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Pricing autocallables under local-stochastic volatility

  • *Corresponding author: Walter Farkas

    *Corresponding author: Walter Farkas 
Abstract / Introduction Full Text(HTML) Figure(29) / Table(8) Related Papers Cited by
  • 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.

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

    Citation:

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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

    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

    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 $

    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

    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} $

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

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

    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

    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

    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 $

    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^*}] $

    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 $

    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^*}] $

    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

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

    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 $

    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

    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 $

    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

    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 $

    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 $

    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^*}] $

    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^*}] $

    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^*}] $

    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^*}] $

    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 $

    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^*}] $

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

    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 $

    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).
     | Show Table
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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%
     | Show Table
    DownLoad: CSV

    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%
     | Show Table
    DownLoad: CSV

    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
     | Show Table
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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
     | Show Table
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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
     | Show Table
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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
     | Show Table
    DownLoad: CSV
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