11.1 Option Fundamentals, Put-Call Parity & Black-Scholes-Merton Pricing

11.1 Option Fundamentals, Put-Call Parity & Black-Scholes-Merton Pricing

Options are derivative contracts granting the holder the legal right, but not the obligation, to buy or sell an underlying asset at a specified strike price within a designated time horizon. For institutional investment consultants and wealth managers, options provide powerful mechanisms to engineer asymmetric return distributions, hedge downside tail risk, monetize volatility forecasts, and generate portfolio income.

                         Options Analytical & Risk Management Framework
                                               │
         ┌─────────────────────────────────────┼─────────────────────────────────────┐
         │                                     │                                     │
Foundational Valuation               Analytical Greeks                 Strategic Overlays
  • Intrinsic vs. Time Value           • Delta (Directional / Hedge)     • Covered Call (Yield / Cap)
  • Put-Call Parity Parity             • Gamma (Delta Curvature)         • Protective Put (Floor)
  • Black-Scholes-Merton Model         • Theta (Time Decay Bleed)        • Collar (Bounded Bracket)
  • Implied Volatility Surface         • Vega (Volatility Exposure)      • Straddles & Strangles

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1. Option Fundamentals, Moneyness & Value Decomposition

Core Contract Parameters

An option contract is defined by four core contractual parameters:

1. Underlying Asset Price ($S$): The current spot market price of the reference instrument (e.g., common stock, equity index, ETF, or commodity).
2. Strike / Exercise Price ($K$): The pre-agreed fixed price at which the option holder can execute the purchase or sale of the underlying asset.
3. Time to Expiration ($T$): The remaining time (expressed in annualized years) until the contract expires.
4. Exercise Style:
   • European Style: Can only be exercised on the definitive expiration date $T$.
   • American Style: Can be exercised on any business day up to and including the expiration date $T$.
   • Bermudan Style: Can be exercised on specific discrete dates prior to expiration.

Moneyness Classifications

Moneyness categorizes the relationship between the underlying spot price ($S$) and the contract strike price ($K$):

Intrinsic Value vs. Extrinsic (Time) Value

The total market price (premium) of an option ($V$) is the sum of two distinct components:

$\text{Option Premium } (V) = \text{Intrinsic Value} + \text{Extrinsic (Time) Value}$

   Call Option Premium ($)
     ▲
     │                                     / Total Market Premium (V)
     │                                    / 
     │                                   /  .-' Time Value (Extrinsic)
     │                                  /.-'
     │                               .-' ═══════════════════════════════
     │                            .-'   / Intrinsic Value: max(0, S - K)
     │                         .-'     /
     │                      .-'       /
     │                   .-'         /
     └──────────────────●───────────┴────────────────────────► Spot Price (S)
                      ATM (S = K)   ITM (S > K)

1. Intrinsic Value: The cash payoff realized if the option were exercised immediately at current market prices: $\text{Call Intrinsic Value} = \max(0, S - K)$ $\text{Put Intrinsic Value} = \max(0, K - S)$ Intrinsic value can never be negative ($V_{\text{intrinsic}} \ge 0$). When an option is ATM or OTM, its intrinsic value is exactly zero.

2. Extrinsic / Time Value: The additional premium that investors pay above intrinsic value, reflecting the probability that the underlying asset price will move favorably prior to expiration. Time value is influenced by:

   • Time to Expiration ($T$): Longer horizons provide greater opportunity for favorable price excursions, increasing time value.
   • Implied Volatility ($\sigma$): Higher expected dispersion expands the probability of extreme favorable payoffs, raising time value.
   • Distance to Strike: Time value is maximized when an option is precisely At-the-Money ($S = K$), where pricing uncertainty is greatest.
   • Interest Rates ($r$) and Dividends ($q$): Affect the cost-of-carry financing and forward price drift.

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2. Put-Call Parity & Synthetic Position Replication

Theoretical Formulation

Put-Call Parity defines a fundamental, no-arbitrage equilibrium relationship linking the prices of European call options, put options, the underlying stock, and a risk-free zero-coupon bond sharing identical strike prices ($K$) and expiration dates ($T$).

$\text{Fiduciary Call} = \text{Protective Put}$

$C + \text{PV}(K) = P + S$

Using continuous-time discounting at risk-free rate $r$:

$C + K e^{-rT} = P + S$

Where:

• $C$ = Current market price of the European call option
• $K e^{-rT}$ or $\text{PV}(K)$ = Present value of the strike price discounted at the risk-free rate
• $P$ = Current market price of the European put option
• $S$ = Current spot price of the underlying asset

                               Put-Call Parity Mechanics
       ┌───────────────────────────────────────┬───────────────────────────────────────┐
       │       Fiduciary Call Portfolio        │       Protective Put Portfolio        │
       │  • Buy European Call (C)              │  • Buy European Put (P)               │
       │  • Invest PV(K) in Zero-Coupon Bond   │  • Buy Underlying Stock (S)           │
       └───────────────────┬───────────────────┴───────────────────┬───────────────────┘
                           │                                       │
                           ▼                                       ▼
         Payoff at Expiration (T):               Payoff at Expiration (T):
         • If S_T ≤ K: 0 + K = K                 • If S_T ≤ K: (K - S_T) + S_T = K
         • If S_T > K: (S_T - K) + K = S_T       • If S_T > K: 0 + S_T = S_T
                           │                                       │
                           └───────────────────┬───────────────────┘
                                               ▼
                   Terminal Payoffs are Identical: max(S_T, K) in All States
                   Therefore, Initial Costs Must Be Identical: C + PV(K) = P + S

Arbitrage Exploitation for Parity Violations

When market prices deviate from Put-Call Parity, arbitrageurs execute riskless, self-financing trades to capture mispricing:

1. Conversion Arbitrage (Call is Overpriced / Put is Underpriced):

   • Condition: $C + \text{PV}(K) > P + S \iff C - P + \text{PV}(K) - S > 0$
   • Arbitrage Execution: Sell (write) the overpriced call, buy the underpriced put, buy the physical underlying stock, and borrow $\text{PV}(K)$ at the risk-free rate.
   • Outcome: Locks in an immediate upfront cash profit with zero net cash flow at maturity $T$.

2. Reversal / Reverse Conversion Arbitrage (Put is Overpriced / Call is Underpriced):

   • Condition: $C + \text{PV}(K) < P + S \iff P + S - C - \text{PV}(K) > 0$
   • Arbitrage Execution: Buy the underpriced call, sell (write) the overpriced put, short the underlying stock, and lend (invest) the proceeds at the risk-free rate $\text{PV}(K)$.
   • Outcome: Locks in an immediate risk-free arbitrage profit.

Synthetic Position Creation

Rearranging the Put-Call Parity formula reveals how institutional investors create synthetic exposures without transacting directly in physical cash assets:

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3. Black-Scholes-Merton (BSM) Pricing Framework & Volatility Surfaces

The BSM Analytical Formulas

Developed by Fischer Black, Myron Scholes, and Robert Merton in 1973, the Black-Scholes-Merton model establishes the closed-form theoretical price for European options on non-dividend-paying assets in continuous time:

$\text{Call Price } (C) = S \cdot N(d_1) - K e^{-rT} \cdot N(d_2)$

$\text{Put Price } (P) = K e^{-rT} \cdot N(-d_2) - S \cdot N(-d_1)$

Where: $d_1 = \frac{\ln(S / K) + \left(r + \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}}$

$d_2 = d_1 - \sigma\sqrt{T} = \frac{\ln(S / K) + \left(r - \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}}$

• $N(d_1)$ = Cumulative standard normal distribution function evaluated at $d_1$, representing the option's theoretical Delta ($\Delta$) and the dynamic replication hedge ratio.
• $N(d_2)$ = Risk-neutral probability that the call option will expire In-the-Money ($S_T > K$).
• $K e^{-rT} N(d_2)$ = Present value of the expected strike payout at expiration under risk-neutral pricing.

Model Inputs & Direct Sensitivities

Core Assumptions & Practical Limitations

1. Continuous Lognormal Distribution: Asset returns follow geometric Brownian motion with constant drift and constant volatility. Limitation: Real asset returns exhibit fat tails (leptokurtosis) and negative skewness (jump-to-default crash risk).
2. Constant Risk-Free Rate & Volatility: Interest rates and volatility are non-stochastic over contract life. Limitation: Volatility is mean-reverting and stochastic.
3. Frictionless Markets: Continuous delta hedging with zero transaction costs, no bid-ask spreads, no taxes, and unrestricted short selling. Limitation: Discrete rebalancing incurs significant market impact and transaction friction.
4. European Exercise Only: No early exercise before maturity. Limitation: American options command an early exercise premium, particularly for deep ITM puts or dividend-paying calls.

Implied Volatility: Smiles and Skews

When market option prices are inverted through the BSM formula, the resulting implied volatility (IV) across strikes is not flat (as BSM assumes), but creates distinct structural curves:

   Implied Volatility (σ)
     ▲
     │   Equity Volatility Skew (Smirk)                FX Volatility Smile
     │   ┌────────────────────────────────┐            ┌────────────────────────────────┐
     │   │ • High IV for OTM Puts         │            │ • High IV for OTM Puts & Calls │
     │   │ • Downside crash protection    │            │ • Bilateral tail jump risk     │
     │   └────────────────────────────────┘            └────────────────────────────────┘
     │
     │    \                                              \            /
     │     \                                              \          /
     │      \                                              \        /
     │       \________                                      \______/
     └──────────┴──────────┴──────────► Strike (K)  ───────────┴──────────┴──────────► Strike (K)
               OTM Put    OTM Call                            OTM Put    OTM Call

• Equity Volatility Skew (Smirk): Implied volatility is significantly higher for OTM puts (low strikes) than for OTM calls (high strikes). Driven by post-1987 institutional demand for downside tail-risk hedges ("crash-phobia") and corporate leverage effects (falling stock prices increase financial leverage, elevating risk).
• Currency Volatility Smile: Implied volatility rises symmetrically for both deep OTM puts and deep OTM calls, reflecting bilateral jump risk in foreign exchange rates.

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