Quanto Adjustment

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1. Basic Concepts of Quanto Adjustment

Quanto adjustment is a method used in asset pricing to account for the difference between the currency of the asset's natural denomination and the currency in which the payoff is settled. Its core purpose is to adjust the drift term and dynamics of the underlying asset to reflect the relationship between the asset's natural currency and the settlement currency, ensuring no-arbitrage conditions are satisfied.

1.1 Scenario Description

In the context of a foreign exchange model $F_x(\text{CCY1}/\text{CCY2})$, suppose the user wants to express the option payoff in a different currency. A typical scenario includes:

• The natural currency is Euro (EUR), but the payoff currency is Japanese Yen (JPY).
• The natural currency is US Dollar (USD), but the payoff currency is Chinese Yuan (CNY).

In such cases, the dynamics of the underlying asset need to be adjusted to account for the relationship between the natural currency and the payoff currency. This adjustment is influenced by:

• Exchange rate volatility;
• Correlation between the underlying asset and the exchange rate;
• Differences in risk-free rates between the currencies.

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2. Dynamics of Quanto Adjustment

2.1 Exchange Rate Dynamics

Let the forward exchange rate $F_x(\text{CCY1}/\text{CCY2})$ in the domestic currency （$\text{CCY2}$） framework follow the dynamics:

$$\frac{dF_x(\text{CCY1}/\text{CCY2})}{F_x(\text{CCY1}/\text{CCY2})} = (r_{\text{dom}} - r_{\text{for}})dt + \sigma(t, F_x)dW_{\text{dom}}(t),$$

where:

• $r_{\text{dom}}$: Risk-free rate of the domestic currency;
• $r_{\text{for}}$: Risk-free rate of the foreign currency;
• $\sigma(t, F_x)$: Implied volatility;
• $W_{\text{dom}}(t)$: Brownian motion in the domestic currency framework.

The dynamics of the reverse exchange rate $F_x(\text{CCY2}/\text{CCY1})$ are:

$$\frac{dF_x(\text{CCY2}/\text{CCY1})}{F_x(\text{CCY2}/\text{CCY1})} = (r_{\text{for}} - r_{\text{dom}})dt + \sigma(t, F_x)dW_{\text{for}}(t).$$

2.2 Correlation Reversal

The correlation between the asset $S$ and the forward exchange rate $F_x(\text{CCY1}/\text{CCY2})$ satisfies the following relationship with the reverse exchange rate $F_x(\text{CCY2}/\text{CCY1})$:

$$\rho(F_x(\text{CCY1}/\text{CCY2}), S) = -\rho(F_x(\text{CCY2}/\text{CCY1}), S).$$

2.3 Asset and Exchange Rate Dynamics

Assume the underlying asset ( S_t ) follows the dynamics in the foreign currency framework:

$$\frac{dS_t}{S_t} = (r_{\text{for}} - q(t))dt + \sigma_S dW_{\text{for}}(t),$$

where:

• $q(t)$: Dividend or convenience yield;
• $\sigma_S$: Volatility of the underlying asset;
• $W_{\text{for}}(t)$: Brownian motion in the foreign currency framework.

Simultaneously, the exchange rate $F_x(t)$ follows the dynamics:

$$\frac{dF_x(t)}{F_x(t)} = (r_{\text{dom}} - r_{\text{for}})dt + \sigma_{\text{FX}} dW_{\text{dom}}(t),$$

where $\sigma_{\text{FX}}$ is the volatility of the exchange rate.

In this context, Quanto adjustment is used to adjust the drift term of the asset, ensuring no-arbitrage pricing in the payoff currency framework.

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3. Calculation Methods for Quanto Adjustment

3.1 Quanto Adjustment in the Black-Scholes Framework

In the Black-Scholes framework, the drift term of the underlying asset is adjusted as:

$$\mu = r_{\text{for}} - q - \rho \sigma_S \sigma_{\text{FX}},$$

where:

• $\rho$: Correlation between the underlying asset and the exchange rate;
• $\sigma_S$: Volatility of the underlying asset;
• $\sigma_{\text{FX}}$: Volatility of the exchange rate.

💡 Drift Term Without Adjustment:

$$\mu = r_{\text{for}} - q.$$

Without Quanto adjustment, the drift term only considers the foreign risk-free rate $r_{\text{for}}$ and the dividend yield $q$, ignoring the correlation $\rho$ and the interaction term $\rho \sigma_S \sigma_{\text{FX}}$.

3.2 Quanto Adjustment Under Log-Normal Assumption

Under the log-normal assumption, the Quanto adjustment can be expressed as a multiplicative adjustment factor:

$$\text{Quanto Adjustment Factor} = \exp(-\rho \sigma_S \sigma_{\text{FX}} T),$$

where $T$ is the time to maturity.

💡 Adjustment Factor Without Quanto Adjustment:

$$\text{Adjustment Factor} = 1.$$

Without Quanto adjustment, the adjustment factor is 1, assuming no correlation between currencies.

3.3 Quanto Adjustment in Local Volatility Models

For cases where volatility varies with time or asset price, the local volatility model can be used. The dynamics of the underlying asset are:

$$\frac{dS_t}{S_t} = \left(r_{\text{for}} - q - \rho \sigma_S(t, S_t) \sigma_{\text{FX}}(t) \right) dt + \sigma_S(t, S_t) dW_{\text{for}}(t).$$

The local volatility model is suitable for scenarios with complex volatility surfaces.

💡 Dynamics Without Quanto Adjustment:

$$\frac{dS_t}{S_t} = \left(r_{\text{for}} - q \right) dt + \sigma_S(t, S_t) dW_{\text{for}}(t).$$

Without Quanto adjustment, the correlation term $\rho \sigma_S(t, S_t) \sigma_{\text{FX}}(t)$ is omitted.

3.4 Quanto Adjustment in Stochastic Volatility Models

In stochastic volatility frameworks (e.g., the Heston model), the adjustment is more complex as volatility itself is a stochastic variable:

$$dv_t = \kappa (\theta - v_t) dt + \eta \sqrt{v_t} dZ_t,$$

where $v_t$ is the stochastic volatility.

💡 Dynamics Without Quanto Adjustment:

$$dv_t = \kappa (\theta - v_t) dt + \eta \sqrt{v_t} dZ_t.$$

In stochastic volatility models, the dynamics of volatility remain unchanged. The Quanto adjustment affects other dynamics rather than the stochastic nature of volatility.

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4. Comprehensive Application of Quanto Adjustment

4.1 Definition of Quanto Adjustment

The relationship between forward prices under the domestic and foreign measures defines the Quanto adjustment:

$$\text{Quanto Adjustment Factor} = \frac{\mathbb{E}^{Q_{\text{dom}}}[S_T]}{\mathbb{E}^{Q_{\text{for}}}[S_T]} = \exp(-\rho \sigma_S \sigma_{\text{FX}} T).$$

Without Quanto adjustment:

$$\text{Adjustment Factor} = 1.$$

In the absence of Quanto adjustment, forward prices under domestic and foreign measures are assumed to be equal.

4.2 Indirect Quanto Adjustment

When direct cross-currency market data is unavailable, Quanto adjustment can be derived indirectly. For example, if the natural currency is $X$ and the payoff currency is $Y$, but the cross-currency rate $X/Y$ is missing, the adjustment can be derived using:

$$\text{Cov}(\ln(X), \ln(X/Y)) = \sigma_X \sigma_{X/K} \rho(X, X/K) + \sigma_X \sigma_{K/Y} \rho(X, K/Y),$$

where $K$ is an intermediate currency.

Without Quanto adjustment:

$$\text{Cov}(\ln(X), \ln(X/Y)) = \sigma_X^2.$$

In the absence of Quanto adjustment, only the asset's own volatility is considered, ignoring interactions with other currencies.

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5. Application Scenarios of Quanto Adjustment

Quanto adjustment is widely used in financial markets for pricing and valuing financial products and derivatives involving cross-currency payoffs. Its core purpose is to address the inconsistency between the natural currency of the underlying asset and the payoff currency, ensuring no-arbitrage pricing across different currency frameworks. Below are typical application scenarios:

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5.1. FX-Related Financial Products

5.1.1 FX Options

• Scenario: The underlying of an FX option is a currency pair (e.g., EUR/USD), but the option premium and payoff are settled in a third currency (e.g., CNY or JPY).
• Role of Quanto Adjustment:
  • Adjusts the drift term of the currency pair to reflect the correlation between the underlying exchange rate and the payoff currency.
  • Ensures the pricing model accounts for both the underlying exchange rate volatility and the payoff currency volatility.

5.1.2 FX-Linked Deposits

• Scenario: The interest rate of a deposit product is determined by the performance of an FX rate (e.g., USD/JPY), but the principal and interest are paid in the local currency (e.g., CNY).
• Role of Quanto Adjustment:
  • Adjusts the deposit's interest calculation to incorporate the volatility of the FX rate and its correlation with the payoff currency.

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5.2. Structured Deposits and Notes

5.2.1 Range Accrual Deposits

• Scenario: The payoff of a range accrual deposit depends on the time an FX rate (e.g., EUR/USD) stays within a specified range, but the payoff is settled in the local currency (e.g., CNY).
• Role of Quanto Adjustment:
  • Addresses the inconsistency between the natural currency of the FX rate and the payoff currency.
  • Adjusts the drift term of the range accrual to ensure no-arbitrage pricing in the local currency framework.

5.2.2 Snowball Structured Products

• Scenario: The return of a snowball product is linked to the performance of an FX rate (e.g., USD/JPY), but the final payoff is settled in another currency (e.g., CNY).
• Role of Quanto Adjustment:
  • Adjusts the dynamics of the FX rate to ensure consistency of the product's return in the payoff currency framework.
  • Incorporates the impact of the correlation between the FX rate and the payoff currency on the final payoff.

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5.3. Commodity-Linked Products

5.3.1 Commodity Options or Swaps

• Scenario: The underlying of a commodity option (e.g., crude oil option) is the commodity price (e.g., in USD), but the option premium and payoff are settled in another currency (e.g., EUR or CNY).
• Role of Quanto Adjustment:
  • Adjusts the drift term of the commodity price to reflect the correlation between the commodity price and the payoff currency.
  • Ensures the pricing model accounts for both the commodity price volatility and the payoff currency volatility.

5.3.2 Commodity-Linked Deposits

• Scenario: The payoff of a deposit product is linked to the performance of a commodity price (e.g., gold or crude oil), but the principal and payoff are paid in the local currency.
• Role of Quanto Adjustment:
  • Incorporates the volatility of the commodity price and its correlation with the payoff currency into the pricing framework.

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5.4. Equity-Linked Products

5.4.1 Foreign Equity Options

• Scenario: The underlying of an option is a foreign stock (e.g., a US stock priced in USD), but the option premium and payoff are settled in the local currency (e.g., CNY).
• Role of Quanto Adjustment:
  • Adjusts the drift term of the stock price to reflect the correlation between the stock price and the payoff currency.
  • Ensures the pricing model accounts for both the stock price volatility and the payoff currency volatility.

5.4.2 Foreign Equity-Linked Deposits

• Scenario: The payoff of a deposit product is linked to the performance of a foreign stock price (e.g., a European stock priced in EUR), but the principal and payoff are paid in the local currency.
• Role of Quanto Adjustment:
  • Adjusts the dynamics of the stock price to incorporate the exchange rate risk of the payoff currency.
  • Ensures the impact of stock price volatility, exchange rate volatility, and their correlation on the product's pricing is accurately reflected.

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5.5. Interest Rate-Linked Products

5.5.1 Foreign Interest Rate-Linked Products

• Scenario: The payoff of a product is linked to a foreign interest rate (e.g., USD Libor or EUR Euribor), but the payoff is settled in the local currency.
• Role of Quanto Adjustment:
  • Incorporates the correlation between the foreign interest rate and the payoff currency into the pricing.
  • Adjusts the drift term to ensure consistency of the foreign interest rate in the payoff currency framework.

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5.6. Multi-Currency Derivatives

5.6.1 Cross-Currency Swaps

• Scenario: A cross-currency swap involves the exchange of cash flows in two currencies, and Quanto adjustment is used to adjust the pricing of one leg of the swap.
• Role of Quanto Adjustment:
  • Adjusts the value of cash flows in the underlying currency to reflect the correlation and volatility of the payoff currency.

5.6.2 FX-Linked Basket Options

• Scenario: The underlying of a basket option is a combination of multiple currencies, but the payoff is settled in a single currency.
• Role of Quanto Adjustment:
  • Adjusts the drift term of each currency in the basket to ensure consistent pricing in the payoff currency framework.

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5.7. Indirect Exchange Rate Pricing

5.7.1 Products with Missing Direct Exchange Rates

• Scenario: If direct cross-currency rates (e.g., EUR/CNY) are unavailable, but indirect rates (e.g., EUR/USD and USD/CNY) are available.
• Role of Quanto Adjustment:
  • Derives the Quanto adjustment indirectly to ensure pricing reflects the volatility and correlation between currencies.

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5.8. Other Scenarios

5.8.1 Risk Management

• Scenario: In risk management, Quanto adjustment is used to hedge cross-currency risks in derivatives.
• Role of Quanto Adjustment:
  • Ensures the dynamics of the underlying asset reflect all relevant currency risk factors.

5.8.2 Valuation of Multi-Currency Assets in Portfolios

• Scenario: A portfolio of multi-currency assets needs to be valued in a single currency.
• Role of Quanto Adjustment:
  • Adjusts the dynamics of assets to ensure consistency in the valuation currency framework.

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Summary

Quanto adjustment is a critical method in asset pricing, widely used in cross-currency financial products. Its core purpose is to adjust the drift term of the underlying asset to address inconsistencies between the natural currency and the payoff currency.

Common methods include:

• Black-Scholes adjustment: Suitable for simple scenarios;
• Log-normal adjustment: Quick calculation for simple products;
• Local and stochastic volatility models: Suitable for complex scenarios;
• Implied market data method: Aligns with market realities.

In practice, the appropriate Quanto adjustment method should be selected based on product characteristics, market data availability, and pricing accuracy requirements to ensure accurate and consistent results.

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References

The literature on Quanto adjustment is extensive, covering theory, practice, and numerical methods. Below are recommended references based on specific needs:

1. Theoretical Foundations:

   • Recommended books: Hull's Options, Futures, and Other Derivatives and Shreve's Stochastic Calculus for Finance II.
   • Recommended papers: Margrabe (1978), Garman and Kohlhagen (1983).

2. Practical Applications:

   • Recommended books: Wilmott's Paul Wilmott Introduces Quantitative Finance and Joshi's The Concepts and Practice of Mathematical Finance.
   • Tools and guides: QuantLib and Bloomberg.

3. Advanced Research:

   • Recommended papers: Hagan (2002), Boyle and Derman (1983).
   • Numerical methods: Glasserman's Monte Carlo Methods in Financial Engineering.