Properties of European Options

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2026

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

Two payoff diagrams side by side. Strategy A combines a dashed long stock line and a dashed short call with strike K; Strategy B combines a dashed long bond paying K and a dashed short put with strike K. Both produce the same solid covered call payoff, which rises with S up to K and is flat at K above it.

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, the 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)

Position Today S_{T} \leq 110 S_{T} > 110
Long Stock -110.00 S_{T} S_{T}
Short Call 13.30 0 110 - S_{T}
Short Bond 105.95 -110 -110
Long Put -9.00 110 - S_{T} 0
Total 0.25 0 0

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 protective 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

Two payoff diagrams side by side. Strategy A combines a dashed long stock line and a dashed long put with strike K; Strategy B combines a dashed long bond paying K and a dashed long call with strike K. Both produce the same solid protective put payoff, flat at K for S below K and rising with S above it.

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

Two payoff diagrams side by side. Strategy A combines a dashed long call and a dashed short put, both with strike K; Strategy B combines a dashed long stock line and a dashed short bond paying K. Both produce the same solid forward payoff, a straight line equal to S minus K.

Option Pricing Bounds

A Simple Lower Bound on Call and Put Options

  • The price of a European call or put option cannot be negative.
  • 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 be 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.

  • Writing the call and buying a share of stock generates the following cash flows:

Position Today S_{T} \leq 110 S_{T} > 110
Short Call 115 0 110 - S_{T}
Long Stock -110 S_{T} S_{T}
Total 5 S_{T} 110
  • 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.
Call price against the stock price S. The shaded region lies between the upper bound, the dashed line where the call price equals S, and the lower bound, the dashed line max of S minus K times e to the minus rT, zero, which starts at K times e to the minus rT on the S axis, just left of K. The solid call payoff starts at K and lies inside the region.

Time Value for European Call Option (r > 0)

Black-Scholes call price against the stock price from 40 to 160, with strike 100, two years to maturity, a 5 percent interest rate and 20 percent volatility. The price curve lies above the dashed lower bound, which lies above the solid payoff. At a stock price of 120, arrows split the price into the intrinsic value of 20 and a positive time value.

Bounds on European Put Options

  • Put-call parity and the fact that C \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 be 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.
Put price against the stock price S. The shaded region lies between the upper bound, a dashed horizontal line at K times e to the minus rT, and the lower bound, the dashed line max of K times e to the minus rT minus S, zero. The solid put payoff starts at K on the vertical axis and reaches zero at K on the S axis.

Time Value of European Put Option (r > 0)

Black-Scholes put price against the stock price from 40 to 160, with strike 100, two years to maturity, a 5 percent interest rate and 20 percent volatility. For low stock prices the price curve dips below the solid payoff, toward the dashed lower bound. At a stock price of 60, an arrow marks the gap between the payoff and the price as a negative time value.

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!

Time Value of European Call Option (r < 0)

Black-Scholes call price against the stock price from 40 to 160, with strike 100, two years to maturity, a minus 5 percent interest rate and 20 percent volatility. The dashed lower bound now lies below the solid payoff, and for high stock prices the price curve falls below the payoff. At a stock price of 140, an arrow marks the gap as a negative time value.

Time Value of European Put Option (r < 0)

Black-Scholes put price against the stock price from 40 to 160, with strike 100, two years to maturity, a minus 5 percent interest rate and 20 percent volatility. The price curve stays above the solid payoff, and the dashed lower bound lies above the payoff. At a stock price of 60, an arrow marks the gap between the payoff and the price as a positive time value.