# 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.

• Have you read "Cooking with collateral" by Pieterbarg? Apr 20, 2017 at 12:50
• No but I have heard about it, it is in my "To Read" list. Does he discuss this issue? Apr 20, 2017 at 13:11
• Quoting: An economy without a risk-free rate has been considered in the past (see Black, 1972) but traditional derivatives pricing theory (see, for example, Duffie, 2001) assumed the existence of such a rate as a matter of course. Until the crisis, this assumption worked well, but now even government bonds cannot be considered credit risk-free. Hence, using a risk-free money-market account or a zero-coupon bond as a foundation for asset pricing theory needs revisiting. (...) What comes closest to a credit risk-free asset in a modern economy, in our view, ... Apr 20, 2017 at 13:42
• is an asset fully collaterised on a continuous basis. Of course, possible jumps in asset values and practicalities of collateral monitoring and posting do not allow for full elimination of credit risk, but we will neglect this here. --- I would say yes, it also goes along what @dm63 explained with "secured" rates being collateral rates. Apr 20, 2017 at 13:42
• @Quantuple thank you for the reference, I have read the article. However, I am a bit puzzled: in the 1st section, Piterbarg presents the model in a simplified framework with 2 assets with the same risk source (same Brownian motion), fair enough; however, in section Many collateralised assets, he introduces an economy with no risk-free rate... but with a risk-free asset, given the dynamics are given by a $N+1$-dimensional price vector and a $N$-dimensional Brownian motion. So he seems nevertheless forced to assume at least the existence of a risk-free asset, is that right? May 27, 2017 at 12:42

• Thank you @dm63. I understand that in practice we use OIS- and repo-type rates for collateralised transactions $-$ and post-crisis Libor-type rates for uncollateralised transactions (?). However, my question is more theoretical/academic than practical: I believe in practice derivative sellers/buyers are not really "seeking" the famous risk free rate to price their derivatives, they rather use cost of funding rates that, post-crisis and in a collateralised framework, happen to be suited as risk free rates. But why do we, in theory, require this reference rate to be risk free? Apr 20, 2017 at 11:13
• Maybe these market practices predated option pricing theory; the same observations you are making were made by Black, Scholes, Merton, Harrison, etc. and as a consequence the theoretical reference rate $(r_t)_{t \geq 0}$ was named "risk free rate" as a consequence? Apr 20, 2017 at 11:23