Question $20$

Consider the following information about a non$-$dividend paying stock:

Current Stock Price

$$27$

Return Standard Deviation

$30\%$ per year

Dividend Rate

$0\%$ per year

Risk$-$free Interest Rate

$4\%$ per year

The dividend rate and risk$-$free rate are expressed as the continuous compounded rates.

a$).$ A call option written on the non$-$dividend paying stock expires in $63$ days and has an exercise price of $$25.$ What should be the Black$-$Scholes price of the call option?

b$).$ Suppose that a put option written on the same stock with the same exercise price and expiry as the call option above is currently trading at $$0\mathrm{.}30.$ Design an arbitrage strategy to exploit the violation of the put$-$call parity, assuming that you could freely buy and short$-$sell the underlying stock, the risk$-$free bond yielding the risk$-$free rate, the call option, and the put option.

c$).$ Suppose another call option written on the same stock with the same expiry but with an exercise price of $$35$ is selling at $$0\mathrm{.}43.$ You are cautiously bullish that the underlying stock will remain at a price range between $$25$ and $$35$ at the expiry, and constructed the bull spread with the $25$ call and the $35$ call options.

Explain what should be the net profit $($payoff net of cost$)$ on your bull spread, if the underlying stock price is $$31$ on the expiration day.