Quanto Options

The general principles of valuing and hedging exotic options carry over to a quanto, with a quanto-specific treatment of the exchange rate and the correlation added.

Converting the payoff into yen at a fixed exchange rate

A quanto option is an option whose payoff is set by converting the price of an underlying asset quoted in a foreign currency into the domestic currency at an exchange rate fixed at the start of the trade (the "fixed rate" below)¹. The payoff is paid in the domestic currency, and the exchange rate at expiry does not enter it. There is no risk from movements in the exchange rate itself². It is a different product from a currency option, whose underlying is the exchange rate.

Consider a hypothetical example in which Company D, in Japan, buys a quanto call on a dollar-denominated stock from a financial institution. The stock price is USD 100, the strike price is USD 100, the option covers 100 shares, expiry is in one year, and the fixed rate is 150 yen per dollar. The call pays the difference between the stock price at expiry and the strike (zero if negative), multiplied by the number of shares and the fixed rate. If the stock price at expiry is USD 120, Company D receives USD 20 × 100 shares × 150 yen = 300,000 yen.

$$\text{Payoff (yen)} = \bar{X} \times \max(S_T - K_f,\ 0) \times \text{number of shares}$$

$\bar{X}$ is the fixed rate (yen per dollar), $S_T$ is the stock price at expiry (dollars), and $K_f$ is the strike in dollars. With the strike $K$ expressed in yen, the payoff can also be written $\max(\bar{X} S_T - K,\ 0)$, where $K = \bar{X} K_f$³.

Figure description

This is the payoff at expiry of a call with a strike of USD 100. The vertical axis is the dollar difference per share; the yen amount received is this multiplied by the fixed rate of 150 yen and the number of shares, 100. If the stock price is at or below the strike, the payoff is 0, and above it the payoff grows by the amount the price exceeds the strike. The exchange rate at expiry does not enter this multiplication.

Comparison with conversion at the rate at expiry

For comparison, an option with the same underlying and strike whose difference is received in dollars and converted into yen at the exchange rate at expiry is called an "ordinary foreign-currency option." The two use the exchange rate differently. There are two ways to convert the value of a foreign-currency asset into the domestic currency: use the exchange rate at the time, or use a fixed exchange rate agreed in advance. The latter is the quanto⁴.

In the example above (strike USD 100, 100 shares, fixed rate 150 yen per dollar), compare the payoffs for each spot exchange rate at expiry. The premium is not included.

With the stock at USD 120 and a stronger yen (weaker dollar) at 140 yen, the quanto pays 20,000 yen more. With a stronger dollar at 160 yen, the ordinary option pays 20,000 yen more. When the stock price is above the strike, the quanto does not include the extra amount from a stronger dollar at expiry (the direction that raises the amount in yen), and a weaker dollar does not reduce it either. If the stock price is at or below the strike, both pay 0 yen, and Company D loses only the premium.

How the correlation enters the premium

The premium of a quanto option changes with the correlation between the price of the underlying asset and the exchange rate⁵. Within the framework of the Black-Scholes model, it can be written as follows. The price of a quanto call is the fixed rate $\bar{X}$ multiplied by the Black-Scholes price of a call with strike $K_f$. In that calculation, the following $q_f$ is used as the dividend yield in place of the actual dividend yield $q$⁵.

$$q_f = q + r_d - r_f + \rho\,\sigma_X\,\sigma$$

$r_d$ is the domestic and $r_f$ the foreign risk-free rate, $\sigma$ is the volatility of the price of the underlying asset, $\sigma_X$ is the volatility of the exchange rate, and $\rho$ is the correlation coefficient between changes in the price of the underlying asset and in the exchange rate. The exchange rate is the value of one unit of the foreign currency in the domestic currency (yen per dollar), and $\rho$ is the correlation under this quotation⁶. If the rate is quoted the other way (dollars per yen), the sign of $\rho$ is reversed.

The effect of the correlation shows up in the forward price. The forward price of the underlying asset, converted at the fixed rate and received at expiry, is given by the following formula⁷.

$$F^q = \bar{X}\, S_0\, e^{(r_f - q - \rho\,\sigma_X\,\sigma)\,T}$$

Consider a hypothetical example in which $S_0$ is USD 100, $q$ is 0, $r_f$ is 3% a year, $\sigma$ is 20%, $\sigma_X$ is 10% and $T$ is 1 year. Compare the forward price divided by the fixed rate (the forward price in dollars) for each $\rho$. The adjustment term is the correlation × the volatility of the exchange rate × the volatility of the underlying asset, and the exponent in the formula is $r_f - q$ minus the adjustment term.

Compared with a correlation of 0, a positive correlation lowers the forward price, which makes the quanto call cheaper and the quanto put more expensive. A negative correlation does the opposite. When the correlation is positive, the dollar tends to strengthen when the price of the underlying asset rises. In that case an ordinary option would pay more yen, but the quanto does not pay more because the rate is fixed. The sign of the adjustment term corresponds to this difference⁸.

This formula holds under the assumptions of the Black-Scholes model (constant volatility and correlation). When the correlation moves, it becomes a risk, as covered in the next section. How the model is used in practice differs by source. A 2013 lecture note says that exotic products such as quantos are not normally valued in this framework⁹. A 2003 paper from the Bank of Japan's Institute for Monetary and Economic Studies says that methods using this framework to price correlation derivatives, including quantos, are widely used in practice¹⁰. The point that the valuation changes with the model used is the same as in the valuation section of Overview of Exotic Options.

The seller's hedge and the remaining risk

When the seller, a financial institution, hedges a quanto option, it needs to limit the risk from movements in the exchange rate. The value $V_t$ (in the domestic currency) of the underlying asset, converted at the fixed rate and received at expiry, can be built from three positions in the Black-Scholes framework. They are buying the foreign-currency underlying asset in an amount equal to $V_t$, borrowing the same amount of foreign currency, and investing the same amount in the domestic currency. The first two cancel the exchange rate position, reproducing a payoff that is not affected by the exchange rate¹¹.

The seller holds the option's ordinary Black-Scholes delta, treating this $V_t$ as the underlying asset¹². Because $V_t$ moves with the price of the underlying asset, the amount held and the amount borrowed are rebalanced over time.

Even with the hedge in place, the following risks remain.

• Correlation risk. A product with several underlying assets has gains and losses from movements in the correlation between the prices of the underlying assets, called correlation-change risk. A product with a single underlying asset does not have this risk¹³. In the premium of a quanto option, the correlation enters through the adjustment term above.
• Difficulty of removing correlation risk in the market. Correlation derivatives have low liquidity, and it can be hard to find a product on the same underlying asset in the market. So the risk is kept, and the overall size is limited by combining correlation derivatives on similar underlying assets¹⁴.
• Large moves in the exchange rate. Many quantos are dynamically hedged by market sellers, and when a currency moves to an extreme, the currency risk can become large¹⁵.
• Conditions in the other country. Price makers such as banks are reluctant to write quantos on assets of high-risk countries¹⁶.

What it is used for

Quantos are traded frequently in the structured products market (2013 lecture note). One example is an option on a basket of three stocks, IBM (in dollars), Toyota (in yen) and Siemens (in euros), set up to be received in dollars so that it is not affected by exchange rates¹⁸. Another is a quanto whose payoff is in euros, so that an investor in the euro area can hold an option on crude oil, which is priced in dollars, without exchange rate risk¹⁸. The buyer can receive a payoff linked to the returns of a foreign stock or stock index, in the buyer's own currency¹⁹.

Look at Company D's gains and losses in the example above. Company D buys because it wants to receive the movement of the dollar-denominated stock in yen, and the premium is 75,000 yen (USD 5 × 100 shares × 150 yen), equal to USD 5 per share. If the stock price at expiry is USD 120, the payoff of 300,000 yen less the premium leaves a gain of 225,000 yen (interest is not considered). If the stock price is USD 90, the payoff is 0 yen and Company D loses only the premium of 75,000 yen. The break-even stock price at expiry is USD 105.

As seen in the comparison section, when the stock price at expiry is above the strike, the quanto does not include the extra amount that comes from the dollar being stronger at expiry than the fixed rate.

Notes

1. Martin Haugh, "Foreign Exchange, ADR's and Quanto-Securities" (lecture note for IEOR E4707, Columbia University, 2013; the "Haugh lecture note" below), Section 4 "Quanto-Securities" (page 8). ↩

2. 中村一岳「相関デリバティブのリスク特性やリスク管理手法に関する考察」(Kazutake Nakamura, a study of the risk characteristics and risk management of correlation derivatives, in Japanese; Bank of Japan Institute for Monetary and Economic Studies Discussion Paper No. 2003-J-7, 2003; the "Nakamura paper" below), Section 2 "Quanto option" (page 2). ↩

3. Haugh lecture note, "Quanto Options" (page 9). $K_f$ in the text corresponds to K/X̄ in the source. ↩

4. Haugh lecture note, opening overview (page 1). ↩

5. The same source, "Quanto Options" (page 9) and equations (17) and (21) (pages 8–9). It uses the dividend yield q_f = q + r_d − r_f + ρσ_xσ. The symbol σ_x in the source is the volatility of the exchange rate. ↩ ↩²

6. The same source, Section 1 (page 1). The exchange rate X_t is the cost in the domestic currency of one unit of the foreign currency, and the correlation is that between the Brownian motions of the underlying price and the exchange rate, as in equation (12) of Section 3 (page 7). ↩

7. The same source, "Quanto Forwards", equation (18) (page 8). ↩

8. An explanation that can be read from the signs in the formula and from the comparison of payoffs at expiry (the table above). It is not wording from a source. ↩

9. Haugh lecture note, opening overview (page 1). The note itself treats these products in this framework because it is easy to introduce. ↩

10. Nakamura paper, Section 3 (page 4). The reason given is that the prices of many correlation derivatives can be obtained as analytical solutions. ↩

11. Haugh lecture note, "Quanto Forwards" (page 9). ↩

12. The same source, "Quanto Options", Remark 5 (page 9). It performs a Black-Scholes delta hedge treating V_t as a non-dividend-paying underlying asset. ↩

13. Nakamura paper, Section 4 (1) b (page 8) and Section 1 (page 1). ↩

14. Nakamura paper, Section 4 (1) c (page 18). ↩

15. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual", Section 4345.1 "Equity Derivatives", "Currency Risk" (April 2003; pages 3–4). ↩

16. Haugh lecture note, Remark 7 (page 9). ↩

17. Nakamura paper, Section 6 (2) "Handling the correlation coefficient" (pages 30–31). ↩

18. Haugh lecture note, Section 4 (page 8). The three stocks are in dollars, yen and euros, and the crude oil example is in the same place. ↩ ↩²

19. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual", Section 4345.1 "Equity Derivatives", footnote 1 (February 1998; page 1). ↩