Pricing vulnerable fader options under stochastic volatility models

Xingchun Wang

doi:10.3934/jimo.2022193

Article Contents

2023, Volume 19, Issue 8: 5749-5766. Doi: 10.3934/jimo.2022193

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Pricing vulnerable fader options under stochastic volatility models

Xingchun Wang

School of International Trade and Economics, University of International Business and Economics, Beijing 100029, China

Received: May 2022

Revised: September 2022

Early access: September 30, 2022

Published online: September 30, 2022

This study was supported by the National Natural Science Foundation of China (No. 11701084).

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Abstract

In this paper, we incorporate default risk into Heston's stochastic volatility model and focus on the valuation of vulnerable fader options. Fader options are path-dependent derivatives, depending on the time the underlying asset price stays inside a given range. We obtain an explicit pricing formula of vulnerable fader options, including fader options and (vulnerable) European options as special cases. Finally, we illustrate the effect of stochastic volatility and default risk on fader option prices. Specially, an inverted U-shaped curve is observed when we keep initial levels of the default intensity constant, but change the relative proportions of two factors in the default intensity.

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Mathematics Subject Classification: Primary: 60G51, 91B70; Secondary: 65K10.

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Figure 1. Fader option prices against strike prices. The solid, dashed, dotted and dot-dashed lines correspond to the prices in Heston's stochastic volatility with $ N = 1 $, Heston's stochastic volatility with $ N = 3 $, the Black-Scholes model with $ N = 1 $, and the Black-Scholes model with $ N = 3 $, respectively

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Figure 2. Fader option prices against the values of $ L $. The solid, dashed, dotted and dot-dashed lines correspond to the prices in Heston's stochastic volatility with $ N = 1 $, Heston's stochastic volatility with $ N = 3 $, the Black-Scholes model with $ N = 1 $, and the Black-Scholes model with $ N = 3 $, respectively

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Figure 3. Fader option prices against the values of $ H $. The solid, dashed, dotted and dot-dashed lines correspond to the prices in Heston's stochastic volatility with $ N = 1 $, Heston's stochastic volatility with $ N = 3 $, the Black-Scholes model with $ N = 1 $, and the Black-Scholes model with $ N = 3 $, respectively

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Figure 4. Fader option prices against strike prices. The dotted, dashed, solid and dot-dashed lines correspond to fader option prices with no default, vulnerable fader option prices with $ \theta = 0.70 $, vulnerable fader option prices with $ \theta = 0.50 $, and vulnerable fader option prices with $ \theta = 0.30 $, respectively

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Figure 5. Fader option prices against the values of $ L $. The dotted, dashed, solid and dot-dashed lines correspond to fader option prices with no default, vulnerable fader option prices with $ \theta = 0.70 $, vulnerable fader option prices with $ \theta = 0.50 $, and vulnerable fader option prices with $ \theta = 0.30 $, respectively

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Figure 6. Fader option prices against the values of $ H $. The dotted, dashed, solid and dot-dashed lines correspond to fader option prices with no default, vulnerable fader option prices with $ \theta = 0.70 $, vulnerable fader option prices with $ \theta = 0.50 $, and vulnerable fader option prices with $ \theta = 0.30 $, respectively

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Figure 7. Fader option prices against the values of $ \beta $. The dotted, dashed, solid and dot-dashed lines correspond to fader option prices with no default, vulnerable fader option prices with $ \theta = 0.70 $, vulnerable fader option prices with $ \theta = 0.50 $, and vulnerable fader option prices with $ \theta = 0.30 $, respectively

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Figure 8. Fader option prices against the values of $ \beta $ when $ \beta Y_0+X_0 $ is kept unchanged. The dotted, dashed, solid and dot-dashed lines correspond to fader option prices with no default, vulnerable fader option prices with $ \theta = 0.70 $, vulnerable fader option prices with $ \theta = 0.50 $, and vulnerable fader option prices with $ \theta = 0.30 $, respectively

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Figure 9. Default probability against the values of $ \beta $ when $ \beta Y_0+X_0 $ is kept unchanged

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Table 1. Vanilla and fader option prices obtained using Monte Carlo simulation methods and the derived pricing formula. The standard errors of the simulations are listed in brackets, and the CPU times are in seconds

	Price	CPU	MC	CPU
Panel A	Vanilla Options
Base case	10.758	0.090	10.862 [0.0411]	122.49
$ K=95 $	13.935	0.090	13.967 [0.0456]	122.38
$ K=105 $	7.9845	0.101	8.1233 [0.0364]	122.62
$ L=85 $	10.758	0.090	10.862 [0.0411]	122.49
$ L=95 $	10.758	0.090	10.862 [0.0411]	122.49
$ H=105 $	10.758	0.090	10.862 [0.0411]	122.49
$ H=115 $	10.758	0.090	10.862 [0.0411]	122.49
Panel B	Fade-in Options
Base case	4.1540	5.448	4.1980 [0.0413]	123.20
$ K=95 $	5.8358	5.357	5.8039 [0.0311]	123.17
$ K=105 $	2.7527	5.557	2.8006 [0.0208]	123.65
$ L=85 $	4.2829	5.671	4.3073 [0.0259]	123.06
$ L=95 $	3.8671	5.387	3.8201 [0.0250]	123.00
$ H=105 $	2.1595	5.391	2.1745 [0.0192]	123.21
$ H=115 $	6.4331	5.692	6.4236 [0.0312]	123.47

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