Covered Interest Parity and FX Forward Pricing

An FX forward rate is a price agreed today for exchanging currencies at a future date. CIP says that price is completely determined by the spot rate and the two interest rates — there is no free parameter. If the forward deviates, an arbitrageur can lock in riskless profit by borrowing in one currency, converting at spot, investing in the other, and covering the return leg with the forward. Markets eliminate this instantly; therefore the forward must satisfy the CIP formula. In QuantLib the relationship is expressed directly via discount factors: forward = spot × dfForeign / dfDomestic. In the synthetic market data stack this means: generate IR curves and FX spot from the GMM, then compute FX forwards analytically — sampling them independently breaks CIP and produces inconsistent NPVs across every FX and cross-currency product.

For a currency pair quoted as CCY1/CCY2 (e.g. EUR/USD, where EUR = CCY1 and USD = CCY2), with \(T\) years to settlement:

\[\text{FX\_fwd}(T) = \text{FX\_spot} \times e^{(r_{\text{dom}} - r_{\text{for}}) \times T}\]

where:

• \(r_{\text{dom}}\) = continuously compounded risk-free rate for CCY2 (the domestic / pricing currency, USD in EUR/USD)
• \(r_{\text{for}}\) = continuously compounded risk-free rate for CCY1 (the foreign currency, EUR in EUR/USD)

In QuantLib discount-factor form (which is what the library actually stores):

\[\text{FX\_fwd}(T) = \text{FX\_spot} \times \frac{DF_{\text{for}}(T)}{DF_{\text{dom}}(T)}\]

where \(DF_{\text{for}}(T) = e^{-r_{\text{for}} \times T}\) is the foreign discount factor and \(DF_{\text{dom}}(T) = e^{-r_{\text{dom}} \times T}\) is the domestic discount factor. Dividing foreign by domestic recovers the rate-differential exponential.

These two forms are identical. Use whichever is more convenient — the discount-factor form is usually what you will see in code.

Suppose EUR/USD spot = 1.10, the 1Y USD rate = 5 %, and the 1Y EUR rate = 3 %. You have $1,000. You can choose one of two paths to hold wealth for one year and end up in USD:

• Path A (stay in USD): Invest $1,000 at 5 % for one year. Result: $1,050.
• Path B (round-trip through EUR):
  1. Convert $1,000 → €909.09 at spot (1,000 / 1.10).
  2. Invest €909.09 at 3 % for one year → €936.36.
  3. Sell €936.36 forward at the 1Y EUR/USD forward rate to get back into USD.

If the 1Y forward is, say, 1.15 instead of the correct 1.1222, Path B returns €936.36 × 1.15 = $1,076.81. That is $26.81 more than Path A with zero risk — because the future USD amount is already locked in at step 3.

Everyone would pile into Path B immediately. This buying pressure pushes the EUR/USD spot up (everyone selling USD for EUR), the EUR interest rate down (everyone investing EUR), and the forward down (everyone selling EUR forward). The market adjusts until both paths return exactly the same amount. That equilibrium is:

\[1.10 \times e^{(0.05 - 0.03) \times 1} = 1.10 \times 1.0202 = 1.1222\]

This is "covered" parity: the forward covers the currency risk of the round-trip. No forecasting required, no model required — just the availability of the three instruments (spot, two deposit rates, and the forward). When all four are tradeable, any deviation is eaten by arbitrage in seconds.

QuantLib expresses CIP directly through its yield term structure machinery. The key calculation is:

// Given two yield term structures and a spot rate, compute the
// T-year forward exchange rate via CIP.
Handle<YieldTermStructure> usdCurve = ...;  // domestic (USD)
Handle<YieldTermStructure> eurCurve = ...;  // foreign (EUR)
Real spot = 1.10;

Real t = 1.0;                          // 1Y horizon
Real dfUSD = usdCurve->discount(t);    // exp(-r_USD * t)
Real dfEUR = eurCurve->discount(t);    // exp(-r_EUR * t)

// CIP: forward = spot * dfForeign / dfDomestic
Real fxForward = spot * dfEUR / dfUSD;
// fxForward == 1.1222 (given the example rates above)

The YieldTermStructure::discount(t) method returns \(DF(t)\) — the present value of 1 unit of that currency at time \(t\). Dividing foreign by domestic is exactly the ratio \(e^{-(r_{\text{for}} - r_{\text{dom}}) t}\), which rearranges to the CIP formula above.

QuantLib's ForwardValueCalculator and ORE's FX forward curve bootstrap both implement this relationship. Any FX product class (FxForward, VanillaOption with FX underlying, CrossCurrencySwap) reaches the same discount-factor ratio internally when it needs to convert a future FX cash flow into a present value.

In ORE, FX forward curves are configured in market.xml and curveconfig.xml. The FXForwardQuote market data type reads FX forward points from market data inputs. During curve building, ORE checks that quoted forwards are consistent with the referenced IR curves; inconsistencies are flagged as calibration errors.

The pricing path is:

1. FxSpotQuote provides the spot rate.
2. Two YieldTermStructure handles (one per currency) provide the discount factors.
3. The pricing engine computes forward rates on demand via spot * dfFor / dfDom — the same formula as the QuantLib snippet above.

Any FX product in ORE — FX forward, FX option, cross-currency swap — uses this CIP relationship internally. There is no separate "forward curve" stored independently of the spot and IR curves; the forward is always derived.

For the ORE market data catalogue, the relevant quote types are:

• FX/RATE — the FX spot rate.
• IR_SWAP/RATE — IR swap rates used to bootstrap the yield curves.
• FXFWD/RATE — FX forward rates (when read from market data rather than derived; ORE cross-checks these against CIP during bootstrap).

In practice, CIP is violated by a small amount in interbank markets. Banks face funding constraints — it is not always possible to borrow freely in a foreign currency and invest freely in the domestic one. This friction creates a residual gap between the CIP-implied forward and the actual traded forward, known as the cross-currency basis.

The basis is typically expressed as an annualised basis-point spread on the foreign leg of a cross-currency swap. For EUR/USD it has ranged from a few basis points in calm markets to several tens of basis points during funding stress (e.g. 2008, 2020).

In ORE the cross-currency basis is captured by the CC_BASIS_SWAP/BASIS_SPREAD quote type. It is added as a spread on top of the CIP-derived forward.

For synthetic market data generation there are two valid approaches:

1. Zero basis (pure CIP): Set CC_BASIS_SWAP/BASIS_SPREAD = 0 for all currency pairs. The generated forwards are exact CIP. This is correct for theoretical analysis where you want a clean no-arbitrage dataset.
2. GMM-generated basis: Generate basis spreads from a separate GMM model and add them to the CIP-derived forward. This produces a more realistic dataset that captures the empirical basis seen in interbank markets, at the cost of introducing a small, controlled CIP violation.

Either approach is internally consistent because the basis spread is added explicitly on top of the CIP formula — the IR curves and spot still determine the base forward.

This is the practical reason the constraint "FX forwards must be derived from CIP" appears throughout the analysis documents.

• Interest Rate Curves — discount factors, bootstrapping, and the QuantLib term structure machinery that feeds the CIP calculation.
• FX Volatility Surface — the implied vol surface that sits on top of the CIP-consistent forward and is used for FX option pricing.
• ORE market data catalogue — the full list of ORE quote types including FX/RATE, FXFWD/RATE, IR_SWAP/RATE, and CC_BASIS_SWAP/BASIS_SPREAD.
• Intermediate analysis: map paper techniques to ORE asset classes — the analysis document where the "derive FX forwards from CIP" constraint originates.