Part 2 — Hedging · BS · SPX call capstone

Carrying forward from Part 1

σ

= 0.1946

d_{1}

= −0.2466

d_{2}

= −0.3873

Φ (d_{1})

= 0.4026

Φ (d_{2})

= 0.3493

C

= $276.7325

e^{- r T}

= 0.9814

§2.1 #

Deriving Δ

(i)

Product rule on C with respect to S₀

\frac{\partial C}{\partial S _{0}} = Φ (d_{1}) + S_{0} φ (d_{1}) \frac{\partial d _{1}}{\partial S _{0}} - K e^{- r T} φ (d_{2}) \frac{\partial d _{2}}{\partial S _{0}} .

(ii)

∂d₁/∂S₀ = ∂d₂/∂S₀

d_{1} = \frac{ln ( S _{0} / K ) + ( r + σ ^{2} /2 ) T}{σ T} ⟹ \frac{\partial d _{1}}{\partial S _{0}} = \frac{1}{S _{0} σ T} .

d_{2} = d_{1} - σ T ⟹ \frac{\partial d _{2}}{\partial S _{0}} = \frac{\partial d _{1}}{\partial S _{0}} = \frac{1}{S _{0} σ T} .

(iii)

Substitute and factor

\frac{\partial C}{\partial S _{0}} = Φ (d_{1}) + \frac{S _{0} φ ( d _{1} ) - K e ^{- r T} φ ( d _{2} )}{S _{0} σ T} .

(iv)

Identity: S₀ φ(d₁) = K e^{-rT} φ(d₂)

ln S_{0} - \frac{d _{1}^{2}}{2} = ? ln K - r T - \frac{d _{2}^{2}}{2} ⟺ ln (S_{0} / K) + r T = ? \frac{d _{1}^{2} - d _{2}^{2}}{2} . (⋆)

d_{1}^{2} - d_{2}^{2} = (d_{1} - d_{2}) (d_{1} + d_{2}) = σ T (2 d_{1} - σ T) = 2 σ T d_{1} - σ^{2} T .

σ T d_{1} = ln (S_{0} / K) + (r + \frac{σ ^{2}}{2}) T ⟹ d_{1}^{2} - d_{2}^{2} = 2 ln (S_{0} / K) + 2 r T .

\frac{d _{1}^{2} - d _{2}^{2}}{2} = ln (S_{0} / K) + r T . ✓ (⋆) .

(v)

Conclude

Δ = \frac{\partial C}{\partial S _{0}} = Φ (d_{1}) = 0.4026.

§2.2 #

Vega and Theta

Setup

φ(d₁) via Taylor

d_{1}^{2} e^{- 0.0304} φ (d_{1}) = 0.0608, d_{1}^{2} /2 = 0.0304, \approx 1 - 0.0304 + 0.0005 - 0.0000 = 0.9701, = \frac{1}{2 π} e^{- d_{1}^{2} /2} = 0.3989 \times 0.9701 = 0.3870.

(i) Vega

∂C/∂σ = S₀ √T · φ(d₁)

S_{0} T Vega (raw) Vega (per 1pp) = 7414 \times 0.7232 = 5361.8048, = 5361.8048 \times 0.3870 = 2075.0185 (per unit rise in σ), = 2075.0185 \times 0.01 = 20.7502.

(i) Theta

∂C/∂t = − S₀ σ φ(d₁) / (2√T) − r K e^{-rT} Φ(d₂)

S_{0} σ S_{0} σ φ (d_{1}) 2 T first term r K r K e^{- r T} r K e^{- r T} Φ (d_{2}) second term Θ_{raw} Θ_{day} = 7414 \times 0.1946 = 1442.7644, = 558.3498, = 1.4464, = - 558.3498/1.4464 = - 386.0272, = 0.036 \times 7900 = 284.4000, = 279.1102, = 279.1102 \times 0.3493 = 97.4932, = - 97.4932, = - 386.0272 - 97.4932 = - 483.5204, = - 483.5204/365 = - 1.3247.

Vega (per 1pp rise in σ)

20.7502

Θ (per calendar day)

−1.3247

(iii) Magnitude vs C = $276.73: $∣ Vega_{1 pp} ∣/ C = 20.7502/276.7325 \approx 7.5%$ per 1pp shift in σ — materially sensitive to volatility. $∣ Θ_{day} ∣/ C = 1.3247/276.7325 \approx 0.48%$ per calendar day; over the ~191 days to expiry the cumulative decay is a sizable fraction of the premium.

§2.3 #

Put–call parity

A no-arbitrage identity that prices the put from the call.

P = C - S_{0} + K e^{- r T} .

(i)

Compute P

C - S_{0} K e^{- r T} P = 276.7325 - 7414.0000 = - 7137.2675, = 7900 \times 0.9814 = 7753.0600, = - 7137.2675 + 7753.0600 = 615.7925 > 0. ✓

Put price (parity-implied)

P = $615.79

(ii)(a) Correct hedging instrument for a long-equity investor

A long put on the index. Its payoff $max (K - S_{T}, 0)$ activates exactly when the long-equity position is losing money, capping the downside while leaving the upside intact. Calls move the wrong way for a holder who fears a decline.

(ii)(b) Non-insurance buyers of OTM SPX calls

Directional speculators. A trader expecting SPX to rally above 7,900 by November can buy this call for $180.30 of premium and obtain convex exposure to ~7,900 index points of notional — capped downside (the premium), unbounded upside.
Volatility traders / dealer desks running long-Γ or long-Vega books. A delta-hedged long call has near-zero directional exposure but profits when realized vol exceeds implied (gamma scalping) and from a rise in implied vol (positive Vega). The underlying delta is hedged away with a short position in SPX futures.

§2.4 #

Hedged portfolio Δ

200 units of SPX exposure + 200 of the K = 7,900 puts.

(i)

Original position Δ

Δ_{unit} = \partial S_{0} / \partial S_{0} = 1 ⟹ Δ_{position} = 200 \times 1 = 200.

(ii)

Δ_put = Φ(d₁) − 1

Δ_{put} = 0.4026 - 1 = - 0.5974.

(iii)

Apply linearity

Δ_{portfolio} = i \sum n_{i} Δ_{i} = 200 (1) + 200 (- 0.5974) = 200 - 119.4800 = 80.4800.

\frac{200 - 80.4800}{200} = \frac{119.4800}{200} = 0.5976 = 59.76%.

Δ hedged portfolio

80.4800

Reduction vs unhedged

59.76%

§2.5 #

Solving N for delta-neutral

Choose N puts so Δ_portfolio = 0.

200 (1) + N (Φ (d_{1}) - 1) = 0 ⟹ 200 + N (- 0.5974) = 0 ⟹ N = \frac{200}{0.5974} = 334.7841.

Delta-neutral N (with Φ(d₁) = 0.4026)

N = 334.7841 round to 335 puts

Δ N = 334.7841 - 200 = 134.7841, \frac{N}{200} = \frac{334.7841}{200} = 1.6739.