Discounting and XVA Exposure Modeling for Securities Lending

# Discounting and XVA for Securities Lending


# Discounting and XVA for Securities Lending












I am working computing exposures for securities lending products and I would like to know how to best incorporate these products in an XVA framework.

Currently I am modelling these as forwards (Loan - Security).

What is the appropriate discount rate to use? I know in a collateralised derivatives pricing framework, one would use the renumeration rate of the collateral. Does the same apply to securities lending?

What about cases where the currency of the collateral is different from the currency of the loan, what is the appropriate discount rate?

What XVAs are applicable to securities lending transactions?

## Answer by almost_surely_ (score 0)

https://quant.stackexchange.com/a/85806

### Short answer

Your forward decomposition is a reasonable starting point for the principal legs, but it hides the thing that actually matters. Two reframings before the specifics:

- In an SFT the collateral secures the market value of the security, not the mark-to-market of the trade. That is why you cannot transplant the collateralised-derivative discounting result mechanically. In a collateralised swap, collateral tracks MTM, so the collateral rate is the discount rate. In securities lending, MTM is roughly zero at inception while collateral sits at 102–105% of the security value. Different object, different treatment.

- The exposure is not driven by the forward drift at all. It is a short-dated, out-of-the-money option on the gap between the security and the collateral over the margin period of risk. Correlation and currency mismatch move it by two orders of magnitude; the discount rate barely moves it.

### 1. Decompose the trade into three pieces

Principal legs — security out, collateral in. This is your forward, and it's the right shape for a term trade. The cash leg accrues at the agreed rebate rate; the security leg carries at the borrow/repo rate, not at OIS.

Fee leg — the lending fee (non-cash collateral) or reinvestment-minus-rebate spread (cash collateral). This is where the economic P&L lives.

The haircut/margin cushion — structurally over-collateralised in the lender's favour, which is why the borrower has positive exposure to you from day one. If you're only computing lender-side exposure you'll miss your own DVA and the borrower's CVA charge back to you.

### 2. Discounting

Yes, the principle carries over — but apply it leg by leg, not to the trade as a whole.

The security's financing rate is the borrow/repo rate. The forward is $F(0,T) = S_0 e^{(b - q)T}$ with $b$ the security-specific borrow rate and $q$ the dividend/coupon yield. For a general-collateral name $b \approx$ OIS; for a special, $b$ sits well below OIS and the spread $(\text{OIS} - b)$ is the specialness. That spread is the entire economics of the trade — if you discount the security leg at OIS you have priced away the reason the trade exists. This is the Duffie (1996) specialness result, and it's the single most common error I see when derivatives infrastructure gets pointed at SFTs.

The cash leg discounts at the rebate rate, because that is the rate at which the cash collateral is actually remunerated. Here the analogy with CSA discounting does hold cleanly.

The fee leg is usually the odd one out. Lending fees typically accrue daily and bill monthly in arrears, and the collateral covers the security value — not the accrued fee. So the accrued-but-unpaid fee is genuine unsecured exposure to the borrower, and should be discounted on your funding curve with CVA applied, even though the principal is over-collateralised. Small in notional, but it's the one piece of the trade with real uncollateralised credit exposure, and it's routinely missed.

### 3. Currency mismatch between collateral and loan

For discounting: the same rule as a cross-currency CSA. A cash flow in currency $D$ secured by collateral in currency $F$ remunerated at $r^c_F$ discounts on the "$D$-collateralised-in-$F$" curve — $r^c_F$ carried into $D$ through the cross-currency basis, which in practice means building the curve from $r^c_D$ plus the $D/F$ basis spread. If your collateral schedule permits a choice of currency (or a choice among eligible securities), the poster holds a cheapest-to-deliver option and you owe yourself a ColVA for it. That optionality is often worth more than the basis adjustment itself.

But the discounting is the small half of the question. The currency mismatch introduces FX directly into the exposure, and for low-volatility loans it dominates completely. The gap exposure collapses to an option on $S/C$ struck at the margin ratio $h$, with effective volatility

$$\sigma_{\text{eff}} = \sqrt{\sigma_S^2 + \sigma_C^2 - 2\rho\,\sigma_S\sigma_C}.$$

Numbers, 5-day MPoR, as a percentage of the lent security value:

| Case | $\sigma_{\text{eff}}$ | EE | 99% PFE |
| Govt bond 5% vol, 102% cash, same ccy | 5.0% | 0.0005% | 0.00% |
| Govt bond 5% vol, 102% cash, FX vol 10% | 11.2% | 0.079% | 1.72% |
| Govt bond 5% vol, 102% cash, FX vol 15% | 15.8% | 0.230% | 3.29% |

Lending a government bond against same-currency cash is essentially riskless at a 2% haircut. Take the collateral in another currency and the expected exposure rises by a factor of 152. The bond's own volatility becomes a rounding error next to the FX. Basel's 8% supervisory haircut for currency mismatch under the comprehensive approach exists for exactly this reason, and it is not conservative padding — it's the dominant term.

### 4. Correlated non-cash collateral

Same formula, and the effect is just as violent in the other direction:

| Collateral | $\sigma_{\text{eff}}$ | EE | 99% PFE |
| Equity 30% vol vs equity collateral, $\rho = 0.9$ | 13.4% | 0.003% | 0.00% |
| $\rho = 0.5$ | 30.0% | 0.266% | 5.23% |
| $\rho = 0.0$ | 42.4% | 0.715% | 9.71% |
| $\rho = -0.3$ | 48.4% | 0.970% | 11.91% |

A factor of ~300 between the best and worst correlation assumption, with identical haircuts. Two consequences:

- Correlation is your dominant model risk, far ahead of the discount curve. It deserves the reserve, the sensitivity, and the stress test.

- Note the $\rho = 0.5$ row reproduces the cash-collateral number exactly. That's not a coincidence: with equal volatilities, $\sigma_{\text{eff}} = \sigma_S$ at $\rho = 0.5$. Useful sanity check when validating an SFT exposure engine — if your model doesn't reproduce it, the correlation isn't entering where you think it is.

### 5. Which XVAs apply

|  | Applies? | Comment |
| CVA / DVA | Yes | Small in normal conditions, MPoR-gap-driven. DVA is real and often ignored: the haircut leaves the borrower structurally exposed to you. |
| FVA | Yes | Funding the haircut, funding the security in a matched book, and the cash reinvestment leg. |
| ColVA | Yes — arguably the primary one | Rebate rate vs your funding rate, plus cheapest-to-deliver optionality across eligible collateral. For SFTs this is usually larger than CVA. |
| MVA | Sometimes | SFTs are generally outside UMR, so there's often no regulatory IM. Where SFTs clear through a CCP, or where haircuts are funded like IM, the economics are identical — book it consistently, wherever it lands. |
| KVA | Yes | SFT exposure under the comprehensive approach or the SFT formula. Whether SFTs attract a CVA capital charge is jurisdiction-dependent and has been moving: historically exempted in the EU under CRR Art. 382 unless material, in scope under Basel where fair-valued. Check your own rules rather than a textbook. |

Wrong-way risk deserves its own line, because securities lending has a structural form of it that derivatives don't. The borrower is typically short the security. If the name squeezes, the security rallies, the borrower's losses mount and your exposure grows — simultaneously, and for the same reason. That is textbook WWR with a mechanical driver rather than a statistical one, and it concentrates in exactly the hard-to-borrow names that generate the best fees. A 60% vol special at a 105% haircut carries an EE of 1.52% and a 99% PFE of 16.3% before any WWR uplift. Independence between exposure and default is not a defensible assumption there.

Taking collateral correlated with the borrower's own credit — same sector, same sovereign — stacks a second WWR layer on top.

### 6. The trap that will cost you the most: open vs term

Most securities lending is open — recallable by the lender and returnable by the borrower at a day or two's notice. Contractual maturity is effectively overnight; behavioural tenor is often months.

- Model it as a term forward to a fixed maturity and you will materially overstate exposure and CVA.

- Model it at contractual maturity (1 day) and you understate it, because the balance rolls.

Neither is right. The workable approach is a behavioural tenor with an explicit rollover assumption, calibrated to observed balance decay by counterparty and collateral type, and then floored at the regulatory MPoR. Whatever you choose, make the assumption explicit and reserve against it — it moves the answer more than any curve decision in this question. Regulatory minimum MPoR for repo-style transactions with daily remargining is 5 business days (against 10 for OTC derivative netting sets), extended for large or disputed netting sets and illiquid collateral.

### Reproducing the numbers

The gap exposure $\max(S_{t+\delta} - C_{t+\delta}, 0)$ with $C_t = h S_t$ at the last margin call is exactly a call on $S/C$ struck at $h$, so there's a closed form and no need to simulate:

```
import numpy as np
from scipy.stats import norm

def gap_exposure(h, vol_S, vol_C=0.0, rho=0.0, mpor_days=5):
    """EE and 99% PFE as % of lent security value, at the moment of default."""
    v = np.sqrt(vol_S**2 + vol_C**2 - 2*rho*vol_S*vol_C)
    T = mpor_days/252
    d1 = (np.log(1/h) + 0.5*v**2*T)/(v*np.sqrt(T)); d2 = d1 - v*np.sqrt(T)
    ee = norm.cdf(d1) - h*norm.cdf(d2)
    q = np.exp(-0.5*v**2*T + v*np.sqrt(T)*norm.ppf(0.99))
    return v, 100*ee, 100*max(q-h, 0)

for lbl, args in [("bond, same ccy",       (1.02, 0.05)),
                  ("bond, FX vol 10%",     (1.02, 0.05, 0.10, 0.0)),
                  ("equity coll, rho=0.9", (1.05, 0.30, 0.30, 0.9)),
                  ("equity coll, rho=-0.3",(1.05, 0.30, 0.30, -0.3))]:
    v, ee, pfe = gap_exposure(*args)
    print(f"{lbl:<24} eff vol {v:6.1%}   EE {ee:7.4f}%   PFE99 {pfe:6.2f}%")
```

```
bond, same ccy           eff vol   5.0%   EE  0.0005%   PFE99   0.00%
bond, FX vol 10%         eff vol  11.2%   EE  0.0792%   PFE99   1.72%
equity coll, rho=0.9     eff vol  13.4%   EE  0.0031%   PFE99   0.00%
equity coll, rho=-0.3    eff vol  48.4%   EE  0.9702%   PFE99  11.91%
```

Use this as the analytic benchmark when you validate a full simulation engine. If your Monte Carlo doesn't reproduce these to within noise on a single flat-vol trade, the discrepancy is in your model, not in the market.

### References

- D. Duffie, Special Repo Rates, Journal of Finance 51(2), 1996 — specialness and why the borrow rate, not OIS, carries the security leg.

- V. Piterbarg, Funding Beyond Discounting: Collateral Agreements and Derivatives Pricing, Risk, February 2010 — the collateral-rate discounting result and, importantly, the assumptions it needs.

- A. Green, XVA: Credit, Funding and Capital Valuation Adjustments, Wiley, 2015 — ColVA and collateral optionality.

- J. Gregory, The xVA Challenge, 4th ed., Wiley, 2020 — MPoR, margined exposure, and wrong-way risk.

- Basel CRE 50–51 for the SFT exposure treatment, MPoR floors and supervisory haircuts, including the 8% currency-mismatch haircut.