Option Greeks (qfn)¶

Background¶

Option Greeks¶

Exploration of the option Greeks (Delta, Gamma, Theta, Vega, Rho) under the Black-Scholes model, with reference to Chapter 6 from Option Volatility & Pricing: Advanced Trading Strategies and Techniques (Natenberg, 1994).

Each Greek is derived as a partial derivative of the option value with respect to an underlying parameter (spot price, time, volatility, interest rate) and is implemented here in closed form for European options.

Source¶

From the "quantitative-finance-notebooks" collection, notebook 4.6.

Code¶

```python """ Option Greeks =============

Exploration of the option Greeks (Delta, Gamma, Theta, Vega, Rho) under the Black-Scholes model, with reference to Chapter 6 from Option Volatility & Pricing: Advanced Trading Strategies and Techniques (Natenberg, 1994).

Each Greek is derived as a partial derivative of the option value with respect to an underlying parameter (spot price, time, volatility, interest rate) and is implemented here in closed form for European options.

Source¶

From the "quantitative-finance-notebooks" collection, notebook 4.6. """

---------------------------------------------------------------------------¶

Imports¶

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import numpy as np import scipy.stats as ss

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Parameters¶

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if name == "main": S0 = 100.0 # spot price K = 100.0 # strike price T = 1.0 # time to expiration (in years) r = 0.04 # annual risk-free rate sigma = 0.2 # annual volatility rate

# ===================================================================
# 1. The Greeks
# ===================================================================
#
# In mathematical finance, the Greeks are the quantities (known in
# calculus as partial derivatives; first-order or higher) representing
# the sensitivity of the price of a derivative instrument such as an
# option to changes in one or more underlying parameters on which the
# value of an instrument or portfolio of financial instruments is
# dependent.
# ===================================================================

# ---------------------------------------------------------------------------
# Black-Scholes closed-form price
# ---------------------------------------------------------------------------
def closed_formula(S0, K, T, r, sigma, payoff='call'):
    d1 = (np.log(S0 / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)
    if payoff == 'call':
        return S0 * ss.norm.cdf(d1) - K * np.exp(-r * T) * ss.norm.cdf(d2)
    elif payoff == 'put':
        return K * np.exp(-r * T) * ss.norm.cdf(-d2) - S0 * ss.norm.cdf(-d1)


# Calculate option prices using the Black-Scholes closed formula
call = closed_formula(S0, K, T, r, sigma, 'call')
put = closed_formula(S0, K, T, r, sigma, 'put')

print(f"Call price: {call:.3f}")
print(f"Put price: {put:.3f}")


# ===================================================================
# 1.1 Delta
# ===================================================================
#
# Delta is the first derivative of the option value V with respect to
# the underlying asset price S:
#
#   Delta = dV / dS
#
# For European options under Black-Scholes:
#
#   Delta_call = N(d1)
#   Delta_put  = N(d1) - 1
#
# The delta of a call is bounded in [0, 1].  A delta of 1.00 means
# the option moves point-for-point with the underlying.  A call with
# delta 0.25 changes value at 25% of the rate of the underlying.
#
# Puts always have negative deltas (range from 0.00 for far OTM to
# -1.00 for deeply ITM) because put values move in the opposite
# direction of the underlying market.
#
# Delta also tells us the proper hedge ratio: the ratio of the
# position in the underlying asset to options required to establish a
# neutral hedge.
#
# As a rough approximation, the delta equals the probability that the
# option will finish in-the-money.
# ===================================================================

def calculate_delta(S0, K, T, r, sigma, payoff='call'):
    d1 = (np.log(S0 / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    if payoff.lower() == 'call':
        return ss.norm.cdf(d1)
    elif payoff.lower() == 'put':
        return ss.norm.cdf(d1) - 1


# Calculate option delta
call_delta = calculate_delta(S0, K, T, r, sigma, payoff='call')
put_delta = calculate_delta(S0, K, T, r, sigma, payoff='put')

print(f"Call Delta: {call_delta:.3f}")
print(f"Put Delta: {put_delta:.3f}")


# ===================================================================
# 1.2 Gamma
# ===================================================================
#
# Gamma is the rate at which an option's delta changes as the price
# of the underlying changes (the second derivative of V w.r.t. S):
#
#   Gamma = d(Delta) / dS = d^2 V / dS^2
#
# For European options under Black-Scholes:
#
#   Gamma = N'(d1) / (S0 * sigma * sqrt(T))
#
# Gamma is usually expressed in deltas gained or lost per one-point
# change in the underlying.  The delta increases by gamma when the
# underlying rises and decreases by gamma when it falls.
#
# Long options (calls or puts) have positive gamma; short options
# have negative gamma.  A large gamma indicates high risk.
#
# Gamma is greatest for at-the-money options and becomes
# progressively smaller as the option moves into- or out-of-the-money.
# ===================================================================

def calculate_gamma(S0, K, T, r, sigma):
    d1 = (np.log(S0 / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    return ss.norm.pdf(d1) / (S0 * sigma * np.sqrt(T))


# Calculate option gamma
call_gamma = calculate_gamma(S0, K, T, r, sigma)
put_gamma = calculate_gamma(S0, K, T, r, sigma)

print(f"Call Gamma: {call_gamma:.3f}")
print(f"Put Gamma: {put_gamma:.3f}")


# ===================================================================
# 1.3 Theta
# ===================================================================
#
# Theta is the rate at which an option loses value as time passes:
#
#   Theta = -dV / d(tau)
#
# For European options under Black-Scholes:
#
#   Theta_call = -S0 N'(d1) sigma / (2 sqrt(T)) - r K e^{-rT} N(d2)
#   Theta_put  = -S0 N'(d1) sigma / (2 sqrt(T)) + r K e^{-rT} N(-d2)
#
# Theta here is computed per year; dividing by 365 gives theta per
# day.  It is usually expressed in points lost per day when all other
# conditions remain the same.
#
# Theta is written as a negative number to remind us that time runs
# in only one direction.
#
# As a general principle, gamma and theta have opposite signs.  Every
# option position is a tradeoff between market movement (gamma) and
# time decay (theta).
# ===================================================================

def calculate_theta(S0, K, T, r, sigma, payoff='call'):
    d1 = (np.log(S0 / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)
    if payoff == 'call':
        return (-S0 * ss.norm.pdf(d1) * sigma / (2 * np.sqrt(T))
                - r * K * np.exp(-r * T) * ss.norm.cdf(d2)) / 365.0
    elif payoff == 'put':
        return (-S0 * ss.norm.pdf(d1) * sigma / (2 * np.sqrt(T))
                + r * K * np.exp(-r * T) * ss.norm.cdf(-d2)) / 365.0


# Calculate option theta
call_theta = calculate_theta(S0, K, T, r, sigma, payoff='call')
put_theta = calculate_theta(S0, K, T, r, sigma, payoff='put')

print(f"Call Theta: {call_theta:.3f}")
print(f"Put Theta: {put_theta:.3f}")


# ===================================================================
# 1.4 Vega
# ===================================================================
#
# Vega is the derivative of the option value with respect to the
# volatility of the underlying asset price:
#
#   Vega = dV / d(sigma)
#
# For European options under Black-Scholes:
#
#   Vega = S0 sqrt(T) N'(d1)
#
# All options gain value with rising volatility, so vega is positive
# for both calls and puts.  If an option has a vega of 0.15, for each
# percentage point increase (decrease) in volatility the option gains
# (loses) 0.15 in value.
# ===================================================================

def calculate_vega(S0, K, T, r, sigma):
    d1 = (np.log(S0 / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    return (S0 * np.sqrt(T) * ss.norm.pdf(d1)) / 100


# Calculate option vega
call_vega = calculate_vega(S0, K, T, r, sigma)
put_vega = calculate_vega(S0, K, T, r, sigma)

print(f"Call Vega: {call_vega:.3f}")
print(f"Put Vega: {put_vega:.3f}")


# ===================================================================
# 1.5 Rho
# ===================================================================
#
# Rho measures the sensitivity of an option's value to a change in
# interest rates:
#
#   rho = dV / dr
#
# For European options under Black-Scholes:
#
#   rho_call =  K T e^{-rT} N(d2)
#   rho_put  = -K T e^{-rT} N(-d2)
#
# Unlike the other Greeks, one cannot generalise about rho since its
# characteristics depend on the type of underlying instrument and the
# settlement procedure for the options.
#
# A call can be thought of as a substitute for purchasing the
# underlying stock.  Rising interest rates make the call more
# attractive (lower carrying cost vs outright purchase), so call
# values increase.
#
# Conversely, selling stock becomes more attractive than buying a put
# in a high-rate environment (the cash credit earns greater interest),
# so rising rates cause put values to fall.
# ===================================================================

def calculate_rho(S0, K, T, r, sigma, payoff='call'):
    d1 = (np.log(S0 / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)
    if payoff == 'call':
        return (K * T * np.exp(-r * T) * ss.norm.cdf(d2)) / 100
    elif payoff == 'put':
        return (-K * T * np.exp(-r * T) * ss.norm.cdf(-d2)) / 100


# Calculate option rho
call_rho = calculate_rho(S0, K, T, r, sigma, payoff='call')
put_rho = calculate_rho(S0, K, T, r, sigma, payoff='put')

print(f"Call Rho: {call_rho:.3f}")
print(f"Put Rho: {put_rho:.3f}")

```

Exercises¶

Exercise 1. Delta can be interpreted as the probability of finishing ITM (approximately). For \(S_0 = K = 100\), \(T = 1\), \(r = 0.04\), \(\sigma = 0.2\), compute call delta and explain why it exceeds the true risk-neutral probability \(N(d_2)\).

Exercise 2. Compute vega for the same parameters. If implied volatility increases from 20% to 22%, estimate the change in call price.

Exercise 3. Rho for a call is \(\rho_C = KTe^{-rT}N(d_2)/100\). Explain why rho is relatively small for short-dated options.

Exercise 4. The gamma-theta relationship implies \(\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma \approx rV\) for delta-neutral portfolios. Verify this numerically for the given parameters.