Chapter 3 W4 Class

3.1 Options

Do you know what kind of option they are?


First: Long Call; Second: Short Put


  1. A call is an option to buy
  2. A put is an option to sell
  3. European option: Exercisable only at expiration
  4. American option: Exercisable at any time

3.1.1 Option positions

  1. Long Call
  2. Long Put
  3. Short Call
  4. Short Put

3.1.2 Specification of Options

  1. Expiration Date
  2. Strike Price X
  3. E or A
  4. C or P

3.1.3 Moneyness:

  • At-the-money
  • In-the-money
  • Out-of-the money option
\(c\): European call option price \(C\) American call option price
\(p\): European put option price \(P\) American put option price
\(S_0\): Stock Price today \(S_T\) Stock price at option maturity
\(K\): Strike price \(D\) PV of dividends during option’s life
\(T\): Life of option \(r\) risk-free rate for maturity T with cts compounding
\(\sigma\): Volatility of stock

3.1.4 Stock Split

Suppose you own options with a strike price of K to buy (or sell) N shares: - No adjustments are made to the option terms for cash dividends - When there is an n-for-m stock split, - the strike price is reduced to mK/n - the no. of shares that can be bought (or sold) is increased to nN/m - Stock dividends are handled in a manner similar to stock splits

3.1.5 Market Makers

  • Most exchanges use market makers to facilitate options trading
  • A market maker quotes both bid and ask prices when requested
  • The market maker does not know whether the individual requesting the quotes wants to buy or sell ### Margin
  • Margin is required when options are sold (Example:A total of 100% of the proceeds of the sale plus 10% of the underlying share price (call) or exercise price (put))

3.1.6 Other information

  • Warrants: options that are issued by a corporation or a financial institution
  • Employee stock options: Employee stock options are a form of remuneration issued by a company to its executives

3.1.7 Effect of Variables on Option Pricing

Var c p C P
\[ S_0 \] + - + -
\[ K \] - + - +
\[ T \] ?? ?? + +
\[ \sigma \] + + + +
\[ r \] + - + -
\[ D \] - + - +


  • Keep other things unchanged, a higher S0 will be more likely to pass K, thus it’ll be price higher. Other logics follow

  • T for European options are murky. Since it’s exercisable only at a certain date, you never know what happens closer or on that date. The story is totally different in American options.

3.1.8 Option Bounds

  • An American option is worth at least as much as the corresponding European option C>=c; P>=p
Case Eur opean call Eur opean put Ame rican Put-call Parity
Basic c>=m ax(So - Xe ^(-r T),0) p> =max( Xe^( -rT)- So,0) S0 - X < C - P < S0 - Xe^ (-rT) c + Xe^(-rT) = p + S0
Discrete Dividend c ≥ m ax(S0 -D-Xe ^(-r T),0) p ≥ m ax(D+ Xe^( -rT)- S0,0) S0 - D - X < C - P < S0 - Xe^( –rT) c + D + Xe^(-rT) = p + S0
Dividend Yield c ≥ max( S0e^ (-qT) - Xe^( -rT), 0) p ≥ max (Xe^ (-rT) – S 0e^( -qT), 0) - c+ X e^(-rT)=p + S0e^(-qT)
Futures Options c ≥ m ax((F 0-X)e ^(-r T),0) p ≥ m ax((X -F0)e ^(-r T),0)

^(-rT )-X< C-P < F0- Xe^( -rT) 

C ≥F0-X and P ≥X-F0

c+ X e^(-rT)=p + F0e^(-rT)

3.2 Option trading strategies