# Call option price relating to Risk-free interest rate

Discussion in 'P1.T4. Valuation & Risk Models (30%)' started by hsuwang, Oct 17, 2009.

1. ### hsuwangMember

Hello David,
According to minimum value So - K((exp)-rt) (call option value), as risk-free rate increases, K decreases, thus increases the value of the call option.
My question, or confusion here, is: if interest rate goes up, wouldn't stock price somehow be affected (decrease), and if S decreases, wouldn't the value of the call also go down? Or maybe I'm mixing the wrong concept here - bond prices will be affected by interest rates but not stock prices?

Thanks!

2. ### David Harper CFA FRMDavid Harper CFA FRM (test)Staff Member

Hi Jack,

I think it's an interesting point ... I am very keen on using the minimum value to keep in mind that higher rate implies higher option value; although, in regard to BSM option value, it's only a sort of short-hand approximation because the N() functions include rate, too. So, Greek Rho = K*T*exp(-rT)*N(d2) gives us the proper linear approximation (itself only approximate but more so!), which is directionally consistent with your observation as the discounting is the overpowering impact ... all i mean is, your use of minimum value is fine, just wanted to remind that it's only a shorthand for the "truer" relationship in the BSM

Re: "if interest rate goes up, wouldn’t stock price somehow be affected"
Keep in mind the BMS does not care about (make any use of) the stock's expected return. Hull says (p280) "the higher the level of interest rates, the higher the expected return required for any given stock. Fortunately, we do not have to concern ourselves with the determinants of [expected return]." This refers to the difficult risk-netural valuation idea that underlies BSM. You can be right about your statement vis a vis impact on stock price but the BSM model, in a sense, is "impervious" because it is a solution based on a riskless portfolio that is robustly riskfree under any future stock price scenario.

Finally, this is all true "within the BSM model" only, right? All we can say is, under the assumptions, the model is invariant to determinants of expected stock return. The model can't account for things outside the model (i.e., model risk). This comes up in Stulz chapter 18, where he asserts that, to the effect that, stock prices and interest rates are inversely correlated. And it gives confusion because it is there merely an empirical assertion based on a study; it's not represented in any model. So, you may be true also empirically, which may be something that requires a different/more powerful model, but our model is not up to explaining it

...it's a long way of reverting to an FRM *theme:* we are dealing generally with simple single-factor, two-factor, etc models. We may omit risk factors and this is an important source of model risk.

..David