Gaussian-mixture and Student-t smiles¶
Author: Artur Sepp
Implemented in stochvolmodels. Software citation: CITATION.cff.
A terminal-distribution model specifies the law of the price at one maturity directly, without dynamics. Its option prices are closed-form expectations under that law, so it fits one smile quickly and gives a density consistent with it. The package has two: a mixture of normal distributions of the log-return, and a Student-t distribution of the simple return, floored at zero. This article explains both, the smiles they can produce, and what they cannot do.
Note
No publication of the author maps to these models; the article rests on the literature cited below and on the package’s code.
Overview¶
Each model has a few parameters per maturity and one constraint, that the expected price equals the forward. A mixture of normal log-returns is a weighted sum of Black-Scholes worlds, so its price is a weighted sum of Black prices (Ritchey, 1990); with states of different means and volatilities it produces skews and smiles of either sign. The Student-t model keeps a fixed shape of the distribution and controls its tails by the degrees of freedom. Neither model has dynamics: the parameters of one maturity say nothing about another, and fitting several maturities gives unrelated sets of parameters.
Inputs, notation, and assumptions¶
Symbol |
Code |
Meaning |
|---|---|---|
\(w_i\), \(\mu_i\), \(\sigma_i\) |
|
Weight, annualised drift and volatility of mixture state \(i\) |
\(\mu\), \(\sigma\), \(\nu\) |
|
Drift, volatility of the simple return and degrees of freedom of the Student-t |
\(T\) |
|
The single maturity the parameters belong to |
Rates are zero in the examples, the forward is \(F = 100\), and the maturity is six months.
Methodology¶
Mixture of normal log-returns¶
The log-return \(X = \ln (S_T / F)\) has the density \(\sum_i w_i n(x; \mu_i T, \sigma_i^2 T)\) with weights summing to one. A call is the weighted sum of Black calls on shifted forwards,
and the price is a martingale if \(\sum_i w_i e^{(\mu_i + \sigma_i^2 / 2) T} = 1\). Brigo and Mercurio (2002) give local-volatility dynamics whose marginal laws are such mixtures. Every moment of the price is finite, so the implied variance grows less than linearly in log-strike in both wings (Lee, 2004).
Student-t simple return¶
The price is \(S_T = F \max(1 + \mu T + X, 0)\), where \(X\) has a Student-t law with \(\nu > 2\) degrees of freedom, scaled so that its variance is \(\sigma^2 T\). The drift \(\mu\) is solved so that \(E[S_T] = F\). Calls and puts are closed-form in the Student-t distribution function and its partial mean; the floor puts a mass at zero price with probability \(P(X \le -(1 + \mu T))\). Because the return, not the log-return, has the Student-t law, the expected price is finite; the exponential of a Student-t variable has no finite mean, a difficulty that log-Student-t pricing has to address (Cassidy, Hamp and Ouyed, 2010). Moments of the price exist only up to order \(\nu\), so the right wing of the smile rises with a slope set by \(\nu\) (Lee, 2004). Student-t laws describe the fat tails of stock returns (Blattberg and Gonedes, 1974).
Calibration¶
GmmPricer.calibrate_model_params_to_chain fits each maturity separately, by sequential least
squares on implied volatilities weighted by vegas, with the two equality constraints on the
weights and the forward; the default has four states, and each slice starts from the previous fit.
TdistPricer.calibrate_model_params_to_chain fits \(\sigma\) and \(\nu\) per maturity and solves the
drift at every trial. Both return a dictionary of parameters by slice.
Worked example¶
Every block below is an excerpt of
examples/docs/terminal_distribution_models.py,
which asserts every number quoted here. A mixture is set by weights, volatilities and log-mean
shifts; one common constant in the drifts makes it a martingale:
def gmm_params(weights, vols, shifts, ttm: float = TTM) -> svm.GmmParams:
"""A mixture of normal log-returns; a common constant in the drifts matches the forward."""
w, s, m = (np.asarray(v, dtype=float) for v in (weights, vols, shifts))
c = -np.log(np.sum(w * np.exp(m * ttm))) / ttm
return svm.GmmParams(gmm_weights=w, gmm_mus=m + c - 0.5 * s ** 2, gmm_vols=s, ttm=ttm)
The Student-t drift is solved by the package:
def tdist_params(vol: float, nu: float, ttm: float = TTM) -> svm.TdistParams:
"""A Student-t simple return floored at zero; the drift is implied so that E[S_T] = F."""
drift = svm.imply_drift_tdist(rf_rate=0.0, vol=vol, nu=nu, ttm=ttm)
return svm.TdistParams(drift=drift, vol=vol, nu=nu, ttm=ttm)
Closed forms. For mixtures of two, three and four states and Student-t laws with \(\nu\) of 3, 5 and 30, the closed-form prices at eleven strikes from 60 to 140 agree with numerical integration of the densities to \(10^{-12}\), and a call struck at zero is worth the forward.
Mixture smiles. The example mixtures add states with lower means and higher volatilities. The log-return then has skewness of -1.60, -1.82 and -2.56 and excess kurtosis of 3.97, 5.32 and 10.40 for two, three and four states. The implied volatility at a strike of 60 rises from 37.32% to 38.20% and 42.11%, with 20.88%, 20.47% and 20.51% at the forward; the smile reaches its minimum of 17.64%, 16.84% and 16.51% above the forward and rises again to 17.92%, 18.10% and 18.95% at 140.
Student-t smiles. With \(\sigma\) chosen so that the implied volatility at the forward is 20%, 0.2518, 0.2170 and 0.2016 for \(\nu\) of 3, 5 and 30, small \(\nu\) adds curvature to both wings: from 39.31% at 60 to 26.73% at 140 for \(\nu = 3\), and from 34.52% to 22.84% for \(\nu = 5\). For \(\nu = 30\) the law is close to normal simple returns and the smile falls across all strikes, from 27.21% to 17.91%. The probability of a zero price is 0.12% for \(\nu = 3\) and 0.019% for \(\nu = 5\).
Synthetic teaching exhibit. Six-month Black implied volatilities against strike, forward 100: (A) mixtures of two, three and four normal log-returns; (B) Student-t simple returns with \(\nu\) of 3, 5 and 30, each with a 20% volatility at the forward.
One maturity only. The four-state mixture of six months, priced at three months, values a call struck at zero at 0.152 below the forward, and at one year 1.063 above it; the Student-t law with \(\nu = 5\) misses by 0.0016 and 0.0161. The constraint that makes each model a martingale holds only at the maturity it was solved for.
Fits to the S&P 500 ETF chain. Fitted slice by slice to the bundled chain of 15 July 2022, four normal states give root-mean-square errors below 30 bp against mid volatilities at two weeks, one, two and six months. The exact errors vary because constrained optimization can reach different local optima. The Student-t law, whose skew is fixed by its two parameters, misses by 231.4, 273.2, 294.2 and 314.5 bp.
Implementation in stochvolmodels¶
Name |
Role |
|---|---|
|
States of the mixture; |
|
Closed-form prices; |
|
Drift, volatility, degrees of freedom and maturity of the Student-t law |
|
Closed-form prices; per-slice calibration of |
|
Drift of the martingale condition, density and distribution function (compatibility tier) |
The provisional terminal models GmmTerminalModel and TdistTerminalModel wrap the same laws and
validate the martingale condition for a slice (see
provisional surfaces). Neither pricer has a Monte
Carlo engine: model_mc_price_chain raises. Run the example from a checkout with
python examples/docs/terminal_distribution_models.py.
Interpretation and limitations¶
No term structure. Each set of parameters belongs to one maturity; the fits of different maturities are unrelated and cannot price a forward-starting or path-dependent payoff.
The mixture fit is not convex. Its states can be relabelled and its objective has several minima; the result depends on the starting point, and extra states may receive zero weight.
The Student-t law is symmetric in the return. Its skew in Black volatility comes from modelling the simple rather than the log-return, and from the floor at zero; with only \(\sigma\) and \(\nu\) free it cannot fit an equity skew.
Mass at zero. The Student-t price is zero with positive probability, which makes the implied volatility of far out-of-the-money puts rise steeply.
See also¶
References¶
Blattberg, R. C. and Gonedes, N. J. (1974). A comparison of the stable and Student distributions as statistical models for stock prices. The Journal of Business 47(2), 244. DOI 10.1086/295634.
Brigo, D. and Mercurio, F. (2002). Lognormal-mixture dynamics and calibration to market volatility smiles. International Journal of Theoretical and Applied Finance 5(4), 427-446. DOI 10.1142/S0219024902001511.
Cassidy, D. T., Hamp, M. J. and Ouyed, R. (2010). Pricing European options with a log Student’s t-distribution: a Gosset formula. Physica A 389(24), 5736-5748. DOI 10.1016/j.physa.2010.08.037.
Lee, R. W. (2004). The moment formula for implied volatility at extreme strikes. Mathematical Finance 14(3), 469-480. DOI 10.1111/j.0960-1627.2004.00200.x.
Ritchey, R. J. (1990). Call option valuation for discrete normal mixtures. Journal of Financial Research 13(4), 285-296. DOI 10.1111/j.1475-6803.1990.tb00633.x.