# Definitions and Math Formulas

Reference doc for the Summer 2026 Project Lab (Fama-Miller funded). Based on the
proposal and supplemental info docs. This repository implements the CAISO slice of
the program.

A typeset PDF of this reference is available: {download}`formulas.pdf <formulas.pdf>`

## Workstream 1: Spreads

Collecting and computing price relationships across ISOs. These are the basic objects.

**ISO (Independent System Operator).** The entity that runs the grid and the wholesale electricity market for a region. Our seven: ERCOT (Texas), CAISO (California), PJM (Mid-Atlantic), NYISO (New York), MISO (Midwest), SPP (Great Plains), ISONE (New England).

**DAM (Day-Ahead Market).** Auction held the day before delivery. Generators and buyers commit to schedules; produces one price per hour per location.

**RT (Real-Time Market).** The balancing market. Every 5 minutes the ISO reprices electricity based on actual conditions. Much spikier than DAM because it absorbs forecast errors, outages, and weather surprises in real time.

**Nodal Prices (LMPs).** Prices are set at thousands of individual grid locations (nodes), not one region-wide price. Two nodes differ because of transmission congestion and line losses between them.

**DAM-RT Spread.** The financial difference between the hourly Day-Ahead Market (DAM) price and the volume or time-weighted average of Real-Time Market (RTM) prices for the corresponding hour and electrical node.

**Location Spread.** Price at node A minus price at node B. Pure congestion and losses.

**Time Spread.** The price difference between two distinct blocks of time within the same operating day at the same electrical node.

**TBx (Top-Bottom x).** You take the x most expensive hours of the day, average them out, and subtract the average of the x cheapest hours at that specific spot on the grid. This represents the absolute maximum cash a storage could theoretically make in a day if it has a crystal ball.

## Workstream 2: Quantos


**HDD / CDD (Heating / Cooling Degree Days).** Daily weather indices against a 65°F base:

- $\text{HDD} = \max(0,\ 65 - T)$: how cold it was, heat (gas) demand
- $\text{CDD} = \max(0,\ T - 65)$: how hot it was, A/C (electricity) demand

Contracts settle on the cumulative sum over a month or season. Note the $\max(0,\cdot)$ structure: each daily value is already an option-style payoff on temperature.

**Heat rate.** A gas generator's efficiency: MMBtu of gas burned per MWh of electricity produced. Lower is better. Physical floor is 3.412 (perfect conversion). A generator's marginal cost is heat rate × gas price.

**Location pairs and the Hub Mapping.** Spark-spread objects are indexed by a market pair $(z, g)$: a power hub $z$ and the gas hub $g$ whose gas a generator at $z$ actually burns. The baseline mapping $g^*(z)$, with the CAISO fuel-region series that carries each gas leg:

| Power hub $z$ | Gas leg $g^*(z)$ | Fuel region series |
|---------------|------------------|--------------------|
| NP15          | PG&E Citygate    | `FRPGE1`           |
| ZP26          | PG&E Citygate    | `FRPGE1`           |
| SP15          | SoCal Citygate   | `FRSCE1`           |
| PACE          | PacifiCorp East  | `FRPACE`           |
| PACW          | PacifiCorp West  | `FRPACW`           |

Shorthand: $S^{(z)}(H) := S^{(z,\,g^*(z))}(H)$. ZP26 sits between gas systems and has no unique hub: PG&E Citygate is the baseline, and Kern River / SoCal Citygate are the admissible alternates for sensitivity runs.

**The gas leg is daily.** Gas is nominated once per gas day, and each CAISO fuel-region price is a daily index carried forward across the day's hours.

**MIHR (Market Implied Heat Rate).** Power price divided by gas price:

$$\text{MIHR}^{(z,g)}_{h,d} = \frac{P^{(z)}_{E,h}}{P^{(g)}_{G,d}}$$

Units work out to MMBtu/MWh, a heat rate. Interpretation: the breakeven efficiency line. Any generator with heat rate below the MIHR is profitable to run right now (undefined when the gas print is non-positive).

**Spark Spread.** The gross margin of a gas generator per MWh, at strike heat rate $H$:

$$S^{(z,g)}_{h,d}(H) = P^{(z)}_{E,h} - H\,P^{(g)}_{G,d} = P^{(g)}_{G,d}\,\big(\text{MIHR}^{(z,g)}_{h,d} - H\big)$$

Same information as MIHR, in dollar form instead of ratio form. A gas plant is economically a real option on the spark spread: it runs when positive, idles when negative. Daily aggregation touches only the power leg (the gas leg is already daily):

$$\bar{S}^{(z,g)}_d(H) = \bar{P}^{(z)}_{E,d} - H\,P^{(g)}_{G,d}, \qquad \bar{P}^{(z)}_{E,d} = \frac{1}{n_d}\sum_{h \in d} P^{(z)}_{E,h}$$

with $n_d$ the number of intervals in Pacific trade day $d$ (23-25 at hourly granularity on DST days).

**Gas basis.** How expensive gas is at hub $g$ relative to the defined reference hub:

$$B^{(g)}_d = P^{(g)}_{G,d} - P^{(\text{PG\&E CG})}_{G,d}$$

Convention: all bases quote relative to PG&E Citygate. The anchor only shifts where zero sits: basis differences between hubs are anchor-invariant, and with the $g^*$ mapping in place the anchor never enters any spread or option value.

**HRCO (Heat Rate Call Option).** The core instrument. Payoff per settlement interval:

$$\text{Payoff} = \max\{0,\ P_E - \text{HR}_{\text{strike}} \times P_G - \text{VOM}\} \times \text{Volume}$$

Equivalently (at zero VOM), a call option on the MIHR itself:

$$\text{Payoff} = P_G \times \max\{0,\ \text{MIHR} - \text{HR}_{\text{strike}}\}$$

VOM is a variable O&M adder (dollars per MWh of non-fuel running cost); the stylized version sets it to 0. The cost is zero for simplicity.

**Profitability and Dispatch.** $\text{ITM}(H) = \Pr[S(H) > 0]$ is the share of intervals a heat-rate-$H$ unit is profitable - it says nothing about how much. A unit forced to run every interval earns the average spread; one that runs only profitable intervals earns the average positive part:

$$\underbrace{E[S(H)]}_{\text{always-on}} \ \leq\ \underbrace{E[\max(0,\ S(H))]}_{\text{perfect dispatch}}$$

**Settlement Frequency Defines the Instrument (Jensen).** $V_\bullet(H)$ is the expected payoff per MWh with $\max(0,\cdot)$ applied at the settlement frequency named in the subscript:

$$V_{\text{hourly}}(H) = E_{h,d}\big[\max(0,\ S_{h,d}(H) - \text{VOM})\big], \qquad V_{\text{daily}}(H) = E_d\big[\max(0,\ \bar{S}_d(H) - \text{VOM})\big]$$

Rule: apply the payoff at the contract's settlement frequency, never at the data's observation frequency. Because $\max(0,\cdot)$ is convex, Jensen's inequality orders the conventions (within one price series; DAM and RT are different prices, not averages of each other):

$$V_{\text{5min}} \geq V_{\text{15min}} \geq V_{\text{hourly}} \geq V_{\text{daily}} \geq V_{\text{monthly}}$$

with equality only when the spread holds one sign inside the averaging window. A physical unit (start costs, minimum up/down times) sits between the brackets, $V_{\text{daily}} \leq V_{\text{unit}} \leq V_{\text{hourly}}$

**Intraday optionality premium.** The headline object:

$$\Pi(H) = V_{\text{hourly}}(H) - V_{\text{daily}}(H) \geq 0$$

The value created purely by the ability to pick profitable hours within the day. With hourly power against daily-indexed gas, $\Pi$ isolates the value of intraday power shape (duck curve, evening ramp) uncontaminated by gas noise.

**Margrabe**: Both legs (Power and Gas) are stochastic; they are constantly changing. The pedagogical pricing model is Margrabe, where the key input is the spread volatility:

$$\sigma^2 = \sigma_E^2 + \sigma_G^2 - 2\rho\,\sigma_E\,\sigma_G$$



**Quanto Option.** Imagine this as a protection contract. It does not just protect you from high prices; it activates when two conditions are met: extreme weather occurs and market prices spike. The pricing consequence: 

$$E[XY] = E[X]E[Y] + \text{Cov}(X,Y)$$

**Cost of carry / the Zero-Carry Assumption.** No-arbitrage forward pricing says

$$F = S \cdot e^{(r\ +\ \text{storage}\ -\ \text{convenience yield})\,T}$$