> [!abstract] Two Worlds, One Alphabet Greek letters appear in two distinct contexts in finance. **Alpha and beta** belong to portfolio theory — they decompose returns into market-driven and skill-driven components. **Delta, gamma, theta, and vega** belong to options pricing — they describe how an option's value responds to changes in underlying price, time, and volatility. --- ## Alpha & Beta: Where Does Your Return Come From? **Beta** ($\beta$) measures how much an asset moves relative to the market as a whole. $\beta = \frac{\text{Cov}(R_{\text{asset}}, R_{\text{market}})}{\text{Var}(R_{\text{market}})}$ - $\beta = 1$: moves in sync with the market - $\beta > 1$: amplifies market movements - $\beta < 1$: dampens market movements Beta captures **systematic risk** — the portion of risk you cannot diversify away. **Alpha** ($\alpha$) is what remains after stripping out beta. $\alpha = R_{\text{portfolio}} - \beta \times R_{\text{market}}$ > [!example] Fruit Merchant Analogy > > Summer arrives. All fruit merchants see business improve — the whole market rises 15%. Your revenue rises 20%. > > - Beta's contribution: $\beta \times 15%$ (say $\beta = 1.2$, so $18%$) > - Alpha: $20% - 18% = 2%$ > > That 2% came from _your_ skill — better sourcing, better pricing, better customer relationships. The 18% came from simply being in the market. ### The Alpha Shrinkage Problem Alpha's definition depends entirely on **which factor model you choose**. ``` CAPM (1 factor) → lots of "alpha" found Fama-French (3 factors) → much alpha disappears (reclassified as size/value beta) Carhart (4 factors) → more alpha disappears (momentum beta) 5-factor, 6-factor... → alpha keeps shrinking ``` **Today's alpha is tomorrow's beta.** The boundary between them is not discovered — it is drawn by the model. > [!tip] Three Tests for "Real" Alpha > > 1. **Persistence** — does it survive across different time periods? > 2. **Crisis behaviour** — does it vanish or reverse during market stress? (If so, it's likely hidden tail-risk beta) > 3. **Capacity constraint** — can it scale indefinitely? (True alpha degrades with size; beta scales freely) --- ## The Options Greeks: Four Dimensions of Risk From here on, we use a single running example. > [!info] Setup: The Mango Option > > - Current mango price: **¥10/斤** > - You buy a **call option** on 1,000 斤 of mango > - Strike price: **¥10/斤** > - Premium paid: **¥500** > - Time to expiry: **30 days** --- ### Delta ($\delta$): Directional Exposure Delta measures how much the option price changes per ¥1 move in the underlying. $\Delta = \frac{\partial V}{\partial S}$ Your call option has $\Delta = 0.5$. Mango rises ¥1 → option value rises ≈ ¥500 ($0.5 \times 1000$). #### Three Interpretations of Delta | Interpretation | Meaning | | ------------------------ | ------------------------------------------------------------------- | | **Hedge ratio** | To delta-hedge, sell $\Delta \times 1000 = 500$ 斤 of physical mango | | **Probability proxy** | ≈ 50% chance the option expires in-the-money | | **Directional exposure** | Positive delta = profit when underlying rises | #### Delta Signs by Position |Position|Delta|You profit when...| |---|---|---| |Buy call|$+$ (0 to 1)|underlying rises| |Buy put|$-$ (-1 to 0)|underlying falls| |Sell call|$-$|underlying falls| |Sell put|$+$|underlying rises| --- ### Gamma ($\gamma$): The Acceleration of Delta Gamma measures how fast delta itself changes when the underlying moves. $\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}$ > If delta is **velocity**, gamma is **acceleration**. #### Positive vs Negative Gamma This is the single most important distinction in options risk. > [!important] Positive Gamma (Option Buyer) > > - Mango rises → delta increases → you gain **faster** > - Mango falls → delta decreases → you lose **slower** > > **Gains accelerate, losses decelerate.** > [!warning] Negative Gamma (Option Seller) > > - Mango rises → delta shifts against you → you lose **faster** > - Mango falls → delta shifts against you → you gain **slower** > > **Gains decelerate, losses accelerate.** #### Concrete Walkthrough: Why Positive Gamma ≠ "Always Profitable" Mango drops from ¥10 to ¥9 to ¥8 to ¥7. You are losing money the entire time. But watch the delta: |Mango Price|Delta|Loss on Next ¥1 Drop| |---|---|---| |¥10|0.50|≈ ¥500| |¥9|0.30|≈ ¥300| |¥8|0.10|≈ ¥100| **You are losing money, but each yuan of decline hurts less than the last.** This is positive gamma at work — it shrinks your exposure as the price moves against you. It does not prevent losses; it decelerates them. --- ### Theta ($\theta$): The Daily Tax of Time Theta measures how much option value erodes per day, all else equal. $\Theta = \frac{\partial V}{\partial t}$ Your option is a right with an expiration date. A right valid for 30 days is worth more than the same right valid for 3 days, because more can happen in 30 days. Each passing day removes a slice of this "time value." - **Option buyer**: negative theta (pays rent daily) - **Option seller**: positive theta (collects rent daily) > [!note] Theta accelerates near expiry > > An option that decays gently over months will decay aggressively in its final week. The last few days are the most expensive for buyers and most profitable for sellers. --- ### Vega ($\nu$): Sensitivity to Expected Volatility Vega measures how much the option price changes per 1 percentage point change in **implied volatility**. $\nu = \frac{\partial V}{\partial \sigma}$ > [!example] Typhoon Season Mango price hasn't moved. But a typhoon warning is issued — supply uncertainty spikes. The market now _expects_ larger future price swings. Your option immediately becomes more valuable, not because mango moved, but because **the probability of large moves increased**. - **Option buyer**: positive vega (benefits from rising IV) - **Option seller**: negative vega (harmed by rising IV) #### Why Vega Follows the Buyer The buyer's payoff structure is **convex**: loss is capped (premium paid), gain is unlimited. Greater expected volatility activates more of the unlimited upside without worsening the capped downside. This is the same asymmetry that produces positive gamma — **convexity makes volatility your friend**, whether that volatility is realised (gamma) or expected (vega). --- ## The Complete Framework ### Signs by Position |Delta|Gamma|Theta|Vega| |---|---|---|---|---| |**Buy Call**|$+$|$+$|$-$|$+$| |**Buy Put**|$-$|$+$|$-$|$+$| |**Sell Call**|$-$|$-$|$+$|$-$| |**Sell Put**|$+$|$-$|$+$|$-$| > [!tip] Pattern > > - **Delta** depends on direction (call vs put) and side (buy vs sell) > - **Gamma, Theta, Vega** depend only on side: buyers get $(+\gamma,\ -\theta,\ +\nu)$, sellers get $(-\gamma,\ +\theta,\ -\nu)$ > - **Gamma and theta are always opposite in sign** — this is structural, not empirical ### What Each Letter Controls | Greek | Dimension | Question It Answers | | -------- | ------------------ | ------------------------------------------------------------- | | $\Delta$ | Price direction | How exposed am I to the underlying moving? | | $\Gamma$ | Convexity | Is price movement helping or hurting my exposure structure? | | $\Theta$ | Time | How much am I paying (or collecting) daily for this position? | | $\nu$ | Implied volatility | How does a shift in market fear/expectation affect me? | ### The Multi-Dimensional Reality > [!warning] No Single Greek Guarantees Profit Your P&L > > on any given day is the **sum** of all Greeks acting simultaneously. Positive vega helps you when IV rises, but delta can overwhelm it if the price drops. Theta drains you every day regardless. > > Each Greek describes **one slice** of the risk landscape. The option lives in all four dimensions at once. --- ## Gamma-Theta Tradeoff & Gamma Scalping This is the central mechanism of delta-hedged options trading. ### Setup You buy the mango call option (¥500 premium, $\Delta = 0.5$, 1000 斤). To delta-hedge, you **sell 500 斤 of physical mango at ¥10/斤**. Net delta = 0. ### Day 1: Mango Rises to ¥11 |Leg|What Happens|P&L| |---|---|---| |Option|$\Delta$ was 0.5, value rises ≈ ¥520|+¥520| |Short 500 斤 mango|500 × (10−11)|−¥500| |**Net**||**≈ +¥20**| Gamma pushes $\Delta$ to 0.7. Your net delta is now +200. To re-hedge → **sell 200 斤 at ¥11** (收入 ¥2,200). ### Day 2: Mango Falls Back to ¥10 Option value returns to roughly starting point. You now hold 700 斤 short. |Leg|What Happens| |---|---| |First 500 斤 short|¥10 → ¥11 → ¥10, round-trip P&L = 0| |Extra 200 斤 short|Sold at ¥11, price now ¥10| To re-hedge → **buy back 200 斤 at ¥10** (支出 ¥2,000). ### The Result $\text{Gamma scalping profit} = 200 \times (11 - 10) = \textbf{¥200}$ Mango returned to ¥10. Position structure returned to its starting state. But ¥200 appeared from the round-trip re-hedging. **Positive gamma forced each rebalance to be a high-sell / low-buy.** ### But Theta Is Running Suppose $\Theta = ¥30/\text{day}$. Over 2 days: $-¥60$. $$$\text{Net P\&L} = 200_{\text{(gamma scalping)}} - 60_{\text{(theta)}} = +¥140$ ### Three Scenarios |Scenario|Mango Movement|Gamma Scalp|Theta Cost|Net| |---|---|---|---|---| |High volatility|¥10 → ¥11 → ¥10|+¥200|−¥60|**+¥140**| |Low volatility|¥10 → ¥10.2 → ¥10|+¥8|−¥60|**−¥52**| |No volatility|¥10 → ¥10 → ¥10|¥0|−¥60|**−¥60**| > [!abstract] The Race > > - **Realised volatility > Implied volatility** → gamma scalping covers theta → buyer profits > - **Realised volatility < Implied volatility** → theta bleeds out the premium → seller profits > - **Realised volatility = Implied volatility** → breakeven: the option was fairly priced This is the mathematical heart of options market-making: **managing the daily tension between gamma income and theta cost (or vice versa), while monitoring vega exposure to shifts in market expectations.** --- ## Higher-Order Greeks (Brief) |Greek|What It Captures| |---|---| |**Vanna**|Cross-effect: how $\Delta$ shifts when IV changes. A deep OTM option's delta is sensitive to vol assumptions — if IV rises, it becomes "less impossible" to finish ITM| |**Volga**|Second-order vega: vega itself is not constant — it changes as IV changes. Matters for very large positions where vega is the dominant risk| > [!note] Core Greeks ($\Delta, \Gamma, \Theta, \nu$) drive the vast majority of daily trading decisions. Higher-order Greeks matter at institutional scale or for exotic positions.