Option Spreads¶

BUSI 722: Data-Driven Finance II¶

Kerry Back, Rice University¶

Hedging Short Positions¶

Protective call¶

  • If you are short an asset, you can protect against the price spiking by buying a call.
  • Your liability cannot exceed the call strike, because the call option can be exercised at the strike, and the asset you acquire can be used to cover the short.
  • Usually use an out-of-the-money call.

Collaring a short position¶

  • Buy an out-of-the-money call for protection
  • Sell an out-of-the-money put to help pay for it
  • Example: short a stock trading at 60, buy an 80 call, sell a 40 put
  • Giving up some potential gain to limit your maximum loss

Spreads for investing/speculating¶

Basics¶

  • Buy a call to speculate on price rising
  • Buy a put to speculate on price falling
  • Out-of-the-money options provide higher potential returns

Bull spread¶

  • Buy a call to speculate on price rising
  • Sell a further-out-of-the-money call to help pay for the first call
  • Reduces potential gain but risk less money

Bear spread¶

  • Buy a put to speculate on price falling
  • Sell a further-out-of-the-money put to help pay for the first put
  • Reduces potential gain but risk less money

Straddles and strangles¶

  • Straddle = call and put with the same strike
  • Win if underlying moves enough in either direction
  • Betting on volatility instead of making a directional bet
  • Strangle = make the put strike lower and the call strike higher
    • Less effective bet but cheaper

Butterflies and condors¶

  • Empirically, selling straddles and strangles is profitable on average, but there is a lot of risk
  • Can buy insurance against the risk by buying options
  • A short straddle or strangle has both upside and downside risk
    • Insure the downside by buying a put (with a lower strike)
    • Insure the upside by buying a call (with a higher strike)
  • Straddle (strangle) w/ insurance = butterfly (condor)

Put-Call Parity and its Implications¶

Underlying + Put = Cash + Call¶

  • Protective put portfolio is same as call with cash, for European options
  • Cash = PV of strike
  • Stock ends up $\Rightarrow$
    • With put, keep underlying
    • With call, exercise to get underlying
  • Stock ends down $\Rightarrow$
    • With put, exercise to get cash
    • With call, keep cash

Implication 1: Prices of puts and calls¶

  • Put-call parity implies that underlying price + put premium should equal PV of strike plus call premium
  • Equivalently, put premium = call premium + PV of strike - underlying price
  • Equivalently, call premium = put premium - PV of strike + underlying price
  • Actually, this is true only for European options on stocks that don't pay dividends before the option maturity.

Implication 2: Calls are better alive than dead¶

  • Never optimal to exercise American calls on non-dividend paying stocks prior to maturity
  • If you exercise, you get underlying price - strike
  • If you hold, value is
    • underlying price + put value - PV of strike
    • $>$ underlying price - PV of strike
    • $>$ underlying price - strike

Can be optimal to exercise American puts early¶

  • Exercise when underlying is very low
  • When it becomes clear that exercise will be optimal, might as well exercise early to get strike and earn a return on it.
  • Still better to close position by selling rather than exercising.
  • Optimality of exercise means that put premium can fall to strike - underlying price.

Implication 3: Two ways to make bets¶

  • Anything you can do with a call, you can also do with a put and the underlying and cash
  • And vice versa
  • Example: bear spread with cash = collared long position
  • Example: make a butterfly spread entirely with calls