Options-Theory
A project exploring various concepts related to European and American Options
Binomial Options Pricing Model
European Options and American Options
![image](https://private-user-images.githubusercontent.com/86458423/256248746-85377e80-fdf7-4983-b23a-ca870e92ba2b.png?jwt=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.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.Y8k_kj8AtG6rKIjm2dwJHg4y82usAU1eFL1GXLYkRgI)
![image](https://private-user-images.githubusercontent.com/86458423/256249115-cf5f4ef4-2b60-4321-889e-7b47bfcf4b61.png?jwt=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.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._d-Sw8E6JvCuo6xG7HtT6BHBT50a0-j2zvG4UnFFY_w)
Black Scholes Model and Implied Volatility
![image](https://private-user-images.githubusercontent.com/86458423/256249858-e24d64e7-8368-4101-9e9a-113d912d345d.png?jwt=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpc3MiOiJnaXRodWIuY29tIiwiYXVkIjoicmF3LmdpdGh1YnVzZXJjb250ZW50LmNvbSIsImtleSI6ImtleTUiLCJleHAiOjE3MDgxMjcwNTMsIm5iZiI6MTcwODEyNjc1MywicGF0aCI6Ii84NjQ1ODQyMy8yNTYyNDk4NTgtZTI0ZDY0ZTctODM2OC00MTAxLTllOWEtMTEzZDkxMmQzNDVkLnBuZz9YLUFtei1BbGdvcml0aG09QVdTNC1ITUFDLVNIQTI1NiZYLUFtei1DcmVkZW50aWFsPUFLSUFWQ09EWUxTQTUzUFFLNFpBJTJGMjAyNDAyMTYlMkZ1cy1lYXN0LTElMkZzMyUyRmF3czRfcmVxdWVzdCZYLUFtei1EYXRlPTIwMjQwMjE2VDIzMzkxM1omWC1BbXotRXhwaXJlcz0zMDAmWC1BbXotU2lnbmF0dXJlPWVkYWZjNWZhNDhkZWU2NDg3ZDM5MzU4MzdmYWFhY2E5ZGUxNTNlZmM2ZDVlYWVmMDgyYTE4YzI5ZDA4YWI0ZjYmWC1BbXotU2lnbmVkSGVhZGVycz1ob3N0JmFjdG9yX2lkPTAma2V5X2lkPTAmcmVwb19pZD0wIn0.YGmH1aAXIyr_S2MgT4bkOcUfBpg4a_Ipb2LQj3dmZVU)
Here we also compared Actual Volatility in the prices and Implied Volatility
Black Scholes Model as a Limiting case for Binomial Model
![image](https://private-user-images.githubusercontent.com/86458423/256252369-2f14c4be-6c94-4df6-bdef-acaec570ea84.png?jwt=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.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.kufiXe5YVTmjccpaSTd60eLGhI4FmudoykangnpnChA)
Delta Hedging, a self-financing strategy
![image](https://private-user-images.githubusercontent.com/86458423/256253956-392b558a-faeb-4abc-9a78-b8c16cd1695c.png?jwt=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.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.Rk0-RXEBeJ5UDpkBEmFX0Yo-7pBd__nKMF6RBnWFd3s)
Comparison of Pricing Models
![image](https://private-user-images.githubusercontent.com/86458423/256254445-6284d5ac-d761-4d9d-adc6-67d8ef8fe18f.png?jwt=eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.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.lxFL1Z-yYCl9BRRqDS_jKcAnoL_KKOcp_rBTVG8Hb8I)