Distribution Approach to Local Volatility for European Options in the Merton Model with Stochastic Interest Rates
The Dupire formula is a very useful tool for pricing financial derivatives. This paper is dedicated to deriving the aforementioned formula for the European call option in the space of distributions by applying a mathematically rigorous approach developed in our previous paper concerning the case of...
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MDPI AG
2025-03-01
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| author | Piotr Nowak Dariusz Gatarek |
| author_facet | Piotr Nowak Dariusz Gatarek |
| author_sort | Piotr Nowak |
| collection | DOAJ |
| description | The Dupire formula is a very useful tool for pricing financial derivatives. This paper is dedicated to deriving the aforementioned formula for the European call option in the space of distributions by applying a mathematically rigorous approach developed in our previous paper concerning the case of the Margrabe option. We assume that the underlying asset is described by the Merton jump-diffusion model. Using this stochastic process allows us to take into account jumps in the price of the considered asset. Moreover, we assume that the instantaneous interest rate follows the Merton model (1973). Therefore, in contrast to the models combining a constant interest rate and a continuous underlying asset price process, frequently observed in the literature, applying both stochastic processes could accurately reflect financial market behaviour. Moreover, we illustrate the possibility of using the minimal entropy martingale measure as the risk-neutral measure in our approach. |
| format | Article |
| id | doaj-art-818c07efcc0b4a9c95548bf6aeb0181d |
| institution | DOAJ |
| issn | 1099-4300 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj-art-818c07efcc0b4a9c95548bf6aeb0181d2025-08-20T02:42:32ZengMDPI AGEntropy1099-43002025-03-0127332010.3390/e27030320Distribution Approach to Local Volatility for European Options in the Merton Model with Stochastic Interest RatesPiotr Nowak0Dariusz Gatarek1Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, PolandSystems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, PolandThe Dupire formula is a very useful tool for pricing financial derivatives. This paper is dedicated to deriving the aforementioned formula for the European call option in the space of distributions by applying a mathematically rigorous approach developed in our previous paper concerning the case of the Margrabe option. We assume that the underlying asset is described by the Merton jump-diffusion model. Using this stochastic process allows us to take into account jumps in the price of the considered asset. Moreover, we assume that the instantaneous interest rate follows the Merton model (1973). Therefore, in contrast to the models combining a constant interest rate and a continuous underlying asset price process, frequently observed in the literature, applying both stochastic processes could accurately reflect financial market behaviour. Moreover, we illustrate the possibility of using the minimal entropy martingale measure as the risk-neutral measure in our approach.https://www.mdpi.com/1099-4300/27/3/320Dupire formulaEuropean optionMerton modellocal volatilitySchwartz distributions |
| spellingShingle | Piotr Nowak Dariusz Gatarek Distribution Approach to Local Volatility for European Options in the Merton Model with Stochastic Interest Rates Entropy Dupire formula European option Merton model local volatility Schwartz distributions |
| title | Distribution Approach to Local Volatility for European Options in the Merton Model with Stochastic Interest Rates |
| title_full | Distribution Approach to Local Volatility for European Options in the Merton Model with Stochastic Interest Rates |
| title_fullStr | Distribution Approach to Local Volatility for European Options in the Merton Model with Stochastic Interest Rates |
| title_full_unstemmed | Distribution Approach to Local Volatility for European Options in the Merton Model with Stochastic Interest Rates |
| title_short | Distribution Approach to Local Volatility for European Options in the Merton Model with Stochastic Interest Rates |
| title_sort | distribution approach to local volatility for european options in the merton model with stochastic interest rates |
| topic | Dupire formula European option Merton model local volatility Schwartz distributions |
| url | https://www.mdpi.com/1099-4300/27/3/320 |
| work_keys_str_mv | AT piotrnowak distributionapproachtolocalvolatilityforeuropeanoptionsinthemertonmodelwithstochasticinterestrates AT dariuszgatarek distributionapproachtolocalvolatilityforeuropeanoptionsinthemertonmodelwithstochasticinterestrates |