Chapter 30: Convexity, Timing, and Quanto Adjustments

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Why Adjustments Are Needed

Standard models (Black, Black–Scholes–Merton) assume the underlying is lognormal under the appropriate measure. When this assumption doesn't hold exactly, or when we need to switch measures, adjustment terms are required.

Convexity Adjustments

A convexity adjustment modifies a forward rate to be the expected future rate under the right measure.

Forward Rate vs. Futures Rate

Forward rates and futures rates differ because futures are settled daily:

$rate−12σ2t1t2\text{Forward rate} = \text{Futures rate} - \frac{1}{2}\sigma^2 t_1 t_2$

where:

$t_1$ = time to maturity of the futures contract
$t_2$ = time to maturity of the underlying rate
$σ\sigma$ = standard deviation of the change in the short rate

In practice, the convexity adjustment can be several basis points, especially for long-dated contracts.

LIBOR-in-Arrears Adjustment

When a rate is paid at the reset date (in arrears) rather than at the end of the accrual period, a convexity adjustment applies. The expected rate in arrears is higher than the forward rate by:

$Adjustment=σ2F2τt1+Fτ\text{Adjustment} = \frac{\sigma^2 F^2 \tau t}{1 + F \tau}$

where $τ\tau$ is the accrual period, $F$ is the forward rate, $t$ is time to the reset date.

CMS (Constant Maturity Swap) Adjustment

For derivative payoffs tied to a longer-term swap rate (e.g., 10-year CMS rate), the convexity adjustment is more complex. The swap rate is not a linear function of zero-coupon bond prices — the nonlinearity creates a convexity effect that can be substantial.

Timing Adjustments

When expected value should be calculated under one measure but for convenience it's calculated under another, a timing adjustment is needed.

Consider a derivative that pays off at time $T$ , but we want to compute the expected payoff under the $T^*$ -forward measure. The adjustment:

$ET∗[VT]=ET[VT]⋅exp⁡(ρσVσP[1−e−a(T−T∗)]a)E^{T^*}[V_T] = E^{T}[V_T] \cdot \exp\left(\rho \sigma_V \sigma_{P} \frac{[1 - e^{-a(T-T^*)}]}{a}\right)$

where $ρ\rho$ = correlation between the payoff and the bond price, and $σV\sigma_V$ , $σP\sigma_P$ are volatilities. The adjustment is zero when the payoff is uncorrelated with interest rates ( $ρ=0\rho = 0$ ).

Quantos

A quanto is a derivative where the payoff is in a different currency from the underlying asset. Example: a call option on the Nikkei 225 index where the payoff is in US dollars, not yen.

For a quanto call:

$S_0 e^{-r_f T} e^{\rho \sigma_S \sigma_X T} N(d_1) - K e^{-rT} N(d_2)$

The quanto adjustment to the drift of the underlying is:

$μadjusted=μ−ρσSσX\mu_{\text{adjusted}} = \mu - \rho \sigma_S \sigma_X$

where:

$σS\sigma_S$ = volatility of the asset (in foreign currency)
$σX\sigma_X$ = volatility of the exchange rate
$ρ\rho$ = correlation between asset return and exchange rate return

Intuition: If the asset and the exchange rate are positively correlated ( $ρ>0\rho > 0$ ), the dollar value of the asset is more volatile, so the quanto call is worth more. The adjustment adds the covariance term to the drift.

Siegel's Paradox

Consider a currency forward: The forward exchange rate $F_0 = S_0 e^{(r-r_f)T}$ . If we reverse the currencies, the forward rate should be exactly the reciprocal ( $1/S_0$ forward), but due to Jensen's inequality and the dynamics of the exchange rate, perfect symmetry does not hold. This is Siegel's paradox, resolved by noting the expectation of $1/S_T$ under the domestic measure is not $1/E[S_T]$ .