2
Oct 22

Strategies for the crashing market

category: FN3142 Quantitative Finance, Quantitative Finance no comments

Strategies for the crashing market

This year is a wonderful time to short the market. During the pandemic the Fed has been pumping money into the market, and it was clear that the huge rally from March 2020 to December 2021 was nothing but a bubble. It was also obvious that the rally would be reversed by the turn from Quantitative Easing to Quantitative Tightening. Since just about all assets are falling, shorting the market is a very low risk play. The question of timing the trades will not be discussed here (January, March, May and August have been the best entry points). We look into how options can be used for shorting.

Put debit spread

When the market crashes, shorting indices (S&P 500, NASDAQ, Dow 30 and Russell 2000) or their proxies (exchange traded funds SPY, QQQ, DIA and IWM) is less risky than shorting individual stocks. That's what I learned (among many other things) from John Carter. Using put options instead of shorting stocks requires less capital. A further reduction in the capital requirement is achieved by using put debit spreads (their effect on buying power is zero).

Denote p(K) the price of a put with a strike K. We know that p\left(K\right) increases with K (you have to pay more for the right to sell at a higher price). A put debit spread strategy has been discussed earlier. It consists of two put options (sorry for the notation change): buy a put p(K_{1}) with a higher strike K_{1} and sell a put p\left( K_{2}\right) with a lower strike K_{2}. The initial outlay is debit=p\left( K_{1}\right) -p\left( K_{2}\right) >0. Let S\left( T\right) be the stock price at expiration T. The max profit is K_{1}-K_{2}-debit and it is positive if K_{1}-debit>K_{2}. The max loss is -debit<0.

The payoff is illustrated in Figure 1.

Comparison of option strategies

Figure 1. Put debit spread on the left, call credit spread on the right

Call credit spread

A call credit spread is a strategy consisting of two call options: buy a call c(K_{1}) with a higher strike K_{1} and sell a call c\left(K_{2}\right) with a lower strike K_{2}. Since you have to pay more for the right to buy at a lower price, we have c\left( K_{2}\right) >c(K_{1}) and you are credited credit=c\left( K_{2}\right) -c\left( K_{1}\right) >0. For the payoff we have 3 cases.

1) Case S\left( T\right) \leq K_{2}. Both calls are out of the money and the max profit is credit.

2) Case K_{2}<S\left( T\right) \leq K_{1}. The c(K_{2}) call, being in the money, is exercised by the buyer and you lose K_{2}-S\left( T\right)<0. The c\left( K_{1}\right) is out of the money and expires worthless. Thus the payoff is credit+K_{2}-S\left( T\right) and the break-even stock price is S\left( T\right) =credit+K_{2}.

3) Case S\left( T\right) >K_{1}. Both calls, being in the money, are exercised. The profit from the long call c\left( K_{1}\right) is S\left( T\right) -K_{1} and the loss from the short call c(K_{2}) is K_{2}-S\left( T\right) , so the payoff is their sum plus the credit K_{2}-K_{1}+credit. Normally, it is a loss.

Comparison of the two strategies

The above discussion is summarized in Figure 1. Both strategies can be used if the outlook is bearish. Here we indicate two situations when one is preferred over the other.

Situation 1. Following unexpected bad news, the market falls a lot in one day and it is clear from macroeconomics that it will continue to go down for some time. If prior to the fall there was a healthy rally, it didn't make sense to buy a put. Right after the fall volatility increases, so puts  become expensive. Buying a put debit spread is appropriate because volatilities from the long and short legs of the spread offset each other and one can increase the potential gain by selecting the strikes K_{1},K_{2} further apart.

Situation 2. A different approach is appropriate if a strong one-day fall is expected and any further developments are hard to predict. In this case selling a call credit spread with close expiration is recommended. This allows the investor to take advantage of the time decay. The strikes are selected above the current price S: S<K_{2}<K_{1}. The time value decay for the short call c\left( K_{2}\right) will be greater than for the long call c\left(K_{1}\right) (because the latter has a higher probability of staying out of the money). Therefore the open profit will quickly approach the max profit credit and the spread can be closed out earlier. This allows the investor to capture most of the premium received from placing the trade.

Mathematical approach to evaluating strategies

This will be explained using call credit spreads

Step 1. The payoff from a long call, neglecting the price c\left( K\right) paid, is \max \left\{ S\left( T\right) -K,0\right\} (if S\left( T\right) \leq K, you throw away the call and get 0; if S\left( T\right) >K, you exercise the right to buy the stock and get S\left( T\right) -K).

Step 2. What the long party gains, the short party looses, so the payoff from the short position (this time neglecting the credit received) is -\max \left\{ S\left( T\right) -K,0\right\} =\min \left\{ 0,K-S\left( T\right)\right\} .

Step 3. Let K_{2}<K_{1}. The payoff from the call credit spread is the sum of the payoffs from the first two steps:

\max \left\{ S\left( T\right) -K_{1},0\right\} -\max \left\{ S\left( T\right) -K_2,0\right\}.

Evaluating this expression for different price intervals gives the next table:

0 if S\left( T\right) \leq K_{2}<K_{1}
K_{2}-S\left( T\right) if K_{2}<S\left( T\right) \leq K_{1}
K_{2}-K_{1} if K_{2}<K_{1}<S\left( T\right)

Step 4. Adding the premium received credit=c\left( K_{2}\right) -c\left(K_{1}\right) we get the total payoff

credit if S\left( T\right) \leq K_{2}<K_{1}
K_{2}-S\left( T\right) +credit if K_{2}<S\left( T\right) \leq K_{1}
K_{2}-K_{1}+credit if K_{2}<K_{1}<S\left( T\right)

Exercise. For a long put the payoff is \max \{K-S\left( T\right) ,0\}, for a short one -\max \{K-S\left( T\right) ,0\}, for the strategy with K_{2}<K_{1} it is \max \{K_{1}-S\left( T\right) ,0\} -\max \{K_{2}-S\left(T\right) ,0\}-debit, where debit is as above.

The pictures have been produced in Mathematica with K_{1}=390; K_{2}=380. I put them side by side for you to better see the difference. The risk-reward ratio is better for the put debit spread (the left chart) than for the call credit spread (the right chart). The latter should be closed earlier and it takes longer for the open profit/loss of the former to approach the max profit. Such strategies could have been used a couple of times on SPY since September 12, 2022.

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20
Apr 21

Put debit spread

category: FN3142 Quantitative Finance, Quantitative Finance no comments

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 p(K) denote the price of the put with the strike K, 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 K_{1}<K_{2} are two strike prices, then for the corresponding put prices (with the same expiration and underlying asset) one has p(K_{1})<p(K_{2}).

Proof.  A put price is higher if the probability of it being in the money at expiration is higher. Let S(T) be the stock price at expiration T. Since T is a moment in the future, S(T) is a random variable. For a given strike K, the put is said to be in the money at expiration if S(T)<K. If K_{1}<K_{2} and S(T)<K_{1}, then S(T)<K_{2}. It follows that the set \{ S(T)<K_{1}\} is a subset of the set \{S(T)<K_{2}\} . Hence the probability of the event \{S(T)<K_{2}\} is higher than that of the event \{S(T)<K_{1}\} and p(K_{2})>p(K_{1}).

Put debit spread strategy. Select two strikes K_{1}<K_{2}, buy p(K_{2}) (take a long position) and sell p(K_{1}) (take a short position). You pay p=p(K_{2})-p(K_{1})>0 for this.

Our purpose is to derive the payoff for this strategy. We remember that if S(T)\ge K, then the put p(K) expires worthless.

Case S(T)\ge K_{2}. In this case both options expire worthless and the payoff is the initial outlay: payoff =-p.

Case K_{1}\leq S(T)<K_{2}. Exercising the put p(K_{2}), in comparison with selling the stock at the market price you gain K_{2}-S(T). The second option expires worthless. The payoff is: payoff =K_{2}-S(T)-p.

Case S(T)<K_{1}. Both options are exercised. The gain from p(K_{2}) is, as above, K_{2}-S(T). The holder of the long put p(K_{1}) sells you stock at price K_{1}. 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 S(T)-K_{1}. The payoff is: payoff =\left(K_{2}-S(T)\right) +\left( S(T)-K_{1}\right) -p=K_{2}-K_{1}-p.

Summarizing, we get:

payoff =\left\{\begin{array}{ll} -p, & K_2\le S(T) \\ K_{2}-S(T)-p, & K_{1}\leq S(T)<K_{2}\\ K_{2}-K_{1}-p, & S(T)<K_{1} \end{array}\right.

Normally, the strikes are chosen so that K_{2}-K_{1}>p. From the payoff expression we see then that the maximum profit is K_{2}-K_{1}-p>0, the maximum loss is -p and the breakeven stock price is S(T)=K_{2}-p. This is illustrated in Figure 1, where the stock price at expiration is on the horizontal axis.

Payoff from put debit spread

Figure 1. Payoff from put debit spread. Source: https://www.optionsbro.com/

Conclusion. For the strategy to be profitable, the price at expiration should satisfy S(T)< K_{2}-p. Buying a put debit spread is appropriate when the price is expected to stay in that range.

In comparison with the long put position p(K_{2}), taking at the same time the short call position -p(K_{1}) 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 p(K_{2})-p(K_{1}) that volatility component partially cancels out.

Remark. There is an important issue of choosing the strikes. Let S denote the stock price now. The payoff expression allows us to rank the next choices in the order of increasing risk: 1) S<K_1<K_2 (both options are in the money, less risk), 2) K_1<S<K_2 and 3) K_1<K_2<S (both options are out of the money, highest risk).  Also remember that a put debit spread is less expensive than buying p(K_{2}) and selling p(K_{1}) in two separate transactions.

Exercise. Analyze a put credit spread, in which you sell p(K_{2}) and buy p(K_{1}).

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