# Administrative Information - **Class Meeting**: Wednesday, October 28, 2026, 1:00–1:50pm — **Coolbaugh 212** - Previous meeting: [[MATH310F26(Day 25) - Cook the diffusivity, then price a call]] - 📄 [[Teaching/MATH310/MATH310F26/Administration/MATH310_F26_Syllabus_1.0.pdf|MATH 310 F26 Syllabus, v1.0]] > [!info]- Admin. Notes > Expect an LR and WB for each lecture. WBs target 15-minute engagements (longer early on); LRs are short, ~5 minutes. > > - 🎯 **[LR] [Day 23](https://elearning.mines.edu/courses/81632/quizzes/121971)** · **[Day 24](https://elearning.mines.edu/courses/81632/quizzes/122140)** · **[Day 25](https://elearning.mines.edu/courses/81632/quizzes/122146)** reflection questions — all due **Friday 10/30, 11:59pm** (statements: [Day 23](https://elearning.mines.edu/courses/81632/quizzes/121972), [Day 24](https://elearning.mines.edu/courses/81632/quizzes/122141), [Day 25](https://elearning.mines.edu/courses/81632/quizzes/122147)) > - 🎯 **[LR] [Day 26 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/122152)** — 4 questions drawn from today's material, unlimited attempts, highest kept — due **Friday 10/30, 11:59pm** > - 🎯 **[LR] [Day 26 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/122153)** — one free response, no wrong answers — due **Friday 10/30, 11:59pm** > - 🎲 **[CA] [Class Activity — Pricing a call option](https://elearning.mines.edu/courses/81632/assignments/591731)**: [[MATH310F26_OptionMarket_Handout.pdf|the handout]] — parts 1–5 are the activity; **don't worry about part 6** (Scott will talk it through) — upload — due **Monday 11/2, 10:00am** > - 📝 **No MA/MP this week.** [MP3](https://elearning.mines.edu/courses/81632/assignments/590872) / [MA2](https://elearning.mines.edu/courses/81632/assignments/590873) were due Monday 10/26, with no follow-ups this Friday; those already in get **feedback on Friday** > - 🎯 **[WB] [Workbook 11](https://elearning.mines.edu/courses/81632/assignments/588368)** · [[MATH310F26_WB11.pdf|blank]] · [[MATH310F26(Workbook 11 Companion) - Three outcomes, two moments|companion]] — Canvas says Friday 10/30; no collections this week. A workbook that runs the option numbers once more is on the way. > - 🍬 The Dum Dums stayed home; Starburst instead > - 🎬 [[MATH310F26 - Random numbers, two videos]] · 🖼️ [[MATH310F26 - Convolution, in pictures]] --- # Deliverables - 🎯 **[LR] [Day 26 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/122152)** — 4 questions drawn from today's material, unlimited attempts, highest kept — due **Friday 10/30, 11:59pm** - 🎯 **[LR] [Day 26 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/122153)** — one free response, no wrong answers — due **Friday 10/30, 11:59pm** - 🎲 **[CA] [Class Activity — Pricing a call option](https://elearning.mines.edu/courses/81632/assignments/591731)** — parts 1–5, then upload — due **Monday 11/2, 10:00am** - 🎯 Day 23, Day 24, and Day 25 reflection pairs — all due **Friday 10/30, 11:59pm** --- # Lecture Boards + Transcript + GenAI **Twenty-sixth meeting, October 28 — call option pricing, finalized.** Board header: *26 | 10/28/26, Call option pricing (finalization), …, discrete time deterministic dynamics.* "It's the 26th lecture on the 28th day of the month." The plan was to finish Monday's handout and then turn to discrete-time deterministic dynamics, "in some ways a pretty chill topic — but of course, the longer you sit with any topic, the less chill it gets." The handout took the hour. It ended where it was meant to end, on the instructor making twenty cents off a desk that priced the call too low, whichever way the die came up, and on a cheeseburger. ## An exam story, and housekeeping A transition by anecdote. In a graduate exam, the instructor was writing as fast as he could the whole time, while the very smart student beside him wrote at a leisurely pace and looked out the window. Walking out: "That was kind of rough." "Yeah, it's all right." "Question four was really throwing me for a loop." And he watched the blood drain from his classmate's face: the exam was one sheet, two sides, and the classmate hadn't known there was a second side. The lesson: after an exam, walk out, enjoy the moment, and don't talk about it with anyone. (A campus cousin of the story: a test printed in the wrong order, so that some copies had every other page missing, with the clock running.) Housekeeping. **No MA/MP this week.** Technically the due date was Monday, but the idea was that there'd be no follow-ups on Friday. Some came in last week, some on Monday, and some are still coming, because "as the chatter tells us, we're in the midterm pressure season." Good luck on the rest of the tests. The MA/MPs already in get **feedback on Friday**. The Dum Dums were forgotten on the way out of the house, but the [Starburst](https://en.wikipedia.org/wiki/Starburst_(confectionery)) came (one orange held back for the instructor). Turnout was a good deal better than Monday's, despite this being hump-day Wednesday, after a scientific-computing exam. ## Why spend a lecture on money Two reasons. First, the language of this material is hard to get hold of, "and the way I can build memories that way is by having these conversations. So I definitely appreciate you enduring this conversation, because it's the first time I'm having to talk about contexts like these." Second, the **continuum limit** of what's on the board, the same shrinking of steps that took the walkers to the diffusion equation, yields the **[Black–Scholes–Merton equation](https://en.wikipedia.org/wiki/Black–Scholes_model)**. The work behind it won the 1997 [Nobel Memorial Prize in Economic Sciences](https://en.wikipedia.org/wiki/Nobel_Memorial_Prize_in_Economic_Sciences), which went to [Myron Scholes](https://en.wikipedia.org/wiki/Myron_Scholes) and [Robert C. Merton](https://en.wikipedia.org/wiki/Robert_C._Merton). [Fischer Black](https://en.wikipedia.org/wiki/Fischer_Black) had died in 1995, and the prize is not awarded posthumously, which is what the instructor half-remembered aloud. "That's pretty modern," and it was awarded when the instructor was "about as old as you are now," which he said made him feel awful. The point is to give the mathematics as many modern contexts as possible. ## Recall: you, the trader; me, the buyer The aim was to get, diagrammatically, to the point where we decide what price should be offered for this option, imagining ourselves inside the construct. There are two parties. **You**, the trader (Monday, everyone forecast a probability, and today builds on that). **Me**, the buyer of the option, who would like to purchase a **[call option](https://en.wikipedia.org/wiki/Call_option)** at [strike](https://en.wikipedia.org/wiki/Strike_price) $K=\$101$. The call is the **right to buy at maturity**, but not the obligation: if the stock went up past the strike, I'd buy; "if the stock went down, you just take my money. I'm not buying anything; I'm not losing more money when the stock goes down." And "if you see me as some sort of rube," the question on your side is what to price the option at, so as to take advantage of me. The structure is the one-step tree: $S_{0}=\$100$, up to $S_{1}=\$102$ with probability $P_{u}$, down to $S_{1}=\$98$ with probability $P_{d}$. The future prices could be judged from the volatility in the market. What's really unknown is the probability mass, and *you were asked to forecast it*. What the analysis demonstrates is that **you don't need to know these probabilities to make a play on this deal**. The forecasts were there so that each desk would see its own outcome. The value of the option at time one is $V_{1}=(S_{1}-K)^{+}=\max(S_{1}-K,\,0),$ the [positive part](https://en.wikipedia.org/wiki/Positive_and_negative_parts) of the stock price minus the strike. Up, $V_{u}=(S_{u}-K)^{+}=(102-101)^{+}=1$, "numero uno." Down, $98-101=-3$, whose positive part is $V_{d}=0$. Andrew's question, partly inaudible on the recording, concerned a calculation of his own. The answer was that what comes next "puts us in a situation where we're neutral," and that his calculation is embedded in it, so keep it in the zeitgeist and compare. ## A price to quote, and the many quotes Still, for a deal between you and me, you have to set a price, $Q$. The die gives probabilities, so it makes sense to quote the [expected value](https://en.wikipedia.org/wiki/Expected_value) at the next moment, built "like when we started [Plinko](https://en.wikipedia.org/wiki/Plinko)": probability up times the outcome up, plus probability down times the outcome down, $Q=\mathbb{E}[V_{1}]=P_{u}V_{u}+P_{d}V_{d}=P_{u},$ since in this "cooked problem" $V_{d}=0$ and $V_{u}=1$. But **you don't know $P_{u}$, so you must guess**, "or in fancier terms, forecast." Then a red **Idea/Point**, from walking the dog that morning (the [American Kennel Club](https://en.wikipedia.org/wiki/American_Kennel_Club)'s breed of the day was the [beagle](https://en.wikipedia.org/wiki/Beagle), and he's "pretty beagley"). Part of what makes this hard, beyond the language, is that with the random walkers we were omniscient, looking down on the walk from on high "like, oh look at those things jiggle around." Now the mathematical structure sits underneath, and two parties are writing contracts on top of it. That is another layer. If I'm the buyer and all of you quote from your own $P_{u}$, **I get many quotes**: on a ten-sided die, $0.1, 0.2, 0.3, \dots, 1$. Whenever numbers rather than letters are needed, the instructor **plays the desk that forecast $0.3$** (in blue on the board). ## The hedge, and Andrew's question Next comes the move that belongs to the [binomial tree option pricing model](https://en.wikipedia.org/wiki/Binomial_options_pricing_model). **Introduce a degree of freedom by holding additional stocks, so that the position is the same whether the walk steps up or down.** The trader's position gets the letter $\Pi$, "quoted with a pi because probabilities are also in play." It is evaluated *after* the roll, tomorrow: $\Pi=\Delta S_{1}-V_{1},$ however many shares you hold at tomorrow's price, minus what you owe on the contract. Demand $\Pi_{u}=\Delta S_{u}-V_{u}\;=\;\Delta S_{d}-V_{d}=\Pi_{d}\quad\Longrightarrow\quad \Delta=\frac{V_{u}-V_{d}}{S_{u}-S_{d}}=\frac{1-0}{102-98}=\frac14 .$ The $S s are known tomorrow, the $V s are built into the contract, so the one unknown is $\Delta$. In the moment, the instructor said this number depends on the $P_{u}$ we set. Andrew asked where the probability shows up in the $\Delta$ equation, and the answer, checked against the notes, is that **it doesn't**. The forecast $Q=0.3=P_{u}$ written beside it on the board is the desk's price, and it gets used next, not in $\Delta$. (Also noted on the board: Monday's roll came up $S_{u}$.) With $\Delta=\tfrac14$, the position is forced to agree in both states: $\Pi=\Pi_{\uparrow}=\Pi_{\downarrow}=\$24.50 .$ ## Borrow for the Δ-neutral position The juicy part, where the cheeseburgers come in. Back to the me-you game: I come up to you "with sucker written all over my face," wanting a call at 101. You'll sell it, but you need the shares on hand, and you don't want to put up your own money. So **borrow** for this [delta-neutral](https://en.wikipedia.org/wiki/Delta_neutral) position, from a bank (a nice one, at zero interest) or from your own coffers. The purchases happen at time zero, at $S_{0}$, and "who's being who at every moment is also a tricky part." You amass $\Delta$ shares on day zero, and I hand you $Q$: $b=\Delta S_{0}-Q=\tfrac14\cdot 100-Q,\qquad\text{with } \Pi \text{ the position either way.}$ "Out of all the abstraction, this is the number that should be the most tangible": money owed to the universe. What I really care about is how much money I make, and knowing the liabilities lets me construct the profit. (While the boards were photographed: an anecdote about a long-retired acquaintance who once factored a seventh-degree polynomial on the fly at the board. Years later the instructor realized how: an integer root must divide the constant term, the [rational root theorem](https://en.wikipedia.org/wiki/Rational_root_theorem) in its integer form. The man held bachelor's and master's degrees in both mathematics and physics, and took his PhD in physics, because he didn't think he could teach himself the physics but could teach himself the math. The instructor agrees. Math books lay every step out, while physics books leave steps out, "because they already knew the answers." Another acquaintance had two PhDs, one in each.) ## Profit, and twenty cents either way The writer's profit: $\text{Profit}=\$24.50-b=\$24.50-(\Delta S_{0}-Q)=24.50-25+Q=Q-0.5 .$ **Analysis.** Say $Q<0.5$; take the desk's $Q=0.3$. Writing his own name on the board ("feels a little weird"), **Scott does the following.** He goes to a bank: "hey, give me $\Delta$ many shares." ("Scott, you seem like a nice guy.") He sells the quarter share immediately, on day zero, for \$25. He pays you the premium, which leaves $\Delta S_{0}-Q=25-0.30=\$24.70$ in his pocket. (Mid-arithmetic, a complaint: typing a life's worth of dollar signs has made the dollar symbol annoying.) Then the roll: - **Up.** The stock goes to 102. "Yo, pay up": you owe him \$1, so the pocket holds \$25.70. But the shares were lent, "and if I keep all this money, eventually somebody will come for my knees." So he buys back $\Delta$ shares, *after* the roll, at $S_{u}$: $\tfrac14\cdot 102=\$25.50$. What's left is **20¢**. "Feels like the end of *Clue*." - **Down.** The stock goes to 98. He won't exercise a right to buy at 101 something worth 98, so he gets \$0 and walks away from the bad deal. Buying back $\Delta$ at $S_{d}$ costs $\tfrac14\cdot 98=\$24.50$ (a calculator was called for, and a student confirmed). He still holds \$24.70, so there is **20¢** to him. "What we're learning today is don't let Scott hold your money." So because you forecast low, the instructor can borrow (or [short](https://en.wikipedia.org/wiki/Short_(finance))) shares, even from another desk in the room, since "there was nothing magical about these calculations, both parties know them." He builds his own stockpile to hedge, holds you to your contract, and **no matter what happens, makes twenty cents off the deal**. His $0.20$ is exactly the $0.5-Q$ that your hedged desk would lose. "This is all pretty complicated." For the class activity, **don't worry about part 6**; it'll be talked through and is no big deal. A workbook that runs through the numbers once more is coming. Then the word Monday promised: **[arbitrage](https://en.wikipedia.org/wiki/Arbitrage)**, which has "shown up in my life a number of times." Here, a deal in which you contractually obligate yourself to me unfavorably, and either way it goes, I make twenty cents. The meme from the summer, at the [Barcelona airport](https://en.wikipedia.org/wiki/Josep_Tarradellas_Barcelona–El_Prat_Airport): cheeseburgers at 1.70 and double cheeseburgers at 5.20. "There should be a way for me to buy two cheeseburgers, assemble a double cheeseburger just over on the side, wrap it back up, and then sell it to you, because the only other place you can get it is for 5.20." It's a profit with no risk to me, and that is the notion of arbitrage. Discrete-time deterministic dynamics didn't get started; it's Friday's. > [!example]- Every premium but one hands someone free money > ![[day26_twenty_cents.png]] > Left: riskless profit against the premium quoted, the same whichever way the die rolls. Blue is the buyer's trade from the board: buy the call at $Q$, borrow and sell $\tfrac14$ share, buy it back after the roll, for a profit of $0.5-Q$. Orange is the hedged writer's from Monday, $Q-0.5$. Dots are the ten quotes a ten-sided die allows. At the desk's $Q=0.30$, twenty cents goes to the buyer. Only at $Q=0.50$ does nobody get free money, and that is the sense in which the call's price "can be nothing else" (handout part 5). Right: the instructor's cash, step by step, at $Q=0.30$. He uses no money of his own and ends with $+0.20$ up or down. > [!note]- Where the handout goes from here (parts 5–6) > The instructor's trade is the **buyer's side** of part 5 run in reverse. The handout's version builds the call instead of buying it: hold $\Delta=\tfrac14$ share and borrow \$24.50. That costs $25-24.50=\$0.50$ today and pays \$1 if up ($25.50-24.50$) and \$0 if down ($24.50-24.50$), exactly what the call pays. It is a **[replicating portfolio](https://en.wikipedia.org/wiki/Replicating_portfolio)**. Any premium below $0.50$ lets a buyer collect $0.5-Q$ (today's twenty cents). Any premium above lets a writer collect $Q-0.5$ by hedging. The price that allows neither is $V_{0}=\tfrac12$, whatever the die's real faces are. Part 6, set aside today, asks which probability of up makes the expected payoff equal $V_{0}$. The answer is $q=\tfrac12$, the **[risk-neutral](https://en.wikipedia.org/wiki/Risk-neutral_measure) probability**, under which $\mathbb{E}[S_{1}]=\tfrac12\cdot102+\tfrac12\cdot 98=100=S_{0}$: the stock is a fair game. The cheeseburger is the [law of one price](https://en.wikipedia.org/wiki/Law_of_one_price) in a paper wrapper. Two singles and a double should cost the same, or someone assembles burgers for a living. (Strictly, the deal is an arbitrage only if someone will actually buy the assembled double for less than 5.20 and more than 3.40.) Shrink the step and the time like the walkers, and the same no-arbitrage argument becomes the Black–Scholes–Merton partial differential equation. > [!success]- Numbers checked in one sandbox run > In exact arithmetic: $V_{u}=(102-101)^{+}=1$, $V_{d}=(98-101)^{+}=0$, and $\Delta=(1-0)/(102-98)=\tfrac14$, with **no $P_{u}$ anywhere in it** (Andrew's question). $\Pi_{u}=\tfrac14\cdot102-1=\tfrac{49}{2}$ and $\Pi_{d}=\tfrac14\cdot 98-0=\tfrac{49}{2}$, so \$24.50 both ways. The writer's profit, $\Pi-(\Delta S_{0}-Q)=Q-\tfrac12$, is $-0.20$ at $Q=0.30$, as on the board. The instructor's trade at $Q=0.30$: $25-0.30=24.70$ in pocket. Up: $24.70+1=25.70$, buy back $25.50$, $+0.20$. Down: $24.70+0=24.70$, buy back $24.50$, $+0.20$. ($98/4=24.5$, the calculator moment, confirmed.) Run over all eleven quotes $Q=0,0.1,\dots,1$, the buyer's up-profit equals the down-profit equals $\tfrac12-Q$, the exact negative of the hedged writer's, so the two sides always sum to zero and only $Q=\tfrac12$ is free of riskless profit. The handout's footnote ($Q=0.40$, buyer makes $0.10$) checks. Part 5's replicating portfolio costs $25-24.50=0.50$. Part 6's risk-neutral $q=(100-98)/(102-98)=\tfrac12$ gives $\mathbb{E}_{q}[S_{1}]=100$ and $\mathbb{E}_{q}[V_{1}]=\tfrac12$. Nobel facts: the 1997 economics prize went to Scholes and Merton, Black died on August 30, 1995, and the prize requires laureates to be living at the October announcement. > [!example]- Board scans — October 28, 2026 > ![[MATH310F26-Day26-Board-1.jpg]] > ![[MATH310F26-Day26-Board-2.jpg]] > ![[MATH310F26-Day26-Board-3.jpg]] > ![[MATH310F26-Day26-Board-4.jpg]] > > 📄 [[MATH310F26-Day26-Boards-2026-10-28.pdf|Full board capture (4 pages, PDF)]]