Question

Explain the basic operation of the Binomial, Trinomial and Replica Method in investing

Answer 1

Answer - Binomial method

The binomial option pricing model is an options valuation method developed in 1979. The binomial option pricing model uses an iterative procedure, allowing for the specification of nodes, or points in time, pduring the time span between the valuation date and the option's expiration date. The model reduces possibilities of price changes, and removes the possibility for arbitrage.

A simplified example of a binomial tree has only one time step. Assume there is a stock that is priced at $100 per share. In one month, the price of this stock will go up by $10 or go down by $10, creating this situation:

Stock Price = $100

Stock Price (up state) = $110

Stock Price (down state) = $90

Next, assume there is a call option available on this stock that expires in one month and has a strike price of $100. In the up state, this call option is worth $10, and in the down state, it is worth $0. The binomial model can calculate what the price of the call option should be today. For simplification purposes, assume that an investor purchases one-half share of stock and writes, or sells, one call option. The total investment today is the price of half a share less the price of the option, and the possible payoffs at the end of the month are:

Cost today = $50 - option price

Portfolio value (up state) = $55 - max ($110 - $100, 0) = $45

Portfolio value (down state) = $45 - max($90 - $100, 0) = $45

The portfolio payoff is equal no matter how the stock price moves. Given this outcome, assuming no arbitrage opportunities, an investor should earn the risk-free rate over the course of the month. The cost today must be equal to the payoff discounted at the risk-free rate for one month. The equation to solve is thus:

Option price = $50 - $45 x e ^ (-risk-free rate x T), where e is the mathematical constant 2.7183

Assuming the risk-free rate is 3% per year, and T equals 0.0833 (one divided by 12), then the price of the call option today is $5.11.

Due to its simple and iterative structure, the binomial option pricing model presents certain unique advantages. For example, since it provides a stream of valuations for a derivative for each node in a span of time, it is useful for valuing derivatives such as American options. It is also much simpler than other pricing models such as the Black-Scholes model.

2. Trinomial method -

An option pricing model incorporating three possible values that an underlying asset can have in one time period. The three possible values the underlying asset can have in a time period may be greater than, the same as, or less than the current value.

The trinomial option pricing model differs from the binomial option pricing model in one key aspect, which is incorporating another possible value in one periods time. Under the binomial option pricing model, it is assumed that the value of the underlying asset will either be greater than or less than, its current value. The trinomial model, on the other hand, incorporates a third possible value, which incorporates a zero change in value over a time period.

This assumption makes the trinomial model more relevant to real life situations, as it is possible that the value of an underlying asset may not change over a time period, such as a month or a year.

3 Replica method -

In the present work, the optimal portfolio minimizing the investment risk with cost is discussed analytically, where this objective function is constructed in terms of two negative aspects of investment, the risk and cost. We note the mathematical similarity between the Hamiltonian in the mean-variance model and the Hamiltonians in the Hopfield model and the Sherrington{Kirkpatrick model and show that we can analyze this portfolio optimization problem by using replica analysis, and derive the minimal investment risk with cost and the investment concentration of the optimal portfolio. Furthermore, we validate our proposed method through numerical simulations.