Question

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 dollar–euro 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.

Answer 1

You have asked multiple unrelated questions in the same post. I have addressed the first two. Please post the balance questions separately one by one.

Part (a)

The delta of a call option is 0.85. This simply implies for a bank that wants to hedge a position in the option, that for every $ 1 change in the price of the underlying, the price of the call option changes by $ 0.85. This shows the sensitivity of the price of the call option. Hence, the bank needs to long 0.85 stock for every option held to obtain a delta hedged portfolio.

Part (b)

Gamma, measures the rate of change in an option’s Delta per $1 change in the price of the underlying stock. It tells us how much the option’s delta should change as the price of the underlying stock or index increases or decreases.

Delta hedging provides protection against small changes in stock price.

Gamma-neutral positions guard against relatively large stock price moves.

Hence, large gama position needs to be hedged immediately. Delta-hedge will provide insulation from small price changes, but large gamma will give a shock in case of large price changes in the underlying. Hence, large gamma position is extremely risky.