black scholes - Why must the risk free rate be free from risk in risk neutral valuation?

I am reading through documentation related to Funding Valuation Adjustments (FVA) which discuss risk free rate and funding matters and the following question came to my mind: in risk neutral valuation theory, why do we require the risk free rate to be risk free?

Indeed, let's assume a Black-Scholes framework but with a stochastic risk free interest rate, whose dynamics are specified by the Hull-White model:

$$\begin{align} & dS_t = \mu S_tdt + \sigma_S S_tdW_t^{(S)} \\[6pt] & dr_t = (\theta_t-\alpha r_t) dt + \sigma_rdW_t^{(r)} \\[6pt] &dW_t^{(S)}\cdot dW_t^{(r)}=\rho_{S,r}dt \end{align}$$

The way I see it is that the risk free rate is supposed to be free of credit risk $-$ indeed, in a stochastic rate framework, this rate has nonetheless market risk. However, nowhere in the specifications of the above model does credit risk appear: it seems to me that $(r_t)_{t \geq 0}$ could represent any rate process. I see 2 situations where it could really make sense to speak about a risk free rate:

• In the original Black-Scholes world, the risk free rate is indeed free of risk because it is the unique process which does not have a random component $-$ it is constant, hence additionally it is also free from market risk.


• If we were modelling asset prices $(S_t)_{t \geq 0}$ with some jump component $-$ to represent default $-$ and the risk free rate was the unique price process free from this credit risk, then it seems it would also make sense to speak about a risk free rate.

Generally speaking, it seems to me that we can speak of risk free rate when the process $(r_t)_{t \geq 0}$ lacks a type of risk that all other assets have $-$ market risk, credit risk. However, I have the impression that in practice the 2 modelling choices above are not common: jump processes are not widely used for pricing, and complex, hybrid and long-dated derivatives tend to be priced with stochastic rates if I am not mistaken.

Hence it seems like $(r_t)_{t \geq 0}$ could very well be anything, for example and importantly the option hedger's cost of funding.

The only characteristic I can think of the risk free rate that might justify its importance is the assumption that any market participant can lend and borrow (without limit) at that rate $-$ hence it represents some sort of "average" or "market" funding rate, like Libor for example. But this does not mean it should be risk free; it does not justify the name of the rate.

Why then stress so much the risk free part, why does the rate need to be free from risk? Couldn't the process $(r_t)_{t \geq 0}$ simply represent the option writer's cost of funding? What am I missing?

P.S.: note that I am not asking why there should be a risk free rate; rather, I am asking why, within the framework of option risk neutral valuation, we have required the "reference" rate under which we discount cash flows in the valuation measure $\mathbb{Q}$ to be free from risk.

Edit 1: my question is a theoretical one mostly. From a practical point of view, my thinking is that the choice of rate used for discounting under $\mathbb{Q}$ $-$ hence to price derivatives $-$ is mostly driven by funding considerations; happily, in a collateralised environment, these funding rates (OIS, Fed Funds) happen to be good proxies for a risk free rate and so there is a matching between theory and practice $-$ maybe my thinking/belief here is wrong.