PutCall Parity and Pricing Options

1
Put/Call Parity and Pricing Options

• Ch. 22 part 2

2
Put-Call Parity

• Consider two combinations (similar setup to the
  homemade leverage discussion earlier)
• Buying a put plus the underlying stock
• Buying a call plus a bond that has a face
  value equal to the strike price
• The strike prices for the options are equal, and
  all the assets mature at the same time.

3
Graphically,
4
The Result

• Since the payoffs are the same, the price must be
  the same. Thus, in terms of prices,
• Call PV Bond Put Stock
• The present value of the bond is knownits
  borrowing at the risk free rate to obtain the
  strike price at maturity.
• The stocks price is knownjust look it up. So
  once the price of the call is known, the price of
  the put will be known as well.

5
Pricing a Call

• There are limits on the price of a call that
  expires in the future.
• Price of call gt 0
• Price of call lt Price of Stock
• Price of call gt Price of Stock Strike Price

6
Limits on Option Prices

• The actual option price will be similar to the
  line inside the grey area above.
• How can we value the call? Build two portfolios
  that have the same payoffs.
• The Call itself
• Stock plus borrowing

7
An Example

• A stock that is currently trading for 30 will be
  trading at 20 or 40 in a year. What is the
  price of a call that expires in a year with a
  strike price of 35 on this stock?

8
Explanation of the Table

• Note We borrowed 4.54 at 10 in order to pay 5
  in one year to balance the last column.
• How did we get the number of shares to buy?
• Since borrowing doesnt change with the state,
  the range of the stock price has to match the
  range of the option price. To make the ranges
  match, divide the range of the call by the range
  of the stock, and use that many shares.

9
Conclusion

• The ¼ Stock plus borrowing will cost ¼ (30) -
  4.54 or a total of 2.96. So a call with a
  strike price of 35 is worth 2.96. How much is a
  put worth with the same strike price?
• Call PV Bond Put Stock
• PV 35 bond 35/1.1 31.82
• Call Price is 2.96, Stock price is 30
• Price of Put 2.96 31.82 30 4.78
• Note The call pricing (but NOT the put-call
  parity) above is with discrete time and prices.
• To go to a continuous time/price setting is
  complicated, but the Black-Scholes formula prices
  options in a continuous setting.