Put-Call Parity





Kerry Back

  • At maturity, protective put portfolio (stock + put) is same as cash + call
  • Cash = strike
  • Stock ends up \(\Rightarrow\)
    • With stock + put, keep stock
    • With cash + call, exercise, using cash, to get stock
  • Stock ends down \(\Rightarrow\)
    • With stock + put, exercise, using stock, to get cash
    • With call + cash, keep cash

Prices of puts and calls

  • Prior to maturity, for European options on a stock that does not pay dividends prior to the option expirations,

\[ \text{price of (stock + put)} = \text{price of (cash + call)}\]

\[ \text{stock price} + \text{put price} = \] \[ \text{PV of strike} + \text{call price}\]

Example

  • One year to maturity, interest rate = 5%, strike = 100
  • Suppose underlying is at 90 and price of call = 10
  • Price of put should be

\[\frac{100}{1.05} + 10 - 90 = 25.24\]

Difference between call and put prices

  • call price - put price = stock price - PV of strike
    • difference in prices (call minus put) goes up 1-for-1 with stock price
    • call price > put price when stock price > PV of strike
  • put price - call price = PV of strike - price of underlying
    • difference in prices (put minus call) goes up 1/(1+r) for 1 with strike
    • e.g., strike \(\uparrow\) 5 \(\Rightarrow\) put - call \(\uparrow\) 5/(1+r)

Apple example on Feb 10, 2023

last AAPL price was $151.01

Apple calls expiring March 17, 2023

lastPrice bid ask
strike
130.0 21.80 22.15 22.40
135.0 17.60 16.90 17.80
140.0 13.45 13.40 13.55
145.0 9.70 9.60 9.70
150.0 6.43 6.35 6.50
155.0 3.87 3.85 3.95
160.0 2.12 2.11 2.13
165.0 1.06 1.03 1.07
170.0 0.50 0.49 0.50

Apple puts expiring March 17, 2023

lastPrice bid ask
strike
130.0 0.66 0.64 0.66
135.0 1.07 1.06 1.08
140.0 1.80 1.80 1.81
145.0 2.99 2.96 2.99
150.0 4.73 4.70 4.80
155.0 7.25 7.15 7.30
160.0 10.55 10.40 10.60
165.0 14.54 14.40 14.65
170.0 19.45 18.95 19.20

Apple put prices minus call prices

Dividends and early exercise

  • Put-call parity is based on values at expiration being equal
  • It may not hold for American options, because the optimal exercise conditions may be different for “stock + put” compared to “cash + call”
  • It will not hold if the stock pays dividends prior to option expiration, because “stock + put” will get the dividends and “cash + call” will not.