Simple example of a funding valuation adjustment?

Question

I'm still a bit confused on how a funding valuation adjustment is actually computed. So I'm looking for a simple example of a funding valuation adjustment, preferably a binomial or discrete model, with terms clearly defined mathematically whenever possible.

Does anyone have such an example to illustrate?

Answer 1

To discuss Funding Valuation Adjustments (FVA) it is first necessary to describe a situation in which such an adjustment would be needed. In here we will take as an example collateral mismatches, which is a common case. For a conceptual treatment of FVA and collateral mismatches refer to Ruiz (2013).

We borrow the modified Black-Scholes framework of Piterbarg (2010). We assume we have sold a European derivative with maturity $T$ which value is given by $V(t)$ at time $t$ and that the transaction is collateralized, meaning that the counterparty we have traded with has to pledge to us a collateral with value $C(t)$ $-$ importantly, we do not assume here that $V(t)=C(t)$ as this would eliminate the need for an FVA. We make two assumptions on this collateral:

• First, it can be rehypothecated;
• Secondly, collateral is remunerated at a rate $r_C(t)$ at time $t$ $-$ the rate can be constant, deterministic or stochastic.

The derivative we have traded is written on an underlying asset which price is given by $S(t)$ at $t$, and this price follows Geometric Brownian Motion (GBM) dynamics. Finally, we assume we can borrow on an unsecured basis $-$ i.e. without collateral $-$ at a rate $r_F(t)$ such that $r_F(t) \geq r_C(t)$.

By hedging the derivative by holding appropriate quantities of the underlying asset and cash, Piterbarg shows that its value is given by:

$$ V(t) = \mathbb{E}_t\left[e^{-\int_t^Tr_C(u)du}V(T)\right]-\mathbb{E}_t\left[\int_t^Te^{-\int_t^ur_C(v)dv}(r_F(u)-r_C(u))(V(u)-C(u))du \right] $$

where $\mathbb{E}_t[\cdot]$ denotes the expectation operator conditional on the $\sigma$-algebra $\mathcal{F}_t$ $-$ i.e. the expectation conditional on market information at $t$.

Now, by designating by $V_{\text{CSA}}(t)$ $-$ where CSA stands for Credit Support Annex $-$ the value of the derivative if $C(t)=V(t)$ for all $t$ until maturity, we have:

$$ \begin{align} V(t) & = \underbrace{\mathbb{E}_t\left[e^{-\int_t^Tr_C(u)du}V(T)\right]}_{\text{CSA-value of derivative}}-\underbrace{\mathbb{E}_t\left[\int_t^Te^{-\int_t^ur_C(v)dv}(r_F(u)-r_C(u))(V(u)-C(u))du \right]}_{\text{FVA for collateral mismatch}} \\[6pt] & = V_{\text{CSA}}(t)-\text{FVA}(t) \end{align} $$

In this case, due to the mismatch between the derivative's value and the collateral the counterparty has to post at any time, we need (respectively can) to borrow (resp. lend) the difference between $V(t)$ and $C(t)$: this constitutes a funding valuation cost (resp. funding valuation benefit).

Now, for simplicity we assume the following:

• Both the collateral rate $r_C(t)$ and the unsecured funding rate $r_F(t)$ are constant for all $t$;
• The counterparty has to post a collateral amount equal to a certain portion $p$ of the derivative's value: $C(t) = pV(t)$.

The FVA can then be rewritten:

$$ \begin{align} \text{FVA}(t) & = (1-p)(r_F-r_C)\int_t^T\mathbb{E}_t[e^{-r_Cu}V(u)]du \\[6pt] & = (1-p)(r_F-r_C)\int_t^TV_{\text{CSA}}(u)du \end{align} $$

To compute the FVA you need the value of the derivative at all time points along its life. You immediately see that computing the funding adjustment $-$ and, in general, any valuation adjustment such as CVA or DVA $-$ is computationally a much harder problem than computing the selling price: as a derivative's dealer you only need the price of the derivative at time $0$ when you actually sell it to your counterparty, but for determining the FVA you need the price at a series of future times. That means that common numerical techniques used by banks to price derivatives, such as Monte Carlo methods or finite differences, cannot be applied to the problem of FVA because they are too computationally expensive.

In practice, a common approach is the following. You first need to simulate your underlying risk factors: in our example here, the derivative only depends on the asset $S(t)$ hence you need to simulate $m$ price paths $\omega_1,\cdots,\omega_m$ from $0$ to $T$ at a discrete time grid $0=t_0 < t_1 < \cdots < t_{n-1} < t_n = T$ $-$ if rates are stochastic then you also need to simulate rates. After having a sample of price paths for the underlying, you proceed as follows:

1. Working backwards, you first record for each price path $\omega_i$, $i \in \{1,\cdots,m\}$, the value of the derivative at maturity $t_n$ $-$ which is trivial since the payoff is known at maturity $-$ which we will designate by $v(\omega_i,t_n)$;
2. You then calculate for each price path the discounted value $v(\omega_i,t_{n-1})$ of the derivative's payoff at the previous grid point, $t_{n-1}$, by simply discounting using rate $r_C$:

$$ v(\omega_i,t_{n-1}) = e^{-r_C(t_n-t_{n-1})}v(\omega_i,t_n)$$

3. You then assume values $v(\omega_i,t_n)$ can be reliably approximated by a "simple" function $f_{n-1,n}(\cdot)$, such as a polynomial, applied to values $v(\omega_i,t_{n-1})$ and estimate that function from the simulated sample $-$ for example by Ordinary Least Squares:

$$ v(\omega_i,t_n) \approx f_{n-1,n}(v(\omega_i,t_{n-1})) \qquad \text{(1)}$$

4. Repeat recursively steps 1 to 3 until you have reached the current time and you have a set of simple pricing functions $f_{0,1},f_{1,2},\cdots,f_{n-1,n}$.

5. Let us designate by $w_0$ the price path for the derivative's value corresponding to the current market state $-$ i.e. such that $v(\omega_0,0)=v(S(0),0)=V_{\text{CSA}}(0)$ which can be obtained either by an analytic formula or numerical techniques. We can then recursively calculate the value of the derivative at each time step $t_1,\cdots,t_n$:

   • $v(\omega_0,t_1)=f_{0,1}(v(\omega_0,0))=f_{0,1}(v(S(0),0))=f_{0,1}(V_{\text{CSA}}(0))$;
   • $v(\omega_0,t_2)=f_{1,2}(v(\omega_0,t_1))=f_{1,2} \circ f_{0,1}(V_{\text{CSA}}(0))$;
   • Etc.

6. Defining $\Delta_{j,j+1}=t_{j+1}-t_j$ for all $j$, you can now calculate FVA with the following formula:

$$ \begin{align} \text{FVA}(t) & = (1-p)(r_F-r_C)\int_t^TV_{\text{CSA}}(u)du \\[6pt] & \approx (1-p)(r_F-r_C)\sum_{j=0}^{n-1}f_{j,j+1}(v(\omega_0,t_j))\Delta_{j,j+1} \qquad \text{(2)} \end{align}$$

References

Piterbarg, V. (2010). "Funding beyond discounting: collateral agreements and derivatives pricing", Risk

Ruiz, I. (2013). "FVA Demystified: CVA, DVA, FVA and their interaction (Part I)", iRuiz Consulting

Ruiz, I. (2014). "FVA Calculation and Management: CVA, DVA, FVA and their interaction (Part II)", iRuiz Consulting

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[Edit 11/09/17]

Note that the procedure outlined above might be problematic for coarse time grids: for example, if the time step is $1$ year, the derivative's price today might not be a good predictor of the derivative's value in $1$ year time.

An alternative methodology would be as follows: instead of predicting the derivative's price at $t_{j+1}$ from the derivative's price at $t_j$, we could predict the derivative's price at $t_{j+1}$ from the underlying price at $t_{j+1}$. Letting $S(\omega_i,t_j)$ be the asset's simulated price on path $\omega_i$ at time $t_j$, equation $\text{(1)}$ above would be replaced by:

$$ v(\omega_i,t_j) \approx f_j\left(S(\omega_i,t_j)\right) $$

Hence equation $\text{(2)}$ would become:

$$ \begin{align} \text{FVA}(t) \approx (1-p)(r_F-r_C)\frac{1}{m'}\sum_{j=0}^n\sum_{i=0}^{m'}f_j(S(\omega_i,t_j))\Delta_{j,j+1} \end{align}$$

where $S(\omega_i,t_0)=S(\omega_i,0)=S(0)$ for all $i$ and $(S(\omega_i,t_j)_{0\leq j\leq n})_{0\leq i\leq m'}$ is a new simulated sample of the asset $S(t)$.