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Replication and Pricing in the Binomial Tree Model
| Content Provider | Scilit |
|---|---|
| Author | Campolieti, Giuseppe Makarov, Roman N. |
| Copyright Year | 2018 |
| Description | Book Name: Financial Mathematics |
| Abstract | By combining the probabilistic framework in Chapter 6 with the main formal concepts of derivative asset pricing presented for the single-period model in Chapter 5, we are now ready to formally discuss derivative asset pricing within the multi-period binomial tree model. Let us begin by recalling the salient features of the standard T-period (recombining) binomial tree model on the space ( Ω , ℙ , ℱ , F ) with two assets, namely, a risky stock S and a risk-free asset B, such as a bank account or zero-coupon bond. The model is specified as follows. The time is discrete: t ∈ {0, 1, 2,…, T}. There are $2^{ T }$ possible market scenarios: Ω ≡ Ω T = { ω = ω 1 ω 2 ⋅ ⋅ ⋅ ω T : ω t ∈ { D, U } , t = 1 , 2 , … , T } , where each scenario can be represented by a path in a multi-period recombining binomial tree. The set of events is the power set ℱ = 2 Ω . The probability function ℙ : ℱ → [ 0 , 1 ] is given by (7.1) ℙ ( E ) = ∑ ω ∈ E ℙ ( ω ) , E ∈ ℱ , where ℙ ( ω ) = ℙ ( { ω } ) = p # U ( ω ) ( 1 − p ) # D ( ω ) , and p ∈ (0, 1) is a probability of the event ${ω_{t}$ = U} = {ω ∈ Ω : $ω_{t}$ = U} for every t = 1, 2,…, T. The flow of information is described by the filtration F = { ℱ t } 0 ⩽ t ⩽ T , where ℱ 0 = 0 and ℱ t is generated by the first t market moves ω$ _{1}$,…, $ω_{t}$ for every t = 1, 2,…, T, i.e., ℱ t = σ ( P t ) , where the partition P t is a collection of atoms of the form A ω 1 * , ω 2 * , … , ω t * = { ω ∈ Ω : ω n = ω n * for all n = 1,2, … , t } , ω 1 * , ω 2 * , … , ω t * ∈ { D, U } (in particular, ℱ T ≡ ℱ = 2 Ω ). The stochastic stock price process, ${S_{t}$ $}_{0⩽t⩽T }$, which is adapted to the filtration F , is given by the recurrence S t ( ω ) = S t − 1 ( ω ) u # U ( ω t ) d # D ( ω t ) , t = 1 , 2 , … , T , or, equivalently, by the relationship S t ( ω ) = S 0 u U t ( ω ) d D t ( ω ) , t = 0 , 1 , 2 , … , T , 258where $U_{ t }$(ω) = #U(ω$ _{1}$, ω$ _{2}$,…, $ω_{t}$ ) and $D_{ t }$(ω) = #D(ω$ _{1}$, ω$ _{2}$,…, $ω_{t}$ ) count, respectively, the number of downward and upward market moves; d and u are, respectively, downward and upward market movement factors which satisfy 0 < d < u; and ω ∈ $Ω_{ T }$ is a market scenario. The initial price of the stock, S$ _{0}$ > 0, is known. The deterministic price process, ${B_{t}$ $}_{0⩽t⩽T }$, for the risk-free asset is given by B t = B 0 ( 1 + r ) t , t = 0 , 1 , 2 , … , T , where r > 0 is a one-period return. With loss of generality, we assume that we deal with a bank account such that B$ _{0}$ = 1. Note that for a unit zero-coupon bond paying $1 at time T, the initial value is B$ _{0}$ = (1 + $r)^{−T }$. |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2011-0-08796-5&isbn=9781315373768&doi=10.1201/9781315373768-7&format=pdf |
| Ending Page | 306 |
| Page Count | 50 |
| Starting Page | 257 |
| DOI | 10.1201/9781315373768-7 |
| 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 Function Model Price Stock Adapted Risk Free Asset |
| Content Type | Text |
| Resource Type | Chapter |
