Options·Option Math

Options Math — Calculations and Worked Examples

Why Options Math Matters

The Series 7 exam tests not just conceptual understanding but also your ability to calculate breakevens, intrinsic value, time value, and profit/loss. These calculations follow consistent formulas. Master the patterns, and any options math question becomes straightforward.

The key principle: always start from the breakeven, then ask "which direction does this position profit?"

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The Four Basic Positions — Complete Formula Reference

Long Call

  • Breakeven: Strike Price + Premium Paid
  • Intrinsic Value at Expiration: Max(Stock Price − Strike, 0)
  • Profit/Loss: Intrinsic Value − Premium Paid
  • Maximum Gain: Unlimited
  • Maximum Loss: Premium Paid
  • Short Call

  • Breakeven: Strike Price + Premium Received (same as long call buyer's breakeven)
  • Profit/Loss: Premium Received − Intrinsic Value
  • Maximum Gain: Premium Received
  • Maximum Loss: Unlimited
  • Long Put

  • Breakeven: Strike Price − Premium Paid
  • Intrinsic Value at Expiration: Max(Strike − Stock Price, 0)
  • Profit/Loss: Intrinsic Value − Premium Paid
  • Maximum Gain: Strike Price − Premium Paid (stock can only fall to zero)
  • Maximum Loss: Premium Paid
  • Short Put

  • Breakeven: Strike Price − Premium Received (same as long put buyer's breakeven)
  • Profit/Loss: Premium Received − Intrinsic Value
  • Maximum Gain: Premium Received
  • Maximum Loss: Strike Price − Premium Received
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    Intrinsic Value and Time Value — Formula Drill

    Intrinsic Value

    The immediately exercisable value of an option. It is never negative.

    `` Call Intrinsic Value = Max(Stock Price − Strike Price, 0) Put Intrinsic Value = Max(Strike Price − Stock Price, 0) `

  • In-the-money options have positive intrinsic value
  • At-the-money and out-of-the-money options have zero intrinsic value
  • Time Value

    ` Time Value = Option Premium − Intrinsic Value ``

    Time value represents the "hope" component — the additional amount investors pay for the possibility that the option moves further in the money before expiration. It is highest when:

  • There is a long time until expiration
  • The underlying stock is highly volatile
  • The option is at or near the money
  • Time value always decays to zero at expiration. An option at expiration is worth exactly its intrinsic value.

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    Worked Examples

    Example 1: Long Call — Profit/Loss at Expiration

    > Setup: You buy a call with a $50 strike price and pay a $3 premium. The stock price at expiration is $57.

    Step 1 — Intrinsic Value: $57 − $50 = $7

    Step 2 — Time Value: At expiration, time value = $0. Premium was $3, but that was at purchase.

    Step 3 — Profit/Loss: Intrinsic Value − Premium Paid = $7 − $3 = $4 profit per share

    Per contract: $4 × 100 shares = $400 profit

    Step 4 — Breakeven check: Breakeven = $50 + $3 = $53. Stock at $57 is above $53 → profitable. ✓

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    Example 2: Long Put — Profit/Loss at Expiration

    > Setup: You buy a put with a $45 strike price and pay a $4 premium. The stock falls to $36 at expiration.

    Step 1 — Intrinsic Value: $45 − $36 = $9

    Step 2 — Profit/Loss: $9 − $4 = $5 profit per share ($500 per contract)

    Step 3 — Breakeven check: Breakeven = $45 − $4 = $41. Stock at $36 is below $41 → profitable. ✓

    Maximum gain scenario: If stock falls to $0: Intrinsic = $45 − $0 = $45. P&L = $45 − $4 = $41 max gain.

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    Example 3: Short Call — When Does the Writer Lose?

    > Setup: You write (sell) a call with a $60 strike and receive a $5 premium. The stock rises to $72 at expiration.

    Step 1 — Option is exercised against you: The call buyer exercises. You must sell stock at $60 when it's worth $72.

    Step 2 — Loss on the short call: The option is worth $72 − $60 = $12 intrinsic value. You collected only $5.

    Step 3 — Net P&L: $5 received − $12 paid = −$7 loss per share (−$700 per contract)

    Step 4 — Breakeven check: $60 + $5 = $65. Stock at $72 is above $65 → writer loses. ✓

    If stock is at $64 at expiration: Option worth $4. P&L = $5 − $4 = $1 profit. Stock is between strike and breakeven — writer is in the profit zone but not at maximum.

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    Example 4: Identifying Intrinsic vs. Time Value Mid-Life

    > Setup: An XYZ $80 put is trading at $9. The stock is currently at $74.

    Intrinsic Value: $80 − $74 = $6 (put is in the money)

    Time Value: $9 − $6 = $3

    > Note: There is still $3 of time premium because expiration has not arrived. An investor deciding whether to exercise early or sell the option in the market should sell — they receive $9 in the market vs. only $6 from exercising now. You never throw away time value by exercising early (for non-dividend situations).

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    Example 5: Exercise vs. Sell Decision

    > Setup: You hold a $100 call that you paid $5 for. The stock is now at $108 and the option has 2 weeks left. The option is trading at $9.50 in the market.

  • Exercise now: You get $108 − $100 = $8 intrinsic value. Your profit = $8 − $5 = $3.
  • Sell the option: You receive $9.50. Your profit = $9.50 − $5 = $4.50.
  • Decision: Sell the option. The option has $1.50 of time value remaining ($9.50 − $8.00 intrinsic). Selling captures both intrinsic and time value. Exercising throws away the time value.

    Rule: Selling the option is almost always better than exercising early when time value remains. The exception is when a large dividend is about to be paid (the dividend goes to the stockholder, not the option holder — so it may make sense to exercise before ex-dividend date to capture the dividend).

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    Spread Math — Bull Call Spread Example

    > Setup: Buy 1 $50 call at $6; sell 1 $55 call at $2. Net debit = $4.

    | Metric | Calculation | Result | |---|---|---| | Net debit (max loss) | $6 − $2 | $4 | | Max gain | ($55 − $50) − $4 | $1 | | Breakeven | $50 + $4 | $54 |

    P&L at stock = $57 (above upper strike):

  • Long $50 call worth: $57 − $50 = $7
  • Short $55 call worth: $57 − $55 = $2 (you owe this)
  • Net option value: $7 − $2 = $5
  • P&L: $5 − $4 cost = $1 profit (maximum — capped at $1)
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    Straddle Math

    > Setup: Buy 1 $70 call at $5; buy 1 $70 put at $4. Total premium = $9.

    | Metric | Calculation | Result | |---|---|---| | Max loss | Total premium at strike | $9 | | Upper breakeven | $70 + $9 | $79 | | Lower breakeven | $70 − $9 | $61 |

    P&L at stock = $83:

  • Call intrinsic: $83 − $70 = $13
  • Put intrinsic: $0
  • Net value: $13
  • P&L: $13 − $9 = $4 profit
  • P&L at stock = $58:

  • Call intrinsic: $0
  • Put intrinsic: $70 − $58 = $12
  • Net value: $12
  • P&L: $12 − $9 = $3 profit
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    Quick Reference Table — All Breakevens

    | Position | Breakeven Formula | |---|---| | Long Call | Strike + Premium | | Short Call | Strike + Premium | | Long Put | Strike − Premium | | Short Put | Strike − Premium | | Long Straddle | Strike + Total Premium (upper); Strike − Total Premium (lower) | | Short Straddle | Same as long straddle | | Bull Call Spread | Lower Strike + Net Debit | | Bear Put Spread | Higher Strike − Net Debit |

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    Key Terms

  • Intrinsic value: Max(exercisable gain, 0); never negative; in-the-money amount
  • Time value: Premium minus intrinsic value; reflects time remaining and volatility
  • Breakeven: Stock price at which the option position neither gains nor loses at expiration
  • At expiration: Time value = $0; option worth intrinsic value only
  • Exercise vs. sell: Selling the option captures time value; early exercise wastes it
  • Net debit: Amount paid in a spread (buy leg costs more than sell leg receives)
  • Net credit: Amount received in a spread (sell leg receives more than buy leg costs)
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    Quiz Questions:

    Q1. An investor buys 1 ABC $45 call for a $3 premium. The stock is at $52 at expiration. What is the investor's profit or loss?

    A) $3 profit B) $4 profit C) $7 profit D) $10 profit

    Answer: B — Intrinsic value at expiration = $52 − $45 = $7. Profit = $7 − $3 premium = $4 per share ($400 per contract). $3 = the premium only (no gain calculation) (A). $7 = intrinsic value before subtracting premium (C). $10 is not derivable from these inputs (D).

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    Q2. A put option on XYZ stock has a $55 strike price and a current premium of $8. The stock is at $49. What are the intrinsic value and time value of this option?

    A) Intrinsic = $8, Time = $0 B) Intrinsic = $0, Time = $8 C) Intrinsic = $6, Time = $2 D) Intrinsic = $4, Time = $4

    Answer: C — The put is in the money (stock $49 < strike $55). Intrinsic = $55 − $49 = $6. Time value = $8 − $6 = $2. If intrinsic were $8, the stock would need to be at $47 (A). OTM options (stock above strike for puts) would have zero intrinsic (B applies when put is OTM, which this is not). $4 intrinsic would require stock at $51, not $49 (D).

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    Q3. An option premium is $11.50 and the option's intrinsic value is $9. What is the time value?

    A) $9.00 B) $11.50 C) $20.50 D) $2.50

    Answer: D — Time Value = Premium − Intrinsic Value = $11.50 − $9.00 = $2.50. Intrinsic alone is $9 (A). Premium alone is $11.50 (B). Adding them together would double-count (C).

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    Q4. An investor holds a $60 put that has 3 weeks until expiration. The stock is at $53 and the put is trading at $8.50 in the market. Should the investor exercise the put or sell it in the market?

    A) Exercise — the put is deep in the money and will lose value rapidly B) Sell the put — exercising now wastes $1.50 in time value C) Exercise — early exercise always locks in the maximum gain D) Sell the put — exercising an in-the-money put is never allowed before expiration

    Answer: B — Intrinsic value = $60 − $53 = $7. The option trades at $8.50, meaning there is still $1.50 in time value ($8.50 − $7.00). Exercising yields only $7 in cash. Selling the option yields $8.50. By selling, the investor captures the extra $1.50 in time value. Exercising early is allowed for American-style options but is suboptimal when time value remains (A). Early exercise is sometimes rational (before a large dividend) but not here (C). Early exercise IS allowed for American-style equity options (D is wrong).

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    Q5. A bear put spread is established: buy 1 $65 put at $7 and sell 1 $60 put at $3. Net debit = $4. At expiration, the stock is at $57. What is the profit or loss?

    A) $4 loss B) $1 profit C) $5 profit D) $7 profit

    Answer: B — At stock = $57:

  • Long $65 put intrinsic: $65 − $57 = $8
  • Short $60 put intrinsic: $60 − $57 = $3 (owe this)
  • Net option value: $8 − $3 = $5
  • P&L: $5 − $4 net debit = $1 profit

This equals the maximum gain for the spread: ($65 − $60) − $4 = $1. Maximum gain is achieved when the stock is at or below the lower strike. $4 loss would occur if both puts expire worthless — stock above $65 (A). $5 = net option value before subtracting premium (C). $7 = long put intrinsic alone, ignoring the sold put and premium (D).