Black–Scholes Interest Rates and Call Option Values
Summary
The document clarifies an assumption behind the Black–Scholes explanation that higher interest rates can increase a call's value by lowering the present value of its strike payment. The questioner worries that rising rates might also depress the stock price and reduce the call payoff. The response distinguishes that real-world relationship from the model's assumptions.
In the standard Black–Scholes setting, the interest rate is constant and deterministic, so it does not move in response to the stock price. A model may allow deterministic rates that vary over time while still leaving them uncorrelated with the stock. Stochastic rates require a multifactor framework, where correlation between rates and the underlying can be represented. The response notes that the benefit of this added complexity may be limited for short-dated options because rates often have modest influence on their prices; actual rate-equity relationships are empirical and are outside the basic model.
Key ideas
- In standard Black–Scholes, the interest rate is constant and deterministic.
- The model does not represent stock price reactions to interest rate changes.
- Time-varying deterministic rates can be included without modeling rate-stock correlation.
- Stochastic rates require a multifactor model to represent possible correlation with the underlying.
- The practical value of adding stochastic rates may be modest for short-dated options.
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Full text
# Why would a lower stock price leads to higher value of a call option? # Why would a lower stock price leads to higher value of a call option? Currently I am reading Basic Black Scholes: Option Pricing and Trading by Timothy Falcon Crack. At page $47,$ the author mentions the following. > Higher interest rates decrease the present value of the strike price. Other things being equal, this increases the value of a call because the strike price you potentially give up has lower present value; conversely for a put. I do not understand the bold sentence. Intuitively, I thought that if interest rate increases, then stock price decreases (which is the same as above). But wouldn't this decrease the value of a call option as the difference between terminal stock price and strike price is smaller. Can someone verify whether my reasoning is correct and explain the bold sentence? ## Answer by Kevin (score 2, accepted) https://quant.stackexchange.com/a/48764 In the B&S world, interest rates are constant and thus deterministic. In particular, they are not correlated to the stock price whatsoever. Thus, firstly interest rates don’t change in the first place in the B&S world. You can generalise the model and allow for time dependent (but still deterministic) interest rates. The interest rates are then still uncorrelated with the stock price. A different generalisation is to allow for stochastic interest rates. So you have a multifactor model. Possible choices are the models from Hull-White (1990) and Cox-Ross-Rubinstein (1985). Here, you can explicitly introduce a correlation between stock prices and interest rates. Due to the short time of expiry of many options and the low influence of interest rates on option prices, it is questionable how much better your model is after this generalisation. It may be true or not that in real life stock prices react to interest rate changes (there are many empirical papers investigating this relationship), but the standard B&S model simply does not incorporate this feature.
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