Add rough Heston stochastic volatility model

docs/api/sp/index.md

@@ -32,6 +32,7 @@ This page gives an overview of all stochastic processes available in the library
 | Process | Description |
 |---|---|
 | [Heston][quantflow.sp.heston.Heston] | Classical square-root stochastic volatility model |
+| [RoughHeston][quantflow.sp.rough_heston.RoughHeston] | Rough (fractional) Heston model with Hurst exponent H < 1/2 |
 | [HestonJ][quantflow.sp.heston.HestonJ] | Heston model with compound Poisson jumps |
 | [DoubleHeston][quantflow.sp.heston.DoubleHeston] | Two independent Heston variance processes |
 | [DoubleHestonJ][quantflow.sp.heston.DoubleHestonJ] | Double Heston with compound Poisson jumps on the first component |

docs/api/sp/rough_heston.md

@@ -0,0 +1,3 @@
+# Rough Heston process
+
+::: quantflow.sp.rough_heston.RoughHeston

docs/bibliography.md

@@ -86,6 +86,14 @@ Wang X., He X., Zhao Y., Zuo Z. (2017) [Parameter Estimations of Heston Model Ba
 
 <div class="bib-entry" markdown>
 
+#### eleuch_rosenbaum
+
+Omar El Euch, Mathieu Rosenbaum. (2019) [The characteristic function of rough Heston models](https://doi.org/10.1111/mafi.12173){target="_blank" rel="noopener"}, Mathematical Finance, 29(1):3-38
+
+</div>
+
+<div class="bib-entry" markdown>
+
 #### gamma-ou
 
 P. Sabino, C. Petroni. (2021) [Gamma Related Ornstein-Uhlenbeck Processes and their Simulation](https://doi.org/10.1080/00949655.2020.1842408){target="_blank" rel="noopener"}, Journal of Statistical Computation and Simulation, 91(6)
@@ -102,6 +110,14 @@ Jim Gatheral, Antoine Jacquier. (2014) [Arbitrage-free SVI volatility surfaces](
 
 <div class="bib-entry" markdown>
 
+#### gatheral_jaisson_rosenbaum
+
+Jim Gatheral, Thibault Jaisson, Mathieu Rosenbaum. (2018) [Volatility is rough](https://doi.org/10.1080/14697688.2017.1393551){target="_blank" rel="noopener"}, Quantitative Finance, 18(6):933-949
+
+</div>
+
+<div class="bib-entry" markdown>
+
 #### gatheral_svi
 
 Jim Gatheral. (2004) [A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives](https://faculty.baruch.cuny.edu/jgatheral/madrid2004.pdf){target="_blank" rel="noopener"}

docs/examples/rough_heston_pricer.py

@@ -0,0 +1,37 @@
+"""Rough Heston: price a short-maturity smile and compare with the Heston model.
+
+The rough model (Hurst H < 1/2) produces a steeper short-maturity skew than the
+classical Heston model with the same parameters, without adding jumps.
+"""
+
+from quantflow.options.inputs import OptionType
+from quantflow.options.pricer import OptionPricer, OptionPricingMethod
+from quantflow.sp.heston import Heston
+from quantflow.sp.rough_heston import RoughHeston
+
+TTM = 0.1
+
+rough = OptionPricer(
+    model=RoughHeston.create(vol=0.5, kappa=1.5, sigma=0.7, rho=-0.6, hurst=0.1),
+    method=OptionPricingMethod.COS,
+)
+heston = OptionPricer(
+    model=Heston.create(vol=0.5, kappa=1.5, sigma=0.7, rho=-0.6),
+    method=OptionPricingMethod.COS,
+)
+
+
+def implied_vol(pricer: OptionPricer, strike: float) -> float:
+    option_type = OptionType.PUT if strike < 100.0 else OptionType.CALL
+    price = pricer.price(option_type=option_type, strike=strike, forward=100.0, ttm=TTM)
+    return float(price.black.iv)
+
+
+print(f"Short-maturity smile at ttm={TTM} (forward=100)")
+print(f"{'strike':>8}{'rough IV':>12}{'heston IV':>12}")
+for strike in (90.0, 95.0, 100.0, 105.0, 110.0):
+    print(
+        f"{strike:>8.1f}"
+        f"{implied_vol(rough, strike):>12.4f}"
+        f"{implied_vol(heston, strike):>12.4f}"
+    )

docs/references.bib

@@ -90,6 +90,17 @@ @article{ekf
   url={https://www.sciencedirect.com/science/article/pii/S2405896317324758},
 }
 
+@article{eleuch_rosenbaum,
+  title={The characteristic function of rough Heston models},
+  author={Omar El Euch and Mathieu Rosenbaum},
+  journal={Mathematical Finance},
+  year={2019},
+  volume={29},
+  number={1},
+  pages={3--38},
+  url={https://doi.org/10.1111/mafi.12173},
+}
+
 @article{gamma-ou,
   title={Gamma Related Ornstein-Uhlenbeck Processes and their Simulation},
   author={Sabino, P. & Petroni, C.},
@@ -111,6 +122,17 @@ @article{gatheral_jacquier
   url={https://doi.org/10.1080/14697688.2013.819986},
 }
 
+@article{gatheral_jaisson_rosenbaum,
+  title={Volatility is rough},
+  author={Jim Gatheral and Thibault Jaisson and Mathieu Rosenbaum},
+  journal={Quantitative Finance},
+  year={2018},
+  volume={18},
+  number={6},
+  pages={933--949},
+  url={https://doi.org/10.1080/14697688.2017.1393551},
+}
+
 @misc{gatheral_svi,
   title={A parsimonious arbitrage-free implied volatility parameterization
          with application to the valuation of volatility derivatives},

mkdocs.yml

@@ -96,6 +96,7 @@ nav:
           - Jump Diffusion: api/sp/jump_diffusion.md
           - Ornstein-Uhlenbeck: api/sp/ou.md
           - Poisson Process: api/sp/poisson.md
+          - Rough Heston Model: api/sp/rough_heston.md
           - Wiener Process: api/sp/wiener.md
           - Copulas: api/sp/copula.md
       - Timeseries Analysis: