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) λ₁ < 0 and buying a call option with the higher strike λ₂ > 0.

B + λ₁C₁ + λ₂C₂ = 0

(a) Construct a pay off diagram for a call spread that offsets a Binary Call with the strike $100.

Note: Use λ₁ = -1 and λ₂ = 1 to see the shape of the payoff.

(b) For the option prices listed in the table below, find symmetric values |λ₁| = |λ₂| 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 λ₁, λ₂.

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 S₀ = 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.