Belief Variance Swaps

• Belief Variance Swaps are financial contracts that derive their payoff from the realized quadratic variation of an asset, utilizing subjective or risk-neutral beliefs.
• They employ model-free pricing methods with Skorokhod embedding techniques like Azéma–Yor and Perkins to establish rigorous no-arbitrage bounds.
• Robust nonparametric estimation of risk-neutral densities from option market data underpins accurate swap pricing and practical hedging strategies.

Belief variance swaps are financial contracts that pay based on the realized variance of an asset price under a chosen probability measure, typically reflecting a "subjective" or "risk-neutral" belief about future price distributions. Central issues in belief variance swaps concern their model-free pricing, realization via Skorokhod embedding techniques, robust nonparametric fitting of risk-neutral densities, and rigorous no-arbitrage bounds. These aspects are rigorously developed in the literature through connections between variance swap pricing, martingale embeddings, and market option data.

1. Theoretical Foundation: Variance Swaps and Skorokhod Embedding

Variance swaps provide a payoff proportional to the realized quadratic variation of an asset, specifically in the form

$V_T = \int_0^T \frac{d[X, X]_t}{(X_{t-})^2},$

where $X$ is a nonnegative martingale representing the discounted asset price, and $[X,X]_t$ is its quadratic variation. Pricing under minimal assumptions ("model-free") leads one to seek extremal expected values of pathwise functionals over all martingale models compatible with option prices at time $T$.

This framework leverages the Skorokhod embedding problem: for Brownian motion $B$, find stopping times $\tau$ such that $B_\tau \sim \mu$, where $\mu$ matches the observed terminal law (via options market quotes). The Dambis-Dubins-Schwarz time change asserts that every martingale (with $X_T \sim \mu$) admits such a representation, reducing the pricing of variance swaps to a problem on Brownian paths and their maxima: $$E[V_T] \geq \inf_{\tau: B_\tau \sim \mu} E\left[\int_0^\tau S_u^{-2}\,du\right],$$ where $S_u = \sup_{s \leq u} B_s$ (Hobson et al., 2010).

2. Extremal Solutions: Azéma–Yor and Perkins Embeddings

The Azéma–Yor (AY) and Perkins solutions to the Skorokhod embedding problem exhibit extremal properties:

• Azéma–Yor embedding ($\tau_{AY}$): Maximizes the law of the stopped process’s maximum and, for increasing functionals $F$, provides

$E[F(S_\tau)] \leq E[F(S_{\tau_{AY}})],$

for all uniform integrable (UI) embeddings (Hobson et al., 2010).

• Perkins embedding ($\tau_P$): Minimizes the law of the maximum,

$E[F(S_\tau)] \geq E[F(S_{\tau_P})],$

for all embeddings (Hobson et al., 2010).

For $g(s) = s^{-2}$, these embeddings yield model-independent bounds on variance swap prices. The minimal UI embedding corresponds to AY, while Perkins yields the maximal bound.

3. Reduction to Bivariate and Path Functionals

Itô's lemma connects variance swap pathwise integrals to expected values of bivariate functions,

$G(w,s) = (s-w)^2 g(s) = \frac{(s-w)^2}{s^2},$

where $w$ is the stopped value and $s$ the running maximum. The main result (Theorem 4.2 from (Hobson et al., 2010)) shows that $G$ satisfies monotonicity conditions (G-MON$\downarrow$), hence the extremal expectations

$$\inf_{\tau} E[G(W_\tau, S_\tau)] = E[G(W_{\tau_{AY}}, S_{\tau_{AY}})], \qquad \sup_{\tau} E[G(W_{\tau_P}, S_{\tau_P})],$$

which recover lower and upper bounds for variance swap pricing in closed form.

4. Model-Free Pricing Bounds and Market Implementation

Market practice, by Breeden–Litzenberger, equates the availability of calls/puts across strikes at maturity $T$ with knowledge of the terminal law $\mu = \mathrm{Law}(X_T)$. This suffices to pin the fair value of a variance swap to the interval

$$\left[ E\left\langle V_T \right\rangle_{\tau_{AY}}, \; E\left\langle V_T \right\rangle_{\tau_P} \right],$$

with $$\langle V_T \rangle_{\tau} := \int_0^{\tau} S_u^{-2} du$$ (Hobson et al., 2010). Ongoing work (Hobson–Klimmek 2012) establishes these bounds as tight and shows their relation to explicit super- and subhedging strategies using market derivatives.

5. Nonparametric Estimation of Belief Densities

Robust empirical pricing of belief variance swaps relies on nonparametric estimation of the risk-neutral or subjective belief density. Jiang et al. (Jiang et al., 2018) develop a piecewise-constant density estimator $f_\Delta$, constructed as:

• Knot placement via observed strike array $K_1 < \dots < K_q$, augmented by "power-tail" knots $K_0 = K_1/c_K$, $K_{q+1} = c_K K_q$.
• Density on log-prices approximated as $f_\Delta(y) = a_\ell$ for $y \in (\log K_{\ell-1}, \log K_\ell]$.
• Enforced via nonnegativity ($a_\ell \geq 0$) and unity mass constraints ($\sum a_\ell \Delta_\ell = 1$, $\Delta_\ell = \log (K_\ell / K_{\ell-1})$).
• Fit to market call/put data by minimizing pricing errors under weighted least squares (WLS) or plain least squares (LS), with the piecewise constant approach outperforming quartic/cubic splines and parametric NIG fits, especially for long maturities.

Once the (a₁,…,a_{q+1}) are fitted, the swap price under belief density becomes: $$VS_{t,T} = e^{-R_{tT}} N_{var} \left[ E_t^Q \left( \frac{A}{T} \sum_{i=1}^T R_i^2 \right) - \sigma_{strike}^2 \right],$$ with moments $E_t^Q[\log S_T]$ and $E_t^Q[(\log S_T)^2]$ available in closed form (Jiang et al., 2018).

6. Application, Comparison, and Robustness

Empirical studies (see Figure 1.1 and Tables 3.2–3.3 of (Jiang et al., 2018)) demonstrate near-100% coverage and minimal pricing errors for the piecewise-constant approach, both LS and WLS, across S&P 500 European option data from 1996–2015. Benchmarking against realized variance, CBOE variance futures, and model-implied prices reveals robust performance and tight alignment for maturities between 365 and 800 days. For very short or very long swaps, increased noise or deviations arise, potentially due to liquidity constraints.

The nonparametric estimator requires only the absence of static arbitrage amongst the option prices and no assumptions about $S_t$ beyond independent log-increments under $Q$. Subjective belief densities (e.g., mixtures, scenario trees) are admissible without loss of generality and can be directly employed to price "belief-based" variance swaps via the same moment calculations.

7. Convergence and Embedding Properties

Convergence of measures $\mu_n \to \mu$ weakly is shown to be sufficient for convergence of Azéma–Yor stopping times, provided the associated potentials at zero converge. For Perkins embeddings, further conditions on the convergence of atoms at zero are required for convergence in probability (and almost surely along a subsequence) (Hobson et al., 2010). This distinction informs the numerical stability and robustness of extremal bounds when the terminal law is inferred from market data.

Belief variance swaps, therefore, are grounded in rigorous model-free bounds obtained from fundamental martingale optimality, with practical pricing and replication methods based on minimal nonparametric assumptions and close connections to observed option markets.