Shallow Representation of Option Implied Information

Abstract

Option prices encode the market's collective outlook through implied density and implied volatility. An explicit link between implied density and implied volatility translates the risk-neutrality of the former into conditions on the latter to rule out static arbitrage. Despite earlier recognition of their parity, the two had been studied in isolation for decades until the recent demand in implied volatility modeling rejuvenated such parity. This paper provides a systematic approach to build neural representations of option implied information. As a preliminary, we first revisit the explicit link between implied density and implied volatility through an alternative and minimalist lens, where implied volatility is viewed not as volatility but as a pointwise corrector mapping the Black-Scholes quasi-density into the implied risk-neutral density. Building on this perspective, we propose the neural representation that incorporates arbitrage constraints through the differentiable corrector. With an additive logistic model as the synthetic benchmark, extensive experiments reveal that deeper or wider network structures do not necessarily improve the model performance due to the nonlinearity of both arbitrage constraints and neural derivatives. By contrast, a shallow feedforward network with a single hidden layer and a specific activation effectively approximates implied density and implied volatility.

Figures (11)

• Figure 1: Contour plot of parameters and moments generated by $X \sim LB(\mu, \varsigma, \alpha, \beta)$ with fixed variance $\mathbb{V}[X] = 0.16$. $(\alpha, \beta) \in [10^{-1}, 10^{1}]^2$. $\varsigma$ in (b) is determined by Equation \ref{['eq:lb_varsigma']}, $\mu$ in (b) is determined by Equation \ref{['eq:lb_mu']}, and $\mathbb{E}[X]$, $\mathbb{S}[X]$, and $\mathbb{K}[X]$ in (c)-(e) are determined by Equation \ref{['eq:lb_moment']}.
• Figure 2: Risk-neutral logistic-beta distributions versus risk-neutral normal distribution. All distributions have a variance of $0.16$. The light violet color corresponds to the normal distribution. $400$ logistic-beta distributions with $(\alpha, \beta) \in \{10^{-1}, 10^{-0.9}, \dots, 10^{1}\}^2$ are plotted, specified in the same manner as Figure \ref{['fig:moment_countor']}, colored by their skewnesses. Negative (positive) skewness is in the red (blue) end of the color spectrum.
• Figure 3: True logistic-beta density versus Black-Scholes densities. In (a)-(e), the consistent blue curve is the PDF $\psi_{LB}$ of $LB(\mu, 0.15, 0.57, 1.15)$, and the green (red) curves are Black-Scholes $\{\psi_{BS}^\omega(\cdot; \kappa)\}_\kappa$ with various implied volatility to match call (put) prices. The blue (purple) shaded areas correspond to the integral area of $\psi_{LB}$ on the supports of calls (puts). The green (red) shaded areas correspond to the integral area of $\{\psi_{BS}^\omega(\cdot; \kappa)\}_\kappa$ on the supports of calls (puts). (f) shows the corresponding implied volatility.
• Figure 4: Implied volatility as pointwise correction. The correct distribution is $X_1 \sim LB(\mu, 0.15, 0.57, 1.15)$. (a) shows the corresponding implied total volatility $\omega$ and its spatial partial derivatives $\partial_\kappa \omega$ and $\partial_{\kappa \kappa} \omega$. In (b) (resp. (c)), blue curve is the correct PDF $\psi_{LB}$ (resp. CDF $\Psi_{LB}$), red curve is the uncorrected Black-Scholes quasi-PDF $\psi_{BS}^\omega$ (resp. quasi-CDF $\Psi_{BS}^\omega$), and green curve is the corrector $\zeta^\omega(1,x)$ (resp. $\xi^\omega)$. We additionally draw the related Black-Scholes PDFs $\{\psi_{BS}(\cdot; \kappa)\}_\kappa$ (resp. CDFs $\{\Psi_{BS}(\cdot; \kappa)\}_\kappa$) in very light red curves.
• Figure 5: Neural representation framework.

Theorems & Definitions (4)

• Proposition 3.1
• Remark 3.2
• Corollary 3.3
• proof : Proof of Proposition \ref{['prop:psi_omega']}