Why Collateral Rates Approximate the Risk-Free Rate in Derivatives Pricing

# Why must the risk free rate be free from risk in risk neutral valuation?


# 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.

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/33778

A non academic answer: In the real world, when dealers or professional counterparties trade options with each other, the option premium is not funded by the dealer's unsecured borrowing. Rather, options and other derivatives are usually subject to Collateral Agreements whereby 'safe' collateral is posted to cover exposure between counterparties. If cash collateral is posted, there needs to be an interest rate specified on the collateral balance. In the US this is usually Fed Funds (an almost risk-free rate). If Treasuries are posted, the implicit rate is the Treasury repo rate (also an almost risk free rate).

Thus, the actual interest rates associated with borrowing and lending against derivatives , stocks and bonds are in fact mostly secured rates that resemble risk-free rates.