Valuation and Risk Models
Valuation and risk models
Master binomial option pricing, Black-Scholes-Merton (BSM), the Greeks, fixed-income sensitivity, VaR, expected shortfall, and credit risk models. In a one-period binomial model, risk-neutral probability p = (e^(rT)-d)/(u-d); the option value is e^(-rT)[p*Cu + (1-p)*Cd]. BSM assumes lognormal prices, constant volatility, no arbitrage, and continuous trading; it can misprice deep out-of-the-money options when volatility skew exists.
Greeks: delta is the change in option value for a $1 change in the underlying; gamma is delta's sensitivity and is highest near the money; vega rises with time to maturity; theta is usually negative for long options. Delta-hedging removes local price risk but leaves gamma and vega risk.
Bond risk: duration approximates percentage price change for a small yield change, dP/P approximately -D*dy; convexity improves the estimate for larger moves. VaR is a loss threshold at a confidence level over a horizon and is not subadditive; expected shortfall (ES) is the average loss beyond VaR and is coherent.
Worked example: A bond has modified duration 6 and convexity 80. For a 50 bp yield increase, dP/P approximately -6(0.005) + 0.5(80)(0.005)^2 = -0.03 + 0.001 = -2.9%.
Traps: using VaR as a worst-case loss; forgetting that the risk-free rate and dividend yield shift call and put values in opposite directions; and mixing modified and Macaulay duration.