Show how exposure measures for equities, fixed income, and options can help gauge and control market risk and volatility risk

For equities, beta times the market return gives the expected return on the position. For bonds, the percentage price change is approximately −modified duration × Δy + ½ × convexity × (Δy)². For options, delta is the value change per unit move in the underlying, gamma the change in delta, and vega the value change per point of implied volatility, so ΔV ≈ delta × ΔS + ½ × gamma × (ΔS)² + vega × Δσ. A delta-hedged book removes first-order price risk but keeps gamma and vega exposure. The trap in calculations is forgetting to square the change in the convexity or gamma term, or dropping the one-half.