How Market Makers Trade Volatility

<style> .math-block { margin: 1.5em 0; text-align: center; } .math-block mjx-container { max-width: 100% !important; overflow-x: hidden !important; } .math-block mjx-container svg { max-width: 100%; height: auto; } .figure-block { margin: 2em 0; text-align: center; } .figure-caption { font-size: 0.9em; color: #666; margin-top: 8px; } .tcolorbox { background: #f7f8fa; border: 1px solid #e2e6ea; border-left: 3px solid #5a4bb2; border-radius: 0 8px 8px 0; padding: 20px 24px; margin: 28px 0; } .ref-list { list-style: none; padding: 0; margin: 0; } .ref-list li { padding-left: 2.5em; text-indent: -2.5em; margin-bottom: 0.8em; } </style> <p>It comes to my attention that many students first meet options through Black-Scholes and treat it as if it were a complete description of option pricing. BSM, however, comes with one big caveat: it assumes volatility stays constant over time. Anyone who has looked at even one simple chart can easily deduce that this assumption does not hold. Markets can have cool and stable phases, and they can have Jerome Powell "Good morning" phases. That is why I have decided to dedicate this AQS article to the idea of changing volatility, volatility surfaces, smiles, and how Systematic Trading Firms actually look at volatility.</p> <h3>What is an Option?</h3> <p>Before we get anywhere, let's first make sure that everyone reading this can accurately understand what an option is at the fundamental level. You can skip if you already know this.</p> <p>Imagine you want to buy a car, and conveniently, your Uncle Sam wants to sell one. You are not fully sure yet, so you agree today on a fixed price for the car you might buy in the future. That is the basic idea of a forward or futures contract: locking in a future price now. Naturally, Sam will not do this for free, he would like some compensation for holding the risk. The price may include inflation, interest he could have earned elsewhere, and the risk of holding the car until the deal date.</p> <p>But you are still unsure, so you ask for one more clause: at expiry, you may choose whether to buy or walk away. If the deal is good, you take it. If the market offers something better, you leave. That flexibility costs money, called the <strong>option premium</strong>. This is a <strong>European call option</strong>: the right, but not the obligation, to buy at a fixed price on expiry. An <strong>American option</strong> allows exercise any time before expiry.</p> <p>In modern markets, this idea becomes standardized into calls and puts. A call gives the buyer the right to buy the asset at a fixed strike price, while a put gives the right to sell it. Buying an option means being long; selling one means being short, receiving the premium but taking on the obligation if the option is exercised. Stock options are listed by expiry dates, commonly the third Friday of the month unless a holiday interferes, because traders are funny like that. One stock option contract usually controls 100 shares, but the buyer only pays the premium for the right to trade those shares at the agreed strike and expiry.</p> <p>For example, an option might be listed as AAPL 150 Call, expiring 19 July 2026, trading at $4.20. That would mean the buyer, you, pay a $4.20 premium per share, or $420 total, because one standard equity option contract controls 100 shares, giving you the right to buy Apple at $150 per share until expiry. Conceptually we can split the value into two parts:</p> <div class="figure-block"> <img src="https://cdn.prod.website-files.com/6913793ee068962f268ed83c/6a145b0c346074da80e193bb_aqs_option_value_breakdown.svg" alt="Option value breakdown: intrinsic vs extrinsic value" style="max-width:100%;height:auto;"/> </div> <p>The value of an option consists of <strong>intrinsic value</strong> and <strong>extrinsic value</strong>. The intrinsic value is what the option would be worth if exercised immediately: for a call option, this is \(\max(S-K,0)\), where \(S\) is the current stock price and \(K\) is the strike price (exercise price). If \(S>K\), the call is <strong>in-the-money (ITM)</strong>; if \(S<K\), it is <strong>out-of-the-money (OTM)</strong>; and if \(S \approx K\), it is <strong>at-the-money (ATM)</strong>. The extrinsic value is everything above intrinsic value, meaning the extra premium paid for time, uncertainty, and the chance that the option becomes more profitable before expiry. This is also where volatility comes into play: higher expected volatility increases the chance of large future moves, which can push extrinsic value, and therefore the option premium, sharply higher.</p> <h3>The Black-Scholes Model and Its Limits</h3> <p>The next natural question now is how will this premium be determined. The Black-Scholes model is often the go-to "truth-teller" in this case. It provides a theoretical price for a European option by linking the option's value to the current stock price, the strike price, the time to expiry, the risk-free interest rate, and the volatility of the underlying asset. If you have read the previous AQS articles, you may recognize the stock price model behind it:</p> <div class="math-block">\[dS_t = \mu S_t dt + \sigma S_t dW_t.\]</div> <p>This is the Geometric Brownian motion model. Here, \(S_t\) is the stock price, \(\mu\) is the average growth rate, \(\sigma\) is the volatility, and \(W_t\) is a Brownian motion term representing random market shocks. The Black-Scholes formula is derived from this type of continuous-time price model, together with no-arbitrage reasoning and dynamic hedging assumptions. We use this together with the Black-Scholes Model (don't pay much attention to the details):</p> <div class="math-block">\[V_{\text{call}} = S_0 N(d_1) - K e^{-rT}N(d_2),\]</div> <p>where</p> <div class="math-block">\[d_1 = \frac{\ln(S_0/K) + \left(r + \frac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}.\]</div> <p>The formula converts the inputs into a fair call option price. The terms \(N(d_1)\) and \(N(d_2)\) come from the normal distribution and reflect the probability-weighted payoff of the option under the model's assumptions. The important point though is not the full derivation, but that volatility \(\sigma\) enters the option price.</p> <p>Volatility matters because options have asymmetric payoffs. If a stock moves against a call buyer, the loss is limited to the premium paid. If the stock moves strongly upward, the upside can be much larger. The same logic holds in reverse for put options. Because the option holder benefits from large favorable moves but is protected from equally large unfavorable moves, more volatility generally makes the option more valuable. Mathematically, this is captured by vega,</p> <div class="math-block">\[\text{Vega} = \frac{\partial V}{\partial \sigma},\]</div> <p>which measures the sensitivity of the option price \(V\) to changes in volatility. We will talk about it further on in the article.</p> <p>The main assumption, however, is that Black-Scholes treats volatility as constant, which is a very strong simplification. In the model, \(\sigma\) does not change over time, across strikes, or across market regimes. Once chosen, the same volatility input is used to price the option from today until expiry.</p> <p>Real markets do not have one stable singular volatility number. Volatility changes after earnings announcements, macro news, crises, Truth Social posts, liquidity shocks, and shifts in investor demand for protection. Even worse for the model, options with different strikes and maturities often imply different volatility levels.</p> <h3>Implied Volatility</h3> <p>In Black-Scholes, volatility enters the model as an input. In real markets, however, traders usually observe the option price first and then work backwards. The <strong>implied volatility</strong> of an option is the value of \(\sigma\) that makes the Black-Scholes price equal to the observed market price:</p> <div class="math-block">\[V_{BS}(S,K,T,r,\sigma_{\text{imp}})=V^{mkt}.\]</div> <p>This means implied volatility is not directly observed volatility, there is not even anything such as "observed volatility". The closest thing to it is realized volatility, which is just historical volatility. IV is an option price translated into Black-Scholes volatility units. It partly reflects the market's expectation of future realized volatility, but it is also affected by risk premia, supply and demand, hedging pressure, liquidity, and the cost of taking the other side of the trade. In that sense, implied volatility is better understood as the market price of uncertainty and risk, expressed as an annualized volatility number.</p> <p>This is important since implied volatility varies across strikes. The market is not saying that the underlying asset has a different "true volatility" at each strike. The underlying has one future path. Instead, different strikes represent different payoff regions and different risks. A far out-of-the-money put, for example, is not just a volatility forecast; it is crash protection, insurance. Since that protection is especially valuable and dangerous to sell, the market may demand a higher premium for it. When that higher premium is translated through Black-Scholes, it appears as a higher implied volatility.</p> <p>So \(\sigma_{\text{imp}}(K)\) should not be read as "the market believes future actual volatility is different at each strike." It is more accurate to read it as "the market assigns different prices to different risks, and Black-Scholes expresses those prices in volatility units." This is the foundation for volatility smiles and skews.</p> <h3>The Volatility Smile and Skew</h3> <p>Options with different strikes usually imply different volatility levels. If we plot implied volatility on the vertical axis and strike price, or more commonly moneyness, on the horizontal axis, the graph often forms a curve. This curve is called the <strong>volatility smile</strong>.</p> <div class="figure-block"> <img src="https://cdn.prod.website-files.com/6913793ee068962f268ed83c/6a145b1005dd7d48b863007a_aqs_volatility_patterns.svg" alt="Three stylized implied volatility patterns" style="max-width:100%;height:auto;"/> <div class="figure-caption">Three stylized implied volatility patterns. A symmetric smile prices large moves in both directions. An equity skew reflects expensive downside protection. A more realistic market smile can combine downside crash premium, lower at-the-money volatility, and a smaller upside premium.</div> </div> <p>Every option strike represents exposure to a different region of the future price distribution. An at-the-money option is mostly exposed to normal movements around today's price. A far out-of-the-money put is exposed to a large downside move. A far out-of-the-money call is exposed to a large upside move.</p> <p>Because these contracts insure against different future scenarios, the market does not price them equally. A low-strike put, for example, is like crash insurance. Even if it is far out-of-the-money today, it pays off in exactly the kind of market state where liquidity disappears, hedging becomes difficult, and investors are desperate for protection. Market makers who sell this protection take on dangerous tail (extreme case/situation) risk, so they demand a higher premium. When that higher option price is translated back through Black-Scholes, it appears as a higher implied volatility.</p> <p>This is why equity markets often show a <strong>volatility skew</strong> rather than a perfectly symmetric smile. Downside puts often trade at higher implied volatilities than at-the-money options or upside calls.</p> <p>In a sense, the smile is a map of how the market prices different outcomes. The middle of the smile reflects the price of ordinary movement around the current stock price. The left side reflects the price of downside tail risk. The right side reflects the price of large upside moves. Black-Scholes starts with one clean volatility number, but the market turns that number into a curve:</p> <div class="math-block">\[\sigma_{\text{imp}} = \sigma_{\text{imp}}(K).\]</div> <p>The volatility smile is therefore one of the first signs that volatility is not just some model input, just another variable. In practice, it becomes an object that traders quote, compare, hedge, and trade across strikes.</p> <h3>The Volatility Surface</h3> <p>A volatility smile describes how implied volatility changes across strikes for one fixed expiry. However, options are not only separated by strike price. They are also separated by expiration date. A one-week option, a three-month option, and a one-year option on the same stock can all imply different volatility levels, even at the same strike. This means that the volatility smile itself can change depending on the maturity of the option.</p> <p>To account for this, most responsible traders extend the idea of the volatility smile into a <strong>volatility surface</strong>. They treat it as a function of both strike and maturity:</p> <div class="math-block">\[\sigma_{\text{imp}} = \sigma_{\text{imp}}(K,T).\]</div> <p>Here, \(K\) is the strike price and \(T\) is the time to expiry. In practice, traders often use moneyness instead of raw strike price, since this makes the graph comparable across different assets. For example, using \(K/S\), where \(S\) is the current stock price, tells us how far the strike is from the current market price in relative terms.</p> <p>Intuitively, a volatility surface is built by stacking many volatility smiles together, one for each expiry date. Each slice of the surface at a fixed maturity is a volatility smile or skew. When these slices are placed next to each other across time, they form a three-dimensional graph. The horizontal axes represent moneyness and time to expiry, while the vertical axis represents implied volatility.</p> <div class="figure-block"> <img src="https://cdn.prod.website-files.com/6913793ee068962f268ed83c/6a145b21990bb25b6b1e38b4_WhatsApp%20Image%202026-05-25%20at%2016.16.55.jpeg" alt="Stylized equity implied volatility surface" style="max-width:100%;height:auto;"/> <div class="figure-caption">A stylized equity implied volatility surface. Implied volatility varies across both moneyness \(K/S\) and time to expiry. The higher left wing reflects the greater price of downside protection, while the changing shape across maturities shows why traders model implied volatility as a surface rather than a single number.</div> </div> <p>The surface gives a more complete picture of how the market prices uncertainty. The left wing, where \(K/S < 1\), often represents lower strikes and therefore downside put protection (the crash insurance). In equity markets, this region frequently has higher implied volatility because investors are willing to pay more for protection against sharp losses.</p> <p>The middle of the surface, near \(K/S = 1\), corresponds to at-the-money options and often acts as the baseline region for implied volatility. The right wing, where \(K/S > 1\), represents higher strikes and upside call exposure.</p> <p>The maturity dimension adds another layer. Short-dated options can have high implied volatility when the market expects an important event soon, such as earnings, a central bank decision, or a major macro announcement. Longer-dated options usually reflect broader uncertainty over a longer horizon, so their implied volatility can behave differently from short-term options. This is why two options with the same strike but different expiries may trade at very different implied volatilities.</p> <p>For market makers and volatility traders, the surface is the object they monitor, quote, hedge, and trade. They care not only about whether volatility is high or low, but also where. A trader, like you, may ask whether downside protection is too expensive, whether short-dated event volatility is overpriced, or whether one maturity looks cheap relative to another.</p> <p>This is the next logical step beyond the Black-Scholes view for most. Black-Scholes began with one volatility input, \(\sigma\). The volatility smile turned it into a curve across strikes. And on top of that, volatility surface turned it into a full map across strikes and maturities:</p> <div class="math-block">\[\sigma_{\text{const}} \quad \longrightarrow \quad \sigma_{\text{imp}}(K) \quad \longrightarrow \quad \sigma_{\text{imp}}(K,T).\]</div> <p>Once volatility became a surface, it formalised into something traders can compare across the entire options market. This is why modern options trading is about understanding how the market prices uncertainty across different future prices and future dates.</p> <h3>Vega: Exposure to Implied Volatility</h3> <p>Once implied volatility becomes a surface, traders need a way to measure how sensitive their positions are to movements in that surface. This is where <strong>vega</strong> comes in the picture. Vega measures how much the value of an option changes when implied volatility changes:</p> <div class="math-block">\[\text{Vega} = \frac{\partial V}{\partial \sigma}.\]</div> <p>Here, \(V\) is the value of the option or option portfolio, and \(\sigma\) is implied volatility. If an option has positive vega, its value increases when implied volatility rises. If it has negative vega, its value decreases when implied volatility rises.</p> <p>Long options are usually <strong>long vega</strong>. This means that buying a call or a put generally benefits from an increase in implied volatility, because higher volatility raises the probability of large future moves. Short options are usually <strong>short vega</strong>. A trader who sells options receives premium upfront, but loses value if implied volatility rises after the trade.</p> <p>This is why volatility trading is not only about choosing a direction for the stock and riding it. A trader can be wrong on direction but still make money if the volatility exposure was correct. At the same time, he can be right on direction but lose money if implied volatility moves against the position. The option price is affected by both the underlying asset and the market's price of uncertainty.</p> <p>For a single option, vega can be written as one number. In a real trading book, however, vega is spread across the volatility surface. A firm may have exposure to short-dated at-the-money volatility, downside put-wing volatility (low-strike put options), longer-dated index volatility (like SPX, STOXX 50, or NASDAQ), or upside call volatility (high-strike call options). These are not all the same risk. A rise in one part of the surface does not necessarily mean the entire surface moves in the same way.</p> <p>This is why market makers and volatility traders often think in terms of <strong>vega buckets</strong>. Instead of asking only whether they are long or short volatility overall, they ask where they are long or short volatility. They may be long vega in one-month options, short vega in six-month options, exposed to downside skew, or concentrated around a specific earnings expiry.</p> <p>More advanced desks also track second-order volatility risks such as <strong>volga</strong>, which measures how vega itself changes when volatility moves. Those second-order risks matter, but they belong to the next layer of the discussion. The central idea is already captured by vega: once volatility becomes tradable, traders need to know how their positions respond when implied volatility changes.</p> <h3>How Market Makers Think About Volatility</h3> <p>For a market maker, the desk observes option prices across strikes and maturities, converts them into implied volatilities, and builds an internal volatility surface that represents how the market is currently pricing uncertainty. In practice, desks often work with moneyness or log-moneyness rather than raw strike. If \(F_T\) is the forward price for maturity \(T\), a common variable is</p> <div class="math-block">\[k = \log\left(\frac{K}{F_T}\right),\]</div> <p>and instead of fitting implied volatility directly, many surface models work with <strong>total implied variance</strong>,</p> <div class="math-block">\[w(k,T) = \sigma_{\text{imp}}^2(k,T)T.\]</div> <p>It mainly comes in handy because implied volatility is quoted as an annualized standard deviation, while variance accumulates over time. The quantity \(w(k,T)\) therefore represents the total variance priced over the lifetime of the option.</p> <p>A professional volatility surface is not supposed to be merely smooth. It also has to be economically and mathematically consistent. Across strikes, call prices for the same expiry should form a convex curve. This matters because the curvature of option prices with respect to strike is linked to the market-implied probability distribution of the future underlying price. If the curve bends the wrong way, the fitted surface can imply impossible negative probabilities for some outcomes. In trading terms, this is related to butterfly arbitrage: a butterfly spread could be priced below zero even though its payoff at expiry is never negative.</p> <p>A similar restriction applies across maturities: the surface must also avoid calendar-spread arbitrage. A longer-dated option should not be incorrectly cheaper than a shorter-dated option with the same strike in a way that violates no-arbitrage logic. In total variance terms, this is why fitted surfaces are often checked so that total implied variance does not decrease incorrectly across maturities.</p> <p>A common way to represent a volatility smile is through a so called "parametric form". For example, an SVI-type (Stochastic Volatility Inspired) smile models total implied variance as a function of log-moneyness:</p> <div class="math-block">\[w(k) = a + b\left(\rho(k-m) + \sqrt{(k-m)^2 + s^2}\right).\]</div> <p>The exact formula is not the main point here. Don't mind it. The important thing to understand, is that a market maker does not want thousands of disconnected option prices. They want a controlled representation of the surface, where the level, skew, curvature, and wings can be understood, adjusted, and compared across maturities.</p> <div class="figure-block"> <img src="https://cdn.prod.website-files.com/6913793ee068962f268ed83c/6a145b20486101a659042abc_WhatsApp%20Image%202026-05-25%20at%2016.16.39.jpeg" alt="SVI-type total implied variance surface" style="max-width:100%;height:auto;"/> <div class="figure-caption">A stylized SVI-type total implied variance surface. Surface models often use total implied variance \(w(k,T)=\sigma_{\text{imp}}^2(k,T)T\) rather than implied volatility directly, because total variance is more natural for modeling across maturities.</div> </div> <p>This is where trading volatility becomes more precise. The desk has an internal view of fair implied volatility, \(\hat{\sigma}_{\text{desk}}(K,T),\) and the market has an observable implied volatility \(\sigma_{\text{mkt}}(K,T).\)</p> <p>The difference between them is a local relative-value signal:</p> <div class="math-block">\[\epsilon(K,T) = \sigma_{\text{mkt}}(K,T) - \hat{\sigma}_{\text{desk}}(K,T).\]</div> <p>If \(\epsilon(K,T)>0\), that part of the surface may be rich relative to the desk's model. If \(\epsilon(K,T)<0\), it may be cheap. However, this does not mean you can blindly sell every rich option and buy every cheap one. The difference must be large enough to compensate for model risk, liquidity, transaction costs, and the possibility that the market is correctly pricing information that the model has not captured. Vega tells the desk how much this difference matters. For a small change in implied volatility,</p> <div class="math-block">\[\Delta V \approx \text{Vega} \cdot \Delta \sigma.\]</div> <p>For a full book, the idea becomes surface-based. The book does not simply have one vega number; it has vega distributed across strikes and maturities. We can write the change in value of the book as:</p> <div class="math-block">\[\Delta V \approx \sum_{K} \sum_{T} \nu(K,T)\Delta \sigma_{\text{imp}}(K,T),\]</div> <p>where \(\nu(K,T)\) represents the vega exposure at each region of the surface.</p> <div class="tcolorbox"> <p>Suppose a market maker is looking at a specific region of the surface, for example a one-month downside put. The market-implied volatility is</p> <div class="math-block">\[\sigma_{\text{mkt}}(K,T)=28\%,\]</div> <p>while the desk's internal fair volatility estimate is</p> <div class="math-block">\[\hat{\sigma}_{\text{desk}}(K,T)=25\%.\]</div> <p>The option therefore looks rich by</p> <div class="math-block">\[\epsilon(K,T)=28\%-25\%=3\text{ vol points}.\]</div> <p>If the position has vega exposure of</p> <div class="math-block">\[\nu(K,T)=10{,}000 \quad \text{dollars per volatility point},\]</div> <p>then the approximate value difference is</p> <div class="math-block">\[\Delta V \approx 10{,}000 \times 3 = 30{,}000.\]</div> <p>This does not automatically mean the trade should be made. The apparent edge still has to be large enough to survive bid-ask spreads, hedging costs, model error, liquidity risk, and the possibility that the market is pricing information the desk has missed.</p> </div> <p>This is also why market makers think in surface movements rather than only in volatility levels. The surface can move through a parallel shift, where most implied volatilities rise together. It can steepen, where downside puts become more expensive relative to at-the-money options. It can change curvature, where the wings move differently from the center. It can also move through the term structure, where short-dated volatility rises while long-dated volatility barely changes. All of these are different volatility risks. The trading problem now becomes deeper: where is the surface mispriced? Is the at-the-money level too high? Is downside protection too expensive relative to the center? Is the short-dated surface pricing too much event risk? Is one maturity inconsistent with neighbouring maturities? The market maker is constantly comparing individual option prices to the shape of the entire surface.</p> <p>This is also how market makers can influence the visible market. If demand repeatedly appears in one region, such as downside puts before a crash scare or short-dated calls before an event, that region of the surface can be repriced upward fast. The surface moves because liquidity, risk transfer, and information arrive unevenly across strikes and maturities. A large enough flow in a specific slice of the surface can force that slice to reprice, especially in less liquid names, short-dated options, or far out-of-the-money wings.</p> <p>So, from a professional standpoint: market makers do not trade "volatility" as one abstract object. They trade the shape, level, and movement of the implied volatility surface. Their edge comes from building a better surface, knowing which parts of the market are rich or cheap relative to that surface, and understanding how changes in implied volatility translate into profit and loss through vega. The surface is the map. Vega tells them where they are exposed on that map. The trade is deciding which parts of the map are wrong.</p> <h3>How Market Makers Trade Volatility II</h3> <p>All that we discussed so far gives us only the first layer of how market makers think about volatility, let alone trading with it. Once implied volatility is treated as a full surface, the next question is how the surface can move, twist, and break.</p> <p>In the second part of this article, we will move beyond vega and study the deeper risks of volatility market making: volga, which measures how vega itself changes; skew risk, which captures the relative pricing of downside protection; term-structure risk, which tracks how implied volatility differs across expiries; and inventory risk, which arises from the positions market makers accumulate while providing liquidity. Together, these ideas can show why professional volatility trading is a constant problem of managing exposure across the entire surface.</p> <h3>References</h3> <ul class="ref-list"> <li>[1] F. Black and M. Scholes, <em>The Pricing of Options and Corporate Liabilities</em>, Journal of Political Economy, 1973.</li> <li>[2] R. C. Merton, <em>Theory of Rational Option Pricing</em>, Bell Journal of Economics and Management Science, 1973.</li> <li>[3] S. Natenberg, <em>Option Volatility and Pricing: Advanced Trading Strategies and Techniques</em>, McGraw-Hill.</li> <li>[4] J. C. Hull, <em>Options, Futures, and Other Derivatives</em>, Pearson.</li> <li>[5] J. Gatheral, <em>The Volatility Surface: A Practitioner's Guide</em>, Wiley.</li> <li>[6] J. Gatheral and A. Jacquier, <em>Arbitrage-Free SVI Volatility Surfaces</em>, Quantitative Finance, 2014.</li> <li>[7] D. T. Breeden and R. H. Litzenberger, <em>Prices of State-Contingent Claims Implicit in Option Prices</em>, Journal of Business, 1978.</li> </ul>