Using Put Call Parity to Calculate Put Option Greeks

Put-call parity is a fundamental relationship in options pricing that allows traders to calculate the price of a put option based on the price of a call option and vice versa. This relationship is particularly useful for determining the Greeks of put options, which measure the sensitivity of an option's price to various factors.

What is Put-Call Parity?

Put-call parity is an economic relationship between European call and put options on the same underlying stock with the same expiration date and strike price. The relationship states that the price of a call option plus the price of the underlying stock should equal the price of a put option plus the present value of the strike price.

Put-Call Parity Formula:

C + S = P + K × e^{-(r × T)}

Where:

C = Price of the call option
S = Current price of the underlying stock
P = Price of the put option
K = Strike price of the options
r = Risk-free interest rate
T = Time to expiration (in years)

This relationship holds true in an arbitrage-free market, meaning that if the relationship is violated, arbitrageurs can profit by buying and selling the options and the underlying stock to exploit the discrepancy.

Calculating Put Greeks Using Parity

The Greeks are measures of an option's sensitivity to various factors. The Greeks for put options can be derived using put-call parity. Here's how to calculate the key Greeks:

Delta

The delta of a put option measures its sensitivity to changes in the underlying stock price. Using put-call parity, the delta of a put option is equal to the negative of the delta of the corresponding call option.

Put Delta:

Δ_put = -Δ_call

Gamma

The gamma of a put option measures its rate of change in delta relative to changes in the underlying stock price. The gamma of a put option is equal to the gamma of the corresponding call option.

Put Gamma:

Γ_put = Γ_call

Vega

The vega of a put option measures its sensitivity to changes in volatility. The vega of a put option is equal to the vega of the corresponding call option.

Put Vega:

V_put = V_call

Theta

The theta of a put option measures its sensitivity to the passage of time. The theta of a put option is equal to the theta of the corresponding call option.

Put Theta:

θ_put = θ_call

By using put-call parity, traders can quickly determine the Greeks of put options without needing to perform complex calculations or simulations.

Example Calculation

Let's walk through an example to illustrate how to use put-call parity to calculate put option Greeks.

Example Scenario

Suppose we have the following information:

Current stock price (S) = $100
Strike price (K) = $100
Risk-free interest rate (r) = 5% (0.05)
Time to expiration (T) = 1 year
Call option price (C) = $10
Call delta (Δ_call) = 0.6
Call gamma (Γ_call) = 0.04
Call vega (V_call) = 0.1
Call theta (θ_call) = -0.05

Step 1: Calculate Put Option Price Using Parity

Using the put-call parity formula:

P = C + S - K × e^{-(r × T)}

P = $10 + $100 - $100 × e^{-(0.05 × 1)}

P = $110 - $100 × 0.9512

P = $110 - $95.12

P = $14.88

Step 2: Calculate Put Greeks

Using the relationships derived from put-call parity:

Δ_put = -Δ_call = -0.6

Γ_put = Γ_call = 0.04

V_put = V_call = 0.1

θ_put = θ_call = -0.05

These calculations show how put-call parity can be used to quickly determine the Greeks of put options based on the Greeks of the corresponding call options.

FAQ

What is the purpose of put-call parity?: Put-call parity is used to establish a relationship between the prices of call and put options on the same underlying stock. It helps traders understand the theoretical relationship between these options and can be used to verify the fairness of option prices.
How are the Greeks of put options calculated using parity?: The Greeks of put options can be derived from the Greeks of the corresponding call options using the relationships provided by put-call parity. For example, the delta of a put option is the negative of the delta of the call option, while the gamma, vega, and theta remain the same.
When is put-call parity most useful?: Put-call parity is most useful in arbitrage-free markets to verify the fairness of option prices. It can also be used to quickly determine the Greeks of put options based on the Greeks of call options, which can be helpful for traders and analysts.
Can put-call parity be used for American options?: Put-call parity is typically applied to European options, which cannot be exercised early. For American options, which can be exercised at any time, put-call parity does not hold because of the early exercise premium.
What happens if put-call parity is violated?: If put-call parity is violated, it indicates an arbitrage opportunity. Traders can exploit this discrepancy by buying and selling the options and the underlying stock to profit from the price difference.