Bull Put Spreads With Limited Risk

What a Bull Put Spread Does

A bull put spread is a two-put position designed to benefit when the underlying price at expiration is not too far below the higher strike. The investor sells a put with the higher strike and buys a put with the lower strike. The short put brings in premium; the long put costs premium and limits the loss if the underlying price falls sharply.

The spread has a defined expiration payoff. Its short put can lose value as the underlying price falls, but once the price reaches the lower strike, the long put’s payoff offsets further losses in the short put. The spread therefore exchanges some premium income for a cap on downside exposure. The exact profit also depends on the premium received and paid.

Definition: A bull put spread consists of selling one put with strike $K_H$ and buying one put with strike $K_L$, on the same underlying and with the same expiration, where $K_H>K_L$. The difference $K_H-K_L$ is the strike width. Unless otherwise stated, assume each option has the same contract multiplier, transaction costs are ignored, and the analysis concerns the expiration payoff.

For one share of underlying exposure, a long put with strike $K$ pays $\max(K-S_T,0)$ at expiration, where $S_T$ is the underlying price at expiration. A short put has the negative of that payoff. These are expiration payoffs, not the options’ prices today.

Deriving the Payoff in Three Regions

Let $K_H>K_L$. The short higher-strike put contributes $-\max(K_H-S_T,0)$, and the long lower-strike put contributes $+\max(K_L-S_T,0)$. Adding these two position payoffs gives the spread payoff per share:

\Pi_T=-\max(K_H-S_T,0)+\max(K_L-S_T,0).

The payoff changes form at the two strike prices. Below or at $K_L$, both puts are in the money; between the strikes, only the short put is in the money; above or at $K_H$, neither put is in the money. Substitution in each region gives:

\Pi_T= \begin{cases} K_L-K_H, & S_T\leq K_L,\\ S_T-K_H, & K_L<S_T<K_H,\\ 0, & S_T\geq K_H. \end{cases}

Why the payoff is capped below the lower strike. If $S_T\leq K_L$, both puts have positive exercise value. The short put loses $K_H-S_T$ per share, while the long put gains $K_L-S_T$. Adding the two amounts gives $(K_L-S_T)-(K_H-S_T)=K_L-K_H$. The $S_T$ terms cancel, so a further fall in the underlying price does not make the spread payoff more negative.

Theorem: The bull put spread’s expiration payoff per share is never less than $-(K_H-K_L)$, and it is zero when $S_T\geq K_H$. Its payoff is negative below $K_H$, although the position’s total profit may still be positive if the initial credit is large enough.

At the boundary $S_T=K_L$, the payoff is $K_L-K_H$, the same as throughout the lower region. At $S_T=K_H$, the payoff is zero, the same as throughout the upper region. Checking both boundaries is a useful way to catch a mistaken region formula.

Payoff, Credit, and Expiration Profit

Suppose the higher-strike put is sold for premium $p_H$ per share and the lower-strike put is bought for premium $p_L$ per share. The initial net credit is $c=p_H-p_L$. When $c>0$, the position receives money at inception. Do not include this credit in the option payoff formula: the payoff describes only the options at expiration.

To calculate profit measured at expiration, state what happens to the initial credit. In the examples below, assume the credit is invested at the effective annual risk-free rate $i$ until expiration, which is $T$ years away. Its accumulated value per share is $c(1+i)^T$. Thus the expiration profit per share is:

\text{expiration profit per share} = \Pi_T+c(1+i)^T.

This is a specified financing convention, not a change to the options’ payoff. If a problem instead defines profit by subtracting the original premium without accumulating it, use that convention. Keep payoff and profit separate so that premiums are not accidentally counted twice.

When the initial credit is positive, and the credit is accumulated as stated, the maximum loss occurs at or below the lower strike. If $w=K_H-K_L$, the expiration loss per share in that region is $w-c(1+i)^T$, provided this amount is nonnegative. With $N$ spreads and contract multiplier $M$, the corresponding dollar loss is multiplied by $NM$. This calculation assumes the premiums and interest rate are consistent with the stated financing convention.

Worked Example: Evaluating All Three Expiration Regions

A trader sells a put with strike $100$ for $ \$8.40$ per share and buys a put with strike $85$ for $ \$3.10$ per share. Both options expire in six months and have a multiplier of 100. The effective annual interest rate is $4\%$, and the initial credit is invested until expiration. Find the payoff and expiration profit if $S_T$ is $80$, $92$, or $108$.

Method. First form the spread payoff from the short and long puts. Then add the accumulated initial credit. The common mistake this avoids is treating the option payoff as if it already included the premiums.

1. Find the strikes’ width and initial credit. The width is $100-85=\$15$ per share. The initial credit per share is $8.40-3.10=\$5.30$. For one contract, the initial credit is $5.30(100)=\$530$.

2. Accumulate the credit. Here $T=0.5$ years and $i=0.04$, so the accumulation factor is $(1.04)^{0.5}=1.0198039$. The accumulated credit is $5.30(1.0198039)=\$5.4049607$ per share, or $\$540.4961$ per contract.

3. Evaluate $S_T=80$. Since $80\leq85$, both puts are in the money. The payoff is $-(100-80)+(85-80)=-20+5=-\$15$ per share. Expiration profit is $-15+5.4049607=-\$9.5950393$ per share, or $-\$959.50$ per contract when rounded to cents.

4. Evaluate $S_T=92$. Since $85<92<100$, only the short put is in the money. The payoff is $-(100-92)=-\$8$ per share. Expiration profit is $-8+5.4049607=-\$2.5950393$ per share, or $-\$259.50$ per contract.

5. Evaluate $S_T=108$. Since $108\geq100$, both puts expire out of the money. The payoff is $\$0$ per share. Expiration profit is $0+5.4049607=\$5.4049607$ per share, or $\$540.50$ per contract.

Check. The most negative option payoff is $-\$15$ per share, exactly the negative of the strike width. The calculated profits in each region equal that payoff plus the same accumulated credit. Dollar amounts are rounded to two decimals only at the final step.

Position Signs and Contract Scaling

A useful way to avoid sign errors is to write down the position and the payoff of each option separately before combining them. “Sell the higher strike” means the higher-strike put contributes a negative payoff; “buy the lower strike” means the lower-strike put contributes a positive payoff. Reversing either sign changes the position.

Write the short-put payoff.
For a short put at $K_H$, write $-\max(K_H-S_T,0)$.

Write the long-put payoff.
For a long put at $K_L$, write $+\max(K_L-S_T,0)$.

Add the payoffs, then add the credit if calculating profit.
Use the same underlying price and expiration for both options. Apply the contract multiplier and number of spreads to convert per-share results into dollars.

For example, with three spreads and a multiplier of 100, a payoff of $-\$4$ per share represents a total option payoff of $-4(3)(100)=-\$1{,}200$. The premium credit must be scaled using the same contract count and multiplier. Applying the multiplier to only one leg or only one part of the calculation gives an inconsistent result.

Worked Example: Scaling a Capped Loss Across Contracts

A trader sells two puts at strike $70$ for $ \$6.75$ per share and buys two puts at strike $62$ for $ \$4.25$ per share. Each contract has a multiplier of 100. The options expire in one year, the effective annual interest rate is $4\%$, and the initial credit is invested until expiration. Find the total expiration profit if $S_T=62$, and verify the spread’s downside limit.

Method. Evaluate the per-share position first, then scale the payoff and credit by the total number of shares represented by the contracts. This avoids mixing per-share amounts with contract-level dollars.

1. Compute the per-share width and credit. The width is $70-62=\$8$. The credit is $6.75-4.25=\$2.50$ per share. There are two contracts on each leg, so the position represents $2(100)=200$ shares.

2. Accumulate the credit. With $T=1$ and $i=0.04$, the factor is $(1.04)^1=1.04$. The accumulated credit per share is $2.50(1.04)=\$2.60$. Across 200 shares, it is $2.60(200)=\$520$.

3. Calculate the option payoff at $S_T=62$. The higher-strike short put pays $-(70-62)=-\$8$ per share. The lower-strike long put pays $\max(62-62,0)=\$0$. The total payoff is $-\$8$ per share, or $-8(200)=-\$1{,}600$.

4. Find expiration profit. Add the accumulated credit: $-1{,}600+520=-\$1{,}080$. Equivalently, the loss is $(8-2.60)(200)=\$1{,}080$.

Check. At the lower strike, the option payoff reaches its minimum, $-\$8$ per share. The resulting loss is capped at $\$1{,}080$ for these contracts under the stated interest assumption. The total option payoff and accumulated credit were both scaled by 200 shares.

Worked Example: Comparing Prices Below, Between, and Above the Strikes

A hypothetical investor sells three puts at strike $48$, receiving $ \$5.60$ per share for each, and buys three puts at strike $42$, paying $ \$2.20$ per share for each. Each contract has a multiplier of 100. The options expire in one year; the effective annual interest rate is $5\%$, and the credit is invested until expiration. Calculate total expiration profit when $S_T=39$, $45$, and $51$.

Method. Use one per-share payoff calculation for each price region, then apply the same accumulated credit and scale by the 300 shares represented. The common mistake this avoids is assuming the middle-region payoff is constant at the full strike width.

1. Find the width and credit. The width is $48-42=\$6$. The net credit is $5.60-2.20=\$3.40$ per share. The position represents $3(100)=300$ shares.

2. Accumulate the credit. The one-year accumulation factor is $1.05$. The accumulated credit is $3.40(1.05)=\$3.57$ per share, or $3.57(300)=\$1{,}071$ in total.

3. Evaluate $S_T=39$. This is below $42$, so the spread payoff is $42-48=-\$6$ per share. Total payoff is $-6(300)=-\$1{,}800$. Expiration profit is $-1{,}800+1{,}071=-\$729$.

4. Evaluate $S_T=45$. This is between $42$ and $48$, so only the short put is in the money. The payoff is $45-48=-\$3$ per share, or $-\$900$ total. Expiration profit is $-900+1{,}071=\$171$.

5. Evaluate $S_T=51$. This is above $48$, so both puts expire out of the money. The payoff is zero, and expiration profit is the accumulated credit, $\$1{,}071$.

Check. The payoff sequence is $-\$6$, $-\$3$, and $\$0$ per share as the underlying price rises across the regions. The profit in the lowest region is $-\$729$, matching the width less accumulated credit: $(6-3.57)(300)=\$729$. Dollar amounts are exact here; no intermediate rounding is needed.

Common Mistakes and a Final Check

Reversing the legs. Selling the lower-strike put and buying the higher-strike put is not the bull put spread defined here. Check that the short option is at $K_H$, the larger strike, before writing the payoff.

Forgetting which position has a minus sign. A short put payoff is negative, even though selling it brings in a premium at inception. The premium cash flow and the expiration payoff are separate parts of the calculation.

Extending the middle formula too far. The expression $S_T-K_H$ applies only when $K_L<S_T<K_H$. Below $K_L$, the long put is also in the money and offsets further loss. Above $K_H$, neither put pays.

Check Your Understanding

In each problem, calculate the requested expiration profit using the stated convention for accumulating the initial credit.

A trader sells a one-year put at strike $50$ for $ \$5.20$ and buys a put at strike $44$ for $ \$2.10$. The effective annual interest rate is $0\%$, the multiplier is 100, and $S_T=47$. What is the expiration profit for one spread? (Answer: $\$10$)
A one-year spread sells the strike-$120$ put for $ \$4.80$ and buys the strike-$110$ put for $ \$2.30$. The effective annual interest rate is $4\%$, the multiplier is 100, and $S_T=105$. What is the expiration profit for one spread? (Answer: $-\$740$)
Three spreads each sell a strike-$90$ put for $ \$6.40$ and buy a strike-$82$ put for $ \$3.10$. The multiplier is 50, the term is one year, the effective annual interest rate is $5\%$, and $S_T=86$. What is total expiration profit? (Answer: $-\$80.25$)
Two spreads each sell a strike-$100$ put for $ \$7.10$ and buy a strike-$90$ put for $ \$3.60$. The multiplier is 100, the term is six months, the effective annual interest rate is $4\%$, and $S_T=94$. What is total expiration profit, rounded to cents? (Answer: $-\$486.14$)
Four spreads each sell a strike-$75$ put for $ \$6.90$ and buy a strike-$68$ put for $ \$2.70$. The multiplier is 100, the term is one year, the effective annual interest rate is $3\%$, and $S_T=70$. What is total expiration profit? (Answer: $-\$269.60$)

What You'll Learn