Question

Please discuss the Black & Scholes model and the binomial model approach to option pricing. What are the advantages and disadvantages of these two approaches? Determine the price of a call and put option assuming that the exercise price is $105, the value of the stock is $101, risk-free rate is 2.05%, standard deviation of returns on the stock is 28%, and the option has 6 months remaining to maturity. What is the price sensitivity of the call and put options to changes in the price of the stock? Would the sensitivity be different if the exercise price in this example was $103? Please explain.

Answer 1

Solution :

Black & Scholes Model

                It is a mathematical model for valuation of options price which calculates both call & put option price as per their respective parameters as both are different from each other. This model shows how price changes random timely & continuous with constant drift and volatility.

• Assumptions of the Black-Scholes Option Pricing Model (BSOPM):
  • No taxes and No transactions costs ……………………….(it is a limitation)
  • Short-selling of stock and full use of short-sale proceeds is not restricted ….(Advantage)
  • Shares are infinitely divisible ………………. (advantage)
  • Risk free rate of loan - constant ………………………( a limitation since in real world interest rate varies based on factors like credit worthiness, collateral etc)
  • No dividends awarded here
  • Its Usable mainly for European options ……………..(Limitation)

Binomial model approach

            It is an approach to value option and it is a more realistic approach as it does not make assumptions not represented by market forces. The model reduces possibilities of price changes and removes the possibility for arbitrage. The model as per its name provides 2 price of stock the up side & the down side. It is mathematically more simpler to use.

Advantage :

• Mathematically correct method and very easy to use
• Roots out arbitrage so as to find correct price of option

Disadvantage :

• Assumes perfect market conditions with no transaction costs
• Permits unlimited borrowing at risk free rate.

Option price :

Given, S = Stock price = $101, T= Time to expiry = 6months i.e.0.5,

            X= Exercise Price = $ 105, σ = standard deviation = 28,

          R= Risk free interest rate = 2.05%

Option price (C_O) = S_O*.N(d₁) – X*N(d₂)^-rt

Substituting D1 & D2

   C_O= 101*196.637 -105* 0.00496

= 19860.8

D_{1=       Ln}(s/x) + (r + σ²/2) t
            σ * square root of t

   = -0.0388 +(2.05+ 784/2)0.5

D1 = 196.637

D2 = d1 - σ * square root of t

     = 196.637 – 19.7989

= 176.8381