Advanced: Models and Numerical Methods

Who this is for

For quants, valuation engine developers and model validators who want to understand models from the inside.

How to read it

Assumes you understand Black-Scholes and the Greeks. Stages 1 and 2 are practitioner pages, included as a check. Start at stage 3 if you know them.

1. Stage 11Probability and the basis of valuationPractitioner

   Probability and statistics, random walks and Brownian motion, risk-neutral valuation, then Black-Scholes and the Black and Bachelier models.

   1. IntroductoryProbability and Statistics BasicsIf a price will be 103 yen, 101 yen, or 95 yen with probabilities of 25%, 50%, and 25%, the expected value is 100 yen and the standard deviation is 3 yen.Core Concepts
   2. PractitionerRandom Walks and Brownian MotionA symmetric random walk moving +1 or −1 at each step has mean 0 and variance n after n steps, so its standard deviation after 100 steps is 10.Core Concepts
   3. PractitionerRisk-Neutral ValuationFor a payoff that can be replicated with the underlying and risk-free investing and borrowing, the expected value under risk-neutral probabilities, discounted, equals the cost of that replication.Valuation
   4. PractitionerThe Black-Scholes ModelThe model assumes that the underlying price follows geometric Brownian motion, that volatility and interest rates are constant, and that trading is continuous and free of costs.Valuation
   5. PractitionerThe Black and Bachelier ModelsIn the Black model, the size of forward moves is proportional to the level of the forward; in the Bachelier model, it is the same at any level.Valuation

2. Stage 22Curves and volatilityPractitioner

   Curve bootstrapping and the multi-curve framework, collateral discounting, implied volatility and the smile: the inputs the models consume.

   1. PractitionerBootstrapping a Yield CurveSwap fixed rates do not show discount factors directly, so you solve for them in order of maturity, starting with the shortest.Core Concepts
   2. PractitionerThe Multi-Curve FrameworkDiscounting uses an OIS curve, projection uses a curve for each reference rate and tenor, and the projection curve is solved from the condition that market swaps have zero value.Core Concepts
   3. PractitionerCollateral and DiscountingThe collateral currency, the interest rate on collateral, the transfer frequency, and the threshold are set in the credit support annex (CSA), and the discount rate changes with these terms.Valuation
   4. PractitionerImplied VolatilityImplied volatility restates an option's price as a volatility, much as a yield restates the price of a bond.Valuation
   5. PractitionerVolatility Smiles and SurfacesEquity options usually show a downward-sloping shape (a skew): implied volatility is higher at lower strikes, and the slope is steeper at shorter expiries.Valuation

3. Stage 33Stochastic calculus and modelsAdvanced

   Ito's lemma and curve interpolation, then Hull-White, the LMM, SABR, local volatility, Heston, credit copulas and calibration.

   1. AdvancedIntroduction to Stochastic Calculus (Ito's Lemma)A stochastic differential equation writes the small movement of a value as a drift term plus a dW term, and writing it in integral form brings in the Ito integral.Core Concepts
   2. AdvancedCurve InterpolationThe quantity you interpolate is the discount factor, its logarithm, or the zero rate, and the quantity you choose sets the shape of the forward rate between the points.Core Concepts
   3. AdvancedThe Hull-White ModelThe short rate moves according to two parameters, the speed of mean reversion and the volatility. It is normally distributed, so it can take negative values.Valuation
   4. AdvancedThe LIBOR Market Model (LMM)The LMM moves the observable forward rate for a given period directly, instead of the unobservable short rate, and the drift of each forward rate follows from the no-arbitrage condition.Valuation
   5. AdvancedThe SABR ModelThe four parameters α, β, ρ, and ν set the level, slope, and curvature of the smile separately.Valuation
   6. AdvancedLocal Volatility ModelsIn a local volatility model, volatility is determined by the underlying price and time, and a single source of randomness drives the price.Valuation
   7. AdvancedThe Heston ModelThe stock price and the variance follow separate stochastic processes, and the two moves are linked by a correlation.Valuation
   8. AdvancedCopulas for CreditA CDS gives only each name's own default probability. The way names cluster is set separately by the copula, which gives the joint distribution of the times to default.Valuation
   9. AdvancedModel CalibrationA model used for valuation is fitted to the current prices of instruments that have observable market prices and are also used for hedging.Valuation

4. Stage 44Numerical methodsAdvanced

   When to use Monte Carlo, finite differences and AAD, and how to think about cost and accuracy.

   1. PractitionerMonte Carlo SimulationPaths are generated under risk-neutral dynamics, in which the expected return of the underlying is set to the risk-free rate.Valuation
   2. AdvancedFinite Difference MethodsReplacing derivatives with differences between grid points lets the explicit method write one time step as a weighted sum of three neighboring values.Valuation
   3. AdvancedAdjoint Algorithmic Differentiation (AAD)Bumping, which moves the inputs one at a time, needs more recomputation as the number of sensitivities grows, and the choice of step size adds error.Valuation

5. Stage 55Applications to productsAdvanced

   Bermudan swaptions, CMS, PRDC, variance swaps and synthetic CDOs: how the models are used to value and hedge products.

   1. AdvancedBermudan SwaptionsFor each exercise date, the swap that starts has the same end date, and the later the start, the shorter the swap.Interest Rate Derivatives
   2. AdvancedConstant Maturity Swaps (CMS)A CMS exchanges a swap rate of fixed tenor, such as the 10-year swap rate, for a short-term floating rate or a fixed rate on each payment date.Interest Rate Derivatives
   3. AdvancedPower Reverse Dual Currency (PRDC)A PRDC swap exchanges an FX-linked coupon for a floating yen rate, over a long term that can reach 30 years or more.FX Derivatives
   4. AdvancedVariance Swaps and Volatility SwapsThe buyer receives the notional amount times the amount by which realized variance exceeds the strike, and pays the same calculation when it falls below.Equity Derivatives
   5. AdvancedSynthetic CDOs and TranchesA tranche is a layer that absorbs portfolio losses from its attachment point to its detachment point, and losses hit the equity, mezzanine, and senior tranches in that order.Credit Derivatives
   6. AdvancedHedging Exotic ProductsNear maturity, a digital seller holds far more underlying than the maximum payment, and hedging losses or costs can exceed the maximum liability (pin risk).Trading and Hedging

6. Stage 66XVAAdvanced

   From an overview of XVA to CVA and DVA, FVA and MVA, KVA and wrong-way risk.

   1. PractitionerOverview of XVACVA, FVA, MVA, and KVA lower the value, and DVA raises it.Valuation
   2. AdvancedCVA and DVAIf the counterparty's default is independent of market moves, CVA is the sum, over time intervals, of the discount factor times expected exposure times default probability times loss given default.Valuation
   3. AdvancedFVA and MVAFVA reflects the cost of funding collateral between an uncollateralized trade and its collateralized offsetting trade.Valuation
   4. AdvancedCapital Valuation Adjustment (KVA)KVA is the present value of the cost of the regulatory capital held until maturity, and it lowers the value.Capital and Profitability
   5. AdvancedWrong-Way RiskIn a CVA valuation, estimating exposure and the counterparty's default separately understates the loss when the risk is wrong-way.Counterparty Credit Risk

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