Spreads

Option spreads and combinations can be useful option strategies. We first consider money spreads, in which the two options differ only by exercise price. The investor buys an option with a given expiration and exercise price and sells an option with the same expiration but a different exercise price. Of course, the options are on the same underlying asset. The term spread is used here because the payoff is based on the difference, or spread, between option exercise prices. For a bull or bear spread, the investor buys one call and writes another call option with a different exercise price, or the investor buys one put and writes another put with a different exercise price.

Bull Call Spread

Regardless of whether someone constructs a bull spread with puts or with calls, the strategy requires buying one option and writing another with a higher exercise price. Because the higher exercise price call is less expensive than the lower strike, a call bull spread involves an initial cash outflow (debit spread). A

A bull spread created from puts also requires the investor to write the higher-strike option and buy the lower-strike one. Because the higher-strike put is more expensive, a put bull spread involves an initial cash inflow (credit spread).

The value of the spread at expiration (VT) depends on the stock price at expiration ST. For a bull spread, the investor buys the low strike option (struck at XL) and sells the high strike option (struck at XH), so that:

Equation (5) 

  • VT = Max(0,ST – XL) – Max(0,ST – XH).

Therefore, the value depends on the terminal stock price ST:

  • VT = 0 − 0 = 0 if ST ≤ XL
  • VT = ST − XL − 0 = ST − XL if XL < ST < XH
  • VT = ST − XL− (ST − XH) = XH − XL if ST ≥ XH

The profit is obtained by subtracting the initial outlay for the spread from the foregoing value of the spread at expiration. To determine the initial outlay, recall that a call option with a lower exercise price will be more expensive than a call option with a higher exercise price. Because we are buying the call with the lower exercise price (for cL) and selling the call with the higher exercise price (for cH), the call we buy will cost more than the call we sell (cL > cH). Hence, the spread will require a net outlay of funds. This net outlay is the initial value of the position, V0 = cL – cH, which we call the net premium. The profit is:

Equation (6) 

  • Π = Max(0,ST – XL) – Max(0,ST – XH) – (cL – cH).

In this manner, we see that the profit is the profit from the long call, Max(0,ST – XL) – cL, plus the profit from the short call, –Max(0,ST – XH) + cH. Broken down into ranges, the profit is as follows:

  • Π = −cL + cH if ST ≤ XL
  • Π = ST − XL − cL + cH if XL < ST < XH
  • Π = XH − XL − cL + cH if ST ≥ XH

If ST is below XL, the strategy will lose a limited amount of money. When both options expire out of the money, the investor loses the net premium, cL – cH. The profit on the upside, if ST is at least XH, is also limited to the difference in strike prices minus the net premium.

In general, for a bull call spread:

  • Maximum loss = net premium paid
  • Breakeven = lower strike + net premium paid
  • Maximum profit = difference between strikes – net premium paid

The bull call spread is an example of a debit spread since it entails a net outlay: the bought call, with a lower strike, is more valuable than the sold call.

The other debit spread is the bear put spread.

Bear Put Spread

To construct a bear put spread, we sell a lower strike put and buy a higher strike put. Because puts with higher exercise prices are (all else equal) more expensive, a put bear spread will result in an initial cash outflow (be a debit spread). For a call bear spread, the investor buys a higher exercise price call and sells the lower exercise price call. Because the higher exercise price call being purchased is less expensive than the lower strike being sold, a call bear spread will result in an initial cash inflow (credit spread).

Mathematically, the value of this bear spread position at expiration is:

  • VT = Max(0,XH – ST) – Max(0,XL – ST).

Broken down into ranges, we have the following relations:

  • VT = XH − ST − (XL − ST) = XH − XL if ST ≤ XL
  • VT = XH − ST − 0 = XH − ST if XL < ST < XH
  • VT = 0 − 0 = 0 if ST ≥ XH

To obtain the profit, we subtract the initial outlay. Because we are buying the put with the higher exercise price and selling the put with the lower exercise price, the put we are buying is more expensive than the put we are selling. The initial value of the bear spread is V0 = pH – pL. The profit is, therefore, VT – V0, which is:

  • Π = Max(0,XH – ST) – Max(0,XL – ST) – (pH – pL).

We see that the profit is that on the long put, Max(0,XH – ST) – pH, plus the profit from the short put, –Max(0,XL – ST) + pL. Broken down into ranges, the profit is as follows:

  • Π = XH − XL − pH + pL if ST ≤ XL
  • Π = XH − ST − pH + pL if XL < ST < XH
  • Π = −pH + pL if ST ≥ XH

The breakeven point, ST* = XH – pH + pL, sets the profit equal to zero between the strike prices. 

Generalized At-Expiration Formulas for Spreads

In these formulas, net premium means the absolute value of the difference between the premiums.

For debit spreads (bull call and bear put):

  • Maximum loss = net premium paid
  • Maximum profit = difference between strikes – net premium paid

For credit spreads (bear call and bull put):

  • Maximum profit = net premium received
  • Maximum loss = difference between strikes – net premium received

For call spreads, breakeven = lower strike + net premium

For put spreads, breakeven = higher strike – net premium

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