Travelling wave solutions of an equation of Harry Dym type arising in the Black-Scholes framework
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| Format: | Preprint |
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2024
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| author | Zubelli, Jorge P. Singh, Kuldeep Albani, Vinicius Kourakis, Ioannis |
| author_facet | Zubelli, Jorge P. Singh, Kuldeep Albani, Vinicius Kourakis, Ioannis |
| contents | The Black-Scholes framework is crucial in pricing a vast number of financial instruments that permeate the complex dynamics of world markets. Associated with this framework, we consider a second-order differential operator $L(x, {\partial_x}) := v^2(x,t) (\partial_x^2 -\partial_x)$ that carries a variable volatility term $v(x,t)$ and which is dependent on the underlying log-price $x$ and a time parameter $t$ motivated by the celebrated Dupire local volatility model. In this context, we ask and answer the question of whether one can find a non-linear evolution equation derived from a zero-curvature condition for a time-dependent deformation of the operator $L$. The result is a variant of the Harry Dym equation for which we can then find a family of travelling wave solutions. This brings in extensive machinery from soliton theory and integrable systems. As a by-product, it opens up the way to the use of coherent structures in financial-market volatility studies. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19020 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Travelling wave solutions of an equation of Harry Dym type arising in the Black-Scholes framework Zubelli, Jorge P. Singh, Kuldeep Albani, Vinicius Kourakis, Ioannis Numerical Analysis Mathematical Finance 35Q51, 35C08, 91G20, 35Q91 The Black-Scholes framework is crucial in pricing a vast number of financial instruments that permeate the complex dynamics of world markets. Associated with this framework, we consider a second-order differential operator $L(x, {\partial_x}) := v^2(x,t) (\partial_x^2 -\partial_x)$ that carries a variable volatility term $v(x,t)$ and which is dependent on the underlying log-price $x$ and a time parameter $t$ motivated by the celebrated Dupire local volatility model. In this context, we ask and answer the question of whether one can find a non-linear evolution equation derived from a zero-curvature condition for a time-dependent deformation of the operator $L$. The result is a variant of the Harry Dym equation for which we can then find a family of travelling wave solutions. This brings in extensive machinery from soliton theory and integrable systems. As a by-product, it opens up the way to the use of coherent structures in financial-market volatility studies. |
| title | Travelling wave solutions of an equation of Harry Dym type arising in the Black-Scholes framework |
| topic | Numerical Analysis Mathematical Finance 35Q51, 35C08, 91G20, 35Q91 |
| url | https://arxiv.org/abs/2412.19020 |