Properties of European Options

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2025

The Put-Call Parity

Building a Covered Call

Consider European call and put options with strike K and maturity T written on a non-dividend paying stock.
There is also a zero-coupon risk-free bond with face value K and same maturity as the options.

Strategy A
Long stock and short call \begin{aligned} \text{Cost} & = S - C \\ \text{Payoff} & = \begin{cases} S_{T} & \text{if $S_{T} \leq K$} \\ K & \text{if $S_{T} > K$} \end{cases} \end{aligned}

Strategy B
Long bond and short put \begin{aligned} \text{Cost} & = K e^{-r T} - P \\ \text{Payoff} & = \begin{cases} S_{T} & \text{if $S_{T} \leq K$} \\ K & \text{if $S_{T} > K$} \end{cases} \end{aligned}

Payoff Diagrams for Both Strategies

Put-Call Parity

Since both strategies have the same payoff, they should have the same price.
- Otherwise, buy the cheapest strategy and sell the most expensive one.
- This would generate a free positive cash flow with zero risk.
For European options written on non-dividend paying stocks, following relationship known as put-call parity must hold: S - C = K e^{-r T} - P

Example 1

Consider a non-dividend paying stock trading at $110 and assume that the continuously-compounded risk-free rate is 5% per year.
A European call option with strike price $110 and maturity 9 months trades for $13.30.
Then, according to put-call parity, we should have that a European put with the same strike and maturity as the call should cost: \begin{aligned} P & = C - S + K e^{-r T} \\ & = 13.30 - 110 + 110 e^{-0.05 \times 0.75} \\ & = 9.25 \end{aligned}

Example 2

What if in the previous example everything stays the same, but you find that the put trades for $9?
- Then we have an arbitrage opportunity!
Let us consider both strategies discussed previously that we know have the same payoff.
Strategy A: Long stock and short call \text{Cost}_{A} = 110 - 13.30 = 96.70
Strategy B: Long bond and short put \text{Cost}_{B} = 110 e^{-0.05 \times 0.75} - 9 = 96.95
Since \text{Cost}_{A} < \text{Cost}_{B}, we should buy A and sell B which generates an instant profit of $0.25 per share.

Example 2 (cont’d)

Put-Call Parity and Protective Puts

We can also express put-call parity in the following way: S + P = K e^{-r T} + C
The left hand-side of this expression is the cost of a covered put, i.e. long stock and long put.
The right hand-side says that a protective put can also be built by buying a bond and a call.

Payoff Diagrams for Protective Put

Put-Call Parity and Forward Contracts

We can express put-call parity in yet a different way: C - P = S - K e^{-r T}
The right hand-side of this expression is the cost of a forward contract with forward price K.
The left hand-side says that a forward contract can be synthesized by buying a call and selling a put.

Payoff Diagrams for Forward Contract

Option Pricing Bounds

A Simple Lower Bound on Call and Put Options

The price of a European call or put option must be positive.
If not, any trader would like to get as many contracts as possible.
- The worst-case scenario is that the options expire out-of-the-money in which case the payoff is zero.
- Otherwise the options expire in-the-money and the option trader gets a positive payoff.
- This would clearly be a nice arbitrage opportunity!
Hence, we must have that C \geq 0 and P \geq 0.

Lower Bound on European Call Options

Remember that we are analyzing an underlying asset that does not pay dividends.
Put-call parity and the fact that P \geq 0 implies that: C = P + S - K e^{-r T} \geq S - K e^{-r T}
Given that we also have c \geq 0, it must the case that: C \geq \max(S - K e^{-r T}, 0)

Example 3

Consider a non-dividend paying stock where S = 110 and r = 5\% per year with continuous compounding.
Consider a call option with strike $110 and maturity 9 months.
It must be the case that: C \geq \max(110 - 110 e^{-0.05 \times 0.75}, 0) = \max(4.05, 0) = 4.05
Hence, no matter how low the volatility is on this European call option, its premium must be higher than $4.05.
- Otherwise it might be possible to synthesize a negative price put.

Upper Bound on European Call Options

On the other hand, the price of a European call on a non-dividend paying asset must cost less than the stock itself: C \leq S
If not, it would make sense to write a call and use part of the proceeds to buy a share of stock.

Example 4

Assume that S = 110, K = 110, T = 0.75 years and C = 115.

This strategy makes money for free at T = 0 and keeps making money at T = 0.75!

Feasible Prices for European Call Options

The picture below describes the region of feasible prices for European call options written on a non-dividend paying asset when the risk-free rate is positive.

Time Value for European Call Option (r > 0)

Bounds on European Put Options

Put-call parity and the fact that p \geq 0 implies that: P = C - S + K e^{-r T} \geq - S + K e^{-r T}
Given that we also have p \geq 0, it must the case that: P \geq \max(K e^{-r T} - S, 0)
Also, the maximum amount of money one can lose by writing a European put is K, which in present value terms is equal to K e^{-r T}, implying that: P \leq K e^{-r T}

Feasible Prices for European Put Options

The graph describes the region of feasible prices for European put options written on a non-dividend paying asset when the risk-free rate is positive.

Time Value of European Put Option (r > 0)

Summary

We have the following bounds for European call and put options written on a non-dividend paying asset: \begin{aligned} \max(0, S - K e^{-r T}) \leq & C \leq S \\ \max(0, K e^{-r T} - S) \leq & P \leq K e^{-r T} \end{aligned}

Example 5

The risk-free rate is r = 10\% per year with continuous compounding.
Furthermore, assume that S = 50, K = 45, and T = 1.20.
Let us compute the bounds for European call and put options.
- First, for the put option we have that: 0 = \max(45 e^{-0.10 \times 1.20} - 50, 0) \leq P \leq 45 e^{-0.10 \times 1.20} = 39.91
- Second, for the call option: 10.09 = \max(50 - 45 e^{-0.10 \times 1.20}, 0) \leq C \leq 50

Impact of Negative Interest Rates

During the last decade, interest rates in many countries became negative, even for maturities longer than 10 years.
For a European call option, when r < 0 its lower bound is less than its intrinsic value, i.e., for a sufficiently high S the option will have negative time value and it might be optimal to early exercise an American call option.
For put options, negative interest rates means that a European put option always has positive time value, i.e., it is not optimal to exercise early an American put option.
Standard results that are usually taught in derivative courses get reversed!