Assume that the interest rate is $3\%,$ the volatility of the gold prices is $30\%,$ the available options strikes are: $1960,1980,2000,2020$ and $2040.$

a$)$ You formed a spread position where you bought the $90$ days $1960$ call and sold the $90$ days $2000$ call.

$(1)$ What is the value of your spread at the following prices of gold: $1960,1980,2000,2020$ and $2040?$

What is the value of the hedge ratio at these prices?

$(2)$ How does the value of your spread change with an increase in volatility from $30\%$ to $40\%?$ How does the hedge ratio change?

$(3)$ How does the spread value change when maturity declines from $90$ days to $10$ days? How does the hedge ratio change?

b$)$ You formed a put spread where you bought the $90$ days $2040$ put and sold the $2000$ put. Repeat $(1),(2),$ and $(3)$ for the put spread. How does the put spread relate to the call spread?

c$)$ You bought a $90$ day $1960-2000$ call spread $($bull spread$)$ and a $90$ days $2040-2000$ put spread $($bear spread$).$

$(1)$ Assuming the price of gold is $2000,$ what is the value of the position and the hedge ratio?

$(2)$ How would the value of the position change with time?

$(3)$ How would the value of the position change with a decrease in volatility from $30\%$ to $20\%?$