Explain how you would hedge a short position in a European (plain vanilla) call with six  weeks to maturity if the spot price is 60, the strike is 65 and σ = 0.3, r=0.1. You rehedge every week. Assume that the stock will follow the following path for the end-of-week prices: 63, 59, 64, 68, 64, 67. Explain how your hedging position changes every week and what trades you should put in to do so. Assume there are no transaction costs. What is the hedging cost?

 2. The following prices are observed in the market for an option: (all options are on the same underlying with same maturity time)

• Stock trades at S0 = 100 

• A straddle with K=100 trades at 7.9 

• A strip with K=100 trades at 12.1 

• A strap with K=100 trades at 11.6 

• A strangle with K1=95 and K2=105 trades at 5.0 

• A butterfly spread with K1=95, K=100 and K2=105 trades at 2.1

a) Describe a risk-free strategy to make money in this market (only trading the above instruments)

3. A European asset-or-nothing option that expired at time T pays its holder the asset value S(T) at time T is S(T) > K and pays 0 otherwise. Determine the no-arbitrage cost of such an option as a function of parameters s,T,K,r,σ. Find its Delta.

4. You buy 1000 six months ATM call options on a non-dividend paying asset with spot price 100, following a lognormal proves with volatility 30%. Assume the interest rates are constant at 5%. 

• How much do you pay for the options?

• What Delta-hedging position do you have to take?

• One the next trading day, the asset opens at 98. What is the value of your position (the option and shares position)?

• Had you not Delta-hedged, how much would you have lost due to the decrease in the price of the asset?