## Put debit spread

This post parallels the one about the call debit spread. A combination of several options in one trade is called a **strategy**. Here we discuss a strategy called a **put 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 price of the put with the strike suppressing all other variables that influence the put 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 put prices (with the same expiration and underlying asset) one has

**Proof**. A put 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 put 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 higher than that of the event and

**Put 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 put expires worthless.

**Case** In this case both options expire worthless and the payoff is the initial outlay: payoff

**Case** Exercising the put , in comparison with selling the stock at the market price you gain The second option expires worthless. The payoff is: payoff

**Case** Both options are exercised. The gain from is, as above, The holder of the long put sells you stock at price Since your position is short, you have nothing to do but comply. The alternative would be to buy at the market price, so you lose 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 put debit spread is appropriate when the price is expected to stay in that range.

In comparison with the long put 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 put 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 put debit spread is less expensive than buying and selling in two separate transactions.

**Exercise**. Analyze a **put credit spread**, in which you sell and buy .

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