**I**n this chapter, we will introduce some basic concepts of equity modeling. We will discuss how the stock price can be modeled in a framework with deterministic interest rates, dividends, and default probabilities and how a given implied volatility surface can be matched with Dupire's “implied local volatility.” We also mention alternatives and how European payoffs whose value depends only on the stock price on a single maturity can be priced independent of further model assumptions by hedging with European options. We also make a few remarks on theoretical aspects of replication. This chapter is the foundation of chapter 2, where we will discuss applications: various stochastic volatility models, pricing of Cliquets, variance swaps, and related products and models to price options on variance. The assumptions of deterministic interest rates and default risk probabilities are then subsequently relaxed in the later chapters of this book.

Since the main focus of this chapter is the modeling of the pure equity risk, we will work with a framework where interest rates, dividends, and default risk are deterministic. We will model the stock price on a stochastic base The measure is the “historic” measure. We denote by *r* = (*r*_{t})_{t≥0} the deterministic ...

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