Why Stochastic Interest Rates Often Matter Little for Equity Options
Summary
The discussion asks whether equity option valuation should account for uncertainty in interest rates, including in a risk-neutral expectation discounted by the path of the short rate. It explains that interest-rate sensitivity, measured by rho, is usually small relative to other option risks, so adding stochastic rates can increase model complexity without materially improving prices or hedging in many settings.
It cites an empirical comparison by Bakshi, Cao, and Chen using S&P 500 call prices from 1988 to 1991. In that study, stochastic volatility and jumps contributed more to model performance than stochastic interest rates; adding rate uncertainty after modeling stochastic volatility did not further improve hedging performance. The discussion says rates may matter more for long-dated options, which it describes as less liquid. These findings are context-specific and do not imply that rates are irrelevant in theory or for every maturity, market, or pricing task.
Key ideas
- Equity option models can account for stochastic interest rates in theory.
- Equity option rho is often small compared with other sensitivities.
- The cited study found stochastic volatility and jumps more consequential than stochastic rates.
- Adding stochastic rates after stochastic volatility did not improve hedging in the cited analysis.
- Rate risk may matter more for long-dated options, though those options may be less liquid.
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Full text
# Why are interest supposed deterministic for equity?
# Why are interest supposed deterministic for equity?
I don't see why would rates be considered as deterministic when trying to price $\mathbb{E}^{Q} \left[ e^{-\int_{0}^{T_{f}}r_{s}ds} \left( S_{T_f} \right) | \mathcal{F}_{0} \right]$
I would like to express this quantity in terms of the rate and as a integral with respect to the density of $S_{T_{f}}$ obtained with Breeden-Litzenberg.
## Answer by Kevin (score 8, accepted)
https://quant.stackexchange.com/a/55422
@Jan Stuller already pointed to Rho, an option's sensitivity to changes in the risk-free rate. This number is indeed very low indicating that a non-flat term structure may not dramatically misprice equity options.
A somewhat dated (but worth reading) empirical analysis was conducted by Bakshi, Cao, and Chen (1997, JF). They investigate how the Black-Scholes model compares to different models featuring stochastic interest rates, volatility and jumps.
Spoiler alert: Stochastic volatility is the most important addition, followed by jumps. Interest rates only play a role for options with long time to maturity. These options are not that liquidly traded though.
I quote from their paper.
> Based on 38,749 S&P 500 call option prices from June 1988 to May 1991, we find that the SI and the SVSI-J models do not significantly improve the performance of the BS and the SVJ models, respectively
This is a first hint that adding stochastic interest rates is not really worth the effort.
> First, judged on internal parameter consistency, all models are misspecified, with the SVJ the least and the BS the most misspecified.
The above result confirms that stochastic volatility and jumps are important model features.
> However, like in the single-instrument hedging case, once stochastic volatility is modeled, adding the SI or the random-jump feature does not enhance hedging performance any further.
Summary: Technically, you're right and equity option prices should account for interest rate risk in theory. But in reality it's negligible. You just make the model more complicated (throw in more parameters) without improving its performance. Never forget: you model the stock price. By definition, you simplify reality.
## Answer by Jan Stuller (score 5)
https://quant.stackexchange.com/a/55419
I think the main reason is that $\rho$ for Equity options is much less significant that the other Greeks: so going into the length of modelling stochastic rates for equity options isn't worth it.
I would be keen to hear what other's have to say if there are other major reasons.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.