M.Sc Student | Kavaler Itay |
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Subject | On Implied Binomial Trees with a Non Constant Interest Rate Dynamics |

Department | Department of Applied Mathematics |

Supervisors | Professor Gershon Wolansky |

Professor Haim Reisman |

The paper examines a market for a stock, discount bonds of all maturities

and European calls and puts on the stock of all strikes and all maturities. It

derives a discrete time arbitrage free model. Said model is implemented in

the binomial framework world in which both stock and bonds dynamics are

determined so that risk neutral prices of the calls, exhibit the above smile

and put prices are determined by the put-call parity. Fitting is done by

deriving an algorithm which allows the interest rate process to be chosen

specifically in order to generate fitting while keeping the stock's volatility

constant. As a result the initial market's smile is allowed to be preserved

forever, independently of time and state. The idea of using both stock and

bonds in order to derive the local fit (and not viewing the bond dynamics

as a given) is new. In this paper we extend the standards implied binomial

models, to obtain a more flexible model which is calibrated with market

data on European puts and calls. Constructing implied trees from a given

smile along with a requirement for the model prices to be fair, generally

involves solving a set of the well known CRR-equations under the restriction

that the probability at each node is risk-neutral. The success of having a

unique solution depends, among others, on the smile's shape and can be

easily violated by a quite steep one. Our model exploits the implied tree

structure, which is extended to incorporate flexible bond process, in order to

achieve a stable risk neutrality procedure.