文档介绍：ASSIGNMENT OF
CF PUTATIONAL METHODS FOR
RISK MANAGEMENT AND FINACIAL ENGINEERING
Lecturer: Olaf Menkens
Group D : Chen, Ying
Lei, Yuanyuan
ABSTRACT
In this paper we explain how to use trinomial trees to price options by different methods and for both European-style option and American-style option. First, we show how trees can be spanned using a set of general branching processes. Secondly, we discuss the advantages and the disadvantages of the trinomial tree in pricing options. Finally, we show what cause pitfalls of trinomial tree. There are two possible reasons, both of them can be solved.
Section one: theoretical background
Model setup
We assume that the market plete and arbitrage opportunities are absent. Then, the expected return from all traded securities is the risk-free interest rate, and future cash flows can be valued by discounting their expected values at the risk-free interest rate. Consider the evaluation of an option on a non-dividend-paying stock. We start by dividing the life of the option into a large number of small time intervals of length T/n. Suppose that the initial value of the stock is S0, in each time interval being at the S0, the price can move to one of three nodes: (i) to the upper node with value S0u with probability q1; (ii) to the lower node with value S0d with probability q2; and (iii) to the middle node with value S0 with probability 1-(q1 + q2) at each node and δt is the length of the time step. Shown as Figure :
S0u²
S0u S0u
S0 S0 S0
S0d S0d
S0d²
Figure Nodes in a Trinomial trees
2. State the Tree
Normally there are five unknown parameters: two transition possibilities q1 and q2 and three prices Si, Si+1 and Si+2 at new nodes as we start from a node Sn,i at each time tn. To determine them we need to introduce the notation and main requirement a tree should satisfy. First of all, let Fi denote the known forward price of the spot price Sn,i and λn,i the known Arrow-Debreu price at node(n,i). The Arrow-