Quanto Options And Cross Currency Derivative Structures

When to Use

Use this skill when an underlying trades in one currency but the option's payoff is cash-settled in another at a rate fixed in the term sheet — a USD-settled option on the Nikkei 225, a EUR-settled option on WTI, a USD-settled option on a KRW-denominated index. The buyer wants the foreign asset's return and explicitly does not want the FX exposure that normally comes with it.

That FX exposure does not disappear; it is transferred to the seller, who prices it in through the drift. Under the domestic risk-neutral measure the foreign asset no longer drifts at $r_f - q$ but at

$$\mu_{\text{quanto}} = r_f - q - \rho,\sigma_S,\sigma_X$$

The engine prices the European call and put on that drift, discounts at the domestic rate $r_d$, scales by the fixed conversion multiplier $F_X$, and returns the Greeks — including the two that a plain Black-Scholes port gets wrong: Vega (which has a second channel through the drift) and $\partial V/\partial\rho$ (which is negative for a call and positive for a put).

Two conventions decide every sign in this model. Read them before passing data in.

• FX direction. $X_t$ is the cost in domestic currency of one unit of the foreign currency — "domestic per foreign". correlation is the correlation between the foreign asset's return (in foreign currency) and the return on $X$ in that direction. A correlation estimated against the inverted quote has the opposite sign, and the drift then moves by $2\rho\sigma_S\sigma_X$ — 1.8 percentage points of annual drift at the engine's defaults. The engine cannot detect this; nothing about a number in $[-1, 1]$ reveals which quote it came from.
• Strike currency. strike_price is in the foreign asset's own units, the same units as spot_price, and the whole payoff is then multiplied by $F_X$: $\text{payoff} = F_X\max(S_T - K, 0)$. If your term sheet fixes the strike in domestic currency, divide it by $F_X$ first.

When NOT to Use

• For a composite ("compo") option. A compo converts the payoff at the prevailing spot FX rate and fixes the strike in domestic currency; a quanto converts at a fixed rate. They are different contracts with different FX risk, not two settings of one model — this engine prices only the quanto. Confusing the two is the most common structuring error in this product family.
• For an American or Bermudan quanto. European exercise only. There is no early-exercise premium anywhere in this model. See american-vs-european-style-option-exercise-handling.
• As a smile-aware pricer. A single flat $\sigma_S$ with no skew and no term structure. Real quanto desks price the skew, and the quanto adjustment itself becomes skew-dependent. Calibrate the surface with options-implied-volatility-surface-construction first and treat this engine's output as a flat-vol reference point, not a mark.
• With a stale or long-horizon correlation. $\rho$ is assumed constant to expiry. Asset-FX correlation is among the least stable inputs in derivatives and reverses sign in stress. See Common Pitfalls and cross-asset-correlation-regime-shifts.
• At or past expiry. $T \le 0$ raises rather than returning intrinsic — an expired contract is a settlement problem, not a pricing one. See physical-vs-cash-settlement-handling.
• On a floating-FX structure of any kind, including dual-currency notes and FX-linked coupons. The fixed multiplier $F_X$ is load-bearing.

Prerequisites

• Contract terms: spot_price and strike_price (both in the foreign asset's currency), time_to_expiry_years $> 0$, fixed_fx_rate $> 0$ (the contractual domestic-per-foreign multiplier), option_type ('CALL' or 'PUT').
• Market data, all continuously compounded annualized decimals: domestic_rate ($r_d$, discounting only), foreign_rate ($r_f$, asset drift only), dividend_yield ($q$), asset_volatility ($\sigma_S > 0$), fx_volatility ($\sigma_X \ge 0$), correlation ($\rho \in [-1, 1]$).
• A correlation estimate whose FX quoting direction you have confirmed is domestic-per-foreign. See currency-pair-quoting-convention-normalization.

Workflow

1. Normalize the inputs before pricing — the engine validates, but the two conventions are yours to get right:
   • Decision point — confirm the FX quote direction of your correlation estimate, not just its magnitude. If $\rho$ was regressed against a foreign-per-domestic series, negate it. There is no downstream check that catches this: a wrong-signed $\rho$ produces a perfectly plausible price.
   • Decision point — convert a domestic-currency strike to foreign units ($K_{\text{foreign}} = K_{\text{domestic}} / F_X$) before passing it in.
   • Reject $|\rho| > 1$, negative volatilities, non-positive $S$/$K$/$T$/$F_X$, and NaN/Inf at the boundary. The engine raises ValueError on all of these; do not wrap that in a retry. A NaN that passes silently produces a report whose every field is NaN and whose status still reads QUANTO_PRICING_SUCCESSFUL.
2. Compute the quanto drift and forward:
   • $\mu_{\text{quanto}} = r_f - q - \rho\sigma_S\sigma_X$, quanto forward $F = S e^{\mu_{\text{quanto}}T}$ (in foreign units).
   • Decision point — $r_d$ discounts, $r_f$ drifts, and they are never interchangeable. $r_d$ appears only in $e^{-r_dT}$; $r_f$ appears only in $\mu_{\text{quanto}}$ and therefore inside $d_1$. Swapping them is a first-order error that survives every plausibility check because the price stays positive and near the money.
3. Price the European payoff on that drift:
   • $d_1 = \dfrac{\ln(S/K) + \left(\mu_{\text{quanto}} + \tfrac{1}{2}\sigma_S^2\right)T}{\sigma_S\sqrt{T}}$, $d_2 = d_1 - \sigma_S\sqrt{T}$.
   • $V_{\text{call}} = F_X e^{-r_dT}\left[F,N(d_1) - K,N(d_2)\right]$; $V_{\text{put}} = F_X e^{-r_dT}\left[K,N(-d_2) - F,N(-d_1)\right]$.
   • Decision point — an unrecognized option_type must raise, not default. A branch of the form if type == "CALL" ... else: <put> prices 'CAL', 'C', and '' as puts. Version 1.0.0 did exactly that and echoed the raw string back into the report, so the audit trail did not record which side was priced.
4. Compute the Greeks — two of them are not the Black-Scholes ones:
   • Delta $= \pm F_X e^{-r_dT} e^{\mu_{\text{quanto}}T} N(\pm d_1)$, Gamma $= F_X e^{-r_dT} e^{\mu_{\text{quanto}}T} n(d_1) / (S\sigma_S\sqrt{T})$.
   • Decision point — Vega has two channels. $\sigma_S$ enters $d_1/d_2$ and the drift, because the adjustment is $\rho\sigma_S\sigma_X$. The total is $\partial V/\partial\sigma_S = \underbrace{F_Xe^{-r_dT}Fn(d_1)\sqrt{T}}{\text{spot}} + \underbrace{(\partial V/\partial\mu)(-\rho\sigma_X)}{\text{drift}}$. Both components are reported separately. If your call and put vegas come out equal, you have only the spot channel — that equality holds in plain Black-Scholes and is false for a quanto.
   • Decision point — $\partial V/\partial\rho$ changes sign between call and put. Higher $\rho$ lowers the drift, so it cheapens the call and enriches the put. Version 1.0.0 returned the correct magnitude with a negative sign for both, which makes a mixed book's correlation exposures add instead of net.
5. Read the report — QuantoOptionPricingReport. Nothing in it is rounded; quantize at the presentation layer, where the notional is known.

Full procedure: see references/workflows.md. Standards reference: see references/standards.md. Printable pre-flight checklist: see assets/checklist.md.

Common Pitfalls

• Feeding in a correlation estimated against the inverted FX quote: the sign flips, the drift moves by $2\rho\sigma_S\sigma_X$, and the price is wrong by roughly twice the whole quanto adjustment — while still looking entirely reasonable. At $\rho = 0.30$, $\sigma_S = 20%$, $\sigma_X = 15%$ that is 1.8 points of annual drift.
• Passing a domestic-currency strike: the model's $K$ is in foreign units and the payoff is scaled by $F_X$ afterwards. A JPY-referenced strike passed against a USD conversion is off by two orders of magnitude and will not raise.
• Reporting the Black-Scholes vega as the quanto vega: it omits the drift channel. At the engine's defaults it overstates call vega by 6.9% (37.91 vs 35.48) and understates put vega by 4.8% (37.91 vs 39.81). The giveaway is identical call and put vega.
• Netting $\partial V/\partial\rho$ across a book without checking the sign per side: calls are negative, puts positive. Signing both the same way turns a partially hedged correlation position into an apparently doubled one, or hides a real one.
• Swapping $r_d$ and $r_f$: $r_d$ discounts, $r_f$ drifts. The resulting price is still positive and still near the money, so nothing downstream flags it.
• Treating $\rho$ as a stable parameter: asset-FX correlation is regime-dependent and inverts in stress — exactly when the quanto book is largest. Re-mark $\partial V/\partial\rho$ against a stressed $\rho$, not just the trailing estimate; the engine gives you the sensitivity precisely so this can be done.
• Assuming the quanto adjustment is small because $\sigma_X$ is small: it scales with the product $\rho\sigma_S\sigma_X$ and with $T$. On a five-year structure at $\rho = 0.5$, $\sigma_S = 30%$, $\sigma_X = 12%$ it is 1.8% per year, 9% of drift over the life.
• Rounding Greeks inside the engine: version 1.0.0 rounded quanto_gamma to 6 decimals. On a Nikkei-scale underlying ($S \approx 38{,}000$) the true gamma is $4.9881789\times10^{-5}$ and that rounding returned $5\times10^{-5}$ — one significant figure, a 0.24% error in the hedge ratio.

Verification

• Degenerate case: with $\sigma_X = 0$ and $r_d = r_f = 5%$, $q = 0$, the quanto adjustment and the rate mismatch both vanish and the price must collapse onto the standard worked example $\text{BS}(100, 100, 1\text{y}, 5%, 20%) = 10.450584$.
• Independent formulation: Haugh shows a quanto call equals $F_X$ times a Merton call with dividend yield $q_f = q + r_d - r_f + \rho\sigma_X\sigma_S$ and strike $K/F_X$. At the defaults that gives $8.156533$ (call) and $7.104404$ (put), matched to $10^{-10}$ across a strike $\times$ tenor $\times$ correlation grid.
• Drift and $d_1$: $\mu_{\text{quanto}} = 0.02 - 0 - 0.30 \times 0.20 \times 0.15 = 0.011$; $d_1 = (0 + 0.011 + 0.02)/0.20 = 0.155$; $d_2 = -0.045$.
• Put-call parity: $C - P = F_X e^{-r_dT}(Se^{\mu_{\text{quanto}}T} - K)$ to $10^{-12}$.
• Monte Carlo: 400,000 antithetic paths of $S_T$ under the domestic measure reproduce the call price to within $0.02$.
• Greeks vs. finite differences: every Greek matches a Richardson-extrapolated central difference of the price to at least 6 decimal places, at both positive and negative $\rho$.
• Vega (regression): total vega is $35.479670$ for the call and $39.807548$ for the put; the spot component alone is $37.910160$ for both. Version 1.0.0 returned $37.910160$ for both sides.
• Correlation sensitivity (regression): $\partial V/\partial\rho = -1.620327$ (call) and $+1.264925$ (put). Version 1.0.0 returned $-1.264925$ for the put — right magnitude, inverted sign.
• Negative checks: non-positive spot_price / strike_price / time_to_expiry_years / asset_volatility / fixed_fx_rate, negative fx_volatility, $|\rho| > 1$, NaN or Inf in any numeric field, and an option_type such as 'CAL', 'C', or '' must all raise ValueError. $\sigma_X = 0$ and $\rho = \pm 1$ must be accepted.
• Run python -m unittest discover -s skills/quanto-options-and-cross-currency-derivative-structures/scripts from the scripts/ directory and confirm a 100% pass rate.

Related Skills

• options-implied-volatility-surface-construction
• currency-pair-quoting-convention-normalization
• multi-currency-pnl-and-fx-conversion
• options-greeks-real-time-portfolio-aggregation
• cross-asset-correlation-regime-shifts
• american-vs-european-style-option-exercise-handling