Call debit spread
A combination of several options in one trade is called a strategy. Here we discuss a strategy called a call debit spread. The word "debit" in this name means that a trader has to pay for it. The rule of thumb is that if it is a debit (you pay for a strategy), then it is less risky than if it is a credit (you are paid). Let denote the call price with the strike
suppressing all other variables that influence the call price.
Assumption. The market values higher events of higher probability. This is true if investors are rational and the market correctly reconciles views of different investors.
We need the following property: if are two strike prices, then for the corresponding call prices (with the same expiration and underlying asset) one has
Proof. A call price is higher if the probability of it being in the money at expiration is higher. Let be the stock price at expiration
Since
is a moment in the future,
is a random variable. For a given strike
the call is said to be in the money at expiration if
If
and
then
It follows that the set
is a subset of the set
Hence the probability of the event
is lower than that of the event
and
Call debit spread strategy. Select two strikes buy
(take a long position) and sell
(take a short position). You pay
for this.
Our purpose is to derive the payoff for this strategy. We remember that if then the call
expires worthless.
Case In this case both options expire worthless and the payoff is the initial outlay: payoff
Case Exercising the call
and immediately selling the stock at the market price you gain
The second option expires worthless. The payoff is: payoff
(In fact, you are assigned stock and selling it is up to you).
Case Both options are exercised. The gain from
is, as above,
The holder of the long call
buys from you at price
Since your position is short, you have nothing to do but comply. You buy at
and sell at
Thus the loss from
is
The payoff is: payoff
Summarizing, we get:
payoff
Normally, the strikes are chosen so that From the payoff expression we see then that the maximum profit is
the maximum loss is
and the breakeven stock price is
This is illustrated in Figure 1, where the stock price at expiration is on the horizontal axis.

Figure 1. Payoff for call debit strategy. Source: https://www.optionsbro.com/
Conclusion. For the strategy to be profitable, the price at expiration should satisfy Buying a call debit spread is appropriate when the price is expected to stay in that range.
In comparison with the long call position taking at the same time the short call position
allows one to reduce the initial outlay. This is especially important when the stock volatility is high, resulting in a high call price. In the difference
that volatility component partially cancels out.
Remark. There is an important issue of choosing the strikes. Let denote the stock price now. The payoff expression allows us to rank the next choices in the order of increasing risk: 1)
(both options are in the money, less risk), 2)
and 3)
(both options are out of the money, highest risk). Also remember that a call debit spread is less expensive than buying
and selling
in two separate transactions.
Exercise. Analyze a call credit spread, in which you sell and buy
.
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