Arbitrage-free pricing under funding costs extends classical no-arbitrage theory to settings where multiple interest rates coexist: risk-free rate rD, funding rates rf^{pm}, repo rates rr^{pm}, and collateral rates rcol^{pm}. The presence of lending-borrowing spreads (rf^- > rf^+, rr^- > rr^+) makes the hedging portfolio value process nonlinear, so classical linear pricing theory does not directly apply.
In the bilateral hedging framework of Sekine and Tanaka (2020), the seller’s replication cost Y^+(0) and the buyer’s replication cost Y^-(0) are defined via nonlinear BSDEs with drivers f^+ and f^- respectively. The arbitrage-free price interval is [Y^-(0), Y^+(0)].
Sufficient conditions for no-arbitrage (Theorem 4): In addition to the natural spread orderings (epsilon_f >= 0, epsilon_r >= 0, rcol^- >= rcol^+), two conditions are required:
- Hazard rate dominance: h_i >= rf^- - rD - (rr^+ - rD)(sigma_i sigma^{-1} 1)^+ + (rr^- - rD)(sigma_i sigma^{-1} 1)^- for i = 1,2. This ensures that the credit spread is large enough to compensate for funding-repo mismatches.
- Funding-collateral ordering: rf^+ >= rcol^-, meaning the lending rate exceeds the collateral receiving rate.
These conditions are rather restrictive (Remark 14): violating them is realistic in practice, suggesting that some degree of “acceptable arbitrage” may be inherent in post-crisis markets. The proof uses the comparison theorem for BSDEs to show monotonicity of Y^+ in the payoff, establishing that Y^+(0) is indeed the minimal superhedging price.
When all rates coincide (rD = rf = rr = rcol), the model collapses to the classical Black-Scholes-Merton setting with a single risk-free rate, and Y^+ = Y^- = V_hat (the unique arbitrage-free price).