17.4 Option Sensitivity Measures: The Greeks

Option Sensitivity Measures: The Greeks

Pricing gives a single number; Greeks explain how that number moves when markets move. VRM–16 is the risk-management layer on top of binomial and BSM pricing: delta hedging, higher-order risks, and portfolio-level neutrality.

Naked Versus Covered Risks

A naked short call or put has no offsetting underlying or long option. Risk is unbounded for naked short calls (stock can rise without limit) and large for naked short puts (stock can fall to zero). A covered call (long stock + short call) caps upside at K and retains downside almost like a short put synthetically. A protective put (long stock + long put) floors downside at K.

“Covered” does not mean Greek-free. Covered calls remain short gamma and short vega relative to holding stock alone.

Stop-Loss Hedging

A crude hedge for a short call: buy the stock when S rises above K; sell when S falls below K (stop-loss). In theory with continuous paths and zero transaction costs this seems to mimic payoff, but:

• Real paths whip across K repeatedly → many round trips and costs.
• Between trades the position is unhedged (gap risk).
• Discrete monitoring misses intraday crossings.

Stop-loss is not the BSM replicating strategy. Exam stance: stop-loss is path-dependent, costly, and inferior to delta hedging as a conceptual replication method.

Delta and Dynamic Delta Hedging

Delta Δ = ∂V/∂S. For a European non-dividend call, Δ = N(d1) ∈ (0, 1); for the put, Δ = N(d1) − 1 ∈ (−1, 0).

Dynamic delta hedging: to replicate a long call, hold Δ shares financed by borrowing; rebalance as Δ changes. To hedge a short call, short Δ shares (or hold −Δ). In the frictionless BSM world, continuous rebalancing plus bond trading replicates the option payoff and earns the risk-free return on the hedged portfolio—hence the unique no-arbitrage price.

Worked delta hedge

Short 10 call contracts (100 shares each = 1,000 option shares) with Δ = 0.60. Hedge: buy 0.60 × 1,000 = 600 shares. If S rises and Δ becomes 0.70, buy another 100 shares. If S falls and Δ becomes 0.50, sell 100 shares. P&L on the hedge offsets option mark-to-market to first order.

Vega, Gamma, Theta, and Rho

Gamma is the curse of delta hedging: large Γ means Δ moves fast, so discrete hedges leave residual P&L ≈ (1/2) Γ (ΔS)². Short gamma desks bleed in volatile markets even if delta-averaged to zero.

Vega (often quoted per 1 vol point) dominates for longer-dated ATM options. Theta is the calendar decay that long option holders usually pay. Rho is secondary for short-dated equity options but material for LEAPs and rate-sensitive underlyings.

Worked gamma P&L sketch

Delta-hedged short option with Γ = −0.05 per $1 of spot (portfolio gamma). Spot jumps +$2. Approximate unhedged curvature P&L ≈ (1/2)(−0.05)(2)² = −0.10 dollars per option share unit—losses from being short gamma on a large move.

Delta-Neutral and Gamma-Neutral

Delta-neutral: portfolio Δ_p = Σ w_i Δ_i = 0. Achieved with the underlying (Γ = 0, ν = 0 for the underlying itself) or with options.

Gamma-neutral: need options (or other convex instruments) because stock/futures have zero gamma. Typical desk trade: stay short customer options (short Γ, short ν), buy liquid ATM options or variance swaps to flatten gamma/vega, then delta-hedge the residual with futures.

Worked two-option neutrality

Portfolio: short 1,000 options with Δ = 0.50, Γ = 0.04. Traded option has Δ = 0.55, Γ = 0.03. To gamma-hedge, buy n contracts where n × 0.03 = 1,000 × 0.04 → n = 1,333 (same multiplier units). Net gamma ≈ 0. Net delta = −1,000×0.50 + 1,333×0.55 ≈ −500 + 733 = +233, so short 233 shares (or futures equivalents) to restore delta neutrality.

Greek Relationships

In BSM, Europeans satisfy the PDE linking Greeks:

Θ + (r − q) S Δ + (1/2) σ² S² Γ = r V

(for the appropriate carry). Intuition for q = 0: a delta-hedged long option portfolio earns financing on the option premium but pays theta and benefits from gamma in volatile moves—the PDE balances expected gamma gains against theta and funding. Rough trader mantra: long gamma / short theta for long options; you hope realized vol exceeds what you paid in decay.

Put–call links: same Γ and same vega for European calls and puts with identical K, T (parity). Deltas differ by e^(−qT). Rho signs flip.

Portfolio Greeks

Greeks aggregate linearly in position amounts (to the model’s first/second derivatives):

Δ_p = Σ n_i Δ_i, Γ_p = Σ n_i Γ_i, ν_p = Σ n_i ν_i, …

Signs follow long/short. Report in consistent units (per share, per contract, or dollar delta = Δ × S × multiplier). Limits are often set on dollar delta, dollar gamma (½ Γ S²), and vega notional.

Worked portfolio aggregation

Δ_p = 500×0.40 − 800×0.60 + 300×(−0.35) = 200 − 480 − 105 = −385

Γ_p = 500×0.05 − 800×0.04 + 300×0.03 = 25 − 32 + 9 = +2

ν_p = 500×0.12 − 800×0.15 + 300×0.10 = 60 − 120 + 30 = −30

Net: short delta, slightly long gamma, short vega—buy stock to flatten delta; address short vega with long options if required by limits.

Portfolio Insurance Versus Delta Hedging

Delta hedging usually means neutralizing the Greek risk of an existing option (or exotics book) by trading the underlying.

Portfolio insurance dynamically trades the underlying (or futures) to synthesize a put on a portfolio—raising exposure as markets rise (buy high) and cutting exposure as markets fall (sell low). It is synthetic long-put / constant-proportion style replication. Both are dynamic strategies; they differ in intent:

Both fail when markets gap and continuous trading assumptions break—exactly when insurance is needed most.

Synthesis

Know definitions and signs of Δ, Γ, ν, Θ, ρ; compute hedge ratios; escalate from delta-neutral to gamma/vega-neutral with options; aggregate portfolio Greeks with signs; contrast stop-loss and portfolio insurance with true replicating delta hedges.