Notwithstanding the discussion in Section 12.1.4, since we forecast based on Libor, the martingale equation only holds
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Notwithstanding the discussion in Section 12.1.4, since we forecast based on Libor, the martingale equation only holds for risky discounting at the creditworthiness of Libor counterparts. As such, OIS discounting induces a quanto correction. In the developed world, it is common for interest rates to be lowered in times of crisis to simulate growth. As such, what should the direction of the quanto correction be? (Note that this is really just a theoretical discussion, since such a quanto correction is accounted for by market quotes which are based on Libor forecasting and OIS discounting.)
Section 12.1.4,
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From our discussion earlier, it can be seen that funding is no longer an afterthought in the pricing of derivatives. It should be clear from the earlier graphs of the OIS spread and Libor basis spreads that funding spreads are not constant. Stochastic funding can lead to increased volatility of an underlying, and thus can increase the price of an option. More interestingly, stochastic funding can even affect the for ward price of an underlying, if the funding spread is correlated with the underlying. This is not a farfetched point. Funding spreads (over the riskfree rate) are likely to be highest in times of finan cial crisis. In safe haven (e.g. G7) economies (that are not on the brink of default), central banks will attempt to stimulate economic growth by cutting interest rates. So, funding spreads can be quite negatively correlated with interest rates. Alternatively, in a nonsafe haven economy on the brink of collapse, interest rates may need to be raised in times of crisis either as borrowers will no longer lend or because international bodies (e.g. the International Monetary Fund) demand such measures as a condition for financial assistance. So, cor relation between funding spreads and interest rates can be positive instead. Piterbarg [Pit10] shows that stochastic funding spreads can lead to a quanto effect on the forward price of an underlying. Let Xt be the value of an underlying, rt be the riskfree rate for collateralised borrowing, r be the (risky) rate for uncollateralised borrowing by some counterparty A (which could but need not be Liboreligible), and sr rt be the spread. Consider a forward contract with maturity T and strike K with a CSAcounterparty. Its value at time t is EleStrudu ¹u (XT  K)], so the fair value CSAforward price of the asset is: FCSA = ELerudu XT]/EX[e² rudu] ET [XT], where we have changed from the riskneutral measure to the Tforward measure (with numeraire asset being the discount bond with maturity T given by D(t, T) = E?[eSerudu], and discounting is based on the OIS curve). Consider instead the same forward contract with counterparty A (for which no CSA arrangement is in place). The value of the forward is EleSrdu (XT  K)]. So the fair value forward price is given by FnoCSA = Eedu XT]/D¹(t, T) = where DA (t, T) E[edu] is the discount curve for counter party A, and we have again changed from the riskneutral measure to the Tforward measure. ET Then the quanto adjustment to the forward rate as a consequence of noncollateralised funding (from our counterparty A) is EleSi ruduſi (r^ru)du XT]/Dª(t,T), D(t, T) DA(t, T) FnoCSA (t,T)  FCSA(t, T) = Correction (%) D(t, T)  B² [(D41²1) X7] = e x XT DA(t, T) since we have Elle ET (D(t, T) 0.40 0.30 0.20 0 10 0.00 0.100 0.20 0.30 0.40 D(t, T)  sdu DA(t, T) e  ST s du XT] Et B² [( DA(1t, T) 5(eli sidu  El (ef² sâ du))) t, T))] X (XT FCSA (t, T))  D(t, T) cov (efsdu, XT), DA(t, T)  S² s^ du] = 5 e² at du _ 1) (XT  FOSA(t, T))]  10 du 1 Thus, we see that the correlation between the funding spread s and the underlying XT induces a quanto adjustment on the fair forward price in the nonCSA case (visàvis the CSA case). Considering that in the interest rates world, 30year swaps and even those with longer maturities are common, this quanto effect can be large. If we posit a short rate model for interest rates (with mean reversion 2% and volatility 0.8%) and a corresponding short rate model for the funding spread (with mean reversion 5% and volatility 0.7%), then we see that over 30 years, a quanto adjustment in excess of 0.25% is quite possible. (This should be compared to a 30year swap rate of just over 3% for euros in 2010.) Figure 12.5 shows the effect of stochastic funding on the fair value Libor rate for varying maturities. and FCSA ETXT] (from earlier). Maturity (yr)  20. 25 40%Corr 20% Corr 0% Corr 20% Corr 40% Corr Finally, we should remark that since stochastic funding influ ences the forward, it also influences the location of the atthemoney volatility. Further, the dynamics of the stochastic spread contributes to the dynamics of the underlying, and in that way influences the smile for options on the underlying.
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