Question

Suppose that call options on ExxonMobil stock with time to expiration 3 months and strike price $90 are selling at an implied volatility of 30%. ExxonMobil stock currently is $90 per share, and the risk-free rate is 4%.

1) If you believe the true volatility of the stock is 32%, how can you trade on your belief without taking on exposure to the performance of ExxonMobil? How many shares of stock will you hold for each option contract purchased or sold?

2) Using the data in the previous problem, suppose that 3-month put options with a strike price of $90 are selling at an implied volatility of 34%. Construct a delta-neutral portfolio comprising positions in calls and puts that will profit when the option prices come back into alignment.

Answer 1

N(d1) =0.5567

Selling at lower implied votality of 30% shows that call options are underpriced. So, in order to hedge, investors must buy options and short 0.5567 shares for each call bought.

2.

The delta is often called the hedge ratio, for example, if you have a portfolio of 'n' shares of a stock, therefore, 'n' divided by the delta gives you the number of calls you would need to be short to create a gedge. In such a "delta neutral" portfolio, a gain whatsoever in the value of the shares held due to an increase in the share price would be exactly offset by a loss on the value of the calls writter, and vice-versa.

Step-by-step explanation:

Stock price So= $90

Interest rate= 4% per year

Exercise price = $90

True volatility= 32%

Implied votality= 30%

Time of expiration= 0.25 per year.

Using Hedge ratio to calculate is shown below.

d1= In(So/X) + (r + σ/2)T/σ√T

d1= ln(90/90) +(0.04+0.32^2/2) 0.25/ 0.32√0.25

= 0 + (0. 04+0.512) × 0.25/ 0.32 × 0.50

= 0.14250

N(d1) = N(0.14250)

= 0.5567

b. An investor taking long position would expect that the underly stock price will become lower in future. He has to pay the put option that makes him have the right to sell his underlying stock at strike price at the expiry of options which will enable him make profit at the strike price when underlying stock decreases. If it turns the other way round and underlying stock increases, he will be at loss. Although, maximum loss is limited to put option.

An investor can short the underlying stock if he is of the opinion that the share value is overvalued and will become lower in short tenure. Therefore, he sell at a higher price in expectation that the underlying stock will decrease.

If the stock decreases, he will buy the share at a lower price and book profit as the difference in stock and selling price and vice-versa, he has to buy his share at higher price and then close his position.

From the above, selling at lower implied votality of 30% shows that call options are underpriced. So, in order to hedge, investors must buy options and short 0.5567 shares for each call bought.

2.

The same approach is applicable for answer 2, by using the implied volatility of 34%.

The delta is often called the hedge ratio, for example, if you have a portfolio of 'n' shares of a stock, therefore, 'n' divided by the delta gives you the number of calls you would need to be short to create a gedge. In such a "delta neutral" portfolio, a gain whatsoever in the value of the shares held due to an increase in the share price would be exactly offset by a loss on the value of the calls writter, and vice-versa.