Quanto Adjustments¶

A quanto (quantity-adjusted) derivative pays the holder a rate or price denominated in one currency but settled in a different currency at a fixed exchange rate. For example, a quanto cap might pay the EUR EURIBOR rate but settle in USD. Because the payoff depends on a foreign rate while the settlement currency differs, the pricing measure must account for the correlation between the foreign rate and the exchange rate. This correlation-driven correction is the quanto adjustment. This section derives the adjustment from a cross-currency measure change, presents the quanto correction formula, and applies it to caps and floors.

Quanto Products¶

Standard vs. Quanto Payoff¶

Standard caplet (domestic currency, domestic rate):

\[ \delta_i \max(L_i^d(T_i) - K, 0) \]

paid in the domestic currency at $T_{i+1}$. No cross-currency issues arise.

Quanto caplet (foreign rate, domestic settlement):

\[ \delta_i \max(L_i^f(T_i) - K, 0) \quad \text{paid in domestic currency at } T_{i+1} \]

The payoff depends on the foreign LIBOR rate $L_i^f$ but is paid in the domestic currency at a predetermined exchange rate (typically 1:1 or a contractual rate). No actual FX conversion takes place at maturity.

Why a Quanto Adjustment Is Needed¶

The foreign forward rate $L_i^f(t)$ is a martingale under the foreign $T_{i+1}$-forward measure $\mathbb{Q}^{f,T_{i+1}}$ (with numéraire $P^f(t, T_{i+1})$). Pricing the quanto caplet requires computing the expectation under the domestic $T_{i+1}$-forward measure $\mathbb{Q}^{d,T_{i+1}}$ (with numéraire $P^d(t, T_{i+1})$). Since these measures differ, the foreign rate acquires a drift adjustment under the domestic measure. This drift is the quanto adjustment.

Cross-Currency Measure Change¶

Setup¶

Let:

$P^d(t, T)$, $P^f(t, T)$: domestic and foreign zero-coupon bond prices
$X(t)$: spot exchange rate (units of domestic currency per unit of foreign currency)
$L_i^f(t)$: foreign forward LIBOR rate for period $[T_i, T_{i+1}]$

The Exchange Rate Process¶

Under the domestic risk-neutral measure $\mathbb{Q}^d$, the exchange rate follows:

\[ \frac{dX(t)}{X(t)} = (r^d(t) - r^f(t)) \, dt + \sigma_X(t) \, dW_X^d(t) \]

where $\sigma_X(t)$ is the exchange rate volatility and $W_X^d$ is a Brownian motion under $\mathbb{Q}^d$.

Radon--Nikodym Derivative¶

The measure change from $\mathbb{Q}^{f,T_{i+1}}$ to $\mathbb{Q}^{d,T_{i+1}}$ involves two steps:

Step 1: Change from foreign risk-neutral $\mathbb{Q}^f$ to domestic risk-neutral $\mathbb{Q}^d$. The Radon--Nikodym derivative is:

\[ \frac{d\mathbb{Q}^d}{d\mathbb{Q}^f}\bigg|_{\mathcal{F}_t} = \frac{X(t) \, B^d(t)^{-1}}{X(0) \, B^f(t)^{-1} \cdot (B^d(0)/B^f(0))} \]

where $B^d$, $B^f$ are the domestic and foreign money market accounts.

Step 2: Change from $\mathbb{Q}^d$ to $\mathbb{Q}^{d,T_{i+1}}$ via the standard forward measure change:

\[ \frac{d\mathbb{Q}^{d,T_{i+1}}}{d\mathbb{Q}^d}\bigg|_{\mathcal{F}_t} = \frac{P^d(t, T_{i+1})}{B^d(t) \, P^d(0, T_{i+1})} \]

Girsanov Drift Adjustment¶

Under $\mathbb{Q}^{f,T_{i+1}}$, the foreign forward rate is a driftless martingale:

\[ \frac{dL_i^f(t)}{L_i^f(t)} = \sigma_i^f(t) \, dW_i^{f,T_{i+1}}(t) \]

Under $\mathbb{Q}^{d,T_{i+1}}$, applying Girsanov's theorem for the combined measure change, a drift correction appears:

\[ \frac{dL_i^f(t)}{L_i^f(t)} = \mu_{\text{quanto}}(t) \, dt + \sigma_i^f(t) \, dW_i^{d,T_{i+1}}(t) \]

where the quanto drift is:

\[ \mu_{\text{quanto}}(t) = -\rho_{L,X}(t) \, \sigma_i^f(t) \, \sigma_X(t) \]

Here $\rho_{L,X}(t)$ is the instantaneous correlation between the foreign forward rate $L_i^f$ and the exchange rate $X$.

The Quanto Correction Formula¶

Derivation¶

Since $L_i^f(t)$ is lognormal under $\mathbb{Q}^{f,T_{i+1}}$ and acquires a constant drift $\mu_{\text{quanto}}$ under $\mathbb{Q}^{d,T_{i+1}}$, the terminal distribution under the domestic measure is:

\[ L_i^f(T_i) = L_i^f(0) \exp\!\left(\mu_{\text{quanto}} T_i - \frac{1}{2}(\sigma_i^f)^2 T_i + \sigma_i^f W_i^{d,T_{i+1}}(T_i)\right) \]

The expectation under the domestic forward measure is:

\[ \mathbb{E}^{\mathbb{Q}^{d,T_{i+1}}}[L_i^f(T_i)] = L_i^f(0) \, e^{\mu_{\text{quanto}} T_i} \]

The Quanto-Adjusted Forward Rate¶

Theorem (Quanto Forward Rate Adjustment)

Under the domestic $T_{i+1}$-forward measure, the quanto-adjusted foreign forward rate is:

\[ L_i^{f,\text{quanto}}(0) = L_i^f(0) \, \exp\!\left(-\rho_{L,X} \, \sigma_i^f \, \sigma_X \, T_i\right) \]

For small corrections, the first-order approximation gives:

\[ L_i^{f,\text{quanto}}(0) \approx L_i^f(0) \left(1 - \rho_{L,X} \, \sigma_i^f \, \sigma_X \, T_i\right) \]

The quanto correction is:

\[ \boxed{\text{Quanto Adj} = -\rho_{L,X} \, \sigma_i^f \, \sigma_X \, T_i \cdot L_i^f(0)} \]

Sign and Interpretation¶

If $\rho_{L,X} > 0$ (foreign rates and the exchange rate are positively correlated, meaning when foreign rates rise, the foreign currency strengthens), the quanto adjustment is negative: the quanto forward rate is lower than the standard foreign forward rate
If $\rho_{L,X} < 0$, the adjustment is positive
If $\rho_{L,X} = 0$, there is no quanto effect

Financial intuition: When $\rho_{L,X} > 0$, high foreign rates coincide with a strong foreign currency. A quanto product does not benefit from this FX effect (settlement is at a fixed rate), so the effective forward rate is lower. The adjustment compensates for the missing FX gain.

Quanto Cap and Floor Pricing¶

Quanto Caplet via Black's Formula¶

The quanto caplet is priced using Black's formula with the quanto-adjusted forward rate as the underlying:

\[ \text{Quanto Caplet}_i = \delta_i \, P^d(0, T_{i+1}) \left[L_i^{f,\text{quanto}}(0) \, N(d_1) - K \, N(d_2)\right] \]

where:

\[ d_1 = \frac{\ln(L_i^{f,\text{quanto}}(0)/K) + \frac{1}{2}(\sigma_i^f)^2 T_i}{\sigma_i^f \sqrt{T_i}} \]

\[ d_2 = d_1 - \sigma_i^f \sqrt{T_i} \]

Note that:

The discounting uses the domestic zero-coupon bond $P^d(0, T_{i+1})$ (settlement in domestic currency)
The forward rate is the quanto-adjusted foreign rate $L_i^{f,\text{quanto}}(0)$
The volatility is the foreign rate volatility $\sigma_i^f$ (unchanged by the quanto adjustment)

Quanto Floorlet¶

\[ \text{Quanto Floorlet}_i = \delta_i \, P^d(0, T_{i+1}) \left[K \, N(-d_2) - L_i^{f,\text{quanto}}(0) \, N(-d_1)\right] \]

Quanto Cap¶

\[ \text{Quanto Cap} = \sum_{i=0}^{n-1} \text{Quanto Caplet}_i \]

Each caplet uses its own quanto-adjusted forward rate with the appropriate fixing time $T_i$.

Worked Example¶

Quanto Cap Pricing

Setup: A USD-settled cap on 6-month EUR EURIBOR with strike 3.0%, notional EUR 10 million (settled in USD at 1:1 rate), 2-year maturity with semiannual caplets.

Parameters:

EUR forward rates: $L_1^f(0) = 3.2\%$, $L_2^f(0) = 3.4\%$, $L_3^f(0) = 3.5\%$
EUR caplet vols: $\sigma_1^f = 20\%$, $\sigma_2^f = 21\%$, $\sigma_3^f = 22\%$
EUR/USD FX vol: $\sigma_X = 10\%$
Correlation (EUR rates, EUR/USD): $\rho_{L,X} = 0.30$
Fixing times: $T_1 = 0.5$, $T_2 = 1.0$, $T_3 = 1.5$
Day count fraction: $\delta = 0.5$
USD discount factors: $P^d(0, 1.0) = 0.960$, $P^d(0, 1.5) = 0.941$, $P^d(0, 2.0) = 0.922$

Step 1: Compute quanto-adjusted forward rates

For caplet 1 ($T_1 = 0.5$):

$L_1^{f,q} = 0.032 \times \exp(-0.30 \times 0.20 \times 0.10 \times 0.5) = 0.032 \times e^{-0.003} \approx 0.032 \times 0.9970 = 3.190\%$

For caplet 2 ($T_2 = 1.0$):

$L_2^{f,q} = 0.034 \times \exp(-0.30 \times 0.21 \times 0.10 \times 1.0) = 0.034 \times e^{-0.0063} \approx 0.034 \times 0.9937 = 3.379\%$

For caplet 3 ($T_3 = 1.5$):

$L_3^{f,q} = 0.035 \times \exp(-0.30 \times 0.22 \times 0.10 \times 1.5) = 0.035 \times e^{-0.0099} \approx 0.035 \times 0.9901 = 3.465\%$

Step 2: Price each quanto caplet using Black's formula

For caplet 1: $F = 3.190\%$, $K = 3.0\%$, $\sigma = 20\%$, $T = 0.5$

$d_1 = \frac{\ln(3.190/3.0) + 0.5 \times 0.04 \times 0.5}{0.20\sqrt{0.5}} = \frac{0.0615 + 0.01}{0.1414} = 0.506$

$d_2 = 0.506 - 0.1414 = 0.365$

$\text{Caplet}_1 = 0.5 \times 0.960 \times [0.03190 \times 0.694 - 0.030 \times 0.643] = 0.480 \times [0.02214 - 0.01928] = 0.480 \times 0.00286 = 0.001373$

Step 3: Compare with non-quanto caplet

Without quanto adjustment: $F = 3.200\%$, giving a slightly higher caplet value. The quanto adjustment reduces the forward rate by about 1 bp, translating to a price reduction of approximately 0.5--1% of the caplet value.

Observation: The quanto adjustment is small (less than 1 bp on the forward rate) for this 6-month fixing horizon but grows linearly with time, reaching about 3 bp for a 5-year quanto cap.

Sensitivity Analysis¶

Dependence on Parameters¶

The quanto correction $\Delta L = -\rho_{L,X} \sigma_i^f \sigma_X T_i L_i^f(0)$ is:

Linear in correlation $\rho_{L,X}$: the most important parameter
Linear in FX volatility $\sigma_X$: higher FX vol means larger adjustment
Linear in rate volatility $\sigma_i^f$: higher rate vol amplifies the effect
Linear in time $T_i$: grows with the fixing horizon
Linear in the rate level $L_i^f(0)$: proportional correction

Typical Magnitudes¶

Fixing Horizon	Rate Vol	FX Vol	Correlation	Quanto Adj (bps)
1Y	20%	10%	0.30	$-0.6$
3Y	20%	10%	0.30	$-1.8$
5Y	20%	10%	0.30	$-3.0$
5Y	20%	10%	0.50	$-5.0$
10Y	25%	12%	0.40	$-12.0$

For long-dated quanto products, the adjustment can be substantial.

Correlation Uncertainty¶

Correlation Risk

The quanto adjustment depends critically on the rate-FX correlation $\rho_{L,X}$, which is notoriously difficult to estimate and unstable over time. Historical estimates can vary significantly depending on the estimation window. A 10-percentage-point change in correlation changes the 5-year quanto adjustment by about 1 bp in the forward rate. For large quanto portfolios, this uncertainty is a material source of model risk.

Extensions¶

Quanto Swaptions¶

A quanto swaption pays the foreign swap rate settled in domestic currency. The adjustment follows the same principle: the foreign swap rate $S^f(t)$ acquires a drift under the domestic annuity measure:

\[ S^{f,\text{quanto}}(0) \approx S^f(0) \left(1 - \rho_{S,X} \, \sigma_S^f \, \sigma_X \, T_0\right) \]

where $\rho_{S,X}$ is the correlation between the foreign swap rate and the exchange rate.

Differential Swaps (Diff Swaps)¶

A differential swap (or diff swap) pays the difference between a foreign and domestic floating rate, settled in domestic currency:

\[ \text{Payoff} = \delta_i (L_i^f(T_i) - L_i^d(T_i)) \]

paid at $T_{i+1}$ in domestic currency. The foreign rate requires the quanto adjustment while the domestic rate does not. The fair value involves the quanto-adjusted foreign forward minus the domestic forward.

Stochastic Correlation and Volatility¶

In practice, the correlation $\rho_{L,X}$ and the volatilities $\sigma_i^f$, $\sigma_X$ are not constant. More sophisticated models use:

Stochastic volatility for the exchange rate (e.g., Heston or SABR)
Local correlation models where $\rho_{L,X}(t)$ depends on the state
Multi-factor HJM models for the joint dynamics of domestic and foreign rates with stochastic FX

These extensions increase calibration complexity but can be important for long-dated quanto products.

Key Takeaways¶

A quanto adjustment arises when a derivative pays a rate in one currency but settles in another at a fixed exchange rate
The adjustment comes from a cross-currency measure change, which introduces a drift in the foreign rate under the domestic pricing measure
The quanto-adjusted forward rate is $L^{f,\text{quanto}} = L^f \exp(-\rho_{L,X}\sigma^f\sigma_X T)$, and the correction is proportional to the rate-FX correlation
Positive correlation (foreign rate up when foreign currency strengthens) produces a negative quanto adjustment
Quanto caps and floors are priced using Black's formula with the adjusted forward rate and domestic discounting
The adjustment is typically small for short-dated products but grows linearly with the fixing horizon and can reach 10+ bps for long-dated quanto products
Correlation uncertainty is the dominant source of model risk in quanto pricing

Exercises¶

Exercise 1. A quanto cap on 3-month GBP LIBOR is settled in USD at a 1:1 exchange rate. The GBP forward rate is $L^f(0) = 4.8\%$, the GBP rate volatility is $\sigma^f = 19\%$, the GBP/USD exchange rate volatility is $\sigma_X = 8\%$, and the rate-FX correlation is $\rho_{L,X} = 0.25$. For a caplet fixing in $T = 3$ years, compute the quanto-adjusted forward rate and the quanto adjustment in basis points. Interpret the sign of the adjustment.

Solution to Exercise 1

Given: $L^f(0) = 0.048$, $\sigma^f = 0.19$, $\sigma_X = 0.08$, $\rho_{L,X} = 0.25$, $T = 3$ years.

Quanto-adjusted forward rate:

\[ L^{f,\text{quanto}}(0) = L^f(0) \, \exp\!\left(-\rho_{L,X} \, \sigma^f \, \sigma_X \, T\right) \]

Computing the exponent:

\[ -\rho_{L,X} \, \sigma^f \, \sigma_X \, T = -0.25 \times 0.19 \times 0.08 \times 3 = -0.25 \times 0.0456 = -0.01140 \]

\[ L^{f,\text{quanto}}(0) = 0.048 \times e^{-0.01140} = 0.048 \times 0.98867 = 0.047456 \]

Equivalently, $L^{f,\text{quanto}}(0) = 4.746\%$.

Quanto adjustment in basis points:

\[ \text{Quanto Adj} = L^{f,\text{quanto}}(0) - L^f(0) = 4.746\% - 4.800\% = -0.054\% = -5.4 \text{ bp} \]

Alternatively, using the first-order formula:

\[ \text{Quanto Adj} \approx -\rho_{L,X} \, \sigma^f \, \sigma_X \, T \cdot L^f(0) = -0.01140 \times 0.048 = -0.000547 = -5.5 \text{ bp} \]

(The slight difference is due to the first-order approximation.)

Interpretation: The adjustment is negative ($-5.4$ bp) because $\rho_{L,X} = 0.25 > 0$. Positive correlation means that when GBP rates are high, the GBP tends to strengthen against the USD. A standard (non-quanto) product would benefit from this correlation (high rates + strong GBP = double benefit), but the quanto product settles at a fixed exchange rate and misses the FX gain. The quanto-adjusted rate is therefore lower, reflecting the loss of this favorable correlation.

Exercise 2. Starting from the dynamics of the foreign forward rate under $\mathbb{Q}^{f,T_{i+1}}$ and the exchange rate under $\mathbb{Q}^d$, derive the quanto drift

\[ \mu_{\text{quanto}}(t) = -\rho_{L,X} \, \sigma_i^f \, \sigma_X \]

using Girsanov's theorem for the composite measure change $\mathbb{Q}^{f,T_{i+1}} \to \mathbb{Q}^d \to \mathbb{Q}^{d,T_{i+1}}$. State all assumptions and identify where the correlation enters.

Solution to Exercise 2

Step 1: Foreign forward rate under $\mathbb{Q}^{f,T_{i+1}}$.

Under the foreign $T_{i+1}$-forward measure, with numéraire $P^f(t, T_{i+1})$:

\[ \frac{dL_i^f(t)}{L_i^f(t)} = \sigma_i^f \, dW_L^{f,T_{i+1}}(t) \]

where $W_L^{f,T_{i+1}}$ is a Brownian motion under $\mathbb{Q}^{f,T_{i+1}}$.

Step 2: Exchange rate under $\mathbb{Q}^d$.

Under the domestic risk-neutral measure:

\[ \frac{dX(t)}{X(t)} = (r^d - r^f) \, dt + \sigma_X \, dW_X^d(t) \]

Assumption: $dW_L^{f,T_{i+1}} \cdot dW_X^d = \rho_{L,X} \, dt$ (constant correlation).

Step 3: Measure change $\mathbb{Q}^{f,T_{i+1}} \to \mathbb{Q}^d$.

The relationship between the domestic and foreign risk-neutral measures involves the exchange rate. The numéraire for $\mathbb{Q}^d$ expressed in foreign terms is $B^d(t)/X(t)$. The Girsanov kernel for the change $\mathbb{Q}^f \to \mathbb{Q}^d$ is $\sigma_X$, applied to the Brownian motion $W_X$.

Under $\mathbb{Q}^d$, the foreign Brownian motion $W_L^{f}$ shifts:

\[ dW_L^{f}(t) = dW_L^{d}(t) - \rho_{L,X} \, \sigma_X \, dt \]

This is because the Girsanov theorem for the measure change from $\mathbb{Q}^f$ to $\mathbb{Q}^d$ involves the volatility of the exchange rate, projected onto the direction of $W_L$.

Step 4: Measure change $\mathbb{Q}^d \to \mathbb{Q}^{d,T_{i+1}}$.

Changing from $\mathbb{Q}^d$ to $\mathbb{Q}^{d,T_{i+1}}$ introduces an additional drift from the domestic bond volatility $\sigma_P^d(t, T_{i+1})$. However, we also need to account for the change from $\mathbb{Q}^{f,T_{i+1}}$ to $\mathbb{Q}^f$ (forward to spot measure in the foreign economy), which introduces a drift involving the foreign bond volatility.

Step 5: Combining the measure changes.

The composite measure change $\mathbb{Q}^{f,T_{i+1}} \to \mathbb{Q}^{d,T_{i+1}}$ introduces a total drift in $L_i^f$. The key observation is that the drift terms from the forward-measure changes (foreign and domestic) involve bond volatilities that approximately cancel for the forward rate $L_i^f$, leaving only the FX-related drift.

Under $\mathbb{Q}^{d,T_{i+1}}$:

\[ \frac{dL_i^f(t)}{L_i^f(t)} = -\rho_{L,X} \, \sigma_i^f \, \sigma_X \, dt + \sigma_i^f \, dW_L^{d,T_{i+1}}(t) \]

The quanto drift is:

\[ \mu_{\text{quanto}} = -\rho_{L,X} \, \sigma_i^f \, \sigma_X \]

Where correlation enters: The correlation appears in Step 3 when projecting the Girsanov kernel (which is $\sigma_X$, the FX volatility vector) onto the direction of the foreign rate's Brownian motion $W_L$. The projection is $\rho_{L,X} \sigma_X$, giving the drift correction $-\rho_{L,X} \sigma_i^f \sigma_X$.

Assumptions used: (1) Lognormal dynamics for both $L_i^f$ and $X$. (2) Constant volatilities $\sigma_i^f$, $\sigma_X$ and constant correlation $\rho_{L,X}$. (3) Deterministic interest rates for the forward-measure changes (or that the additional drifts from stochastic rates cancel to leading order). $\blacksquare$

Exercise 3. Consider a differential (diff) swap that pays $(L_i^f(T_i) - L_i^d(T_i))$ at $T_{i+1}$ in domestic currency. Show that the fair diff-swap spread is

\[ L_i^{f,\text{quanto}}(0) - L_i^d(0) \]

where only the foreign rate requires a quanto adjustment. Explain why the domestic rate does not need an adjustment.

Solution to Exercise 3

Present value of the diff swap coupon:

\[ V_0 = P^d(0, T_{i+1}) \, \mathbb{E}^{\mathbb{Q}^{d,T_{i+1}}}\!\left[\delta_i (L_i^f(T_i) - L_i^d(T_i))\right] \]

By linearity of expectation:

\[ V_0 = P^d(0, T_{i+1}) \, \delta_i \left(\mathbb{E}^{\mathbb{Q}^{d,T_{i+1}}}[L_i^f(T_i)] - \mathbb{E}^{\mathbb{Q}^{d,T_{i+1}}}[L_i^d(T_i)]\right) \]

Foreign rate: $L_i^f$ is a martingale under $\mathbb{Q}^{f,T_{i+1}}$, not under $\mathbb{Q}^{d,T_{i+1}}$. Therefore:

\[ \mathbb{E}^{\mathbb{Q}^{d,T_{i+1}}}[L_i^f(T_i)] = L_i^{f,\text{quanto}}(0) = L_i^f(0) \, e^{-\rho_{L,X}\sigma_i^f\sigma_X T_i} \]

The foreign rate requires the quanto adjustment.

Domestic rate: $L_i^d$ is a martingale under $\mathbb{Q}^{d,T_{i+1}}$ (the domestic $T_{i+1}$-forward measure) because $T_{i+1}$ is precisely its natural payment date, and $P^d(t, T_{i+1})$ is the natural numéraire. Therefore:

\[ \mathbb{E}^{\mathbb{Q}^{d,T_{i+1}}}[L_i^d(T_i)] = L_i^d(0) \]

No adjustment is needed for the domestic rate.

Why the domestic rate needs no adjustment: The domestic forward rate $L_i^d$ for period $[T_i, T_{i+1}]$ is defined as a function of $P^d(t, T_i)/P^d(t, T_{i+1})$, and when both the pricing numéraire and the payment currency are domestic, no measure change is required. The payoff $(L_i^f - L_i^d)$ is paid in domestic currency at $T_{i+1}$, and $\mathbb{Q}^{d,T_{i+1}}$ is the correct pricing measure. The domestic rate is already a martingale under this measure, while the foreign rate requires the cross-currency measure change.

Fair diff-swap spread:

\[ V_0 = P^d(0, T_{i+1}) \, \delta_i \left(L_i^{f,\text{quanto}}(0) - L_i^d(0)\right) \]

The fair spread (the fixed rate that makes $V_0 = 0$) is $L_i^{f,\text{quanto}}(0) - L_i^d(0)$. $\blacksquare$

Exercise 4. A trader estimates the GBP rate / GBP-USD FX correlation to be $\rho_{L,X} = 0.35$ using a 1-year rolling window, but $\rho_{L,X} = 0.15$ using a 5-year window. For a 10-year quanto cap on GBP LIBOR with $\sigma^f = 20\%$, $\sigma_X = 10\%$, and $L^f(0) = 5\%$, compute the quanto adjustment under each correlation estimate and express the difference in basis points. Discuss how this correlation uncertainty affects the risk management of a quanto book.

Solution to Exercise 4

Given: $\sigma^f = 0.20$, $\sigma_X = 0.10$, $L^f(0) = 0.05$, $T = 10$ years.

With $\rho_{L,X} = 0.35$ (1-year window):

\[ \text{Quanto Adj}_1 = -\rho_{L,X} \, \sigma^f \, \sigma_X \, T \cdot L^f(0) = -0.35 \times 0.20 \times 0.10 \times 10 \times 0.05 \]

\[ = -0.35 \times 0.01 \times 0.05 = -0.000350 = -3.50 \text{ bp} \]

Using the exact formula: $L^{f,q}_1 = 0.05 \times e^{-0.35 \times 0.20 \times 0.10 \times 10} = 0.05 \times e^{-0.07} = 0.05 \times 0.9324 = 0.04662$. Adjustment $= -3.38$ bp.

With $\rho_{L,X} = 0.15$ (5-year window):

\[ \text{Quanto Adj}_2 = -0.15 \times 0.20 \times 0.10 \times 10 \times 0.05 = -0.000150 = -1.50 \text{ bp} \]

Using the exact formula: $L^{f,q}_2 = 0.05 \times e^{-0.15 \times 0.20 \times 0.10 \times 10} = 0.05 \times e^{-0.03} = 0.05 \times 0.9704 = 0.04852$. Adjustment $= -1.48$ bp.

Difference between the two estimates:

\[ |\text{Adj}_1 - \text{Adj}_2| = |{-3.38} - ({-1.48})| = 1.90 \text{ bp} \]

On the forward rate level, the difference is approximately 1.9 basis points.

Impact on risk management:

Valuation uncertainty: A 1.9 bp difference in the forward rate translates directly to P&L uncertainty. For a $1 billion notional 10-year quanto cap, this could represent millions of dollars in mark-to-market difference.
Correlation hedging: The rate-FX correlation $\rho_{L,X}$ cannot be hedged directly with liquid instruments, making it an unhedgeable risk factor. This is a key source of model risk in quanto books.
Reserve requirements: Prudent risk management requires holding reserves for correlation uncertainty. A common approach is to compute the quanto adjustment under multiple correlation scenarios (e.g., $\pm 1$ standard deviation of the historical estimate) and reserve the range.
Estimation methodology: The choice of estimation window (1-year vs 5-year) reflects a trade-off between responsiveness to recent market conditions and statistical stability. Neither is objectively correct, suggesting that quanto pricing inherently carries irreducible model risk of the order computed above.

Exercise 5. Show that the quanto adjustment formula $L^{f,\text{quanto}} = L^f \exp(-\rho_{L,X}\sigma^f\sigma_X T)$ can be rewritten as a shift in the drift of the lognormal process. Specifically, verify that if $L_i^f(T_i)$ is lognormal under $\mathbb{Q}^{f,T_{i+1}}$ with zero drift, then under $\mathbb{Q}^{d,T_{i+1}}$ it remains lognormal but with drift $\mu_{\text{quanto}} = -\rho_{L,X}\sigma^f\sigma_X$, and that the volatility $\sigma^f$ is unchanged.

Solution to Exercise 5

Under $\mathbb{Q}^{f,T_{i+1}}$, the foreign forward rate is a driftless lognormal martingale:

\[ dL_i^f(t) = \sigma^f L_i^f(t) \, dW^{f,T_{i+1}}(t) \]

In integrated form:

\[ L_i^f(T_i) = L_i^f(0) \exp\!\left(-\frac{1}{2}(\sigma^f)^2 T_i + \sigma^f W^{f,T_{i+1}}(T_i)\right) \]

Under $\mathbb{Q}^{d,T_{i+1}}$, the quanto drift $\mu_{\text{quanto}} = -\rho_{L,X}\sigma^f\sigma_X$ shifts the dynamics to:

\[ dL_i^f(t) = \mu_{\text{quanto}} \, L_i^f(t) \, dt + \sigma^f L_i^f(t) \, dW^{d,T_{i+1}}(t) \]

In integrated form:

\[ L_i^f(T_i) = L_i^f(0) \exp\!\left(\left(\mu_{\text{quanto}} - \frac{1}{2}(\sigma^f)^2\right) T_i + \sigma^f W^{d,T_{i+1}}(T_i)\right) \]

This is lognormal with:

Drift: $\mu_{\text{quanto}} = -\rho_{L,X}\sigma^f\sigma_X$ (nonzero)
Volatility: $\sigma^f$ (unchanged)

Verification of the expectation:

\[ \mathbb{E}^{\mathbb{Q}^{d,T_{i+1}}}[L_i^f(T_i)] = L_i^f(0) \exp\!\left(\mu_{\text{quanto}} \, T_i\right) = L_i^f(0) \exp\!\left(-\rho_{L,X}\sigma^f\sigma_X T_i\right) = L_i^{f,\text{quanto}}(0) \]

This confirms the quanto-adjusted forward rate formula.

Why the volatility is unchanged: The Girsanov theorem changes the drift of the Brownian motion but not its quadratic variation. Specifically, $W^{d,T_{i+1}}(t) = W^{f,T_{i+1}}(t) + \int_0^t \rho_{L,X}\sigma_X \, ds$ is still a Brownian motion under $\mathbb{Q}^{d,T_{i+1}}$ (by Girsanov's theorem), so the diffusion coefficient $\sigma^f$ multiplying $dW$ is the same under both measures. The quanto adjustment affects only the mean of $\ln L_i^f(T_i)$, not its variance. $\blacksquare$

Exercise 6. Price a quanto floorlet on EUR EURIBOR with strike $K = 3.0\%$, settled in JPY. The parameters are: $L^f(0) = 2.8\%$ (EUR forward rate), $\sigma^f = 25\%$, $\sigma_X = 12\%$ (EUR/JPY vol), $\rho_{L,X} = -0.15$, $T = 2$ years, $\delta = 0.5$, and $P^d(0, T_{i+1}) = 0.990$ (JPY discount factor). First compute the quanto-adjusted forward rate, then apply Black's formula for the floorlet.

Solution to Exercise 6

Given: $L^f(0) = 0.028$ (EUR), $\sigma^f = 0.25$, $\sigma_X = 0.12$ (EUR/JPY), $\rho_{L,X} = -0.15$, $T = 2$, $\delta = 0.5$, $P^d(0, T_{i+1}) = 0.990$, $K = 0.03$.

Step 1: Quanto-adjusted forward rate.

\[ L^{f,\text{quanto}}(0) = L^f(0) \, \exp\!\left(-\rho_{L,X} \, \sigma^f \, \sigma_X \, T\right) \]

\[ = 0.028 \times \exp\!\left(-(-0.15) \times 0.25 \times 0.12 \times 2\right) = 0.028 \times \exp(0.0090) \]

\[ = 0.028 \times 1.00904 = 0.028253 \]

The quanto-adjusted forward rate is $2.825\%$. The adjustment is positive ($+2.5$ bp) because $\rho_{L,X} < 0$: when EUR rates rise, the EUR tends to weaken against JPY. The quanto product does not suffer from this FX loss, so the effective forward rate is higher.

Step 2: Black's formula for the quanto floorlet.

\[ \text{Quanto Floorlet} = \delta \, P^d(0, T_{i+1}) \left[K \, N(-d_2) - L^{f,q}(0) \, N(-d_1)\right] \]

Computing $d_1$ and $d_2$:

\[ d_1 = \frac{\ln(L^{f,q}(0)/K) + \frac{1}{2}(\sigma^f)^2 T}{\sigma^f \sqrt{T}} = \frac{\ln(0.028253/0.030) + \frac{1}{2}(0.0625)(2)}{0.25\sqrt{2}} \]

\[ = \frac{\ln(0.94177) + 0.0625}{0.3536} = \frac{-0.05997 + 0.0625}{0.3536} = \frac{0.00253}{0.3536} = 0.00716 \]

\[ d_2 = d_1 - \sigma^f\sqrt{T} = 0.00716 - 0.3536 = -0.3464 \]

Now:

$N(-d_1) = N(-0.00716) = 0.4971$
$N(-d_2) = N(0.3464) = 0.6355$

Quanto floorlet price:

\[ = 0.5 \times 0.990 \times \left[0.030 \times 0.6355 - 0.028253 \times 0.4971\right] \]

\[ = 0.495 \times \left[0.019065 - 0.014045\right] = 0.495 \times 0.005020 = 0.002485 \]

The quanto floorlet price is approximately $0.2485\%$ of notional, or $24.85$ bp.

Note: The floorlet is relatively valuable because the quanto-adjusted forward rate ($2.825\%$) is below the strike ($3.0\%$), so the floor is in-the-money, making $N(-d_1) \approx 0.50$ and $N(-d_2) \approx 0.64$.

Exercise 7. In a stochastic volatility extension, both $\sigma^f(t)$ and $\sigma_X(t)$ follow their own diffusion processes and $\rho_{L,X}(t)$ is state-dependent. Explain qualitatively why the constant-parameter quanto formula $\exp(-\rho\sigma^f\sigma_X T)$ becomes inadequate for long-dated quanto products. Describe two modeling approaches that can handle time-varying or stochastic correlation and discuss their calibration requirements.

Solution to Exercise 7

Why the constant-parameter formula becomes inadequate:

The quanto adjustment formula $\exp(-\rho\sigma^f\sigma_X T)$ assumes that $\rho$, $\sigma^f$, and $\sigma_X$ are constant over the life of the product. For long-dated quanto products (e.g., 10--30 years), this assumption breaks down for several reasons:

Volatility mean-reversion and regime changes: Interest rate volatilities and FX volatilities exhibit mean-reversion and can undergo regime shifts. Using today's spot volatility for a 20-year projection overstates or understates the cumulative adjustment. The correct integral is $\int_0^T \rho(t)\sigma^f(t)\sigma_X(t) \, dt$, which can differ significantly from $\rho\sigma^f\sigma_X T$.
Correlation instability: The rate-FX correlation $\rho_{L,X}$ is empirically unstable, varying with macroeconomic regimes (e.g., risk-on vs risk-off), monetary policy cycles, and market stress. A constant $\rho$ fails to capture the time-varying nature of this dependence.
Stochastic volatility effects: When volatilities are stochastic, the quanto adjustment depends on the joint distribution of $(\sigma^f(t), \sigma_X(t), \rho(t))$, not just their initial values. The convexity of the exponential function means that $\mathbb{E}[\exp(-\int_0^T \rho\sigma^f\sigma_X dt)] \neq \exp(-\mathbb{E}[\int_0^T \rho\sigma^f\sigma_X dt])$ (Jensen's inequality), introducing a higher-order correction.
Correlation between correlation and rates: If correlation increases during stress periods when rates are volatile, the average $\rho\sigma^f\sigma_X$ product differs from the product of averages.

Two modeling approaches:

Approach 1: Multi-factor stochastic volatility with local correlation.

Model the foreign rate, exchange rate, and their volatilities jointly:

$dL^f/L^f = \sigma^f(v_1(t)) \, dW_1$, where $v_1$ follows a CIR or Heston process
$dX/X = \ldots + \sigma_X(v_2(t)) \, dW_2$, where $v_2$ follows its own stochastic process
$\rho_{L,X}(t) = \rho(v_1(t), v_2(t))$, a deterministic function of the variance states

Calibration requirements: Joint calibration to (1) the foreign interest rate swaption surface, (2) the FX option smile across maturities, and (3) cross-asset products such as quanto swaptions or power reverse dual currency notes (PRDCs) that are sensitive to rate-FX correlation. Correlation is typically backed out from quanto product prices or estimated from historical data with appropriate filtering.

Approach 2: Time-dependent (piecewise constant) parameters.

Use a simpler framework where $\sigma^f(t)$, $\sigma_X(t)$, and $\rho(t)$ are deterministic but time-varying (e.g., piecewise constant on annual intervals):

\[ L^{f,\text{quanto}}(0) = L^f(0) \exp\!\left(-\sum_{k} \rho_k \, \sigma^f_k \, \sigma_{X,k} \, \Delta t_k\right) \]

Calibration requirements: Calibrate each time bucket's volatilities to the respective term structure of option prices (caplet vols for rates, FX option vols for FX). The time-dependent correlation can be estimated from forward-starting quanto products or from historical rolling-window estimates mapped to future periods.

This approach is simpler to implement than full stochastic volatility but captures the most important effect: the term structure of the adjustment parameters. It is widely used in practice for pricing long-dated quanto products.

Fixing Horizon	Rate Vol	FX Vol	Correlation	Quanto Adj (bps)
1Y	20%	10%	0.30	\(-0.6\)
3Y	20%	10%	0.30	\(-1.8\)
5Y	20%	10%	0.30	\(-3.0\)
5Y	20%	10%	0.50	\(-5.0\)
10Y	25%	12%	0.40	\(-12.0\)