On Implied Binomial Trees With A Non Constant Interest Rate Dynamics

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 suggests a different approach by considering an optimization procedure. The model uses Brent method which incorporates other well known optimization methods. It is very stable for convergence and although complex it can easily be applied through Matlab program.