{"id":1439,"date":"2021-06-17T09:28:00","date_gmt":"2021-06-17T09:28:00","guid":{"rendered":"https:\/\/nag.com\/?post_type=insights&p=1141"},"modified":"2023-07-05T14:56:09","modified_gmt":"2023-07-05T14:56:09","slug":"optcorner-calibrate-the-heston-model-faster-using-derivative-free-optimization-techniques","status":"publish","type":"insights","link":"https:\/\/nag.com\/insights\/optcorner-calibrate-the-heston-model-faster-using-derivative-free-optimization-techniques\/","title":{"rendered":"Optcorner: Calibrate the Heston Model Faster Using Derivative-Free Optimization Techniques"},"content":{"rendered":"
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In this post, we demonstrate how DFO can speed up a practical calibration problem arising frequently in the finance industry<\/strong>: calibrating an option pricing model to market data.<\/p>\n

The following material is based on a recent webinar we presented, a recording is available\u00a0here<\/a>.<\/p>\n

Recent improvements in DFO solvers<\/h3>\n

DFO has seen enormous activity in the last years since our previous\u00a0DFO post<\/a>. n<\/i><\/span>AG collaborated on a research project with\u00a0Coralia Cartis<\/a>\u00a0and\u00a0Lindon Roberts<\/a>\u00a0from the University of Oxford. This research led to significant improvements, such as noise resilience and enhancements to interpolation models for structured problems, and brought the state of the art of DFO techniques [1] into the n<\/i><\/span>AG Library. We now offer a bound-constrained nonlinear least squares solver suitable for data fitting problems,\u00a0handle_solve_dfls<\/a>\u00a0(e04ff<\/a>), as well as one that is aimed at general nonlinear functions,\u00a0handle_solve_dfno<\/a>\u00a0(e04jd<\/a>). We also provide alternative interfaces (reverse communication<\/a>) to pass the function values to the solvers,\u00a0handle_solve_dfls_rcomm<\/a>\u00a0(e04fg<\/a>) and\u00a0handle_solve_dfno_rcomm<\/a>\u00a0(e04je<\/a>), respectively.<\/p>\n

Pricing European Options: calibrating the Heston model<\/h3>\n

A European option is a contract giving the buyer the option to buy (call) or sell (put) an underlying at a given price (strike) and expiration date (maturity). Pricing the option consists in attributing it a price based on the probability that the buyer will want to exercise the option at maturity.<\/p>\n

Many numerical pricing methods have been introduced throughout the years, the most common ones still being based on the Black-Scholes equations. However, the Black-Scholes model has some shortcomings, among which the assumption that the volatility is constant over time. It is in fact observed that market implied volatilities are smile-shaped. Stochastic volatility models such as Heston\u2019s were introduced to try to explain this shape. The model consists in the following set of equations:<\/p>\n <\/div>\n <\/div>\n<\/div>\n\n\n\n \n \n \n d<\/mi>\n \n \n S<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n <\/msub>\n <\/mtd>\n \n =<\/mo>\n μ<\/mi>\n \n \n S<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n <\/msub>\n \n \n d<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n <\/msub>\n +<\/mo>\n σ<\/mi>\n \n \n \n \n v<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n <\/msub>\n <\/mrow>\n <\/msqrt>\n \n \n S<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n <\/msub>\n d<\/mi>\n \n \n W<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n \n 1<\/mn>\n <\/mrow>\n <\/msubsup>\n <\/mtd>\n \n <\/mtr>\n \n \n d<\/mi>\n \n \n v<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n <\/msub>\n <\/mtd>\n \n =<\/mo>\n λ<\/mi>\n \n (<\/mo>\n \n 1<\/mn>\n –<\/mo>\n \n \n v<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n <\/msub>\n <\/mrow>\n )<\/mo>\n <\/mrow>\n d<\/mi>\n t<\/mi>\n +<\/mo>\n α<\/mi>\n \n \n \n \n v<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n <\/msub>\n <\/mrow>\n <\/msqrt>\n d<\/mi>\n \n \n W<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n \n 2<\/mn>\n <\/mrow>\n <\/msubsup>\n <\/mtd>\n <\/mtr>\n \n \n d<\/mi>\n \n \n W<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n \n 1<\/mn>\n <\/mrow>\n <\/msubsup>\n ⋅<\/mo>\n d<\/mi>\n \n \n W<\/mi>\n <\/mrow>\n \n t<\/mi>\n <\/mrow>\n \n 2<\/mn>\n <\/mrow>\n <\/msubsup>\n <\/mtd>\n \n =<\/mo>\n ρ<\/mi>\n d<\/mi>\n t<\/mi>\n <\/mtd>\n <\/mtr>\n <\/mtable>\n<\/math>\n\n\n
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where \\(S_{t}\\) is the spot price of the underlying \\(vt\\) is the time-dependent volatility and \\( (dW_{t}^1 , dW_{t}^2) \\) are two Brownian motions.<\/p>\n

This model still depends on 4 parameters that are not easy to directly observe in the market: the volatility scaling \\( \u03c3 \\) the mean reversion rate \\( \u03bb \\) the volatility of volatility \\(a\\) and the Brownian motion correlation \\(\u03c1\\). These parameters, therefore, need to be calibrated. This can be done by observing historical data and trying to optimize the parameters to make our model prediction match it as closely as possible. This is a great candidate for DFO due to the lack of readily available derivatives and relatively expensive evaluation.<\/p>\n

Introducing term structure in the Heston model<\/h4>\n

The Heston equations still assume that our 4 parameters are constant with respect to time, which does not necessarily correspond to market reality. To alleviate this problem, a possible extension is to add term structure: \\( \u03c3, \u03bb, \u03b1 \\) and \\(\u03c1\\) are only considered constant over fixed time periods. This allows the model to better mimic market behaviour but also multiplies the number of parameters to calibrate by the number of time periods considered.<\/p>\n

An implementation of the Heston model with term structure is available in the n<\/i><\/span>AG Library (opt_heston_term<\/a>,\u00a0s30ncf<\/a>) and is the one used in the following numerical experiments.<\/p>\n

Setting up the calibration problem<\/h4>\n

We have access to historical data in the foreign exchange market for the currency pair EUR-USD ranging from 2012 to 2016. For each date, we have data for 7 maturities (2m, 3m, 6m, 1y, 2y, 3y and 5y):<\/p>\n