Find the Black-Scholes value of a put option on the following non-dividend paying stock:
Time to maturity: 6 months
Standard Deviation: 50% per year
Exercise price:$50
Current stock price:$50
Interest rate:10%
You found that the put option is trading at the fair value you calculated in part a) but the call option with the same exercise price and maturity is currently selling for 9 dollars. Is there an arbitrage opportunity? If so, identify a trading strategy. Explain what actions you would take at the inception and expiry of the strategy to capture an arbitrage profit and calculate the amount of this profit. Show that your strategy eliminates all risk irrespective of the value of the underlying stock at the maturity of the put and call options.
You decided to establish a position (unrelated to part a and b) by buying a share of the stock for $50, buying a 6-month put option with exercise price $45, and writing a 6-month call option with exercise price $55. Draw a payoff (not profit and loss) graph to illustrate the outcome of the combined position at expiry. Clearly label all axes and important points, showing relevant numbers on the axes.