoffs in the calibration process and fit the parameters such that market prices of the caps / floors / swpations are met (using the external risk free discount curve). However, I struggle to incorporate the risk-free rate in the calibration process of the second vasicek model. Do you have any ideas? Side note: modeling the credit spread (via vasicek?) instead of the risky short rate would also be a solution in this case. But in this approach, I face the same problems when I try to find a proper risk-neutral calibration of the model. Thankful for any of your thoughts. Best, Frank
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