NOTE: In the questions which follow reference may be made to the following definitions:

$(x)=\frac{1}{\sqrt[\phantom{2}]{2}}{\displaystyle \underset{-}{\overset{x}{}}}$exp$(\frac{-t^2}{2})dt,(x)=\frac{1}{\sqrt[\phantom{2}]{2}}$exp$(\frac{-x^2}{2})$

$d_1=\frac{log(\frac{S}{K})+(r+\frac{1}{2}{}^2)t}{\sqrt[\phantom{2}]{t}},d_2=\frac{log(\frac{S}{K})+(r-\frac{1}{2}{}^2)t}{\sqrt[\phantom{2}]{t}}$

where $K$ denotes the exercise price, $r$ the riskless rate, $$ the volatility and $t$ is the time to expiry. The Black$-$Scholes formula for pricing a European call is

$C=S(d_1)-K{e}^{-rt}(d_2)$

Describe Put$-$Call Parity and calculate the Black$-$Scholes formula for a European put option from a European call option struck at $K$ with expiration $T.$

Show that $d_2^2=d_1^2-2log(S\frac{e^{rt}}{K}).$ Hence, or otherwise, show that the delta of a European call option is

$\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{S}}=(d_1)$

Explain how an option trader can use delta$-$hedging to manage their risk.

What is the delta of a European put option?