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Risk-Neutral Pricing in a Multi-Asset Economy
| Content Provider | Scilit |
|---|---|
| Author | Campolieti, Giuseppe Makarov, Roman N. |
| Copyright Year | 2018 |
| Description | Book Name: Financial Mathematics |
| Abstract | In the previous chapter we considered continuous-time risk-neutral derivative pricing in the simplest so-called (B, S) economy consisting of only one risky asset (stock) S and a risk-free bond or money market account B. We now extend the (B, S) economy to a continuoustime model of an economy consisting of an arbitrary number n ⩾ 1 of tradable risky base assets as well as a money market account. The simplest way to extend the classical (B, S) model to include multiple risky base assets is to assume that they are all independent of one another. For example, we can let each base asset price process be an Itô process, such as a geometric Brownian motion (GBM), driven by an independent Brownian motion (BM). Of course, this is a trivial and not very interesting extension. A more realistic extension is a model where the base assets are correlated stochastic processes. In particular, in this chapter, we shall remain within the framework of multidimensional correlated Itô processes. The interdependence among the base asset price processes arises by forming correlations among BMs that drive each price process. As was learned in Chapter 11, such correlations can be constructed by taking linear combinations of several, say d, independent BMs. The multidimensional GBM process is one such model for describing n stock price processes driven by d independent BMs. In all derivations of explicit pricing formulae for multi-asset derivatives considered in this chapter, we assume the “classical” multidimensional GBM model. This model offers fairly simple analytical tractability for many standard options written on multiple stocks. However, we shall first develop the pricing and hedging theory in a more general multidimensional continuous-time framework where all base asset price processes are quite general correlated Itô processes. We then simplify the framework to the classical multidimensional GBM model for the risky assets and also assume nonrandom interest rates and stock dividends. This allows us to price several standard (as well as some path dependent) European-style derivatives whose payoffs are dependent on the prices of multiply correlated assets. As in the case of a single asset, for non-path-dependent European-style derivatives the Markov property reduces the pricing problem to a conditional expectation where the multidimensional (discounted) Feynman–Kac formula leads to the Black–Scholes–Merton PDE for multi-asset contracts. Section 11.10 of Chapter 11 gives us this connection between the SDEs of Itô processes and the corresponding PDE problem. |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2011-0-08796-5&isbn=9781315373768&doi=10.1201/9781315373768-13&format=pdf |
| Ending Page | 592 |
| Page Count | 40 |
| Starting Page | 553 |
| DOI | 10.1201/9781315373768-13 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2018-10-24 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Financial Mathematics Operations Research and Management Science Economy Continuous Time Risky Asset Correlated Assets Correlated Itô |
| Content Type | Text |
| Resource Type | Chapter |
