week 3

• Option


  A type of derivative whose payoff is a function of another asset
• Types of options


  • Calls
  • Puts
  • American
  • European
• Strike Price (K)


  The predetermined price at which the option can be exercised
• Time to Maturity (τ)


  The time until the option expires
• Call Option Payoff


  Max[0, ST - K]
• Put Option Payoff


  Max[0, K - ST]
• Options Clearing Corporation handles options trading on the Chicago Board Options Exchange, Mercantile Exchange, e-trade
• Put Option


  Provides insurance against a decline in the value of an asset
• House insurance


  Is like a put option on the house
• Starbucks coupon


  Is essentially a call option with a strike price of £0
• Exercise price of a put option


  The money you owe the bank
• Options can be combined in various ways to create an unlimited number of payoff profiles
• Option trading strategies


  • Buy a stock and a put
  • Buy a call with one strike price and sell a call with another
  • Buy a call and a put with the same strike price

  
• Calculating call option profit/loss


  1. If S < K, Payoff = 0, Profit = -C

  2. If S = K, Payoff = 0, Profit = -C

  3. If S > K, Payoff = S - K, Profit = S - K - C

  
• Calculating put option profit/loss


  1. If S < K, Payoff = K - S, Profit = K - S - P

  2. If S = K, Payoff = 0, Profit = -P

  3. If S > K, Payoff = 0, Profit = -P

  
• A straddle comprising a call and put option with the same strike price will break even if the stock price is either $71 or $89
• Exploiting mispriced American call option


  Buy the call and exercise it, then sell the stock
• Put-Call Parity


  Defines a price relationship between a call option, put option and the underlying stock

  
• Binomial Option Pricing Model


  1. Considers one-period call option on a stock

  2. Current stock price S0

  3. Strike price K

  4. Option expires tomorrow, C_T = Max[S_T - K, 0]

  5. Calculates today's option price C_0

  
• Upside gross change (u)

  The factor by which the stock price increases

  
• Downside gross change (d)

  The factor by which the stock price decreases

  
• Put-call parity payoff diagram


  Both call and put options are worth max(ST , K ) at the maturity of the options

  
• The Put-Call Parity Result


  Both call and put options must be worth the same today, so c + PV(K) = p + S0 (value of call + present value of exercise price = value of put + share price)

  
• The Put-Call Parity Result only holds for European options
• The Put-Call Parity Result can be expressed in several ways, each implying two investment strategies that give identical results
• If the Put-Call Parity Result does not hold, there will be an arbitrage profit opportunity
• Binomial Option-Pricing Model of Cox, Ross, and Rubinstein (1979)


  1. Consider one-period call option on stock A

  2. Current stock price S0

  3. Strike price K

  4. Option expires tomorrow, C1 = Max [S1 - K , 0 ]

  5. What is today's option price C0?

  
• Binomial Option-Pricing Model


  • u is the upside gross change
  • d is the downside gross change
  • r is the gross interest rate, which is 1 plus the net interest rate
  • The risk-neutral probability of a rise in value p = (r - d) / (u - d)

  
• If we can't agree on the u & d for one day, we can firstly agree on the outcomes on stock price between now and five minutes from now, 1 minute from now, 1 second from now

  
• If we continue to apply the same way and let the number of periods go to infinity, we can control the u and d and make them smaller and smaller so as to reasonably agree with them, then we get the Black-Scholes formula
• Options


  • Have nonlinear payoffs
  • Can be viewed as insurance contracts
  • Allow investors to take more sophisticated bets
• Option valuations are typically derived via arbitrage arguments (e.g. binomial)
• Essential Reading: Brealey, R.A., Myers, S.C., Allen, F., Principles of corporate finance. Global edition.11th Edition. McGraw-Hill Education. Chapter 20, 21.2,21.3