9.4 The Black-Scholes-Merton (BSM) Model, Option Greeks & Dynamic Hedging

9.4 The Black-Scholes-Merton (BSM) Model, Option Greeks & Dynamic Hedging

Core Insight: The Black-Scholes-Merton (BSM) model extends contingent claims analysis into continuous time, providing closed-form valuation formulas for European options by constructing a continuously rebalanced, riskless portfolio. For CFA Level II candidates, mastery extends beyond formula memorization to deconstructing the economic meaning of the terms $N(d_1)$ and $N(d_2)$, analyzing the Option Greeks (Delta, Gamma, Vega, Theta, Rho) across moneyness and time-to-expiration, interpreting implied volatility skews, and executing Dynamic Delta Hedging strategies to neutralize portfolio risk.

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1. The Black-Scholes-Merton (BSM) Formula

For a European option on an underlying asset with current spot price $S_0$, strike price $K$, time to maturity $T$, risk-free rate $r$, continuous dividend yield $q$, and annualized volatility $\sigma$:

$\mathbf{c = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)}$ $\mathbf{p = K e^{-rT} N(-d_1) \quad \text{or} \quad p = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1)}$

where: $\mathbf{d_1 = \frac{\ln(S_0 / K) + \left(r - q + \frac{\sigma^2}{2}\right)T}{\sigma \sqrt{T}}}$ $\mathbf{d_2 = d_1 - \sigma \sqrt{T}}$

                          DECONSTRUCTING THE BSM CALL FORMULA
                 c = [ S_0 e^(-qT) N(d_1) ] - [ K e^(-rT) N(d_2) ]
                       │           │              │          │
                       │           │              │          └─► Risk-Neutral Prob of
                       │           │              │              Finishing ITM (S_T > K)
                       │           │              └────────────► Present Value of Strike
                       │           └───────────────────────────► Delta of Call (Share Exposure)
                       └───────────────────────────────────────► Present Value of Asset

Economic Interpretation of Components

• $N(d_1)$: The sensitivity of the call option price to changes in the underlying spot price (the Delta of the call). $S_0 e^{-qT} N(d_1)$ represents the expected present value of receiving the stock at expiration conditional upon the option expiring in-the-money.
• $N(d_2)$: The risk-neutral probability that the option will expire in-the-money ($S_T > K$). Therefore, $K e^{-rT} N(d_2)$ is the present value of the expected cash strike payment upon exercise.
• $N(-d_1)$ and $N(-d_2)$: For put options, $N(-d_2) = 1 - N(d_2)$ is the risk-neutral probability that the put expires in-the-money ($S_T < K$), and $-e^{-qT} N(-d_1) = e^{-qT}[N(d_1) - 1]$ is the put delta.

Core Assumptions and Limitations

1. Price Dynamics: The underlying asset price follows geometric Brownian motion with constant drift and constant volatility $\sigma$ (asset returns are lognormally distributed).
2. Interest Rates: The risk-free rate $r$ is constant and known across all maturities.
3. Trading Friction: Continuous trading is possible with zero transaction costs, no taxes, and no restrictions on short sales.
4. Dividends: Dividend yield $q$ is paid continuously.
5. Exercise Style: European style only (no early exercise permitted).

The Black Model (Options on Futures / Forwards)

Fischer Black adapted the BSM model for options on futures contracts. Because entering a futures contract requires no initial cash outlay, spot price $S_0$ is replaced by the discounted futures price $F_0(T) e^{-rT}$:

$\mathbf{c = e^{-rT} [F_0(T) N(d_1) - K N(d_2)]}, \qquad \mathbf{p = e^{-rT} [K N(-d_2) - F_0(T) N(-d_1)]}$ $\mathbf{d_1 = \frac{\ln(F_0 / K) + \left(\frac{\sigma^2}{2}\right)T}{\sigma \sqrt{T}}}, \qquad \mathbf{d_2 = d_1 - \sigma \sqrt{T}}$

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2. The Option Greeks

The Option Greeks measure the sensitivity of option prices to changes in underlying parameters.

                                  THE FIVE OPTION GREEKS
  ┌──────────────┐ ┌──────────────┐ ┌──────────────┐ ┌──────────────┐ ┌──────────────┐
  │    DELTA     │ │    GAMMA     │ │     VEGA     │ │    THETA     │ │     RHO      │
  │   (∂V/∂S)    │ │   (∂²V/∂S²)  │ │   (∂V/∂σ)    │ │   (∂V/∂t)    │ │   (∂V/∂r)    │
  ├──────────────┤ ├──────────────┤ ├──────────────┤ ├──────────────┤ ├──────────────┤
  │ Call: 0 to 1 │ │ Always > 0   │ │ Always > 0   │ │ Usually < 0  │ │ Call: > 0    │
  │ Put: -1 to 0 │ │ Peaks ATM    │ │ Peaks ATM    │ │ Time Decay   │ │ Put: < 0     │
  └──────────────┘ └──────────────┘ └──────────────┘ └──────────────┘ └──────────────┘

1. Delta ($\Delta = \frac{\partial V}{\partial S}$)

• Definition: Change in option price per $1.00 change in underlying asset price.
• Call Delta: $\Delta_c = e^{-qT} N(d_1) \in [0, 1]$. Deep in-the-money calls approach $+1.0$; deep out-of-the-money calls approach $0.0$.
• Put Delta: $\Delta_p = -e^{-qT} N(-d_1) = e^{-qT}[N(d_1) - 1] \in [-1, 0]$. Deep ITM puts approach $-1.0$; deep OTM puts approach $0.0$.

2. Gamma ($\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}$)

• Definition: Rate of change of delta per $1.00 change in underlying price. Measures the convexity of the option price curve.
• Sign: $\Gamma > 0$ for both long calls and long puts. Short option positions have negative gamma ($-\Gamma$).
• Moneyness Profile: Gamma is highest for at-the-money (ATM) options and accelerates sharply as time to expiration approaches zero ($T \to 0$). Deep ITM and deep OTM options have near-zero gamma.

3. Vega ($\nu = \frac{\partial V}{\partial \sigma}$)

• Definition: Sensitivity of option price to a 1 percentage point (0.01) change in volatility.
• Sign: $\nu > 0$ for both long calls and long puts. Volatility increases the probability of extreme favorable outcomes without increasing downside loss (which is capped at zero).
• Profile: Vega is highest for ATM options with long times to expiration.
• Volatility Smile / Smirk: In equity markets, deep OTM puts trade at higher implied volatilities than ATM or OTM calls (volatility smirk/skew) due to market downside crash protection demand and corporate leverage effects. In FX markets, both OTM puts and calls show elevated IV (volatility smile) reflecting fat-tailed exchange rate distributions.

4. Theta ($\Theta = \frac{\partial V}{\partial t} = -\frac{\partial V}{\partial T}$)

• Definition: Rate of option price decline as calendar time passes (time decay).
• Sign: $\Theta < 0$ for long options (option values decay over time). Time decay accelerates rapidly for ATM options in the final 30 days before expiration.

5. Rho ($\rho = \frac{\partial V}{\partial r}$)

• Definition: Sensitivity of option price to changes in the risk-free interest rate.
• Sign: $\rho_{call} > 0$ (higher interest rates reduce the present value of the strike price $K$, increasing call value). $\rho_{put} < 0$ (higher rates reduce the present value of strike received, decreasing put value).

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3. Comprehensive Option Greeks Reference Matrix

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4. Dynamic Delta Hedging & Gamma Risk

A market maker or institutional portfolio manager who sells option contracts faces non-linear directional risk. To immunize the position against price fluctuations, they create a Delta-Neutral Portfolio.

Delta-Neutral Hedge Ratio

To hedge a short position of $N_{options}$ call options, the hedger must purchase shares of underlying stock:

$\mathbf{N_{shares} = - N_{options} \times \Delta_{option}}$

$\text{Portfolio Delta } \Delta_P = N_{shares} + (N_{options} \times \Delta_{option}) = 0$

                         DYNAMIC DELTA HEDGING FEEDBACK LOOP
  Stock Price Rises ────► Call Delta Increases (e.g., 0.50 -> 0.65) ────► Buy More Shares
                                                                           (Buy High)
  Stock Price Falls ────► Call Delta Decreases (e.g., 0.50 -> 0.35) ────► Sell Shares
                                                                           (Sell Low)
  * Net result: Rebalancing creates ongoing trading losses ('Cost of Gamma'), funded by Option Premium.

The Cost of Dynamic Hedging (Gamma Risk)

• As stock price rises, call delta increases $\implies$ the hedger must buy more shares at higher prices.
• As stock price falls, call delta decreases $\implies$ the hedger must sell shares at lower prices.
• This inherent "buy high, sell low" dynamic produces trading losses known as the cost of gamma. The initial option premium received by the option writer compensates for these expected dynamic hedging losses.

Gamma-Neutral Hedging

Stock shares have $\Delta = 1.0$ and $\Gamma = 0.0$. Therefore, buying or selling shares cannot alter portfolio gamma. To achieve both Delta Neutrality and Gamma Neutrality, an investor must introduce a second option contract ($Opt_2$):

$\mathbf{N_{options, 2} = - \frac{N_{options, 1} \times \Gamma_1}{\Gamma_2}}$ $\mathbf{N_{shares} = - [N_{options, 1} \Delta_1 + N_{options, 2} \Delta_2]}$

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5. Comprehensive Worked Numerical Calculation: Delta Hedging & Rebalancing

Initial Setup: An options dealer sells 10,000 European call options on Titan Energy stock ($S_0 = $100.00$, $K = $100.00$, $\sigma = 25%$, $r = 4.00%$, $T = 0.50$ years, $q = 0$).

• Current call delta $\Delta_0 = +0.5800$
• Current call gamma $\Gamma_0 = +0.0220$
• Option premium $= $8.50$ per share

Step 1: Establish Initial Delta-Neutral Hedge

$\text{Total Option Delta} = -10,000 \times 0.5800 = -5,800$ $\mathbf{N_{shares\,to\,buy}} = -(-5,800) = \mathbf{+5,800\text{ shares of stock}}$ $\text{Capital Required for Stock} = 5,800 \times \$100.00 = \$580,000$ $\text{Cash Received from Option Sale} = 10,000 \times \$8.50 = \$85,000$ $\text{Net Borrowing to Finance Hedge} = \$580,000 - \$85,000 = \$495,000$

Step 2: Stock Price Shifts to $104.00 (Rebalancing Event)

Next week, Titan Energy stock rises to $104.00. The call option delta increases to +0.6650.

1. New Option Position Delta: $\text{New Short Option Delta} = -10,000 \times (+0.6650) = -6,650$
2. Shares Currently Held: $+5,800$ shares
3. Rebalancing Action Required: $\Delta N_{shares} = 6,650 - 5,800 = \mathbf{+850\text{ additional shares to buy}}$ $\text{Additional Cash Spent} = 850 \times \$104.00 = \mathbf{\$88,400}$
4. Total Position After Rebalancing:
   • Short 10,000 Calls (Delta = $-6,650$)
   • Long 6,650 Shares (Delta = $+6,650$)
   • Net Portfolio Delta $= 0.0$ (Delta-neutral restored)