Option Pricing and Call-Put Parity: A Guide
Options
Kinds of Options
Calls and puts are the two simplest forms of options, often referred to as vanilla due to their commonality. There are many other types of options, some of which will be discussed later. Other terms used to describe contracts with some dependence on a more fundamental asset are derivatives or contingent claims.
Vocabulary
- Premium: The initial amount paid for the contract. Determining this value is a significant focus of this resource.
- Underlying (asset): The financial instrument on which the option value depends (e.g., stocks, commodities, currencies, indices). These will be denoted by S. The option payoff is a function of the underlying asset at expiry.
- Strike (price) or exercise price: The price at which the underlying can be bought (call) or sold (put). This will be denoted by K.
- Maturity time or expiration (date) or expiry (date): The date when the option can be exercised or ceases to exist, denoted by T.
- Intrinsic value: The payoff if the underlying is at its current level at option expiry.
- Time value: Any value the option holds beyond its intrinsic value. Uncertainty about the underlying asset’s future value typically makes the option value different from the intrinsic value.
- In the money: An option with positive intrinsic value. A call option when the asset price is above the strike, a put option when the asset price is below the strike.
- Out of the money: An option with only time value. A call option when the asset price is below the strike, a put option when the asset price is above the strike.
- At the money: A call or put with a strike close to the current asset level.
Effect of Parameters on Option Pricing
If the underlying price increases:
| Variable | Call | Put |
|---|---|---|
| S0 | + | – |
| K | – | + |
| T | ? | ? |
| Volatility | + | + |
| r | + | – |
Speculation Example
On March 14, 2018, OurCompany Inc. stock was €90.38, and a 100 call option with a September 21, 2018 maturity cost €5.
Assume we anticipated a price rise by September 21, 2018.
One strategy was to buy 1 asset. The final profit would be (106.8 – 90.38) * 100 / 90.38 = 18.17% of the initial investment.
Another option was buying a call option. The final profit would be (106.8 – 100 – 5) * 100 / 5 = 36%.
Speculation and Gearing
This illustrates gearing or leverage. Out-of-the-money options have high gearing, potentially yielding high payoffs for small investments.
However, the call option may expire worthless, losing your entire investment. If OurCompany Inc. remains at €90.38, the stock investment retains its value, but the call option suffers a 100% loss.
Call-Put Parity
Consider a call option with strike K and maturity T, and a put option (on the same underlying) with the same K and T.
If we buy the call and write the put, the portfolio payoff is: max(ST – K, 0) – max(K – ST, 0) = ST – K
This portfolio replicates a long asset and short cash position. This cashflow equality holds regardless of future stock behavior.
Let the call price be c, the put price be p, and the asset price be S.
Locking in a payment of E at time T requires a cash flow of Ke-r(T-t) at time t.
Therefore, the previous equality implies: c – p = S – Ke-r(T-t).
Examples
- Assume S0 = 100. A European call option with K = 100 and T = 1/12 is priced at 3.062, while a European put option with the same K and maturity is priced at 2.652. The volatility is 2%. What is the corresponding (continuously compounded) interest rate? This can be deduced from the call-put parity: 3.062 – 2.652 = 100 – 100e-r/12. Then, r = 4.9%.
- To construct a strategy with a payoff equal to ST using a call, a put, and a risk-free investment, buy one at-the-money call, sell one at-the-money put, and invest S0e-rT at the risk-free rate. This strategy’s price must be S0.