Call option: right (not obligation) to buy the underlying at strike price K by expiration.
Put option: right (not obligation) to sell the underlying at strike price K by expiration.
Buyer’s loss is limited (premium paid). Seller (writer) can face large losses.
Minimum standard for this chapter: you must be able to explain the difference between calls and puts and interpret strike price K, premium, and expiration.
2) Payoffs at expiration (must be able to draw) ▾
At expiration (T), with underlying price S_T and strike K:
Call payoff (long): max(S_T − K, 0)
Put payoff (long): max(K − S_T, 0)
Profit = payoff − premium (ignore fees for class unless stated)
Graph rule: Payoff graphs use S_T on the x-axis and payoff/profit on the y-axis. Mark the strike K.
Generates income (premium) but caps upside above strike.
Straddle
Long call + long put (same K, same T)
Bet on volatility: profit if big move up or down; loses if price stays near K.
For strategy questions, you will often be asked to (i) draw the payoff, and (ii) interpret who should use it and why.
4) Binomial model: one-period case study (class example) ▾
Case (1-period):
S0 = 40
Possible prices at T=1 year: Su = 50, Sd = 30
Strike K = 35
Risk-free rate r = 8%
(Assume European call for this example.)
Step A — Up/Down factors:
u = Su / S0 = 50/40 = 1.25
d = Sd / S0 = 30/40 = 0.75
Step B — Option payoffs at expiration:
Vu = max(Su − K, 0) = max(50 − 35, 0) = 15
Vd = max(Sd − K, 0) = max(30 − 35, 0) = 0
Step C — Risk-neutral probability (1-step):
p = (e^(rT) − d) / (u − d)
= (e^(0.08*1) − 0.75) / (1.25 − 0.75)
≈ (1.083287 − 0.75) / 0.50
≈ 0.6666
Step D — Option value today:
C0 = [p*Vu + (1 − p)*Vd] / e^(rT)
= [0.6666*15 + 0.3334*0] / 1.083287
≈ 9.17
Binomial logic: price the option by discounting its risk-neutral expected payoff.
(In multi-step trees, you work backward node-by-node.)
5) Binomial hedging interpretation (the “replicating portfolio” idea) ▾
Replication concept (1-step):
Choose Δ shares of stock and borrow/lend B so that:
Δ*Su + B*(1+r) = Vu
Δ*Sd + B*(1+r) = Vd
Solve for Δ and B, then:
Option value today = Δ*S0 + B
Interpretation:
A properly chosen stock + bond portfolio replicates the option payoff,
so by no-arbitrage it must have the same price.
6) Black–Scholes–Merton (BSM) overview ▾
BSM prices (European) options using inputs: S, K, r, T, σ (volatility), and dividend yield (if applicable).
It is a continuous-time model (contrast with discrete-time binomial trees).
In practice: binomial is flexible (American options, early exercise logic); BSM is fast and widely used for European-style benchmarks.
For exams: know what each input means, what volatility does to option values, and how BSM differs from binomial.
Chapter 19 resources
Core materials ▾
Chapter 19 PPT (upload/link here)
Self-produced video 1 (add link)
Self-produced video 2 (add link)
Self-produced video 3 (add link)
Games (practice) ▾
Game 1 — Call and Put Options (Basic): learn to draw payoff graphs
Practical workflow: compute with a calculator, then confirm you can explain the payoff and the sensitivity (especially volatility and time).
Binomial demonstrations ▾
Binomial calculation demonstration (add link)
Binomial model demonstration (add link)
Black–Scholes model demonstration (add link)
Case study (Final exam)
Due with final ▾
Chapter 19 Case Study Part I — due with final
Chapter 19 Case Study Part II — due with final
In-class case video Part I (4/9/2024) — Black–Scholes–Merton model
In-class case video Part II (4/11/2024) — Binomial model
Expectation: You must be able to (i) compute a one-step binomial price, (ii) explain risk-neutral pricing,
and (iii) interpret strategy payoffs (protective put, covered call, straddle).
Disclaimer: Educational content for jufinance.com. Not investment, legal, or tax advice.