Module 13: Options & Derivatives

1. What Are Derivatives?

A derivative is a financial contract whose value derives from the price of an underlying asset. The underlying can be a stock, bond, commodity, interest rate, currency, or even another derivative. Derivatives are the most mathematically rich instruments in finance, and for a statistician, they are where stochastic calculus, probability theory, and estimation all converge.

This module focuses on options, the most commonly traded and conceptually rich type of derivative.

1.1 Why Derivatives Exist

Derivatives serve three fundamental purposes:

2. Call and Put Options

2.1 Call Options

A call option gives the holder the right, but not the obligation, to buy the underlying asset at a specified price (the strike price K) on or before a specified date (the expiration date T). The payoff at expiration is:

If S_T > K, you exercise the option and profit S_T − K. If S_T ≤ K, the option expires worthless and your payoff is zero.

2.2 Put Options

A put option gives the holder the right to sell the underlying at the strike price. The payoff at expiration is:

If S_T < K, you exercise and profit K − S_T. If S_T ≥ K, the put expires worthless. The put is like insurance against a price decline.

2.3 Option Terminology

2.4 Profit Diagrams

The profit (not payoff) accounts for the premium paid upfront. For a call buyer with premium c:

For a put buyer with premium p:

The breakeven point for a call is S_T = K + c, and for a put is S_T = K − p. Note the fundamental asymmetry: the buyer's maximum loss is the premium paid (bounded), but the potential gain is unlimited for calls and large for puts.

3. The Black-Scholes Formula

The Black-Scholes formula (1973) is one of the most important equations in finance. It provides a closed-form solution for the price of a European call or put option on a non-dividend-paying stock.

3.1 The Model Assumptions

Black-Scholes rests on several strong assumptions. A statistician should know each one and its implications:

3.2 The Black-Scholes Formulas

For a European call option:

For a European put option:

where:

and N(·) is the standard normal CDF.

4. The Greeks: Partial Derivatives of Option Price

The "Greeks" are the partial derivatives of the option price with respect to various inputs. They measure how the option price changes when one parameter changes while all others are held constant. For a statistician, these are simply a sensitivity analysis.

4.1 Delta in Depth

Delta is the most important Greek. For a call option under Black-Scholes:

This means delta is always between 0 and 1 for calls, and between −1 and 0 for puts. An at-the-money call has delta ≈ 0.5: the option price moves about 50 cents for every $1 move in the stock.

4.2 Gamma: The Curvature

Gamma measures how fast delta changes. It is highest for at-the-money options near expiration, when small moves in the stock can flip the option from worthless to valuable.

where φ is the standard normal PDF. Gamma is always positive for both calls and puts. High gamma means the option's delta is changing rapidly — the position is becoming more or less sensitive to the underlying price.

4.3 Theta: Time Decay

Theta measures how much value the option loses as time passes, all else equal. Options are wasting assets — they lose value every day due to the decreasing time until expiration. The rate of decay accelerates as expiration approaches.

4.4 Vega: Sensitivity to Volatility

Vega measures sensitivity to volatility. Since volatility is the key input to option pricing, vega is arguably the most economically important Greek for traders:

Vega is always positive for both calls and puts: higher volatility increases option values because it increases the probability of large moves, which benefits the option holder (due to the truncated payoff).

5. Implied Volatility

All inputs to Black-Scholes are directly observable (stock price, strike, time to expiration, risk-free rate) except one: volatility (σ). The implied volatility is the value of σ that makes the Black-Scholes price equal to the observed market price:

5.1 Why Implied Volatility Matters

Implied volatility has become the market's preferred way of quoting option prices. Instead of saying an option costs $5.30, traders say the implied vol is 25%. This has several advantages:

• Comparability: Implied vol normalizes for strike, maturity, and underlying price, making options on different stocks comparable.
• Forward-looking: Unlike historical volatility (computed from past returns), implied vol reflects the market's expectation of future volatility.
• Market sentiment: High implied vol signals fear or uncertainty; low implied vol signals complacency.

6. The Volatility Smile and Skew

If Black-Scholes were perfectly correct, implied volatility would be the same for all strikes and maturities. In reality, it is not. When you plot implied volatility against strike price, you see a distinctive pattern:

6.1 The Volatility Smile

For short-dated options (especially on currencies and indices), implied vol is higher for both deep out-of-the-money calls and puts than for at-the-money options. This creates a U-shaped curve — the volatility smile.

6.2 The Volatility Skew

For equity index options (like S&P 500), the pattern is typically a skew: implied vol is higher for low strikes (OTM puts) and lower for high strikes (OTM calls). This creates a downward-sloping curve rather than a symmetric smile.

7. Put-Call Parity

Put-call parity is one of the most elegant results in finance. It states:

or equivalently:

This is not a model — it is an arbitrage identity that must hold regardless of which option pricing model you use. It says that a call plus cash equals a put plus stock.

7.1 Proof by No-Arbitrage

Consider two portfolios at time 0:

• Portfolio A: Buy a call + invest K e^−rT in the risk-free asset
• Portfolio B: Buy a put + buy one share of stock

At expiration T:

Both portfolios have identical payoffs in every state of the world. By no-arbitrage, they must have the same price today. Therefore: C + K e^−rT = P + S₀.

8. Python: Black-Scholes Pricer, Greeks, and Payoffs

8.1 Black-Scholes Implementation

Pythonimport numpy as np
from scipy.stats import norm
from scipy.optimize import brentq

class BlackScholes:
    """
    Black-Scholes option pricer with Greeks computation.
    All formulas assume European options on non-dividend-paying stocks.
    """

    def __init__(self, S, K, T, r, sigma):
        """
        S     : current stock price
        K     : strike price
        T     : time to expiration (years)
        r     : risk-free rate (annualized)
        sigma : volatility (annualized)
        """
        self.S = S
        self.K = K
        self.T = T
        self.r = r
        self.sigma = sigma

        # Precompute d1 and d2
        self.d1 = (np.log(S / K) + (r + sigma**2 / 2) * T) / (sigma * np.sqrt(T))
        self.d2 = self.d1 - sigma * np.sqrt(T)

    def call_price(self):
        return (self.S * norm.cdf(self.d1)
                - self.K * np.exp(-self.r * self.T) * norm.cdf(self.d2))

    def put_price(self):
        return (self.K * np.exp(-self.r * self.T) * norm.cdf(-self.d2)
                - self.S * norm.cdf(-self.d1))

    def delta(self, option_type='call'):
        if option_type == 'call':
            return norm.cdf(self.d1)
        return norm.cdf(self.d1) - 1

    def gamma(self):
        return norm.pdf(self.d1) / (self.S * self.sigma * np.sqrt(self.T))

    def theta(self, option_type='call'):
        # Per calendar day (divide by 365)
        term1 = -(self.S * norm.pdf(self.d1) * self.sigma) / (2 * np.sqrt(self.T))
        if option_type == 'call':
            term2 = -self.r * self.K * np.exp(-self.r * self.T) * norm.cdf(self.d2)
        else:
            term2 = self.r * self.K * np.exp(-self.r * self.T) * norm.cdf(-self.d2)
        return (term1 + term2) / 365

    def vega(self):
        # Per 1% change in volatility
        return self.S * norm.pdf(self.d1) * np.sqrt(self.T) / 100

    def rho(self, option_type='call'):
        if option_type == 'call':
            return self.K * self.T * np.exp(-self.r * self.T) * norm.cdf(self.d2) / 100
        return -self.K * self.T * np.exp(-self.r * self.T) * norm.cdf(-self.d2) / 100


# === Example: Price an option and compute all Greeks ===
bs = BlackScholes(S=100, K=100, T=0.25, r=0.05, sigma=0.20)

print("=== ATM Option (S=100, K=100, T=3mo, vol=20%) ===")
print(f"  Call price:  ${bs.call_price():.4f}")
print(f"  Put price:   ${bs.put_price():.4f}")
print(f"  Call delta:   {bs.delta('call'):.4f}")
print(f"  Put delta:    {bs.delta('put'):.4f}")
print(f"  Gamma:        {bs.gamma():.4f}")
print(f"  Theta (call): ${bs.theta('call'):.4f}/day")
print(f"  Vega:         ${bs.vega():.4f} per 1% vol")
print(f"  Rho (call):   ${bs.rho('call'):.4f} per 1% rate")

# Verify put-call parity
parity_lhs = bs.call_price() - bs.put_price()
parity_rhs = bs.S - bs.K * np.exp(-bs.r * bs.T)
print(f"\nPut-Call Parity Check:")
print(f"  C - P = {parity_lhs:.6f}")
print(f"  S - Ke^(-rT) = {parity_rhs:.6f}")
print(f"  Difference: {abs(parity_lhs - parity_rhs):.2e}")

8.2 Implied Volatility Solver

Pythondef implied_vol(market_price, S, K, T, r, option_type='call'):
    """
    Find implied volatility using Brent's method (root finding).
    """
    def objective(sigma):
        bs = BlackScholes(S, K, T, r, sigma)
        if option_type == 'call':
            return bs.call_price() - market_price
        return bs.put_price() - market_price

    # Brent's method: find sigma in [0.01, 5.0] where objective = 0
    return brentq(objective, 0.01, 5.0)

# Example: extract implied vol from market prices
print("=== Implied Volatility Extraction ===")
market_prices = {
    90:  12.50,   # Deep ITM call
    95:  8.20,    # ITM call
    100: 4.80,    # ATM call
    105: 2.40,    # OTM call
    110: 1.00,    # Deep OTM call
}

print(f"{'Strike':>8} {'Market Price':>14} {'Implied Vol':>13}")
for strike, price in market_prices.items():
    iv = implied_vol(price, S=100, K=strike, T=0.25, r=0.05)
    print(f"  {strike:>5}   ${price:>10.2f}    {iv:>10.1%}")

print("\nNote: if implied vol varies with strike, you're seeing")
print("the volatility smile/skew -- evidence of non-lognormal tails.")

8.3 Payoff and Profit Diagrams

Pythonimport numpy as np

def print_payoff_table(S_range, K, call_premium, put_premium):
    """
    Display call and put payoff/profit at various stock prices.
    """
    print(f"{'S_T':>6} {'Call Payoff':>13} {'Call Profit':>13} "
          f"{'Put Payoff':>12} {'Put Profit':>12}")
    print("-" * 60)
    for S in S_range:
        call_pay = max(S - K, 0)
        call_prof = call_pay - call_premium
        put_pay = max(K - S, 0)
        put_prof = put_pay - put_premium
        print(f"${S:>5.0f}  ${call_pay:>10.2f}   ${call_prof:>+10.2f}  "
              f"${put_pay:>9.2f}   ${put_prof:>+9.2f}")

# ATM option at K=100
print("=== Payoff/Profit Table: K=100, Call=$4.80, Put=$3.56 ===")
S_values = np.arange(80, 121, 5)
print_payoff_table(S_values, K=100, call_premium=4.80, put_premium=3.56)

print("\nBreakeven (call): S_T = K + premium = 100 + 4.80 = 104.80")
print("Breakeven (put):  S_T = K - premium = 100 - 3.56 = 96.44")
print("Max loss (buyer):  limited to premium paid")
print("Max gain (call):   unlimited")
print("Max gain (put):    K - premium = 96.44 (if stock goes to 0)")

8.4 Greeks Across Strike Prices

Python# Show how Greeks vary across strike prices for a fixed option
print("=== Call Greeks Across Strikes (S=100, T=0.25, vol=20%) ===")
print(f"{'Strike':>8} {'Price':>8} {'Delta':>8} {'Gamma':>8} "
      f"{'Theta':>8} {'Vega':>8}")
print("-" * 50)

for K in range(80, 121, 5):
    bs = BlackScholes(S=100, K=K, T=0.25, r=0.05, sigma=0.20)
    print(f"  {K:>5}  ${bs.call_price():>6.2f}   {bs.delta('call'):>.4f}  "
          f" {bs.gamma():>.4f}  {bs.theta('call'):>+.4f}   {bs.vega():>.4f}")

9. Summary

This module has covered the statistical foundations of options and derivatives:

In the next module, we leave the world of pure financial instruments and enter the realm of physical assets: commodities, gold, and oil. We will see how supply and demand for real goods create unique pricing dynamics that differ fundamentally from stocks and bonds.