a$.$ You are currently holding a delta$-$hedged portfolio consisting of a single unit of an underlying and a short position in some at$-$the$-$money

call options. If the underlying price rises significantly tomorrow, describe what you would expect to happen to your portfolio delta and

why you would expect this. What action might you need to take tomorrow to regain delta$-$neutrality.

b$.$ As time passes we expect long option portfolios to lose value. Explain

why this is$.$ How and why does this effect depend on the money$-$ness

of the options?

c$.$ You are receiving floating $($and paying fixed$)$ in a $10$ year plain vanilla

interest rate swap. What is the key factor causing changes in the

market value of this swap position? In which circumstances would

you expect to see the value of your positions increase and decrease

respectively. $[$Write no more than $1/2$ a page of A$4]$

d$.$ A plain vanilla interest rate swap has a remaining lifetime of $8$ months,

pays cashflows every $6$ months and has a notional principal of $50$

million. The fixed rate in the swap is $11\mathrm{.}00\%$ per annum and the

floating rate in the swap is a market$-$determined UK interest rate, denoted by L$.$ At the most recent reset date for this swap $($which was $4$

months ago$),$ L was equal to $10\mathrm{.}25\%$ per annum. At present the UK

term structure of interest rates is flat at $10\mathrm{.}00\%$ per annum.

i$.$ Pretend that the notional principal actually changes hands between counterparties. Under this assumption compute the present

value of the fixed payments in the swap. $($

$($a$)$ Assuming that the notional principal changes hands, what is the

value of the swap$$s floating payment stream? $($

$($b$)$ Thus what is the market value of this swap? Give an intuitive explanation of why this value is positive or negative.