Overview of Volatility Models

In financial markets, various models are used to capture and interpolate the implied volatility surfaces across different strike prices and maturities. These models provide a framework for estimating the prices of options based on the volatility smile and are essential tools for traders, quants, and analysts. The models below—RFV, SLV, SABR, and SVI—are some of the commonly used models for such purposes, each with its own structure, assumptions, and parameters.

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    • RFV Model

    The RFV (Rational Function Volatility) model is a flexible model designed for interpolation of volatility smiles. It is particularly useful for fitting implied volatility surfaces where simplicity and computational efficiency are desired. The model represents the implied volatility as a rational function of log-moneyness, making it adaptable to various market conditions and different strike prices. It is parameterized by five variables: aa, bb, cc, dd, and ee.

    The formula for the RFV model is:

     extRFV(k)=\ raca+b⋅k+c⋅k21+d⋅k+e⋅k2\ ext{RFV}(k) = \ rac{a + b \cdot k + c \cdot k^2}{1 + d \cdot k + e \cdot k^2}

    where:

    - kk is the log-moneyness
    - aa, bb, cc, dd, and ee are model parameters.

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    • SLV Model

    The SLV (Simple Linear Volatility) model represents the implied volatility surface using a polynomial function with log-moneyness. This model is ideal for simpler markets where volatility behavior is smooth and can be well-approximated by a polynomial curve, making it a popular choice in less volatile markets. The model uses parameters aa, bb, cc, dd, and ee to describe the curvature.

    The formula for the SLV model is:

     extSLV(k)=a+b⋅k+c⋅k2+d⋅k3+e⋅k4\ ext{SLV}(k) = a + b \cdot k + c \cdot k^2 + d \cdot k^3 + e \cdot k^4

    where:

    - kk is the log-moneyness
    - aa, bb, cc, dd, and ee are model parameters.

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    • SABR Model

    The SABR (Stochastic Alpha Beta Rho) model is a well-established model in financial markets for pricing options. It is particularly popular in interest rate derivatives and FX markets. The model assumes that the asset's volatility follows a stochastic process, making it suitable for markets where volatility dynamics are complex and driven by random factors. The SABR model has been widely adopted due to its ability to capture the volatility smile in options pricing, using parameters like alphaalpha, betabeta, rhorho, nunu, and f0f_0.

    The formula for the SABR model is:

    SABR(k)=α⋅(1+β⋅k+ρ⋅k2+ν⋅k3+f0⋅k4)\text{SABR}(k) = \alpha \cdot (1 + \beta \cdot k + \rho \cdot k^2 + \nu \cdot k^3 + f_0 \cdot k^4)

    where:

    - kk is the log-moneyness
    - alphaalpha, betabeta, rhorho, nunu, and f0f_0 are model parameters.

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    • SVI Model

    The SVI (Stochastic Volatility Inspired) model is one of the most widely used models for fitting implied volatility surfaces, particularly in equity and commodity markets. Its popularity stems from its ability to accurately capture the volatility smile using a relatively simple five-parameter form. The SVI model is highly flexible and can model complex volatility behavior across different strike prices. Its parameters—aa, bb, rhorho, mm, and sigmasigma—allow traders to easily calibrate the model to market data, making it an industry standard.

    The formula for the SVI model is:

    SVI(k)=a+b⋅(ρ⋅(k−m)+(k−m)2+σ2)\text{SVI}(k) = a + b \cdot (\rho \cdot (k - m) + \sqrt{(k - m)^2 + \sigma^2})

    where:

    - kk is the log-moneyness
    - aa, bb, rhorho, mm, and sigmasigma are model parameters.

    Overview of Pricing Models

    Pricing models are essential for determining the fair value of options, incorporating factors such as the stock price, strike price, time to expiration, volatility, and dividend yield. These models allow traders to evaluate American options with early exercise features.

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    • Barone-Adesi Whaley

    The Barone-Adesi Whaley model is used for pricing American options, accounting for dividends and early exercise. It approximates the price of an American option and adjusts the classic Black-Scholes model to handle dividend-paying assets.

    The formula for the Barone-Adesi Whaley model (for calls) is:

     extPricecalls=S⋅e−qT⋅N(d1)−K⋅e−rT⋅N(d2)\ ext{Price}_{calls} = S \cdot e^{-qT} \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2)

    and for puts is:

     extPriceputs=K⋅e−rT⋅N(−d2)−S⋅e−qT⋅N(−d1)\ ext{Price}_{puts} = K \cdot e^{-rT} \cdot N(-d_2) - S \cdot e^{-qT} \cdot N(-d_1)

    where:

    - SS is the current stock price
    - KK is the strike price
    - TT is the time to expiration
    - rr is the risk-free interest rate
    - qq is the continuous dividend yield
    - N()N() represents the cumulative normal distribution function.