"300 years of calculus and nobody saw this. Why is that? Well, my guess is nobody ever thought about using the hockey sticks, right? Who would use that? Option traders would use that." - Doug Costa [00:41:20]
"The exciting thing about this is that the financial engineering technology that was used to create the new VIX... was invented in the 1990s to model variance swaps." - Doug Costa [00:02:11]
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"If you believe that disjoint intervals of time produce independent changes in the index, then the variances should add... the total variance should be proportional to time." - Doug Costa [00:08:20]
"One thing we know about Black-Scholes implied volatilities is the Black-Scholes model is really not correct. And so there's what's called a volatility smile." - Doug Costa [00:11:08]
"We require that these integrals exist for each path followed by sigma t and those paths will be continuous because Brownian motion is a continuous process." - Doug Costa [00:18:58]
"This says f of x can be represented as a constant plus a linear function plus linear combinations of those functions... the call and put payoff functions are the hockey sticks." - Doug Costa [00:36:45]
"If you want to price some weird derivative security with a weird payoff function that's a European style security, all you need to know is the market prices of puts and calls." - Doug Costa [00:45:00]
Doug Costa: Quantitative Researcher and Educator at Susquehanna International Group (SIG) [00:00:19]. He holds a B.A. in Mathematics and earned a Ph.D. in Abstract Algebra in 1974 [00:00:34]. Following his doctoral studies, Costa served as a mathematics professor at the University of Virginia [00:00:41]. He began consulting for Susquehanna in 1993, formally joined as the Head of Quantitative Research in 1997, and transitioned into an intensive educational role for traders and quants in mid-2015 [00:01:11].
The presentation serves as a rigorous historical and mathematical autopsy of the Chicago Board Options Exchange (CBOE) Volatility Index (VIX), tracing its evolution from a flawed 1980s construct to an advanced financial engineering marvel implemented in 2002 [00:02:34].
The foundational failure of the "Old VIX" is thoroughly deconstructed, specifically its over-reliance on the Black-Scholes model's "at-the-forward" implied volatilities, which ignored the reality of the volatility smile and deterred the creation of derivative markets [00:11:08].
The architectural transition to the "New VIX" is presented as a paradigm shift anchored by two distinct breakthroughs: the 1987 Lewis Scott Log Contract for stochastic volatility pricing and the 1990s Carr-Madan formula [00:13:06].
By applying stochastic calculus rules (e.g., $dt^2=0$) and Taylor expansions, the briefing demonstrates how risk-neutral expected future realized variance can be accurately modeled independently of predictive pathing assumptions [00:26:28].
The Carr-Madan theorem fundamentally changed derivative pricing by proving that any twice continuously differentiable payoff function can be synthesized using a static, linear combination of standard market puts and calls (the "hockey sticks"), weighted by $1/K^2$ [00:45:00].
Ultimately, the SIBO/CBOE execution is analyzed, detailing the pragmatic numerical approximations (Trapezoidal rule, put-call parity forward pricing) required to translate pure quantitative theory into the tradable VIX futures (launched in 2004) and options (launched in 2006) [00:52:46].
[00:55:14] - Market Expansion: VIX Futures and VIX Options
3. Detailed Thematic Summary
Introduction & The Architecture of the Old VIX [00:00:19]
Historical Baseline: Doug Costa achieved his Ph.D. in abstract algebra in 1974 [00:00:34] before joining Susquehanna as a consultant in 1993, becoming the firm's Head of Quantitative Research in 1997 [00:01:01], and later moving to education in 2015 [00:01:11].
Defining the Asset: The CBOE (SIBO) VIX was built to measure standard deviation of returns for the S&P 500 index [00:01:37]. The modern technology defining the "New VIX" was initially invented in the 1990s to model variance swaps [00:02:11].
Pre-2002 Construct: In the 1980s, the CBOE rolled out the original iteration aiming to nominally predict 30 days of future volatility [00:02:34].
The Black-Scholes Dependency: The Old VIX was calculated via European style S&P 500 futures options, employing the Black-Scholes model inputs: Index Price ($S_0$), Strike Price ($K$), Maturity ($T$), Risk-Free Rate ($R$), Dividend Yield ($Q$), and constant Volatility ($\Sigma$) [00:03:34].
The Implied Volatility Binary Search: Because the actual constant $\Sigma$ is unknown, quants use a simple binary search to identify the exact $\Sigma$ that matches the current market price of the option [00:05:00].
At-the-Forward Averaging: To derive the index, CBOE identified strikes closest to the forward price for a given maturity, and then averaged the implied volatilities of both calls and puts ($(\Sigma_{call} + \Sigma_{put}) / 2$) [00:06:07].
Time-Linear Interpolation & The Old VIX's Demise [00:06:44]
The Maturity Problem: Options rarely expire exactly 30 days in the future. To solve this, CBOE found two expirations framing the target date ($T_1$ and $T_2$) and interpolated the variances [00:06:44].
Variance Mathematics: Black-Scholes outputs an annualized standard deviation. Total variance realized over time period $T_1$ equals $\Sigma^2 * T_1$ [00:07:35].
The Independence Assumption: Time-linear interpolation mathematically hinges on the belief that disjoint time intervals generate independent index changes. Due to independence, variances sum up smoothly, making total variance proportional to elapsed time [00:08:20].
The VIX Multiplier: The final Old VIX number was expressed in volatility points by multiplying the final interpolated standard deviation by 100 (e.g., $0.35$ becomes $35$ Vol) [00:10:11].
The Fatal Flaw (The Volatility Smile): The Old VIX was essentially an "at-the-forward" implied volatility. Because the Black-Scholes constant volatility assumption is objectively incorrect, options price along a "volatility smile" where different strikes yield different implied volatilities [00:11:08].
Commercial Failure: Because of this arbitrary mathematical footing, institutional demand for trading futures/options on the Old VIX completely failed to materialize [00:10:41]. It only survived as a generalized "Fear Index" for Wall Street commentators [00:11:52].
The Transition to the New VIX & Stochastic Calculus [00:13:06]
The Two Pillars: The modern 2002 VIX rests on the 1987 Lewis Scott Log Contract and the 1990s Peter Carr & Dilip Madan formula [00:13:06].
The Log Contract: Lewis Scott hypothetically valued a European derivative paying the natural logarithm of the stock price at maturity ($\log(S_T)$) [00:14:11].
Risk-Neutral Pricing: The no-arbitrage price equates to the discounted expected payoff operating strictly under a risk-neutral probability distribution ($\hat{E}$) [00:14:53].
Stochastic Differential Equations (SDEs): To define the risk-neutral distribution, the S&P 500 stochastic process is modeled where $dS_T/S_T = (r-q)dt + \Sigma_t dW_t$ (Standard Brownian Motion) [00:16:26].
The Power of Infinite Variance Paths: Unlike Black-Scholes, Lewis Scott assumed stochastic volatility, requiring an entirely separate equation for $d\Sigma_t$. Critically, the explicit model of $d\Sigma_t$ doesn't matter; the math holds infinitely, provided the sample paths remain square integrable and continuous [00:18:58].
Calculus Discard Rules: Using Taylor Expansions to unspool the logarithm, a fundamental stochastic calculus rule is applied: Any term multiplied by $dt$ to a power strictly greater than 1 approaches absolutely zero in the continuous limit [00:21:00].
Standard Brownian Motion Mechanics: The L2 norm (Standard Deviation) of a Brownian increment $dW_t$ is the square root of $dt$ ($dt^{1/2}$). Therefore, terms containing $dW_t$ survive the $dt > 1$ wipeout rule [00:23:41].
Chi-Squared Simplification: When analyzing the standard variable $Z^2$, it acts as a Chi-squared distribution with exactly 1 degree of freedom, possessing a mean of 1 and a variance of 2 [00:25:25].
Variance Collapses to dt: Because the variance of $dW_t^2$ calculates to zero, it behaves strictly as a constant. Meaning in stochastic calculus, $dW_t^2$ is perfectly equal to $dt$ [00:26:28].
Fubini's Theorem: To manipulate expected values, Costa cites Fubini's Theorem, allowing quants to seamlessly switch the order of integration when moving expectations inside an integral [00:29:55].
Non-Anticipating Volatility: A critical leap required assuming $\Sigma_t$ is independent of the random shock $dW_t$ (non-anticipating). This zeros out highly complex integrals [00:30:30].
The Variance Revelation: Once simplified, Lewis Scott's model proves that Risk-Neutral Expected Future Realized Variance precisely equals $2 * \log(F) - \hat{E}[\log(S_T)]$. It offered a mathematically sound variance metric entirely stripped of implied volatility distortions [00:32:02].
The Carr-Madan Formula & Synthetic Replication [00:34:20]
The "Calculus 2" Oversight: Despite 300 years of calculus, it took 1990s option quants to realize this formula, largely because mathematicians had no practical use for "hockey stick" functions [00:41:20].
The Theorem Mechanics: Carr-Madan proves that any twice piecewise continuously differentiable function $f(x)$ can be rewritten precisely as $f(x_0) + f'(x_0)(x - x_0)$ plus an integration of $f''(y)$ against the positive parts of $(y-x)$ and $(x-y)$ [00:34:20].
The Hockey Sticks: When graphed, $(y-x)^+$ strictly resembles the payoff profile of a Put option, and $(x-y)^+$ mirrors a Call option. Peter Carr natively referred to these shapes as "hockey sticks" [00:37:16].
Integration by Parts: The fundamental proof of this reality runs through basic integration by parts ($\int u , dv = uv - \int v , du$), manipulating intervals around the fixed strike threshold [00:39:20].
The Pricing Epiphany: By applying Carr-Madan to risk-neutral pricing, quants demonstrated that you can price any weird European derivative simply by taking a linear combination of existing market put and call prices [00:45:00].
Logarithmic Weighting ($1/K^2$): Plugging the Lewis Scott Log Contract into Carr-Madan requires the second derivative of the logarithm ($-1/K^2$). The resulting synthesis demands holding a portfolio of puts and calls weighted precisely by $1/K^2$ [00:46:20].
The Square Root Conundrum: The New VIX publishes the final volatility value as $100 * \sqrt{\hat{E}[Variance]}$. Mathematically, the square root of an expectation is technically not the expectation of a square root. This was a pragmatic choice CBOE accepted to standardize the metric output [00:49:21].
Handling Continuous Integrals in a Discrete Market: Integrals assume infinite strike granularity. CBOE addresses this by applying numerical approximation, explicitly using the Trapezoidal Rule ($[K_{i+1} - K_{i-1}] / 2$) to bridge discrete option strikes [00:52:46].
Taylor Approximation of the Logarithm: SIBO condenses the leading calculation terms by deploying a truncated Taylor series of $\log(1+x) \approx x - x^2/2$, compressing mathematical drag [00:50:20].
Bid-Ask Dynamics & Forward Prices: CBOE utilizes pure mid-quotes across the options chain to generate $Q(K)$. Furthermore, instead of calculating future dividend streams to establish the Forward Price, SIBO leverages the market's efficiency by backing into the forward via Put-Call Parity ($F = K + e^{rt}(Call - Put)$) [00:53:16].
Trimming the Tails (Zero Bids): Any option containing a Zero Bid is instantly discarded. If the algorithm hits two consecutive Zero Bids marching upward (Calls) or downward (Puts), the entire remaining tail is truncated [00:54:20].
The Tradable Architecture: Because of this mathematically rigorous foundation, the "New VIX" succeeded where the old failed. VIX Futures officially launched in 2004 with Susquehanna installed as the primary specialist. VIX Options successfully launched in 2006, eventually enabling recursive constructs like the "VIX of the VIX" [00:55:14].
The Black-Scholes Model: The original baseline pricing formula that assumes constant volatility and log-normal asset prices. Its fatal flaw—failing to account for the "volatility smile"—necessitated the invention of the modern VIX framework. [00:03:28]
No-Arbitrage Pricing via Risk-Neutral Probability: The core doctrine asserting that to find the true, arbitrage-free price of a derivative, one must strictly calculate the discounted expected payoff operating under a mathematical "risk-neutral" probability distribution, divorcing the equation from subjective real-world directionality expectations. [00:14:53]
Time-Linear Interpolation of Variance: A mental model governing how to stitch timelines together. Based on the assumption that market movements in disjoint, separate time intervals are completely independent. Because of this assumed independence, variances are strictly additive, allowing quants to proportionally linearly interpolate 30-day horizons between differing expirations. [00:08:20]
Stochastic Calculus Size Discard Rule ($dt > 1 \to 0$): A computational filtering framework. When dealing with infinitesimally small increments of time ($dt$), any variable containing an exponent greater than one becomes so exceptionally small that it is erased from the physics of the calculation. It simplifies massive multi-term Taylor expansions down to only their first two critical nodes. [00:21:00]
Fubini's Theorem: A mathematical theorem allowing the interchange of the order of integration. Essential for Lewis Scott's model to move the expectation operator inside complex integrals seamlessly. [00:29:55]
The "Hockey Stick" Synthesis (Carr-Madan Theorem): A structural framework proving that highly complex, non-linear problems (any European payoff function) can be mathematically perfectly replicated using an infinite linear stack of simple, fundamental building blocks (puts and calls). The "Hockey stick" is the visual mental model for these payoff profiles. [00:36:45]
Put-Call Parity Forwards Synthesis: Rather than executing complex, high-friction predictive models (like estimating thousands of independent S&P 500 company dividend schedules to find the index forward price), this framework outsources the processing to the collective wisdom of the market makers. By forcing $F = K + e^{rt}(Call - Put)$, CBOE ensures their index remains perpetually tethered to real-world market friction. [00:53:16]
6. Anecdotes
The 300-Year Blindspot of Calculus: Costa highlights the sheer chronological magnitude of the Carr-Madan formula. Despite Sir Isaac Newton and Gottfried Leibniz developing calculus in the late 17th century, nobody documented the exact mathematical breakdown of functions via $(x-y)^+$ limits until the 1990s. Costa notes this happened purely because pure mathematicians had no use for "hockey stick" functions—only option traders dealing with calls and puts forced the discovery. [00:41:20]
The "In the Air" Discovery: Costa details that while he named the breakthrough formula "Carr-Madan" based on a journal article he read by Peter Carr and Dilip Madan, Carr himself humbly corrected him. When asked about it, Carr admitted, "Oh, everybody knew about that in the '90s. It was just in the air." The anecdote illustrates how parallel discovery happens in quantitative finance when market needs arise simultaneously. [00:13:44]
The Failure of Old VIX Futures: CBOE heavily anticipated institutional demand for VIX futures when they launched the Old VIX in the 80s. However, because quants and institutional traders recognized the "Volatility Smile" flaw—where single at-the-forward Black-Scholes implied volatilities broke down structurally—nobody was willing to risk capital trading it. It became merely a TV pundit "Fear Index" until the 2002 revamp. [00:10:41]
Peter Carr's "Hockey Sticks" Moniker: Costa shares a brief, humanizing anecdote regarding Peter Carr, one of the co-authors of the Carr-Madan formula. Carr was Canadian and an enormous hockey fan. Whenever reviewing the asymmetric payoff charts for Put and Call options ($y-x$ positive limits), Carr instinctively called them "hockey sticks", embedding the visual shorthand into quant culture. [00:37:16]
7. References & Recommendations
Institutions & Companies:
Susquehanna International Group (SIG): Costa's firm; acted as the primary specialist/market maker for VIX Futures in 2004. Explains the pragmatic application of theory. [00:00:51]
Chicago Board Options Exchange (CBOE / SIBO): The foundational exchange that constructed, publishes, and houses derivatives for the Volatility Index. [00:01:37]
University of Virginia: The academic institution where Doug Costa taught prior to his career at Susquehanna, providing his mathematical pedigree. [00:00:41]
Literature, Theories & Publications:
Options, Futures, and Other Derivative Securities by John Hull: Referred to natively by Costa as a standard baseline textbook for Black-Scholes terminology and equations ($d_1$, $d_2$). [00:04:34]
Lewis Scott's Log Contract Paper (c. 1987): Foundational quantitative paper outlining stochastic volatility and risk-neutral log contract expectations. [00:13:06]
The Carr-Madan Paper (1990s): Peter Carr and Dilip Madan's journal publication formally documenting the integral synthesis of twice continuously differentiable functions using put/call profiles. [00:13:32]
Fubini's Theorem: Classic mathematical theorem referenced to justify swapping the order of integration when dealing with expected values in stochastic calculus. [00:29:55]
Chi-Squared Distribution: Statistical model used to prove that the squared Brownian motion increment strictly evaluates to an exact variance. [00:25:25]
CBOE/SIBO VIX White Papers: The exchange's published documentation breaking down the strict, exact numerical approximations utilized in calculating real-time VIX. Recommended for further reading. [00:54:53]
Variance Swap White Papers: Mentioned alongside the CBOE docs as recommended reading to fully grasp the implementation of the $1/K^2$ portfolio weighting. [00:55:58]
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