This project builds upon the foundational Black-Scholes model to explore the valuation of complex, path-dependent derivatives, specifically Barrier Options (Up-and-Out, Down-and-Out).
It bridges the gap between pure probability theory and financial engineering, establishing the duality between two quantitative approaches:
- Stochastic Calculus: Leveraging the Strong Markov Property, Girsanov's Theorem, and the Reflection Principle to determine the joint distribution of Brownian motion and its running maximum.
- Partial Differential Equations (PDEs): Using the Feynman-Kac theorem to derive the pricing equations and specific boundary conditions for knock-out events.
- The Reflection Principle: Mathematical determination of stopping times and the law of the maximum for standard and drifted Brownian motions.
- Risk-Neutral Valuation: Closed-form analytical pricing formulas for Up-and-Out (UAO) and Down-and-Out (DAO) call options.
- Model Limitations: Critical analysis of the constant volatility assumption (volatility smile), discrete price jumps, and Delta-hedging challenges near the barrier (Pin Risk).
- Real Market Application: Theoretical findings are confronted with real market dynamics using LVMH option data.
├── PRI_Barrier_Options_Report.pdf # Full mathematical and quantitative report
└── README.md
## 👥 Authors
* **Eliott Oster**
* **Addi Raphaël**
* **Tirard Thomas**
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*Developed as part of the GMM Department curriculum at INSA Toulouse.*