Options Final • Calculation Study Guide

Options Calculation Study Guide

Options only • no dividend calculations • no M&A calculations

Focus only on the calculation part for options

This page is for the options calculation section. It focuses on long call, long put, short call, short put, one-step binomial pricing, and plug-in Black-Scholes.

Graph part: be ready to draw the long call payoff graph and the long put payoff graph. You do not need short-call or short-put payoff graphs, and you do not need long-call or long-put profit graphs.

T/F section50 questions total
60 points
Calculation section8 questions total
40 points
Open book / notesThe calculation part is open book and open notes.

Expected calculation structure

1
4 payoff / profit questionsLong call, long put, short call, short put. Use payoff first, then connect payoff and premium to profit.
2
1 simple one-step binomial questionUse stock up/down states, strike, and risk-free rate to find the option value today.
3
1 Black-Scholes plug-in questionNo derivation. Just identify inputs and plug them into the formula.
4
2 graph questionsDraw the payoff graph for a long call and the payoff graph for a long put.

Exam format for the final

Important: The T/F part has 50 questions worth 60 points. The calculation part has 8 questions worth 40 points. The calculation part is open book and open notes.

Question mix

  • 6 calculation questions
  • 2 graph-drawing questions
  • 8 questions total in the calculation section
  • 40 points total for calculation part

Options topics included

  • Long call
  • Long put
  • Short call
  • Short put
  • One-step binomial
  • Black-Scholes plug-in

Not needed here

  • No dividend calculations
  • No M&A calculations
  • No option combinations / spreads
  • No short-call or short-put graphs
  • No long-call or long-put profit graphs

Key formulas to know

The biggest idea is: payoff is the value at expiration. Profit adjusts payoff for the premium.

Long Call payoff = max(S_T - X, 0) Long Call profit = max(S_T - X, 0) - Premium Long Put payoff = max(X - S_T, 0) Long Put profit = max(X - S_T, 0) - Premium Short Call payoff = -max(S_T - X, 0) Short Call profit = Premium - max(S_T - X, 0) Short Put payoff = -max(X - S_T, 0) Short Put profit = Premium - max(X - S_T, 0)

Simple memory rule

  • Long position: profit = payoff - premium paid
  • Short position: profit = premium received - intrinsic value paid later
  • Call: right to buy
  • Put: right to sell

Symbols

  • ST = stock price at expiration
  • X = strike price
  • Premium = option price paid or received
  • Intrinsic value = the exercise value at expiration

Graphs you need to draw

Graph requirement: only be ready to draw the long call payoff graph and the long put payoff graph. Do not spend time memorizing short-call or short-put graphs for the drawing part. Also, no profit graphs are required for drawing.
Long Call Payoff
X S_T Payoff slope = +1 after X

Starts flat at zero until the strike price, then rises upward.

Long Put Payoff
X S_T Payoff slope = -1 before X

Starts high when the stock price is very low, then declines to zero at the strike and stays flat after that.

Payoff and profit practice

The most important relationship is shown below. For long positions, profit equals payoff minus premium. For short positions, premium is received up front, so profit equals premium minus the exercise loss later.

Long Call
Payoff − Premium
profit = max(ST − X, 0) − premium
Long Put
Payoff − Premium
profit = max(X − ST, 0) − premium
Short Call
Premium − Exercise Loss
profit = premium − max(ST − X, 0)
Short Put
Premium − Exercise Loss
profit = premium − max(X − ST, 0)

Quick calculator

Choose a position and click calculate.

Worked-example reminder

Position Rule Example idea
Long Call Benefit when ST > X If X = 50, premium = 4, ST = 60, payoff = 10, profit = 6
Long Put Benefit when ST < X If X = 50, premium = 3, ST = 40, payoff = 10, profit = 7
Short Call Keep premium if stock does not rise above X If X = 50, premium = 4, ST = 60, profit = 4 − 10 = −6
Short Put Keep premium if stock does not fall below X If X = 50, premium = 3, ST = 40, profit = 3 − 10 = −7

Simple one-step binomial model

Keep it simple: one-step binomial means the stock has only two possible future prices: an up-state price and a down-state price. Then discount the expected option payoff using the risk-neutral probability.
For a one-step call option: C_u = max(S_u - X, 0) C_d = max(S_d - X, 0) p = ((1 + r)S_0 - S_d) / (S_u - S_d) C_0 = [pC_u + (1 - p)C_d] / (1 + r)

Worked example

Given: S0 = 40, Su = 50, Sd = 30, X = 35, r = 8%

  • Cu = max(50 − 35, 0) = 15
  • Cd = max(30 − 35, 0) = 0
  • p = ((1.08)(40) − 30) / (50 − 30) = 13.2 / 20 = 0.66
  • C0 = [0.66(15) + 0.34(0)] / 1.08 = 9.17

Try your own numbers

Enter values and calculate.

Black-Scholes: plug in the numbers

For this study guide: treat Black-Scholes as a plug-in formula. You should know what each symbol means and how to substitute the values. You do not need a long derivation here.
European Call: C = S_0 N(d_1) - X e^(-rT) N(d_2) European Put : P = X e^(-rT) N(-d_2) - S_0 N(-d_1) d_1 = [ln(S_0 / X) + (r + sigma^2 / 2)T] / [sigma sqrt(T)] d_2 = d_1 - sigma sqrt(T)

What to identify quickly

  • S0 = current stock price
  • X = strike price
  • r = risk-free rate
  • T = time to expiration in years
  • σ = volatility
  • N(d) = cumulative standard normal probability
Simple plug-in steps
  1. Write down S0, X, r, T, and σ.
  2. Compute d1.
  3. Compute d2.
  4. Find N(d1) and N(d2).
  5. Plug into the call or put formula.

Plug-in calculator

Enter values and calculate.

8 question types to prepare for

# Likely question type Main skill Notes
1Long call payoff and profitKnow payoff first, then subtract premiumMay also ask for break-even idea
2Long put payoff and profitKnow payoff first, then subtract premiumGood when stock falls below strike
3Short call payoff and profitPremium received minus exercise lossWatch sign carefully
4Short put payoff and profitPremium received minus exercise lossWatch sign carefully
5One-step binomialFind Cu, Cd, p, and C0Keep it very mechanical
6Black-Scholes plug-inIdentify inputs and substitute correctlyNo big derivation needed
7Draw long call payoff graphFlat until strike, then risingPayoff graph only
8Draw long put payoff graphDeclining to strike, then flat at zeroPayoff graph only
Point structure reminder: these 8 questions together are worth 40 points. The T/F part has 50 questions worth 60 points.

Major confusion areas

Common mistakes

  • Confusing payoff with profit.
  • Using the call formula when the problem is a put.
  • Forgetting that short positions receive the premium first.
  • Forgetting to use max( , 0).
  • Drawing a profit graph when the question asks for a payoff graph.
  • Mixing up current stock price S0 with expiration stock price ST.

Fast checklist before the exam

  • Can you write all four payoff and profit formulas from memory?
  • Can you explain why profit = payoff − premium for long options?
  • Can you explain why short-option profit starts with the premium received?
  • Can you do one simple binomial problem step by step?
  • Can you plug numbers into Black-Scholes without panicking?
  • Can you draw the long call and long put payoff graphs neatly?
Built for FIN435 final review. This page focuses only on the options calculation section: basic call/put payoff and profit, one-step binomial, and plug-in Black-Scholes.