How does a quanto option eliminate currency risk, and what adjustment is made to the drift rate in pricing?

A quanto (quantity-adjusted) option converts the payoff of a foreign-asset option into the investor's domestic currency at a predetermined fixed exchange rate, eliminating FX risk entirely. The pricing trick is an adjustment to the underlying asset's drift rate.\n\nQuanto Call Payoff:\n\nPayoff = Q x max(S_T - K, 0)\n\nwhere Q is the fixed (pre-agreed) exchange rate and S_T is the foreign asset price at expiration, both S_T and K denominated in the foreign currency.\n\nThe Quanto Drift Adjustment:\n\nIn a standard risk-neutral framework, the foreign asset drifts at the foreign risk-free rate r_f. Under the domestic risk-neutral measure, the quanto-adjusted drift becomes:\n\nmu_quanto = r_f - rho x sigma_S x sigma_X\n\nwhere:\n- rho = correlation between the foreign asset and the exchange rate\n- sigma_S = volatility of the foreign asset\n- sigma_X = volatility of the exchange rate\n\nWhy the Adjustment Exists:\n\nConsider a Japanese investor holding a quanto call on a US stock. If the stock rises and the dollar simultaneously strengthens (positive correlation), a non-quanto holder would benefit doubly. The quanto removes this second benefit, so the drift must be reduced. Conversely, if correlation is negative, the quanto actually increases the drift because it removes a natural drag.\n\nWorked Example:\nTakara Investments (JPY-based) buys a 1-year quanto call on Westfield Industries (USD stock). Parameters:\n\n- Westfield current price: $120\n- Strike: $120 (ATM)\n- Fixed FX rate: 150 JPY/USD\n- USD risk-free rate (r_f): 5.0%\n- JPY risk-free rate (r_d): 0.5%\n- Westfield volatility (sigma_S): 25%\n- USD/JPY volatility (sigma_X): 12%\n- Correlation (rho): 0.35\n\nQuanto drift adjustment: rho x sigma_S x sigma_X = 0.35 x 0.25 x 0.12 = 0.0105 (1.05%)\n\nAdjusted drift: 5.0% - 1.05% = 3.95%\n\nThe quanto call is then priced using BSM with:\n- Drift: 3.95% (not 5.0%)\n- Volatility: 25% (unchanged — only the drift is adjusted)\n- Discount rate: 0.5% (domestic JPY rate, since payoff is in JPY)\n\nQuanto call value (in JPY): 150 x BSM(S=120, K=120, r=0.5%, mu=3.95%, sigma=25%, T=1)\n\nCompared to a vanilla USD-denominated call discounted at 5%, the quanto is cheaper when rho > 0 and more expensive when rho < 0.\n\nKey Exam Points:\n- Positive correlation reduces the quanto drift, lowering call value\n- The discount rate switches to the domestic (investor's) risk-free rate\n- Quanto volatility uses only the asset volatility, not the FX volatility\n- Popular in cross-border index products (e.g., Nikkei futures traded in USD on CME)\n\nMaster cross-currency derivatives in our FRM course materials.