# 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.

@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.

• Thank you a lot , It was never introduced to me at shcool why they would theyuse deterministic rates for pricing equity options. Jul 3 '20 at 18:15
• It’s just to make your life a little easier. Quant finance is difficult enough. Interest rates don’t particularly impact short term equity options. So, they are often just ignored. And there’s empirical evidence supporting this simplification. In recent years, interest rates have been particularly low and flat. Jul 3 '20 at 18:16
• And if i had an hybrid of a CMS and an equity , it means the interest rates would only be linked to the CMS or should i account the interest rate for both ? Jul 3 '20 at 18:20
• @Kupocallahoui it depends on your precise product/portfolio. If you include interest rate elements, then the assumption of a flat term structure may be too heroic. But it depends how one links equity products with fixed income products. Jul 3 '20 at 23:26

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.

• Thank you a lot , the comment below nicely developped your answer. Jul 3 '20 at 18:16