Binomial Tree for Options

Summaries

A. This is by assuming the stock future price will

1)      move up (U) or down (D) (Also assume U=1/D, just like what we leant in the interest rate tree i_1,U=i_1,L(e^2sigma))

2)      with probability (risk neutral probability) of pi_u or pi_d, where

3)      pi_u = (1+Rf-D)/(U-D) and pi_d=1-pi_u

4)      then calculate the expected stock price and thus the option price and discount to the present value

If the options are overpriced (underpriced), we may short (long) options and long (short) stocks to form a riskfree hedge portfolio. Amount of stocks to long (short) for each option is the Hedge ratio:

H = (C+-C-)/(S+-S-)

H is the fractional share need to hold for arbitrage trade for each call option. This is the ratio that the both downside and upside change result in the same portfolio – thus risk free. C+, C- are the upside and downside call price respectively. And the S+ and S- are the upside and downside stock price respectively.

For 2 period-binomial tree, starts with the 2^nd period and treat each node as 1 period and calculate backward. First thing to do is always to find the U, D, pi_u and p_d.

B. When pricing options for bonds, we need interest rate tree. Note that, that risk-neutral probability of interest rate tree is always 50%.

*** The value of the on-the-run issues generated by interest rate tree should prohibit arbitrage opportunities.

*** If it is American Option, make sure the values intermediate nodes are larger than the intrinsic values!

C. Value of Caplet and Floorlet

Caps or Floors are a bundle of European Interest Rate options (Caplets or Floorlets)

E.g. a 2-year Cap is composed of 1 year caplets and 2 year caplets. The value of a cap is the sum of the value of all the caplets.

Value of a caplet = max(0, (rate – cap rate)x notional amount))/(1+rate)

Value of a floorlet = max(0, (floor rate – rate) x notional amount))/(1+rate)