Valuation and Risk
Valuation
Pricing theory and practice: no-arbitrage, Black–Scholes, the Greeks, XVA, pricing models and numerical methods.
Introductory3 pagesPractitioner9 pagesAdvanced11 pages
1Valuation Foundations
- IntroductoryNo-Arbitrage and ReplicationWith no arbitrage opportunities, two things whose future cash flows are the same in every outcome have the same price today (the law of one price).
- IntroductoryBinomial TreesIn a one-period model, buying the underlying and borrowing money reproduces the payoff of a call option (call), and the cost of doing so is the call's price.
- 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.
2Option Valuation
- 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.
- IntroductoryThe GreeksA hypothetical call (price 8.43) has a delta of 0.5596, so its price is estimated to rise about 0.56 when the underlying price rises by 1.
- PractitionerImplied VolatilityImplied volatility restates an option's price as a volatility, much as a yield restates the price of a bond.
- 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.
- 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.
3Collateral and XVA
- 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.
- PractitionerOverview of XVACVA, FVA, MVA, and KVA lower the value, and DVA raises it.
- PractitionerISDA SIMM: The Standard Model for Initial MarginInitial margin for OTC derivatives not cleared through a central counterparty uses the standardized schedule or, in Japan, a notified quantitative model.
- 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.
- AdvancedFVA and MVAFVA reflects the cost of funding collateral between an uncollateralized trade and its collateralized offsetting trade.
4Valuation Models
- 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.
- 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.
- AdvancedThe SABR ModelThe four parameters α, β, ρ, and ν set the level, slope, and curvature of the smile separately.
- 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.
- AdvancedThe Heston ModelThe stock price and the variance follow separate stochastic processes, and the two moves are linked by a correlation.
- 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.
- 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.
5Numerical Methods
- PractitionerMonte Carlo SimulationPaths are generated under risk-neutral dynamics, in which the expected return of the underlying is set to the risk-free rate.
- 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.
- 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.