a) Using relevant algebra and a hypothetical example, explain what the statement the delta of a call option is 0.85 implies for a bank that wants to hedge a position in the option.

b) Using relevant algebra, explain what the risks for option writers facing a large position gamma while their portfolio is delta hedged? c) A hedge fund owns a portfolio of options on the US dollareuro exchange rate. The delta of the portfolio is 65. The current exchange rate is 1.100. Derive an approximate linear relationship between the change in the portfolio value and the percentage change in the exchange rate. The daily volatility of the exchange rate is 0.6%. Assuming the normal distribution of the exchange rate returns, estimate the 30-day 99% VaR of the portfolio.

d) What is called a volatility smile in financial options? Is the existence of volatility smile consistent with the assumptions of Black-Scholes model for option prices? Explain what type of volatility smile is typically observed for equity options and propose some possible explanations for this type of smile.