Worked Vertical Spread And Collar Payoff Problems

From Option Positions to Payoff and Profit

A vertical spread combines options on the same underlying asset with the same expiration date but different strike prices. A collar combines a stock position with a long put and a short call. In either strategy, a reliable solution starts by recording each position and its cash flow, then writing the expiration payoff by price region. Only after that should premiums and any stock basis be included to obtain profit.

The calculations below use a single share of the underlying unless a contract multiplier is stated. Let $S_T$ be the underlying price at expiration, and write $x^+=\max(x,0)$. A call with strike $K$ has expiration payoff $(S_T-K)^+$; a put with strike $K$ has payoff $(K-S_T)^+$. These are payoffs to the option holder. A short option has the negative of the corresponding payoff.

Model and conventions: The options in each position have the same underlying and expiration. Premiums are paid or received at the outset, and profit means expiration payoff or value minus the initial net cost, without interest on that initial cash flow. Dividends, transaction costs, taxes, and early exercise are ignored. If a contract multiplier applies, multiply per-share amounts by the number of shares per contract and the number of contracts.

A payoff describes the expiration value of the positions; profit also accounts for what they cost or generated initially. For a spread, the net premium is usually the simplest way to track that initial cash flow. For a collar, include both the stock’s purchase basis and the net option premium.

Vertical Spreads: Build the Regions First

A bull call spread buys the lower-strike call and sells the higher-strike call. It is established for a net debit when the purchased call’s premium exceeds the written call’s premium. A bear put spread buys the higher-strike put and sells the lower-strike put; it is also commonly established for a net debit. The long option provides the favorable exposure, while the short option limits the payoff once the underlying passes the second strike.

For a debit spread, profit per share equals the combined expiration payoff minus the net debit. The strike interval determines where the payoff changes; the premium determines where profit crosses zero. That distinction is central: the spread’s payoff is not its profit.

Worked Example: A Bull Call Spread With Two Contracts

An investor buys two call contracts with strike $K_1=\$50$, paying a premium of $\$4.80$ per share, and sells two calls with strike $K_2=\$62$, receiving $\$1.80$ per share. Each contract represents 100 shares. Find the net initial cost, the piecewise expiration payoff and profit, the breakeven, and the maximum gain and loss.

Method and initial cash flow. The spread is long the lower-strike call and short the higher-strike call, so its payoff per share is $(S_T-50)^+-(S_T-62)^+$. The net debit per share is $4.80-1.80=\$3.00$. Across $2(100)=200$ shares, the initial debit is $200(3.00)=\$600$.

Payoff by region. If $S_T\leq50$, both calls expire without value, so payoff is $\$0$. If $50<S_T<62$, only the lower-strike call has value, so payoff is $S_T-50$. If $S_T\geq62$, the difference between the call payoffs is $(S_T-50)-(S_T-62)=\$12$ per share.

\text{Payoff per share}= \begin{cases} 0, & S_T\leq 50,\\ S_T-50, & 50<S_T<62,\\ 12, & S_T\geq 62. \end{cases}

Profit and key values. Subtract the $\$3$ debit per share in each region. Below or at $\$50$, profit is $-\$3$ per share; between strikes, it is $S_T-53$; at or above $\$62$, it is $12-3=\$9$ per share. The breakeven solves $S_T-53=0$, giving $\$53$, which lies between the strikes. Maximum loss is the debit, $200(3)=\$600$. Maximum gain is $200(9)=\$1{,}800$.

Check. At $S_T=\$53$, the payoff is $\$3$ per share and profit is zero. At $S_T=\$62$, payoff is capped at $\$12$, so profit is $\$9$ per share, or $\$1{,}800$ for the position. The maximum-gain calculation also agrees with the spread width less the debit: $(62-50)-3=\$9$ per share.

Result: For a bull call spread with $K_1<K_2$ and net debit $D$, maximum loss per share is $D$, maximum gain per share is $K_2-K_1-D$, and the breakeven is $K_1+D$, provided the debit is between zero and the strike width.

Worked Example: A Bear Put Spread and Its Breakeven

A trader buys four put contracts at strike $\$80$, paying $\$9$ per share, and sells four puts at strike $\$65$, receiving $\$3$ per share. Each contract represents 100 shares. Determine the payoff and profit regions, the maximum gain and loss, and the profit if the underlying finishes at $\$70$.

Method and initial cash flow. This is a bear put spread: long the higher-strike put, short the lower-strike put. The net debit is $9-3=\$6$ per share. The position covers $4(100)=400$ shares, so its initial cost is $400(6)=\$2{,}400$.

If $S_T\geq80$, neither put has value. If $65<S_T<80$, only the long put pays, giving $80-S_T$ per share. If $S_T\leq65$, both puts are in the money and their payoff difference is $(80-S_T)-(65-S_T)=\$15$ per share.

\text{Profit per share}= \begin{cases} 15-6=9, & S_T\leq65,\\ 80-S_T-6=74-S_T, & 65<S_T<80,\\ -6, & S_T\geq80. \end{cases}

The strike width is $\$15$, so maximum payoff is $\$15$ per share and maximum profit is $15-6=\$9$ per share. Maximum loss is the $\$6$ debit per share. In the middle region, breakeven satisfies $74-S_T=0$, so $S_T=\$74$.

At $S_T=\$70$, the long put pays $80-70=\$10$ per share and the short put pays zero. Profit is $10-6=\$4$ per share. Across 400 shares, profit is $400(4)=\$1{,}600$. As checks, the breakeven $\$74$ is between the strikes, and the maximum profit and loss for the full position are $400(9)=\$3{,}600$ and $400(6)=\$2{,}400$, respectively.

Why the Spread Is Capped

The maximum-payoff result follows directly from the option payoffs. For a bull call spread with $K_1<K_2$, the payoff is zero below $K_1$. Between the strikes, the short call has no payoff, so the spread payoff rises with $S_T$. Above $K_2$, both calls are in the money, and their price increases cancel in the difference:

Below the first strike.
If $S_T\leq K_1$, both call payoffs are zero, so spread payoff is zero.

Between the strikes.
If $K_1<S_T<K_2$, the payoff is $S_T-K_1$; it increases from zero toward $K_2-K_1$.

At or above the second strike.
If $S_T\geq K_2$, subtracting the short-call payoff gives $(S_T-K_1)-(S_T-K_2)=K_2-K_1$, a fixed cap.

A bear put spread has the analogous cap. Below the lower strike, both puts are in the money and their underlying-price terms cancel: $(K_2-S_T)-(K_1-S_T)=K_2-K_1$. Between strikes, only the higher-strike put pays. Above the higher strike, neither put pays. In both spread types, the net premium shifts profit up or down without changing the shape of the expiration payoff.

Collars: Combine a Stock Basis With Two Options

A standard collar consists of long stock, a long put with strike $K_P$, and a short call with strike $K_C$, where $K_P<K_C$. The put establishes a floor on the stock-and-option value, while the written call establishes a cap. To calculate profit, use the stock’s initial purchase basis and the net option premium. The collar’s expiration value alone does not include those initial costs.

Worked Example: A Collar’s Floor, Cap, and Breakeven

An investor buys a share of stock for $\$48$, buys a put with strike $\$42$ for a premium of $\$2.50$, and writes a call with strike $\$60$, receiving $\$1.25$. Find the payoff and profit across the three expiration regions, plus the breakeven and the maximum gain and loss.

Method and initial cash flow. The options produce a net debit of $2.50-1.25=\$1.25$ per share. Including the stock basis, total initial cost is $48+1.25=\$49.25$ per share. At expiration, the position’s value is $S_T+(42-S_T)^+-(S_T-60)^+$.

If $S_T\leq42$, the stock and put together are worth $S_T+(42-S_T)=\$42$; the call expires worthless. If $42<S_T<60$, both options expire without value and the position is worth $S_T$. If $S_T\geq60$, the call is exercised against the investor, and the stock-plus-call value is $S_T-(S_T-60)=\$60$.

\text{Profit per share}= \begin{cases} 42-49.25=-7.25, & S_T\leq42,\\ S_T-49.25, & 42<S_T<60,\\ 60-49.25=10.75, & S_T\geq60. \end{cases}

Breakeven occurs in the middle region where $S_T-49.25=0$, so it is $\$49.25$. The maximum loss is $\$7.25$ per share, or $100(7.25)=\$725$ for one 100-share contract-sized position. The maximum gain is $\$10.75$ per share, or $\$1{,}075$ for 100 shares.

Check. Below the put strike, the position’s value stays at $\$42$, giving $42-49.25=-\$7.25$. Above the call strike, value stays at $\$60$, giving $60-49.25=\$10.75$. At $\$49.25$, which is between the strikes, the stock value exactly matches the total initial cost.

Common Traps in Multi-Step Problems

The most frequent error is to report a payoff as though it were profit. In the bull call example, the spread payoff at or above $\$62$ is $\$12$ per share. Calling that the maximum gain is wrong: the investor paid a $\$3$ net debit, so maximum profit is $12-3=\$9$ per share. For the collar, the value floor is $\$42$, not a loss of $\$42$; compare it with the total initial cost of $\$49.25$ to obtain the $\$7.25$ maximum loss.

Another common mistake is to subtract premiums with the wrong sign. A premium paid is a cost; a premium received reduces cost. Thus, buying an option for $\$5$ and writing another for $\$2$ creates a $\$3$ debit, not a $\$7$ debit or a $\$3$ credit. Finally, do not apply a contract multiplier halfway through the calculation: first find per-share payoff, profit, and breakeven, then scale dollar gains and losses by the total number of shares.

Key takeaway: For each expiration region, calculate the position’s payoff or value from the option and stock positions, then subtract the complete initial cost. Check that any breakeven lies in the region where you solved for it, and scale to contract dollars only after the per-share result is established.

This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus because it connects option position signs, premiums, expiration payoffs, and profit measures in a single calculation.

Check Your Understanding

For each problem, work per share first and then apply any stated contract multiplier.

A bull call spread buys a $\$30$ call for $\$5.20$ and writes a $\$38$ call for $\$2.20$. Find the maximum gain and loss per share and the breakeven. (Answer: Maximum gain $\$5$, maximum loss $\$3$, breakeven $\$33$.)
A bear put spread buys a $\$100$ put for $\$11$ and writes an $\$88$ put for $\$4$. Find maximum gain, maximum loss, and breakeven. (Answer: Maximum gain $\$5$, maximum loss $\$7$, breakeven $\$93$.)
A collar has stock basis $\$72$, a long $\$65$ put costing $\$3$, and a short $\$82$ call receiving $\$1.50$. Find the breakeven, maximum loss, and maximum gain per share. (Answer: Breakeven $\$73.50$, maximum loss $\$8.50$, maximum gain $\$8.50$.)
An investor buys five contracts of a $\$45$ call for $\$6.40$ and writes five contracts of a $\$57$ call for $\$2.40$. Each contract represents 100 shares. Find the profit if $S_T=\$54$ and the maximum gain for the position. (Answer: Profit $\$2{,}500$; maximum gain $\$4{,}000$.)
A collar consists of stock bought for $\$95$, a long $\$90$ put costing $\$2.40$, and a short $\$105$ call receiving $\$4.10$. Give the profit in the three price regions and identify maximum gain, maximum loss, and breakeven. (Answer: Profit is $-\$3.30$ for $S_T\leq\$90$, $S_T-\$93.30$ for $\$90<S_T<\$105$, and $\$11.70$ for $S_T\geq\$105$; maximum loss $\$3.30$, maximum gain $\$11.70$, breakeven $\$93.30$.)

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