Pricing European Options with a Vasicek Short-Rate Model
Summary
The document discusses pricing a European option when the short interest rate follows a stochastic process such as the Vasicek model. It points to affine pricing methods associated with Bakshi, Cao, and Chen: when the risk-neutral conditional characteristic function is exponentially affine, option values can be computed by Fourier inversion rather than by Monte Carlo simulation. The cited paper is presented as a source of equations for implementing the approach.
The response cautions that modeling short-rate uncertainty may add little to ordinary option pricing and that a chosen rate process can be misspecified. It distinguishes this issue from the term structure: the maturity-matched risk-free rate matters, so a rate appropriate to the option's expiry should be used. It also notes that practitioners may infer forward prices using put-call parity instead of estimating dividend yields directly. The post is an informal answer rather than a worked calibration or valuation example; it gives no data or empirical comparison demonstrating the size of the effect.
Key ideas
- An exponentially affine risk-neutral characteristic function can enable European option pricing through Fourier inversion.
- The cited approach can avoid Monte Carlo simulation for the described pricing problem.
- A suitable risk-free rate should reflect the option's maturity along the yield curve.
- The response warns that short-rate uncertainty may have limited pricing impact and that the rate model may be misspecified.
- Put-call parity can be used to infer futures prices when dividend yields are not modeled directly.
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# Black & Scholes under stochastic interest rate (Vasicek) # Black & Scholes under stochastic interest rate (Vasicek) I'm a beginner in Quantitative finance and I'd like to ask you for help about this exercise. I have to price a put option on a risky asset by working under stochastic interest rate, so I have to calibrate this process with simulate data, hence price this derivative. Could someone help me with a practical exemple, in order to understand? Thanks. ## Answer by Stéphane (score 2) https://quant.stackexchange.com/a/52934 If you're looking to price a European option under a stochastic short rate, you can take a look at the Bakshi, Cao and Chen (1997) paper. Some of their model combine stochastic volatility, jumps in the price process, as well as a stochastic short rate process -- such as what you ask. The convenience of their approach is that all of their models imply that the conditional characteric function of the price process under the risk-neutral measure is exponentially affine meaning you can price European options using the inverse Fourrier Transform -- no Monte Carlo simulation required! You'll find all the equations you need to code everything on your own in that paper. Now, if you aren't obligated to do this, I would advise against wasting time on this issue. The reason is that while it is important to recognize that interest rates obey a term structure (i.e., depending on the time to maturity of the option you price, you should definitely use a different interest rate), the uncertainty surrounding changes in the short rate process is insignificant when it comes to pricing an option, not to mention the fact that your short rate process is bound to be misspecified. In all the academic papers you will find, here is what people do: ``` when you price an option with X days to maturity, you look for something like a US Treasury bond with as close a maturity as possible and you use the yield on that bond as your interest rate. ``` Most of them will also not bother trying to find adequate proxies for the expected dividend yield. They'll just impose put-call parity, derive implies futures prices and use these to price options.
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