Option Valuation and Properties of Options

In this question, you need to price options with different valuation approaches and comment on your results. You will consider puts and calls on a share with a spot price of $$44\mathrm{.}21.$ Strike price is $$42\mathrm{.}5.$

The risk$-$free interest rate is $4\mathrm{.}34\%$ per annum with continuous compounding.

Binomial trees:

Furthermore, assume that over each of the next two three$-$month periods, the share price is expected to go up by $7\%$ or down by $7\%.$

a$.$ Draw a two$-$step binomial tree using the template provided and populate the individual nodes with the share price values at each node.

b$.$ Use the two$-$step binomial tree from a$.$ to calculate the value of a six$-$month European call option using risk$-$neutral valuation.

c$.$ Use the two$-$step binomial tree from a$.$ to calculate the value of a six$-$month European put option using risk$-$neutral valuation.

d$.$ Use the two$-$step binomial tree from a$.$ to calculate the value of a six$-$month European call option using the no$-$arbitrage approach.

e$.$ Use the two$-$step binomial tree from a$.$ to calculate the value of a six$-$month European put option using the no$-$arbitrage approach.

f$.$ Use the two$-$step binomial tree from a$.$ to calculate the value of a six$-$month American put option.

g$.$ Show whether the put$-$call$-$parity holds for the European call and the European put prices you just calculated in d$.$ and e$.$

h$.$ Verify whether the no$-$arbitrage approach and the risk$-$neutral valuations lead to the same results.

Notes: When you use no$-$arbitrage arguments, you need to show in detail how to set up the riskless portfolios at the individual nodes of the binomial tree.

Black$-$Scholes$-$Merton model:

Furthermore, assume that the volatility of the underlying asset is $14\%$ per annum.

i$.$ What is the Black$-$Scholes$-$Merton price of a six$-$month European call option?

j$.$ What is the Black$-$Scholes$-$Merton price of a six$-$month European put option?

k$.$ Verify whether the put$-$call parity holds for the option prices you just calculated in i$.$ and j$.$

l$.$ What is the price of a six$-$month American call and why?

Comparison across models:

m$.$Compare the call option prices you just calculated in d$.$ and i$.$ Compare also the put option prices you just calculated in e$.$ and j$.$ Do you expect these prices to be the same? Why$/$Why not?