Pricing autocallables under local-stochastic volatility

Walter Farkas; Francesco Ferrari; Urban Ulrych

doi:10.3934/fmf.2022008

Article Contents

2022, Volume 1, Issue 4: 575-610. Doi: 10.3934/fmf.2022008 open access

This issue Previous Article Optimal stopping contract for Public Private Partnerships under moral hazard Next Article

Pricing autocallables under local-stochastic volatility

1.
Department of Banking and Finance, University of Zurich, Switzerland
2.
College of Management of Technology, EPFL Lausanne, Switzerland
3.
Department of Mathematics, ETH Zurich, Switzerland
4.
Swiss Finance Institute, Switzerland

^*Corresponding author: Walter Farkas
^*Corresponding author: Walter Farkas

Received: September 2022

Early access: December 2022

Published: December 2022

Abstract / Introduction Full Text(HTML) Figure(29) / Table(8) Related Papers Cited by

Abstract

This paper investigates the pricing of single-asset autocallable barrier reverse convertibles in the Heston local-stochastic volatility (LSV) model. Despite their complexity, autocallable structured notes are the most traded equity-linked exotic derivatives. The autocallable payoff embeds an early-redemption feature generating strong path- and model-dependency. Consequently, the commonly-used local volatility (LV) model is overly simplified for pricing and risk management. Given its ability to match the implied volatility smile and reproduce its realistic dynamics, the LSV model is, in contrast, better suited for exotic derivatives such as autocallables. We use quasi-Monte Carlo methods to study the pricing given the Heston LSV model and compare it with the LV model. In particular, we establish the sensitivity of the valuation differences of autocallables between the two models with respect to payoff features, model parameters, underlying characteristics, and volatility regimes. We find that the improved spot-volatility dynamics captured by the Heston LSV model typically result in higher prices, demonstrating the dependence of autocallables on the forward-skew and vol-of-vol risk. Moreover, we show that the parameters of the stochastic component of LSV models enable controlling for the autocallables price while leaving the fit to European options unaffected.

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Mathematics Subject Classification: Primary: 91G20, 91G60; Secondary: 62P05.

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Figure 1. Implied volatilities in forward-moneyness $ K/F(0, T) $ space computed from SPX mid option prices across all the listed regular expiries as of January 23, 2020

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Figure 2. eSSVI smiles fit to regular S & P 500 index options as of January 23, 2020, in log-forward-moneyness $ k $ and implied volatility $ \hat{\sigma} $ space. Red and blue circles are bid and offer implied volatilities, respectively, and the orange line presents the eSSVI fit

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Figure 3. $ T-t = 180 $ and 365 days IV smile dynamics in the Heston LSV model calibrated to SPX options as of January 23, 2020. In black the current IV smile. In orange, green, and red the forward IV smiles resulting from different levels at time $ t = 1/365 $

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Figure 4. 180 and 365 days spot- and forward-starting implied volatility smiles generated by the Heston local-stochastic volatility model calibrated to SPX options as of January 23, 2020. $ T_1 $ denotes the forward start date and $ T_2-T_1 $ the residual maturity

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Figure 5. Different ABRC payoff scenarios for $ T^E = 12 $, $ T^O = 3 $, $ K = 1.00 $, $ H = 0.80 $ European, $ Y = 7.19\% $ p.a., $ \mathbf{H^{Y}} = \mathbf{0} $, $ \mathbf{Y^{AC}} = \mathbf{0} $ and $ \mathbf{H^{AC}} = \mathbf{1} $

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Figure 6. Payoff diagram, price, and sticky-moneyness delta at the inception of the benchmark ABRC for different $ S $ at $ t = 0 $

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Figure 7. Vega computed with a parallel shift of the IVS, and survival rate of the benchmark ABRC for different $ S $ at $ t = 0 $

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Figure 8. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ and $ \mathbb{E}^\mathbb{Q}[T_{i^*}] $ for different observation tenors $ T^O $ and across different expiry tenors $ T^E $. In the right panel, the solid line is for LV and the dashed line for Heston LSV

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Figure 9. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ and $ \mathbb{Q}\left[L\leq S(0)H\right] $ for different barriers $ H $ and across different expiry tenors $ T^E $. In the right panel, the solid line is for LV and the dashed line for Heston LSV

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Figure 10. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ in bps for different coupon levels $ Y^\prime $, left panel, and coupon barrier levels $ \mathbf{H^{Y}} $ without memory, right panel, across various expiry tenors $ T^E $

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Figure 11. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for different volatility of variance levels $ \eta $ across various expiry tenors $ T^E $. In the right panel, the corresponding Heston LSV expected expiries $ \mathbb{E}^\mathbb{Q}[T_{i^*}] $

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Figure 12. $ T_1 = $ 3 months forward-starting IV smiles generated by the Heston-like LSV model calibrated to SPX option as of January 23, 2020, for different levels of Heston parameters $ \eta $ and $ \rho $

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Figure 13. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for different correlation levels $ \rho $ across various expiry tenors $ T^E $. In the right panel, the corresponding Heston LSV expected expiries $ \mathbb{E}^\mathbb{Q}[T_{i^*}] $

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Figure 14. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ and $ \mathbb{Q}\left[L\leq S(0)H\right] $ for different dividend yields $ q $ and across different $ T^E $. In the right panel, the solid line is for LV and the dashed line for Heston LSV

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Figure 15. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for the indices SPX, NDX, DJX, and RUT across expiry tenors $ T^E $

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Figure 16. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for SPX, NDX, DJX, and RUT across expiry tenors $ T^E $, in low (subplots a and b) and high (subplots c and d) volatility regimes and for different $ Y $

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Figure 17. Evolution of $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ over the period January 3, 2017 to December 31, 2020, for the benchmark ABRC. The green and red lines are the VIX and $ Y $ level in percentage, respectively, see right $ y $-axis. In blue is $ \Delta^{LSV} $ for $ Y $ such that $ U^{LV}(0) = 1 $ and in black is $ \Delta^{LSV} $ for $ Y = 0 $ in bps, see left $ y $-axis

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Figure 18. Approximation of the conditional expectation $ \mathbb{E}^\mathbb{Q}\left[V(t)\mid S(t) = S\right] $, where the joint dynamics of $ (S(t), V(t))_{t\geq0} $ is given by the Heston model with $ \kappa = 3.7764 $, $ \theta = 0.0365 $, $ \eta = 0.9555 $, $ \rho = -0.7946 $, and $ V_0 = 0.0141 $. $ T_{max} = 1057 $ days, $ \ell = 100 $, $ n = 2^{14} $, and $ m = 1057 $

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Figure 19. Approximation of the conditional expectation $ \mathbb{E}^\mathbb{Q}\left[V(t)\mid S(t) = S\right] $, where the joint dynamics of $ (S(t), V(t))_{t\geq0} $ is given by the system of SDEs from Eqs. (1) to (3). The LV function is obtained as of January 23, 2020, $ T_{max} = 1057 $ days, $ \ell = 100 $, $ n = 2^{14} $, and $ m = 1057 $. The Heston parameters are as in Fig. 18

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Figure 20. Leverage function approximation calibrated to SPX options as of January 23, 2020. The parameters for the approximation are $ T_{max} = 1057 $ days, $ \ell = 200 $, $ n = 2^{18} $, and $ m = 1057 $

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Figure 21. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for different observation tenors $ T^O $ and autocall coupon levels $ \mathbf{Y^{AC}} $ across various expiry tenors $ T^E $

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Figure 22. In the left panel, $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for different constant autocall barrier levels $ \mathbf{H^{AC}} $ and across different $ T^E $. In the right panel, the LV implied $ \mathbb{E}^\mathbb{Q}[T_{i^*}] $

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Figure 23. In the left panel, $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for different step-down autocall barrier levels $ \mathbf{H^{AC}} $ and across different $ T^E $, $ H_1^{AC} = 1 $. In the right panel, the LV implied $ \mathbb{E}^\mathbb{Q}[T_{i^*}] $

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Figure 24. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for different volatility of variance levels $ \eta $ across various expiry tenors $ T^E $. In the right panel, the corresponding Heston LSV expected expiries $ \mathbb{E}^\mathbb{Q}[T_{i^*}] $

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Figure 25. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for different speed of mean reversion levels $ \kappa $ across various expiry tenors $ T^E $. In the right panel, the corresponding LSV expected expiries $ \mathbb{E}^\mathbb{Q}[T_{i^*}] $

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Figure 26. $ T_1 = $ 3 months forward-starting IV smiles generated by the Heston-like LSV model calibrated to SPX option as of January 23, 2020, for different levels of $ \kappa $ and $ \theta $

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Figure 27. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for different long-term variance levels $ \theta $ across various expiry tenors $ T^E $. In the right panel, the corresponding LV expected expiries $ \mathbb{E}^\mathbb{Q}[T_{i^*}] $

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Figure 28. SPX, NDX, DJX, and RUT ATMF total implied variance and forwards term structure as of January 23, 2020

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Figure 29. $ \hat{U}^{LSV}(0)-\hat{U}^{LV}(0) $ for SPX, NDX, DJX, and RUT across expiry tenors $ T^E $, in medium-low and medium-high volatility regimes and for different coupon amounts $ Y $

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Table 1. Description of the main features determining the ABRC. $ T^E $, $ T^O $, $ K $, $ H $, and $ Y $ are scalars. $ \mathbf{H^{Y}} $, $ \mathbf{Y^{AC}} $, and $ \mathbf{H^{AC}} $ are $ N\times 1 $ vectors, where $ N = T^E/T^O $ is the number of observation dates

Feature	Notation	Description
Expiry tenor	$ T^E $	Tenor corresponding to the final valuation date of the SN expressed in months, e.g., 12 months.
Observation tenor	$ T^O $	Tenor corresponding to the length of the observation periods expressed in months, i.e., the time intervals between the dates at which the coupon barrier and autocall barrier are observed, e.g., 3 months.
Strike	$ K $	Strike of the short put option expressed in % of the current spot $ S(0) $, e.g., 90%. The gearing of the put option is given by $ 1/K $.
Knock-in barrier	$ H $	D & I barrier level expressed in % of $ S(0) $, e.g., 80%. If hit, it knocks-in the short put option. It can be observed only at the final valuation date (European) or continuously (American).
Coupon	$ Y $	Coupon level paid at the end of each observation period if the coupon barrier is not hit, e.g., 3% per annum (p.a.).
Coupon barrier	$ \mathbf{H^{Y}} $	D & O barrier levels expressed in % of $ S(0) $, e.g., 80%. If hit, they invalidate or delay the coupon payment. They can be constant or variable and are observed at the end of each observation period.
Autocall coupon	$ \mathbf{Y^{AC}} $	Coupon levels paid upon early redemption expressed in percentage of the notional, e.g., 3%. They can be constant or "snowballing".
Autocall barrier	$ \mathbf{H^{AC}} $	U & O barrier levels expressed in % of $ S(0) $, e.g., 100%. If hit, they trigger the early redemption of the SN. They can be constant or "step-down" and are observed at the end of each period (Bermudan).

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Table 2. $ Y $ for different $ T^E $ computed as of January 23, 2020

$ {\boldsymbol{T^E }}$	3	6	9	12	15	18	21	24	30	36	42
$ Y $	3.99%	5.94%	6.85%	7.19%	7.18%	7.21%	7.12%	7.01%	6.80%	6.55%	6.31%
$ {\boldsymbol{T^E}} $	48	54	60	66	72	78	84	90	96	108	120
$ Y $	6.08%	5.86%	5.73%	5.55%	5.40%	5.26%	5.13%	5.01%	4.87%	4.68%	4.53%

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Table 3. $ Y $ for different underlyings and $ T^E $ computed as of January 23, 2020, for the benchmark ABRC

	$ {\boldsymbol{T^E}} $	3	6	9	12	15	18	21	24	30	36	42
NDX	$ Y $	5.15%	7.25%	8.02%	8.31%	8.27%	8.13%	7.93%	7.78%	7.39%	7.06%	6.81%
DJX	$ Y $	3.56%	5.61%	6.70%	7.18%	7.32%	7.29%	7.20%	7.08%	6.85%	6.55%	6.31%
RUT	$ Y $	4.65%	6.70%	7.56%	7.84%	7.83%	7.71%	7.55%	7.45%	7.15%	6.84%	6.55%
	$ {\boldsymbol{T^E}} $	48	54	60	66	72	78	84	90	96	108	120
NDX	$ Y $	6.48%	6.27%	6.05%	5.82%	5.66%	5.50%	5.34%	5.23%	5.09%	4.86%	4.68%
DJX	$ Y $	6.09%	5.92%	5.71%	5.56%	5.43%	5.28%	5.14%	5.03%	4.92%	4.71%	4.54%
RUT	$ Y $	6.27%	6.07%	5.87%	5.69%	5.53%	5.36%	5.23%	5.10%	4.98%	4.77%	4.57%

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Table 4. $ \bar{Y}^{AC} $ for different $ T^E $, computed as of January 23, 2020

$ {\boldsymbol{T^E}} $	3	6	9	12	15	18	21	24	30	36	42
$ \bar{Y}^{AC} $	6.24%	9.03%	10.39%	10.96%	11.06%	11.24%	11.23%	11.13%	11.00%	10.8%	10.67%
$ {\boldsymbol{T^E}} $	48	54	60	66	72	78	84	90	96	108	120
$ \bar{Y}^{AC} $	10.37%	10.15%	10.13%	9.92%	9.74%	9.69%	9.52%	9.37%	9.17%	8.99%	8.96%

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Table 5. eSSVI parameters calibrated to SPX, NDX, DJX, and RUT traditional, weekly, and end-of-month options as of January 23, 2020

	$ \boldsymbol{\eta} $	$ \boldsymbol{\lambda} $	$ \boldsymbol{\rho_m} $	$ \boldsymbol{\rho_0} $	$ \boldsymbol{a} $
SPX	0.6066	0.5929	-0.7564	-0.4245	651.0968
NDX	0.5692	0.5741	-0.7593	-0.3328	724.3588
DJX	0.5752	0.5771	-0.7715	-0.4323	200.0267
RUT	0.5706	0.5971	-0.6823	-0.3437	387.2297

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Table 6. Heston parameters calibrated to SPX, NDX, DJX, and RUT traditional options as of January 23, 2020

	$ \boldsymbol{\kappa} $	$ \boldsymbol{\theta} $	$ \boldsymbol{\eta} $	$ \boldsymbol{\rho} $	$ \boldsymbol{V_0} $
SPX	3.7764	0.0365	0.9555	-0.7946	0.0141
NDX	3.5723	0.0463	0.8791	-0.7784	0.0242
DJX	2.0432	0.0388	0.5728	-0.7919	0.0159
RUT	4.6137	0.0409	1.0805	-0.7206	0.0205

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Table 7. VIX levels, ATMF implied volatilities, and forwards versus different tenors $ T $ as of November 3, 2017, June 23, 2020, March 31, 2020, and March 17, 2020, respectively

		VIX		$ \boldsymbol{\sigma(F(0, T), T)} $					$ \boldsymbol{F(0, T)/S(0)} $
			$ T $	1/4	1/2	1	2	3	1/4	1/2	1	2	3
2017-11-03	SPX	9.14		0.0923	0.1098	0.1309	0.1525	0.1628	0.9995	0.9993	0.9990	1.0013	1.0052
	NDX			0.1350	0.1497	0.1647	0.1755	0.1905	1.0008	1.0028	1.0072	1.0188	1.0317
	DJX			0.1009	0.1167	0.1325	0.1436	0.1485	0.9971	0.9957	0.9955	0.9936	0.9933
	RUT			0.1397	0.1541	0.1708	0.1827	0.1911	1.0001	1.0006	1.0017	1.0057	1.0118
2020-06-23	SPX	31.37		0.2652	0.2717	0.2545	0.2381	0.2298	0.9969	0.9938	0.9876	0.9754	0.9640
	NDX			0.2809	0.2848	0.2666	0.2536	0.2485	0.9995	0.9984	0.9967	0.9924	0.9887
	DJX			0.2945	0.2914	0.2632	0.2434	0.2378	0.9955	0.9911	0.9839	0.9709	0.9564
	RUT			0.3575	0.3445	0.3009	0.2691	0.2543	0.9982	0.9966	0.9940	0.9878	0.9848
2020-03-31	SPX	53.54		0.3920	0.3469	0.2974	0.2554	0.2331	0.9942	0.9933	0.9945	0.9853	0.9784
	NDX			0.3948	0.3388	0.2921	0.2579	0.2415	0.9969	0.9969	0.9976	1.0003	1.0002
	DJX			0.4571	0.3874	0.3301	0.2938	0.2631	0.9919	0.9892	0.9805	0.9595	0.9464
	RUT			0.4430	0.3849	0.3273	0.2849	0.2650	0.9909	0.9890	0.9858	0.9837	0.9840
2020-03-17	SPX	75.91		0.5872	0.4779	0.3763	0.2944	0.2875	0.9861	0.9815	0.9771	0.9690	0.9600
	NDX			0.5776	0.4679	0.3682	0.3000	0.2696	0.9902	0.9886	0.9835	0.9774	0.9741
	DJX			0.6313	0.5026	0.3960	0.3357	0.3340	0.9832	0.9769	0.9659	0.9441	0.9278
	RUT			0.5941	0.4874	0.3834	0.3195	0.2931	0.9817	0.9786	0.9716	0.9591	0.9483

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Table 8. eSSVI and Heston parameters calibrated to options data in different volatility regimes for SPX, NDX, DJX, and RUT

		eSSVI					Heston
		$ \boldsymbol{\eta} $	$ \boldsymbol{\lambda} $	$ \boldsymbol{\rho_m} $	$ \boldsymbol{\rho_0} $	$ \boldsymbol{a} $	$ \boldsymbol{\kappa} $	$ \boldsymbol{\theta} $	$ \boldsymbol{\eta} $	$ \boldsymbol{\rho} $	$ \boldsymbol{V_0} $
2017-11-03	SPX	0.7303	0.5981	-0.7126	-0.3635	676.32	2.8555	0.0401	1.0806	-0.7473	0.0046
	NDX	0.6122	0.5816	-0.6759	-0.1902	513.85	4.4025	0.0428	1.0791	-0.6989	0.0135
	DJX	0.6499	0.6135	-0.6243	-0.0435	589.12	3.9711	0.0325	1.0894	-0.6900	0.0052
	RUT	0.6504	0.5918	-0.6141	-0.2397	364.16	4.3905	0.0464	1.2418	-0.6534	0.0173
2020-06-23	SPX	0.9664	0.4963	-0.7633	-0.2269	340.68	2.0230	0.0964	1.3275	-0.8176	0.1052
	NDX	0.9439	0.4900	-0.6838	-0.2543	306.51	1.7808	0.1040	1.2268	-0.7328	0.1126
	DJX	0.7798	0.5500	-0.7816	-0.3627	83.62	1.7819	0.0860	1.1351	-0.8111	0.1284
	RUT	1.1080	0.4078	-0.7628	-0.3608	120.76	1.7505	0.1032	1.3331	-0.7843	0.1911
2020-03-31	SPX	0.7710	0.5231	-0.8321	-0.4816	137.92	4.6048	0.0805	2.2566	-0.8245	0.3137
	NDX	0.7687	0.5227	-0.7706	-0.2908	212.56	4.2299	0.0766	2.0220	-0.7809	0.2971
	DJX	1.1234	0.4049	-0.6938	-0.5099	4854.88	3.8431	0.0882	2.1734	-0.7513	0.4058
	RUT	1.1092	0.4148	-0.7463	-0.4252	125.98	4.6056	0.1118	2.8350	-0.7620	0.4234
2020-03-17	SPX	1.1246	0.3460	-0.9227	-0.6585	120.92	6.0662	0.0940	3.4464	-0.8911	0.8440
	NDX	0.6885	0.4514	-0.9641	-0.9273	3841.28	6.4882	0.0780	2.4367	-0.8998	0.7813
	DJX	1.2104	0.2882	-0.8558	-0.8092	1449.55	5.8442	0.0938	3.7716	-0.8190	1.0221
	RUT	0.9665	0.3153	-0.9951	-0.9929	1449.54	5.5594	0.1057	2.6639	-0.9176	0.7730

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