## 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.

**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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