Question

Consider a 3-month put option. Suppose that the underlying stock price is $25, the strike $26, the interest rate is 5% p.a., stock volatility is 6% per month. Use the same data to answer questions a) – h).

a) What is the level of annual volatility (compute)?

b) Define in your own words implied volatility.

c) How would you compute implied volatility? Explain (no need to compute).

d) What is the probability of stock price going down (Note: use annual volatility, number of steps in a tree is N=3)?

e) Build the binomial tree for the underlying asset (stock). Note: the tree nodes can be edited. Show computations for first up and first down nodes.

f) Compute the price of the European put option using a 3-step binomial tree. Show computations for terminal and two non-terminal nodes.

g) If the market price on the European put option is $1.5, what should be the price of the European call option of the same strike and maturity to prevent arbitrage?

h) Compute the price of the American put option using a 3-step binomial tree. Show computations for two non-terminal nodes.

Answer 1

a) Annual Volatility: Monthly Volatility * (12)^(1/2)

= 6 * 12^(1/2) = 20.78% (aaprox)

b) Implied Volatility:

It refers to the volatility in returns of the stock over the life of the option. It is impacted by the chnages in the demand and supply of the underlying options and the expectations of the market eith regards to the stock price. If the implied volatility is high, it shows the market has an opinion that the stock might move in large swings in any direction, whereas low implied volatility means that the market thinks that stock will not move too much due to option expiration.

c) Implied Volatility is calculated by doing reverse calculation in Black Sholes Model formula, by taking the market price of the option as the intrinsic value of the option. Steps to calculate are numerated below:

1. Take inputs required for BSM formula like Market price of stock, market price of option, strike price of stock, time to expire and risk free rate.

2. Apply the inputs in the formula:

C = SN (d1) – N (d2) Ke -rt

Where,

• C: Option Premium
• S: price of the stock
• K: Strike Price
• r: risk-free rate
• t: time to maturity
• e: exponential term

3. Now start trial error method to find the implied volatility which will render LHS= RHS

4. It can also be calculated by interpolating between 2 rates

  

d) Probability of stock price going down:

Upmove and downmove factor:

U=size of the up move factor=e^(σ√t) = e^((0.2078)*(3)^(1/2))= 1.365

D=size of the down move factor=e^(−σ√t) = 1/U = 1/1.365= 0.733

σ is the annual volatility of the underlying asset’s returns and t is the length of the step in the binomial model.

Probablity of upmove= (1+ rf- D) / (U-D) = (1+ 0.05- 0.733) / (1.365- 0.733) = 50.16%

Probablity of downmove= 100- 50.16= 49.84% (approx)