Pricing, Risk and Volatility in Subordinated Market Models
Abstract
:1. Introduction
1.1. Time Subordination in Financial Modelling
1.2. Contributions of the Paper
- (a)
- demonstrate that the recent pricing formulas for the VG, NIG and FD models are precise and fast converging, and can be successfully used for other applications (e.g., calculations of volatility curve);
- (b)
- provides efficient closed-form formulas for first and second-order risk sensitivities (Delta, Gamma) and compare them with numerical techniques; and,
- (c)
- deduce from these formulas several practical features regarding delta-hedging policies and portfolio performance.
1.3. Structure of the Paper
2. Exponential Lévy Processes
2.1. Basics of Lévy Processes
2.2. Exponential Lévy Motions
2.3. Option Pricing
3. Subordinated Models
3.1. Exponential VG Model
3.1.1. Model Characteristics
3.1.2. Financial Applications
3.2. Exponential NIG Model
3.2.1. Model Characteristics
3.2.2. Financial Applications
3.3. Fractional Diffusion Model
3.3.1. Lévy-Stable Processes and Fractional Derivatives
3.3.2. Model Characteristics
3.3.3. Financial Applications
4. Pricing and Volatility Modelling
- -
- (OTM price) If ,
- -
- (ITM price) If ,
- -
- (ATM price) If ,
4.1. At-the-Money Forward Approximations
4.2. Implied Volatility
5. First-Order Sensitivities
- -
- (OTM sensitivity) If ,
- -
- (ITM sensitivity) If ,
- -
- (ATM sensitivity) If ,
5.1. Delta Hedging
5.2. Comparisons with Numerical Techniques
6. Second-Order Sensitivities and Portfolio Performance
6.1. Gamma, Dollar Gamma
- -
- (OTM sensitivity) If ,
- -
- (ITM sensitivity) If ,
- -
- (ATM sensitivity) If ,
6.2. Properties and Particular Cases
7. Concluding Remarks
- (a)
- The pricing formulas are smooth and fast converging, and provide excellent agreement with efficient numerical techniques (such as the PROJ method). Moreover, these formulas can provide useful approximations for at-the-money options, and allow for the construction of volatility curves.
- (b)
- We have derived several analytical formulas for risk sensitivities and shown that they also provide excellent agreement with standard numerical (Fourier) evaluations.
- (c)
- Thanks to these formulas, we were able to show that the presence of a time subordination in the VG, NIG, and FD models has a minimal impact on the delta hedging policy of an at-the-money option, but, on the contrary, has a direct impact on the P&L of the delta hedged portfolio.
Author Contributions
Funding
Conflicts of Interest
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Subordinated Model | Limiting Regimes |
---|---|
VG | VG BS |
NIG | NIG BS |
FD | FD FMLS BS |
sub-BS(,) FD | sub-BS BS |
ATMF Implied Volatility (European Options) | |
---|---|
Exponential VG | |
Low variance regime (): | |
Exponential NIG | Solve |
Large steepness regime (): | |
At order : | |
sub-BS | |
Non-fractional regime (): | |
Exponential VG Model [, ] | |||||
---|---|---|---|---|---|
Formula (2) | Lewis (74) | ||||
Deep OTM () | 2.1823 | 0.6347 | 0.0941 | 0.0940 | 0.0940 |
OTM () | 0.4113 | 0.2567 | 0.2455 | 0.2455 | 0.2455 |
ATM () | 0.5703 | 0.5718 | 0.5719 | 0.5719 | 0.5719 |
ITM () | 0.7569 | 0.8113 | 0.8134 | 0.8134 | 0.8134 |
Deep ITM () | 0.4729 | 0.8589 | 0.9206 | 0.9206 | 0.9206 |
Exponential NIG Model [, ] | |||||
Formula (2) | Lewis (74) | ||||
Deep OTM () | 0.2921 | 0.2722 | 0.2747 | 0.2748 | 0.2748 |
OTM () | 0.4289 | 0.4309 | 0.4311 | 0.4311 | 0.4311 |
ATM () | 0.6336 | 0.6410 | 0.6412 | 0.6412 | 0.6412 |
ITM () | 0.6936 | 0.7030 | 0.7033 | 0.7033 | 0.7033 |
Deep ITM () | 0.7827 | 0.7966 | 0.7971 | 0.7971 | 0.7971 |
1st Order () | 2nd Order () | |
---|---|---|
Exponential VG | ||
Low variance regime (): | ||
Exponential NIG | ||
Large steepness regime (): | ||
FD | ||
sub-BS | ||
Non fractional regime (): | ||
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Aguilar, J.-P.; Kirkby, J.L.; Korbel, J. Pricing, Risk and Volatility in Subordinated Market Models. Risks 2020, 8, 124. https://doi.org/10.3390/risks8040124
Aguilar J-P, Kirkby JL, Korbel J. Pricing, Risk and Volatility in Subordinated Market Models. Risks. 2020; 8(4):124. https://doi.org/10.3390/risks8040124
Chicago/Turabian StyleAguilar, Jean-Philippe, Justin Lars Kirkby, and Jan Korbel. 2020. "Pricing, Risk and Volatility in Subordinated Market Models" Risks 8, no. 4: 124. https://doi.org/10.3390/risks8040124