The differential rates problem is a canonical example of nonlinear pricing in quantitative finance, where the borrowing rate r^b exceeds the lending rate r^l for cash balances in a self-financing trading strategy. This asymmetry in funding costs introduces a nonlinearity in the pricing PDE because the effective discount rate depends on the sign of the cash position, which in turn depends on the unknown portfolio value.
The BSDE generator for differential rates takes the form f(t, X_t, Y_t, Pi_t) = -r^l Y_t + (r^b - r^l)(sum_i pi_i(t) - Y_t)^+, where Pi_t is the vector of risky portfolio holdings and Y_t is the total portfolio value. When the portfolio is leveraged (sum pi_i > Y_t, so the cash position is negative), the negative cash balance accrues interest at the higher borrowing rate r^b; when the portfolio has excess cash (sum pi_i ⇐ Y_t), cash grows at the lending rate r^l. This max(., 0)^+ operator makes f non-differentiable and nonlinear in Y, so standard linear pricing breaks down: price(t, X; g) is not equal to -price(t, X; -g), producing distinct upper and lower prices.
The differential rates problem serves as a key test case for deep BSDE methods because it is one of the simplest genuinely nonlinear pricing problems with known benchmark solutions. Yu, Hientzsch, Ganesan (2020) derived exact analytical backward step solutions by case-splitting on the borrowing/lending condition and showed these agree with Taylor approximations to within 10^{-5}. Their results on call combinations and straddles match the HJB PDE benchmarks of Forsyth and Labahn (2007), validating the deep BSDE approach for nonlinear financial problems that are intractable by standard Monte Carlo.
Key Details
- Generator: f(t,X,Y,Pi) = -r^l Y + (r^b - r^l)(sum pi_i - Y)^+
- Two regimes: borrowing (sum pi_i > Y, effective rate r^b) and lending (sum pi_i ⇐ Y, effective rate r^l)
- Nonlinearity source: the max operator creates a kink; the effective discount rate depends on the solution Y itself
- Upper vs lower price: pricing the long position g(X_T) and the short position -g(X_T) gives different prices due to nonlinearity
- Test cases: call combination (long call K=120, short 2 calls K=150, T=0.5) from Han, Jentzen, E (2017); straddle (K=100, T=1) from Forsyth and Labahn (2007)
- Benchmark: Forsyth-Labahn fully implicit and Crank-Nicolson HJB PDE solvers with 101 spatial nodes
- Relevance to XVA: differential funding rates are a component of FVA (funding valuation adjustment), connecting to the broader Deep BSDE methods for XVA programme
Textbook References
The xVA Challenge (Gregory, 2020)
- Section 18.2.4 (pp. 535—537): Symmetric FVA assumes borrowing and lending at the same unsecured rate; the adjustment reduces to simple discounting at the funding rate. This is the special case r^b = r^l of the differential rates problem.
- Section 18.3.1 (pp. 551—552): Asymmetric FVA arises from differential borrowing and lending rates: borrow at unsecured rate, lend at OIS (fully asymmetric) or at a shorter-term unsecured rate (partially asymmetric). The four cases listed (no FVA, symmetric, asymmetric, partially asymmetric) correspond directly to different parameter regimes of the differential rates BSDE.
- Section 18.2.6 (pp. 546—548): The FVA debate connects to the differential rates problem. Hull and White (2012a) argue that FCA is cancelled by a “DVA2” benefit from defaulting on general funding liabilities, recovering a CVA + DVA framework with no funding asymmetry. The symmetric FVA (shareholder value) view and the CVA + DVA (total firm value) view correspond to different assumptions about the effective lending rate.