Valuation and Risk Models
Valuation and Risk Models covers derivative pricing, the Greeks, fixed-income sensitivity, and the mapping of positions into risk measures. The exam expects you to price a claim from a tree or closed-form formula, then interpret how value responds to the underlying, volatility, time, and rates.
Core rules
- One-step binomial: p = (e^{rT} − d)/(u − d); value = e^{−rT}[p·Cu + (1−p)·Cd]. p is a risk-neutral probability, not a forecast.
- Black-Scholes-Merton: C = S e^{−qT}N(d1) − K e^{−rT}N(d2); call delta = e^{−qT}N(d1); put delta = −e^{−qT}N(−d1).
- Gamma peaks at-the-money near expiry; vega peaks at-the-money with long maturity; theta is usually negative for long options.
- Bond price change ≈ −D × Δy + ½ × C × (Δy)²; DV01 is the price change for a 1bp move; duration hedging assumes a parallel shift.
- Implied volatility is the σ that reproduces the market price; the smile and skew reflect non-lognormal tails.
Worked example
S0 = 100, u = 1.2, d = 0.8, r = 5%, T = 1, K = 100, European call. Payoffs: Cu = 20, Cd = 0. p = (e^{0.05} − 0.8)/(1.2 − 0.8) = (1.05127 − 0.8)/0.4 = 0.6282. Value = e^{−0.05}(0.6282 × 20) = 0.95123 × 12.564 = 11.95. Delta = (Cu − Cd)/(S0(u − d)) = 20/40 = 0.50, so the replicating portfolio holds half a share.
Traps
- Treating the risk-neutral p as the real-world probability of an up move.
- Dropping the negative sign on put delta.
- Relying on duration alone for large yield changes where convexity dominates.
- Assuming BSM is valid when the implied volatility smile is steep.