Strategies and Pricing:
1) A European put option is traded at $4 with the underlying price at $95. Time to expiry is one month and the strike is $100. The risk-free interest rate is 3% per annum. Devise a trading strategy to explore the arbitrage opportunity. What is the minimum profit of the strategy?
2) To hedge a Binary Call option B, a trader creates a call spread by short-selling a vanilla call option with the lower strike (below Binary's) λ_{1} < 0 and buying a call option with the higher strike λ_{2} > 0.
B + λ_{1}C_{1} + λ_{2}C_{2} = 0
(a) Construct a pay off diagram for a call spread that offsets a Binary Call with the strike $100.
Note: Use λ_{1} = -1 and λ_{2} = 1 to see the shape of the payoff.
(b) For the option prices listed in the table below, find symmetric values |λ_{1}| = |λ_{2}| that offset a payoff of exactly one Binary Call quoted in the market at B = $0:5.
Option Strike Maturity Price
C1 $90 1 $14.81
C2 $110 1 $4.94
Construct a payoff diagram for a basket of all three options using the updated values of λ_{1}, λ_{2}.
3) Implement the multi-step Binomial Method to price a European put with the following parameters:
Strike K = 100 and maturity T = 1. Asset price level S_{0} = 100 and interest rate r = 0:05.
a) For the constant number of time steps in the tree NTS=4, find out the value of the option for a range of volatilities and plot the result.
b) Then, x volatility at = 0.2 and plot the value of the option as a function of the number of time steps in the tree, NTS = 1, 2… 50. You will need a different tree for each NTS value.
Note: This is a computational task. Preferred solution method is a function written in VBA but Excel spreadsheets with binomial trees will be accepted given the plots are correct. For simplicity, use compound rate as a substitute to simple (discrete) rate.