Interest Rates in Option Pricing: Curves, Carry, and Rho

Convert rate quotes into discount factors, build consistent forwards, scale Rho correctly, and full-reprice curve, dividend, exercise, and execution scenarios.

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

Interest rates enter option values through discount factors, the underlying forward, funding and the timing of exercise and settlement cash flows. There is no universal scalar rate: valuation requires a currency- and maturity-consistent curve under a stated collateral or funding convention. A policy target, overnight fixing, Treasury par yield and zero rate are different objects.

For a controlled European equity-option comparison, higher rates commonly raise a call and lower a put when spot, dividends, borrow, volatility and every other input are fixed. That sign is not a market-profit rule or a universal result for American claims, bond-price options, yield options, futures options, caps, floors or swaptions. Rho is a local model derivative, not a forecast or a complete curve-risk measure.

A controlled workflow

Lock the claim and clocks: underlying, call or put, strike, currency, valuation timestamp, expiry, exercise and settlement dates, American or European style, cash or physical settlement, multiplier and signed position.
Define the curve mandate: discount, collateral, funding or borrow; record source, quote side, timestamp and instruments. Do not substitute the policy rate or one constant-maturity yield for a complete curve.
Convert simple, discount, nominal and continuously compounded quotes with their day count into positive discount factors D(0,t). Bootstrap and interpolate cash-flow dates under documented rules; negative zero rates can validly imply D(0,t)>1.
Build carry consistently. With constant continuous rates, D(0,T)=exp(-r*T) and F=S*exp((r-q)*T); with deterministic cash dividends, use FP=S-PV(dividends) and F=FP/D(0,T). Keep dividends, borrow, foreign carry and corporate actions separate.
Match model to style and payoff. Use European parity or closed form only for compatible claims; use an exercise-aware tree or PDE for American options and product-specific models for bond, yield, futures or rate payoffs.
State risk units: raw Rho_raw=partial V/partial r, Rho_100bp=Rho_raw*0.01, Rho_1bp=Rho_raw*0.0001, each per share, contract or signed position. Rebuild the curve and full-reprice parallel, steepening, flattening and key-rate shocks.
Keep model attribution separate from executable bid and offer, fills, fees, funding, borrow, margin, collateral and tax. Reconcile actual cash dates, exercise or assignment, settlement and P/L against the saved curve version.

Worked examples

Exact European parity: For S=100, K=100, T=2, q=0 and continuous r=2%, the discount factor is 0.960789439152, discounted strike is 96.078943915232, and C-P=3.921056084768. At r=5%, these are 0.904837418036, 90.483741803596 and 9.516258196404; call minus put changes by 5.595202111636 per share. This is a controlled relative-value identity, not predicted profit.
Compounding and day count: A 182-day ACT/360 simple quote of 5.25% gives D=1/(1+0.0525*182/360)=0.974144579291. With T=182/365=0.498630136986, the equivalent continuous zero rate is 5.2535026897%. If continuous q=1% and S=100, the consistent forward is 102.143576091945; inserting 5.25% directly into a different convention changes the result.
Rho units and full repricing: A European option with S=K=100, T=2, q=1%, sigma=25% and r=2% has call 14.609621487800, put 12.668698072357, raw call Rho 86.885943346376 and raw put Rho -105.271944484089. Per 100 bp these are 0.868859433464 and -1.052719444841. Full repricing at r=3% gives call 15.491134163794 and put 11.647720191543, changes +0.881512675994 and -1.020977880814; old-Rho linear errors are +0.012653242530 and +0.031741564027 per share.
American exercise floor: For S=80, K=100, T=1, r=5%, q=0 and sigma=20%, a European put is 16.982362022884, below immediate exercise value 20. An American put must be worth at least 20; the 3.017637977116 gap shows why a European formula cannot price that exercise right. Separately, r=-1% gives valid D=1.010050167084; do not clip negative rates to zero.

Risks and validation

Currency risk: A curve in the wrong currency discounts the wrong cash claim.
Mandate risk: Collateral, funding, discount and borrow curves are not interchangeable.
Quote risk: Par yield, zero rate, forward rate and policy target are different objects.
Compounding risk: Simple, discount, nominal and continuous quotes require conversion.
Day-count risk: ACT/360, ACT/365F and 30/360 change year fractions and values.
Clock risk: Valuation, expiry, exercise, delivery and cash settlement dates differ.
Timestamp risk: Stale or unsynchronized curve and option inputs create false Rho.
Bootstrap risk: Instruments, interpolation and extrapolation affect node discount factors.
Shock risk: One parallel shift misses steepening, flattening and localized moves.
Negative-rate risk: Clipping rates or discount factors destroys valid economics.
Dividend risk: Amount, ex-date and payment-date errors move the forward.
Double-count risk: Continuous yield and the same discrete dividends must not both be applied.
Borrow risk: Rebate and hard-to-borrow costs alter carry and exercise incentives.
Coordinate risk: Spot and forward moneyness can move differently after a curve shock.
Style risk: European and American claims need different parity and exercise treatment.
Lifecycle risk: Holder exercise and writer assignment create different actions and cash flows.
Unit risk: Raw, 100-bp and 1-bp Rho or share and contract units can be confused.
Greek risk: Local Rho does not replace curve rebuild and full repricing.
Joint-move risk: Spot, IV, skew, dividends and funding often move with rates.
Execution risk: Bid-offer, fees, liquidity, margin and tax separate model change from P/L.

Common misconceptions

“The policy rate is the right input for every expiry.” Each cash date needs a consistent discount factor.
“Higher rates make every call or position gain.” Claim type, other inputs and exercise rights can dominate.
“Raw Rho times 100 gives a 100-bp change.” A 100-bp decimal shock is 0.01, and units must match.
“Rates and dividends are interchangeable.” They arise from different cash flows and must not be double-counted.
“Equity-option Rho signs apply to every rate product.” Bond, yield, futures and rate-option payoffs have different directions.

Authoritative sources

Federal Reserve H.15 — observed selected rates, not a universal option discount curve.
Treasury Yield Curve Methodology — par-curve construction and indicative inputs, not executable zero or funding rates.
New York Fed SOFR Averages — backward-looking overnight compounding conventions, not a complete forward curve mandate.
OIC Rho — educational local sensitivity and common scaling, not full curve risk or P/L.
OCC Characteristics and Risks — contract and lifecycle risks, not numerical curve construction.
Cboe Theoretical Options Methodology — published style, curve and dividend conventions, not universal desk truth or fills.
Black and Scholes — the European constant-rate model foundation, not American exercise or live execution.
Merton, Theory of Rational Option Pricing — carry, dividends and no-arbitrage theory under assumptions, not current contract rules.