Option Pricing Methods: Black-Scholes, Binomial, And Trinomial Models Essay.

Overview of Option Pricing Models

Investors should have a comprehensive knowledge about various variables which are essential for determining the price of an option. There are several methods of option pricing which use parameters like volatility, price of the underlying, risk free rate and time to expiration to arrive at a theoretical fair value of options. The most commonly used option pricing model is the Black-Scholes option pricing model which was developed by Fisher Black and Myron Scholes (Feunou and Okou 2018). Fisher Black and Myron Scholes were awarded the Nobel Prize for their achievement regarding the pricing model. The other commonly used models are binomial model and trinomial model. The details of the models mentioned above are discussed below:

The Black Scholes and Merton model is statistical model which is used for pricing options that takes into account a variety of criteria. The BSM model, which was awarded the Nobel Prize in Economics, was one of the first models used to price options. For pricing options, the model considers dividends, current stock prices, option strike prices, interest rates, volatility, and maturity date (Shirvani, Rachev and Fabozzi 2019). Only one type of option is valued using the BSM model: European options that cannot be exercised before the expiration date. The BSM model is built on a set of assumptions that are detailed below:

• The underlying asset price follows a geometric Brownian motion – this mean that asset price is log-normally distributed. i.e. Ln(St) follows a normal distribution – this means that Ln (St/St-1) which stands for continuous return on the underlying asset follows normal distribution.
• Volatility of the underlying asset continuous return is known and constant.
• The annualized continuous compounded dividend yield on the stock is known and constant.
• Volatility of the underlying asset has been found to be understated than the implied volatility of the traded option.

The BSM model states that the price of the underlying asset usually follows a Brownian motion which indicates that the asset price is log normally distributed, which suggests that the underlying stock's continuous returns follow a normal distribution. The Black Scholes formula's reasoning is that the option value today equals the present value of the payoff that is expect reward from the option, where the payoff is computed using risk neutral probability and then discounted using the risk free rate of return (Derman and Miller 2016).

The following is the formula for the BSM model:

The following are the variable components:

• S = is the current stock price, which can be anywhere between 0 and infinity, but both extremes are exceedingly rare.
• T = reflects the option's expiration date; • t = current date
• X = denotes the exercise price.
• R = This denotes the risk free rate of return
• N = Cumulative distribution function, which denotes the probability of a variable being equal to the input with distribution which is normal.
• e^r = represents the current value of K's bond at some time in the future. Using the rate 'r' for the length of T, this term calculates the bond's present value.

The delta of the option is represented by D1 in the calculation, and the likelihood of the underlying price reaching the strike price of the option is represented by D2.

S0 e –qt *N (d1) can be regarded as the underlying's future value, with the requirement that it exceeds the option's exercise price. The second term in the formula is the amount of the known payment K multiplied by the chance that the target cost will be paid N(d2).

It's easy to understand if we think about it like this:

S0(1st term) - PV of exercise price should be the value of a call option (2nd term)

As a result, S0 – X/e rt.

This method uses an iterative process making the use of multiple periods to value American options. It allows for incorporating multiple specified nodes or time points which lies between the date of the valuation or the date of expiration of the option (Kim et al. 2019). In this model it is estimated that there can be two possible outcomes i.e. the price can either go upwards or downwards. The primary importance of this method of option valuation is that it is immensely simple but can be complex in a multi period setting. It's a discrete model, after all. It splits the whole time of the choice into 30 pieces. Based on the risk neutrality assumption, trees are created and values are determined accordingly. In this case, the price is set so that the expected return in both the up and down states is equal, leaving no room for arbitrage. It does not take stock price volatility into account.

Due to its understated properties, the binomial method is deemed more acceptable in practice:

• It is capable of appreciating the American option. Such possibilities are available at any moment.
• You can determine whether or not exercising an option is lucrative at each stage based on the results of the computation.
• This technique may be used to value a variety of non-standard choices. For example exotic options which are different from traditional options based on their payoff structure, date of expiry and the strike or exercise prices. It can be customized based upon the risk tolerance and profit required by the investor and they provide flexibility.
• Its value may be made almost equivalent to the analytical technique by lowering the time intervals.
• It may be used to invest in stocks that produce dividends.
• It is rather simple to comprehend for regular investors.

The binomial model has a number of flaws, which are explored in detail below:

The binomial model of option valuation has a fundamental flaw in that it is extremely sluggish and takes a long time to arrive at the notional fair value of an option. Because trading is a continuous operation, the binomial model's sluggish pace provides a hurdle for investors. In a multi-period scenario, the computations needed in the binomial option pricing model can be complicated and time-consuming. A proper and accurate binomial model for pricing a single stock option might be time-consuming and wasteful (Rambaud and Perez 2016). 

4) Calculation of current price of the call:

C = [(.75)(9.93) +(.25) (3.32)] / 1.05 = $7.88

5) Hedge ratio = (9.93 – 3.32) ÷ (38.50 – 34.65) = 1.72

References

Shirvani, A., Rachev, S.T. and Fabozzi, F.J., 2019. A rational finance explanation of the stock predictability puzzle. arXiv preprint arXiv:1911.02194.

Feunou, B. and Okou, C., 2018. Risk?neutral moment?based estimation of affine option pricing models. Journal of Applied Econometrics, 33(7), pp.1007-1025.

Derman, E. and Miller, M.B., 2016. The volatility smile. John Wiley & Sons.

Kim, Y.S., Stoyanov, S., Rachev, S. and Fabozzi, F.J., 2019. Enhancing binomial and trinomial equity option pricing models. Finance Research Letters, 28, pp.185-190.

Cruz Rambaud, S. and Sánchez Pérez, A.M., 2016. Assessing the Option to Abandon an Investment Project by the Binomial Options Pricing Model. Advances in Decision Sciences.

Kirkby, J.L. and Deng, S., 2019. Static hedging and pricing of exotic options with payoff frames. Mathematical Finance, 29(2), pp.612-658.