Due to the huge supply of yen, the funding bias for USD/JPY is such that it is cheaper to borrow yen versus dollars (i.e.s^JPY BRL ≠ 0) Consider now a cross currency swap with one leg paying floating in yen (with no spread) and the other paying floating in Brazilian reais plus a spread s^*. What would the fair value spread s^* be?

Section 12.2.1

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When pricing cross-currency structures (especially involving the Japanese yen or emerging market currencies), it may be necessary to take funding bias into account. This means we need to con- struct curves that reprice cross-currency basis swaps. Standard cross- currency basis swaps are quoted vis-à-vis the US dollar. Cross-Currency Swap Spread (bp) 20.0 10.0 0.0 -10.09 -20.0 -30.0 -40.0 -50.0 5 10 -DCCY Maturity (yr) Further, it is also necessary that we can reprice standard swaps in the domestic currency. For simplicity, for now, let us consider the world prior to 2008, where we can fund at Libor. Specifically, for US dollars (i.e. the base currency), we can let the forecasting curve be the same as the discounting curve, and boot- strap our discounting curve DUSD (0, T) by sequentially matching the prices of deposits, futures, and swaps as before. For example, hav- ing obtained prior discount factors, for a swap with maturity T; and coupon Rj, we require D(0, T;) to satisfy j Rj Σ¹¡D(0, T;) = D(0, To) — D(0, T;) i=1 15 as we did earlier. For other currencies, it is necessary to bootstrap by simultane- ously matching the prices of swaps and cross-currency swaps for the same expiry. Specifically, we build a discounting curve DCCY (0, T) and a forecasting curve DCCY (0,7). For maturity T; swap constraint given swap rate RCCY is TM, the - DUSD 20 j M CCY ROCYTDCCY (0, T₁) = Σ £ccY (0, T-1, T )T DCCY (0, Ti ), i=1 i=1 25 where the Libor rate is obtained from the forecasting curve via fccy (0, T₁-1, T) bycx (0, 17) - 1) / 2, DCCY and the cross-currency basis swap constraint given cross-currency basis spread SCCY is (0, To) + M (0, To) + (CCY (0, T₁-1, Tr) + SCY) T₂ DCCY (0, T*) i=1 + DCCY (0, TM) M -USD/JPY --GBP/US -EUR/US i=1 ƒUSD (0, Ti-1, Ti* ) Ti DUSD D(0, Ti) + DUSD (0, TM). Notice that there is no need to take account of the spot exchange rate Xo (number of units of currency CCY per dollar) because we would do the swap based on notional 1 for dollars and notional Xo for the other currency. (See Section 2.3.1 for a more detailed explanation.) We have enough degrees of freedom since we have two con- straints RCCY and SCCY at each time point T; = TM, and two free parameters DCCY (0,T;) and DCCY (0, Tj). The above methodology extends to the deposit part of the domestic curve (which depends on DCCY (0,T;) only) and the futures part of the domestic curve (which depends on DCCY (0,T;) only), whilst we still have to match cross-currency basis swaps for the same maturities. In this way, we have constructed a forecasting and discounting curve that could simultaneously reprice domestic interest rates instruments and cross-currency swaps against the dollar. Since standard cross- currency basis swaps are quoted against the dollar, our curves will now also reprice all other cross-currency swaps quoted consistently.