ABSTRACT
Let us denote F T an European payoff depending on the value S T of an asset at T. The main questions, that we will focus on, are
At which price should we sell this option? Is there a unique price?
After having sold this option, what should we do in order to reduce (i.e., hedge) our potential losses?
By working on a discrete-time setting, we will try to answer these two questions without relying on knowledge of stochastic analysis, but by using classical tools in optimization and in probability. In particular, we will formulate the problem of pricing and hedging of derivative products as some linear, quadratic and convex optimization problems. This consists in minimizing an utility function written on the profit and loss wealth value of a delta-hedged portfolio. In particular, we will insist on convex duality from which risk-neutral models emerge as dual variables. This convex (linear) duality will appear naturally in the next chapter and will be a key tool when we will discuss (M)OT. We will also highlight the model-dependence of our approach.
