Chapter 17 Option Pricing

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Chapter 17 Option Pricing
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Framework

• Background
• Definition and payoff
• Some features about option
• strategies
• One-period analysis
• Put-call parity,
• Arbitrage bound,
• American call option
• Black-Scholes Formula
• Price using discount factor
• Derive Black-Scholes differential equation

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Background (1)

• Option
• Call/Put
• Strike Price
• Expiration Date
• Underlying Asset
• European/ American Option
• Payoff/Profit

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Background (2)
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Background (3)

• Some Interesting Features of Options
• High Beta (High Leverage)
• Trading
• Hedging
• Shaping Distribution of Returns
• OTM Put Stock
• But Short OTM Put Option and Long Index
• Return Distribution Extremely Non-normal
• The Chance of Beating the Index for one or even
  five years is extremely high, but face the
  catastrophe risk
• So what kind of investments can and cannot be
  made is written in the portfolio management
  contracts.

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Background (4)

• Strategies
• By combining options of various strikes, you can
  buy and sell any piece of the return
  distribution.
• A complete set of option is equivalent to
  complete markets.
• Forming payoff that depends on the terminal stock
  price in any way

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One-period analysis

• The law of one price
• existence of a discount factor
• No arbitrage
• existence of positive discount factor
• How to pricing option
• Put-Call Parity
• Arbitrage Bounds
• Discount Factors and Arbitrage Bounds
• Early Exercise

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Put-call parity
In the book of John C. Hull,

• Strategies
• (1) hold a call, write a put ,same strike price
• (2) hold stock, borrow strike price X

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Put-call parity

• According to the Law of One Price,
• applying to both sides for any m,
• We get??

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Arbitrage bounds

• Portfolio A dominates portfolio B
• Arbitrage portfolio

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Arbitrage bounds
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Discount factors and arbitrage bounds
This presentation of arbitrage bound is
unsettling for two reasons, First, you many
worry that you will not be clever enough to dream
up dominating portfolios in more complex
circumstances. Second, you may worry that we have
not dream up all of the arbitrage portfolios in
this circumstance.
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Discount factors and arbitrage bounds

• This is a linear program. In situations where you
  do not know the
• answer, you can calculate arbitrage
  bounds.(Ritchken(1985))

• The discount factor method lets you construct the
  arbitrage bounds

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Early exercise?

• By applying the absence of arbitrage, we can
  never exercise an American call option without
  dividends before the expiration date.
• S-X is what you get if you exercise now. the
  value of the call is greater than this value,
  because you can delay paying the strike, and
  exercising early loses the option value

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Black-Scholes Formula (Standard Approach Review)
Portfolio Construction
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Black-Scholes Formula (Standard Approach Review)
Risk Neutral Pricing
Where
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Black-Scholes Formula (Discount Factor)

• Write a process for stock and bond, then use
  
• to price the option. the Black-Scholes
  formula results,
• (1) solve for the finite-horizon discount factor
• and find the call option price by taking the
  expectation
• (2) find a differential equation for the call
  option and solve it backward.

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Black-Scholes Formula (Discount Factor)

• The call option payoff is
• The underlying stock follows
• The is also a money market security that pays the
  real interest rate
• In continuous time, all such discount factors are
  of the form

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Method 1 price using discount factor

• Use the discount factor to price the option
  directly

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• How to find analytical expressions for the
  solutions of equations of the form (17.2)

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Applying the Solution to (17.2)
Ignoring the term of And Proof Later
We get
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Evaluate the call option by doing the integral
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Proof not Affect
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Where
This is the integral under the normal
distribution, with mean of and,
standard variance of 1,so the integral is 1.we
multiply both sides without any change.
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Method 2derive Black-Scholes Differential
Equation

• Guess that solution for the call option is a
  function of stock price and time to expiration,
  CC(S,t). Use Itos lemma to find derivatives of
  C(S,t)

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• This is the Black-Scholes differential equation
• for the option price

• This differential equation has an analytic
  solution, one
• standard way to solve differential equation is to
  guess and
• check, and by taking derivatives you can check
  that (17.7)
• does satisfy (17.8).

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