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Extending the Heston Model to Forecast Motor Vehicle Collision Rates


  • Darren Shannon
  • Grigorios Fountas


We present an alternative approach to the forecasting of motor vehicle collision rates. We adopt an oft-used tool in mathematical finance, the Heston Stochastic Volatility model, to forecast the short-term and long-term evolution of motor vehicle collision rates. We incorporate a number of extensions to the Heston model to make it fit for modelling motor vehicle collision rates. We incorporate the temporally-unstable and non-deterministic nature of collision rate fluctuations, and introduce a parameter to account for periods of accelerated safety. We also adjust estimates to account for the seasonality of collision patterns. Using these parameters, we perform a short-term forecast of collision rates and explore a number of plausible scenarios using long-term forecasts. The short-term forecast shows a close affinity with realised rates (over 95% accuracy), and outperforms forecasting models currently used in road safety research (Vasicek, SARIMA, SARIMA-GARCH). The long-term scenarios suggest that modest targets to reduce collision rates (1.83% annually) and targets to reduce the fluctuations of month-to-month collision rates (by half) could have significant benefits for road safety. The median forecast in this scenario suggests a 50% fall in collision rates, with 75% of simulations suggesting that an effective change in collision rates is observed before 2044. The main benefit the model provides is eschewing the necessity for setting unreasonable safety targets that are often missed. Instead, the model presents the effects that modest and achievable targets can have on road safety over the long run, while incorporating random variability. Examining the parameters that underlie expected collision rates will aid policymakers in determining the effectiveness of implemented policies.

Suggested Citation

  • Darren Shannon & Grigorios Fountas, 2021. "Extending the Heston Model to Forecast Motor Vehicle Collision Rates," Papers 2104.11461,, revised May 2021.
  • Handle: RePEc:arx:papers:2104.11461

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    References listed on IDEAS

    1. Najaf, Pooya & Thill, Jean-Claude & Zhang, Wenjia & Fields, Milton Greg, 2018. "City-level urban form and traffic safety: A structural equation modeling analysis of direct and indirect effects," Journal of Transport Geography, Elsevier, vol. 69(C), pages 257-270.
    2. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    3. Arismendi, Juan C. & Back, Janis & Prokopczuk, Marcel & Paschke, Raphael & Rudolf, Markus, 2016. "Seasonal Stochastic Volatility: Implications for the pricing of commodity options," Journal of Banking & Finance, Elsevier, vol. 66(C), pages 53-65.
    4. Claus, Stefan & Wiltshire, Chris & Silk, Nicholas, 2017. "Potential impacts of autonomous vehicles on the UK insurance sector," Bank of England Quarterly Bulletin, Bank of England, vol. 57(1), pages 40-48.
    5. Clifford Winston & Vikram Maheshri & Fred Mannering, 2006. "An exploration of the offset hypothesis using disaggregate data: The case of airbags and antilock brakes," Journal of Risk and Uncertainty, Springer, vol. 32(2), pages 83-99, March.
    6. Fabian Pütz & Finbarr Murphy & Martin Mullins, 2019. "Driving to a future without accidents? Connected automated vehicles’ impact on accident frequency and motor insurance risk," Environment Systems and Decisions, Springer, vol. 39(4), pages 383-395, December.
    7. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    8. Grzelak, Lech & Oosterlee, Kees, 2009. "On The Heston Model with Stochastic Interest Rates," MPRA Paper 20620, University Library of Munich, Germany, revised 18 Jan 2010.
    9. Kröger, Lars & Kuhnimhof, Tobias & Trommer, Stefan, 2019. "Does context matter? A comparative study modelling autonomous vehicle impact on travel behaviour for Germany and the USA," Transportation Research Part A: Policy and Practice, Elsevier, vol. 122(C), pages 146-161.
    10. Albert C. Bemmaor & Nicolas Glady, 2012. "Modeling Purchasing Behavior with Sudden "Death": A Flexible Customer Lifetime Model," Management Science, INFORMS, vol. 58(5), pages 1012-1021, May.
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    Cited by:

    1. Darren Shannon & Grigorios Fountas, 2022. "Amending the Heston Stochastic Volatility Model to Forecast Local Motor Vehicle Crash Rates: A Case Study of Washington, D.C," Papers 2203.01729,
    2. Nemanja Deretić & Dragan Stanimirović & Mohammed Al Awadh & Nikola Vujanović & Aleksandar Djukić, 2022. "SARIMA Modelling Approach for Forecasting of Traffic Accidents," Sustainability, MDPI, vol. 14(8), pages 1-18, April.
    3. Bianca Reichert & Adriano Mendonça Souza, 2022. "Can the Heston Model Forecast Energy Generation? A Systematic Literature Review," International Journal of Energy Economics and Policy, Econjournals, vol. 12(1), pages 289-295.

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