Valuation models
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Every instrument type and the models that price it
Rates · 30
| Instrument type | Pricing models |
|---|---|
Fixed-rate bond (risk-free or on a credit curve)fixed_bond | Discounted cash flowsBootstrapped hazard curve |
Zero-coupon bondzero_bond | Discounted cash flows |
Amortizing bondamortizing_bond | Discounted cash flows |
Floating rate notefrn | Projected cash flows (forward rates) |
Capped or floored FRNcapped_frn | Projected cash flows (forward rates)Black-76 (and shifted Black)Bachelier (normal) |
Inverse floaterinverse_frn | Projected cash flows (forward rates)Black-76 (and shifted Black)Bachelier (normal) |
Callable bondcallable_bond | Hull-White trees and finite differences |
Putable bondputable_bond | Hull-White trees and finite differences |
Interest rate swapirs | Projected cash flows (forward rates) |
Overnight index swapois | Projected cash flows (forward rates) |
Basis swapbasis_swap | Projected cash flows (forward rates) |
Zero-coupon swapzero_coupon_swap | Projected cash flows (forward rates) |
Step-up swapstep_up_swap | Projected cash flows (forward rates) |
In-arrears swap (convexity adjusted)in_arrears_swap | Projected cash flows (forward rates) |
Cancelable (callable) swapcancelable_swap | Hull-White trees and finite differencesBachelier (normal)Black-76 (and shifted Black) |
Extendible swapextendible_swap | Hull-White trees and finite differencesBachelier (normal)Black-76 (and shifted Black) |
Dual currency bonddual_currency_bond | Discounted cash flows |
Cross-currency swapxccy_swap | Cross-currency discounting |
Forward rate agreementfra | Projected cash flows (forward rates) |
Interest rate futureir_future | Projected cash flows (forward rates) |
Bond forwardbond_forward | Cost of carry |
Bond futurebond_future | Cost of carry |
Interest rate capcap | Black-76 (and shifted Black)Bachelier (normal) |
Interest rate floorfloor | Black-76 (and shifted Black)Bachelier (normal) |
Interest rate collarcollar | Black-76 (and shifted Black)Bachelier (normal) |
Swaption (European, Bermudan)swaption | Black-76 (and shifted Black)Bachelier (normal)Hull-White trees and finite differences |
CMS swapcms_swap | CMS convexity and timing adjustment |
CMS capcms_cap | CMS convexity and timing adjustmentBlack-76 (and shifted Black)Bachelier (normal) |
CMS floorcms_floor | CMS convexity and timing adjustmentBlack-76 (and shifted Black)Bachelier (normal) |
Range accrual noterange_accrual | Digital replication on the smile |
Inflation · 3
| Instrument type | Pricing models |
|---|---|
Zero-coupon inflation swapinflation_zc_swap | Inflation index projection |
Year-on-year inflation swapinflation_yoy_swap | Inflation index projection |
Inflation-linked bondinflation_linked_bond | Inflation index projectionBlack-76 (and shifted Black) |
Credit · 5
| Instrument type | Pricing models |
|---|---|
Credit default swapcds | Hazard rate (ISDA-style CDS)Bootstrapped hazard curve |
CDS index (CDX, iTraxx)cds_index | Bootstrapped hazard curve |
CDS index optioncds_index_option | Black on the forward credit spread |
Term loan (fixed or floating, amortizing)term_loan | |
Revolving credit facilityrevolver |
Securitized · 5
| Instrument type | Pricing models |
|---|---|
Mortgage pass-through (agency MBS)mbs_passthrough | Prepayment projection (PSA, CPR)OAS on Hull-White rate paths |
TBA (to-be-announced) forwardtba | Prepayment projection (PSA, CPR)OAS on Hull-White rate paths |
Interest-only (IO) stripio_strip | Prepayment projection (PSA, CPR)OAS on Hull-White rate paths |
Principal-only (PO) strippo_strip | Prepayment projection (PSA, CPR)OAS on Hull-White rate paths |
Adjustable-rate mortgage (ARM) poolarm_pool | Prepayment projection (PSA, CPR)OAS on Hull-White rate paths |
FX · 4
| Instrument type | Pricing models |
|---|---|
FX spotfx_spot | Mark to market |
FX forwardfx_forward | Covered interest parity |
FX futurefx_future | Covered interest parity |
FX optionfx_option | Garman-KohlhagenBinomial tree (Cox-Ross-Rubinstein) |
Equity · 26
| Instrument type | Pricing models |
|---|---|
Equity holdingequity | Mark to market |
Equity forwardequity_forward | Cost of carry |
Equity index futureequity_future | Cost of carry |
Vanilla option (European, American, Bermudan)option | Black-ScholesBlack-76 (and shifted Black)Bachelier (normal)Binomial tree (Cox-Ross-Rubinstein) |
Digital optiondigital_option | Closed-form exotics (Reiner-Rubinstein, Kemna-Vorst, Turnbull-Wakeman) |
Barrier optionbarrier_option | Closed-form exotics (Reiner-Rubinstein, Kemna-Vorst, Turnbull-Wakeman) |
Asian option (fixed or average strike)asian_option | Closed-form exotics (Reiner-Rubinstein, Kemna-Vorst, Turnbull-Wakeman) |
Chooser optionchooser_option | Closed-form exotics (Reiner-Rubinstein, Kemna-Vorst, Turnbull-Wakeman) |
Double barrier optiondouble_barrier_option | Double barrier eigenfunction series |
Double touch or no-touchdouble_touch_option | Double barrier eigenfunction series |
Lookback option (floating, fixed)lookback_option | Lookback closed forms (Goldman-Sosin-Gatto, Conze-Viswanathan) |
Compound optioncompound_option | Geske compound options |
Warrantwarrant | Black-Scholes with dilution (Galai-Schneller) |
Rainbow option (best of, worst of)rainbow_option | Monte Carlo (correlated lognormal)Stulz two-asset rainbow |
Option under stochastic volatilityheston_option | Heston stochastic volatility |
Option on the SABR smilesabr_option | SABR (Hagan) |
Option under local volatilitylocal_vol_option | Local volatility (Dupire PDE) |
Autocallable noteautocallable | Local volatility Monte CarloMonte Carlo (correlated lognormal) |
Reverse convertiblereverse_convertible | Local volatility Monte CarloMonte Carlo (correlated lognormal) |
Cliquet notecliquet | Local volatility Monte CarloMonte Carlo (correlated lognormal) |
Basket optionbasket_option | Monte Carlo (correlated lognormal) |
Spread optionspread_option | Monte Carlo (correlated lognormal) |
Variance swapvariance_swap | Variance replication |
Volatility swapvolatility_swap | Variance replication |
Total return swaptotal_return_swap | Total return discounting |
Dividend swapdividend_swap | Dividend discounting |
Commodity · 4
| Instrument type | Pricing models |
|---|---|
Commodity holdingcommodity | Mark to market |
Commodity forwardcommodity_forward | Cost of carry |
Commodity futurecommodity_future | Cost of carry |
Commodity swapcommodity_swap | Average of the forward curve |
Money market and cash · 6
| Instrument type | Pricing models |
|---|---|
Depositdeposit | Money market conventions |
Certificate of depositcd | Money market conventions |
Commercial papercommercial_paper | Money market conventions |
Bankers' acceptancebankers_acceptance | Money market conventions |
Treasury billt_bill | Money market conventions |
Reporepo | Money market conventions |
Hybrid · 4
| Instrument type | Pricing models |
|---|---|
Convertible bondconvertible_bond | Tsiveriotis-Fernandes tree |
Convertible preferredconvertible_preferred | Tsiveriotis-Fernandes tree |
Mandatory convertible (PEPS, PERCS)mandatory_convertible | Black-Scholes |
Preferred sharepreferred_share | Discounted cash flows |
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Rates
Linear rates products on a zero curve, including today's government curves.
Interest rate swap
Fixed for floating, overnight index and zero-coupon swaps, with an optional amortizing notional.
IRSOISzero-couponPV01ValueCallable and putable bonds
Bonds with call or put dates on a Hull-White tree: OAS, option value and effective duration.
callableOASHull-WhiteValueConvertible bonds
Bonds convertible into shares, with calls and puts, on a Tsiveriotis-Fernandes tree.
convertibleparitycredit spreadValueLoans
Term loans and revolvers: amortization, prepayments, floors, commitment fees and discount margin.
loanrevolverdiscount marginValuePreferred shares
Fixed or floating, perpetual or dated, callable preferreds: price, spread, yield and CS01.
preferredperpetualcallableValueMandatory convertibles
PEPS and PERCS by Black-Scholes replication: equity and coupon parts, premium, delta and vega.
mandatory convertiblePEPSPERCSValueSwaptions
European swaptions on Black-76 or Bachelier, and Bermudans on a calibrated Hull-White model.
swaptionBermudanHull-WhiteValueCaps, floors and collars
Strips of caplets and floorlets with Black-76 or Bachelier, a flat cap vol or a surface.
capfloorcollarvegaValueCaplet vol stripping
Caplet vols bootstrapped from flat cap vols, ready to use as a surface.
caplet volcap volbootstrapValueCMS swaps, caps and floors
Constant-maturity swap rates with convexity and timing adjustments, off a swaption cube.
CMSconvexity adjustmentswaption cubeValueRange accruals
Notes paying a coupon on each day the index sets in range, priced as digitals on the smile.
range accrualdigitalstructured noteValueBasis and cross-currency swaps
Float against float in one currency, or two currencies with notional exchanges, with the par spread.
basisxccypar spreadValueCancelable and extendible swaps
A swap with the right to end it early or extend it: the swap plus a Bermudan or European swaption.
cancelable swapcallable swapextendibleValueDual currency bonds
Coupons in one currency, principal in another: value, FX forward, breakeven and FX delta.
dual currencyFXbondValueFRA and forward rates
Forward rate agreement value and settlement, with the forward curve implied by the zero curve.
FRAforward curveValueBond pricer
Price from yield or yield from price, with duration, convexity and DV01.
bonddurationDV01ValueFloaters and amortizing bonds
FRNs, capped and inverse floaters with Black-76 caps and floors, and amortizing bonds.
FRNdiscount marginamortizingValueYield toolkit
YTM, yield to call and worst, current and simple yield, rate conversion and T-bill yields.
YTMyield to worstT-billsValueZero curve and bond off a curve
Bootstrap discount factors, zero and forward rates, then price a bond off the curve with a z-spread.
bootstrapz-spreadValue
Money market and cash
Deposits, bills, commercial paper and repos on the curve, and cash holdings at market.
Futures and forwards
Cost of carry, forward rates and bond forwards; futures margined, forwards discounted.
FX
Forwards and options on currency pairs, using both currencies' curves.
Options
Closed-form and lattice models for options on assets, futures and rates.
Black-76
Options on futures and forwards, and the building block for caps, floors and swaptions.
futurescapsswaptionsValueBachelier (normal)
Normal-volatility options for low or negative rates, quoted in basis points of vol.
normal volnegative ratesValueBinomial tree
American, Bermudan and European options on a Cox-Ross-Rubinstein lattice, with the exercise boundary.
AmericanBermudanCRRValueBlack-Scholes
European calls and puts with a dividend yield and all first-order Greeks.
EuropeanGreeksValueExotic options
Digital, barrier, double barrier, Asian, lookback, compound and chooser options and warrants in closed form.
barrierdouble barrierdigitalAsianlookbackcompoundwarrantValueHeston stochastic volatility
European options with a stochastic variance, given parameters or calibrated to a vol surface.
Hestonstochastic volcalibrationsmileValueSABR smile
Options on the SABR smile, fitted to a vol surface or given, shifted for negative rates.
SABRsmilecalibrationshifted SABRValueLocal volatility
Dupire local vol from the surface: vanillas, American, digital and barrier options by PDE.
local volDupirebarrierPDEValueBasket, spread and rainbow options
Options on a weighted basket, the spread between two assets, or the best or worst of several.
basketspreadrainbowMonte CarlocorrelationValue
Inflation
Index-linked cash flows projected off zero-coupon inflation swap rates.
Equity swaps and notes
Swaps on volatility, total return and dividends, and structured notes on a stock or index.
Variance and volatility swaps
Fair variance replicated from the smile, realized to date, and the volatility swap convexity.
variance swapvolatility swapskewvegaValueTotal return swaps
Price return and dividends against floating funding: value, par spread and equity exposure.
TRSequity swapfundingsyntheticValueAutocallables, reverse convertibles and cliquets
Structured equity notes by Monte Carlo under local volatility from the smile.
autocallablereverse convertiblecliquetlocal volValueDividend swaps
Dividends over a period against a fixed strike, from listed dividends or an implied yield.
dividend swapdividend futureimplied dividendsValue
Mortgages
Agency pass-throughs with prepayment models, OAS on rate paths and effective duration.
Credit
Default risk priced from credit spreads and bootstrapped hazard curves.
Credit default swap
Hazard rate, default probabilities, upfront, CS01 and jump-to-default from a CDS spread.
CDShazard ratePDValueCDS indices
CDX and iTraxx: upfront, price, CS01 and the basis to the single names.
CDXiTraxxindex basisValueCDS index options
Payer and receiver swaptions on CDX and iTraxx, Black on the forward spread with front-end protection.
credit swaptionpayerreceiverValueBonds on a credit curve
Price a corporate bond off the issuer's CDS curve: default probabilities, expected loss and z-spread.
corporate bondhazard curveCS01Value
Government curves stand in for swap and OIS curves, so rates-product values differ from dealer quotes by the swap spread. Results are for analysis and education, not investment advice.