Definitions

Option - is a contract that gives the holder the right to buy or sell a specific underlying instrument at a predetermined price (strike price) before the specified date (expiration date).

Main options type based on underlying asset

  • Stock Options
  • Foreign Currency Options
  • Index Options
  • Futures Options
  • Commodity Options

Types of options

A call option gives the holder of the option the right to buy an asset by a certain date for a defined price.
A put option gives the holder the right to sell an asset by a certain date for a defined price.

American options can be exercised at any time up to the expiration date.
European options can be exercised only on the expiration date itself.

Specification

  • Strike price
  • Underlying price
  • Option premium
  • Expiration date
  • Exercise date

Option position

Long position an investor who has bought the option (holder)
Short position an investor who has sold the option (writer)

Basic positions

  • Long call - long position in call option
  • Short call - short position in call option
  • Long put - long position in put option
  • Short put - short position in put option

Profit profiles

Payoff profiles

Long call, \(Payoff = max(S_T - K, 0)\)
Short call, \(Payoff = -max(S_T - K, 0)\)
Long put, \(Payoff = max(K - S_T, 0)\)
Short put, \(Payoff = -max(K - S_T, 0)\)

Example for long call:

Options are called in the money if one can generate positive payoff when exercising immediately. For call option S > K, for put option K > S.
Options are called at the money if the strike is equal spot price: S = K.
Options are called out of the money if one cannot generate positive payoff when exercising immediately. For call option S < K, for put option K < S.

Value of the option

The premium of any option consists of two components: its intrinsic value and its time value.

The intrinsic value of an option is defined as the value it would have if there were no time to maturity, so that the exercise decision had to be made immediately. For a call option, the intrinsic value is therefore max(S - K, 0). For a put option, it is max(K - S; 0).

The time value of the option is related to the possibility of the movements in the price of the underlying instrument on time for expiry of the option contract and decreases as the option approaches the expiry date.

Premium = intrinsic value + time value

Pricing of options

stock options

Parameters influencing the price of an option:

  • strike price
  • current underlying price
  • time to expiration
  • volatility of the underlying price
  • risk-free rate
  • dividend rate

Basic formula for european option pricing was defined by Fischera Black and Myrona Scholes in 1970s. The formula was enriched by Robert C. Merton by incorporation of dividend rate for valuation of option:

\[Call =Se^{-qT} N(d_1) - Ke^{-rT}N(d_2)\] \[Put =Ke^{-rT}N(-d_2) - Se^{-qT} N(-d_1)\] \[d_1 = \frac{ln\frac{S}{K} + (r - q + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] \[d_2 = \frac{ln\frac{S}{K} + (r - q- \frac{\sigma^2}{2})T}{\sigma\sqrt{T}} = d_1 - \sigma\sqrt{T}\] Where:

  • \(K\) - strike price
  • \(S\) - current underlying price
  • \(T\) - time to expiration
  • \(\sigma\) - volatility of the underlying price
  • \(r\) - risk-free rate
  • \(q\) - dividend rate

Assumptions used to derive the Black–Scholes–Merton
1. The stock price follows the log-normal process with \(\mu\) and \(\sigma\) constant.
2. The short selling of securities with full use of proceeds is permitted.
3. There are no transaction costs or taxes. All securities are perfectly divisible.
4. There are no dividends during the life of the derivative.
5. There are no riskless arbitrage opportunities.
6. Security trading is continuous.
7. The risk-free rate of interest, r, is constant and the same for all maturities.

Relation between parameters increase and european option premium
Strategy call put
current underlying price \(+\) \(-\)
strike price \(-\) \(+\)
time to expiration \(+/-\) \(+/-\)
volatility of the underlying price \(+\) \(+\)
risk-free rate \(+\) \(-\)
dividend rate \(-\) \(+\)

Exercises’ note: all options in the exercises are european type

Exercise 1

Trader bought a 6-month option contract for the purchase of 100 Intel shares. The option premium was $ 400. At the time of the purchase, the share price was $ 55 and the price of the underlying instrument was equal to the contract price (strike). What profit / loss will be when at expiration if the underlying instrument price is:
a) $ 54,
b) $ 62.

Exercise 2

Investor owns 1,000 shares of IBM. Current share price is $51. Investor is afraid that the price will fall in the near future. Therefore, he purchased 10 put options with an exercise price of $ 50 and a premium of $ 3 per share. Option contract size is 100 shares. What is the result of the hedging strategy in the following cases:
a) the share price falls to $ 50,
b) The share price goes up to $ 60,
c) The share price falls to $ 40?

Exercise 3

Jim believed that Apple’s shares would fall in the near future. Therefore he purchased 4 put options for Apple shares. The price of the option was $ 3 per share. The exercise price of the option was $ 40, and the expiry date was 3 months. The market price of the company’s share was $ 39. Jim was right and Apple’s stock price is $ 30 after three months. What profit / loss has he achieved on the above transaction?

Exercise 4

What is the price of the call option according to the Black-Scholes formula for Z shares with the following characteristics: S = 120, K = 130, σ = 0.35, T = 0.25, r = 0.07?

Exercise 5

Calculate the price of the 3-month put option for shares of company ABC paying the dividend at the rate of 2% per annum. The other characteristics are: S = 13.3, K = 14, σ = 0.45, r = 0.07.

Exercise 6

Mark has 500 shares of Microsoft and is afraid that in the near future their price will fall. Therefore, he buys 2 put options with a strike price of $ 55 and a premium of $ 2 per share (each option is issued for 100 shares of Microsoft) and also opens a short position in three futures contracts for Microsoft shares (each contract is issued for 100 shares of the company). The futures price of the contract is $ 57, and the futures transaction cost is $ 20 (for one contract). The current share price is $ 56. What is the result of the hedging strategy at the expiration date, if:
a) the share price drops to $ 40,
b) if the price increases to $ 70?
Please present the situation on the spot, option and futures markets separately.