See the project here svi_calibration_understanding
The Black–Scholes model assumes an arbitrage-free and perfectly efficient market. In theory, this means that if an option’s market price deviates from its theoretical price, traders could exploit the difference through arbitrage, quickly restoring equilibrium. Under such conditions, implied volatility would remain constant across strikes — exactly as the model suggests.
However, real markets are far more complex. Market makers may quote prices above or below the theoretical value due to inventory imbalances, risk management needs, or hedging costs.
Meanwhile, informed traders (market takers) attempt to profit from perceived mispricings, and overall market sentiment drives buying and selling behavior.
These forces cause option prices to diverge from the ideal Black–Scholes prediction, producing a volatility skew — or, when symmetric, a volatility smile (named because the curve resembles a smile when plotted).
Now that the motivation to model the volatility smile is clear, it’s time to dive into the broader details of the Stochastic Volatility Inspired (SVI) model.
This model was first developed at Merrill Lynch in 1999 and was later presented by Jim Gatheral at the Global Derivatives Conference in Madrid in 2004. Gatheral’s book, The Volatility Surface: A Practitioner’s Guide, explores the model in depth in its opening chapter.
Before introducing the model’s formulation, it’s worth understanding why stochastic volatility models are useful. In Gatheral’s own words:
“Stochastic volatility (SV) models are useful because they explain in a self-consistent way why options with different strikes and expirations have different Black–Scholes implied volatilities — that is, the ‘volatility smile.’”
This statement captures the core motivation: traditional constant-volatility models cannot explain the shape of the implied volatility surface observed in real markets. SV models, including SVI, address this by allowing volatility itself to evolve as a random process.
The next important question is: what motivates the derivation of the SVI model? Gatheral explains this in the context of modeling variance as a stochastic, mean-reverting quantity:
“Having motivated the description of variance as a mean-reverting random variable, we are now ready to derive the valuation equation.”
This lays the groundwork for the SVI approach — a simplified, analytically inspired representation of more complex stochastic volatility dynamics.
By the end of this project, we’ll visualize the result of fitting the Stochastic Volatility Inspired (SVI) model to our own Bitcoin options data. The figure below shows the calibrated implied volatility surface obtained from that process.
The Black–Scholes model assumes an arbitrage-free and perfectly efficient market. In theory, this means that if an option’s market price deviates from its theoretical price, traders could exploit the difference through arbitrage, quickly restoring equilibrium. Under such conditions, implied volatility would remain constant across strikes — exactly as the model suggests.
However, real markets are far more complex. Market makers may quote prices above or below the theoretical value due to inventory imbalances, risk management needs, or hedging costs.
Meanwhile, informed traders (market takers) attempt to profit from perceived mispricings, and overall market sentiment drives buying and selling behavior.
These forces cause option prices to diverge from the ideal Black–Scholes prediction, producing a volatility skew — or, when symmetric, a volatility smile (named because the curve resembles a smile when plotted).
Now that the motivation to model the volatility smile is clear, it’s time to dive into the broader details of the Stochastic Volatility Inspired (SVI) model.
This model was first developed at Merrill Lynch in 1999 and was later presented by Jim Gatheral at the Global Derivatives Conference in Madrid in 2004. Gatheral’s book, The Volatility Surface: A Practitioner’s Guide, explores the model in depth in its opening chapter.
Before introducing the model’s formulation, it’s worth understanding why stochastic volatility models are useful. In Gatheral’s own words:
“Stochastic volatility (SV) models are useful because they explain in a self-consistent way why options with different strikes and expirations have different Black–Scholes implied volatilities — that is, the ‘volatility smile.’”
This statement captures the core motivation: traditional constant-volatility models cannot explain the shape of the implied volatility surface observed in real markets. SV models, including SVI, address this by allowing volatility itself to evolve as a random process.
The next important question is: what motivates the derivation of the SVI model? Gatheral explains this in the context of modeling variance as a stochastic, mean-reverting quantity:
“Having motivated the description of variance as a mean-reverting random variable, we are now ready to derive the valuation equation.”
This lays the groundwork for the SVI approach — a simplified, analytically inspired representation of more complex stochastic volatility dynamics.
By the end of this project, we’ll visualize the result of fitting the Stochastic Volatility Inspired (SVI) model to our own Bitcoin options data. The figure below shows the calibrated implied volatility surface obtained from that process.
