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Interest Rate Swaps and Their Derivatives: A Practitioner's Guide by AMIR SADR

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CHAPTER 5
Derivatives Pricing: Risk-Neutral Valuation
While derivatives (forwards, options) have been around for a long time, and many attempts were made to value them, the first successful pricing formula was derived by Fisher Black, Myron Scholes, and Robert Merton.1 The resulting formula is the celebrated Black-Scholes-Merton Formula, and was derived via an application of stochastic calculus by setting up and solving a partial differential equation relating the price of a derivative to the underlying. While the techniques used are rather daunting, the basic idea is simple and powerful: You can replicate an option payoff by taking a position in the underlying asset and financing this position. Therefore the value of an option is the value of its replicating portfolio. The only nuance is that the portfolio is not static, and needs to be dynamically rebalanced (delta-hedged) in response to changes in the underlying.
The original derivation of Black-Scholes-Merton Formula and its variants (Black′s Formula) somewhat obscured this dynamic. In a 1979 paper,2 Cox, Ross, and Rubenstein (CRR) distilled the replication argument to a simple binomial tree model, and showed that the Black-Scholes-Merton Formula can be obtained as the number of time steps in the tree tends to infinity. This constructive algorithm did away with the stochastic calculus machinery and highlighted the dynamic replicating portfolio.
Extensions of the CRR binomial tree model to more general settings were swift. ...

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