Early Exercise





Kerry Back

Intrinsic value

  • The “intrinsic value” of an option is what it would be worth if it were immediately expiring.
    • The intrinsic value of a call is max(0, stock price - strike)
    • The intrinsic value of a put is max(0, strike - stock price)
  • Prior to maturity, the value of an option usually exceeds its intrinsic value. It is “better alive than dead.”
  • No one should exercise when value > intrinsic value

Early exercise

  • Sometimes, prior to maturity, the value of an option equals its the intrinsic value.
  • All options should be exercised, or positions should be closed by trading.

Calls are better alive than dead

  • A call option on an underlying asset that does not pay dividends prior to the option maturity should never be exercised early.
  • Proof is by put-call parity:

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

\[ \text{call price} > \text{stock price} - \text{PV of strike}\]

\[ \text{call price} > \text{stock price} - \text{strike}\]

Early exercise of puts is often optimal

  • If the stock price has fallen very low, you can be pretty sure you will not regret exercising a put.
  • Exercising the put early gets the cash early, which can be invested.
  • There is always a critical stock price (threshold) such that it is optimal to exercise a put when the stock falls below the threshold.
  • The threshold depends crucially on the stock volatility.