Question

In: Finance

You just have engaged in a trading strategy (called DUMBO COMBINATION) that requires going long a...

You just have engaged in a trading strategy (called DUMBO COMBINATION) that requires going long a call and short a put. Both options are written on the same stock, have the same expiry of one year and the same strike. The call is currently $2 out-of-the-money. You know that the risk-free rate is 10% per annum continuously compounded and that the stock, which trades today for $50, is expected to declare a dividend in 8 months and go ex-dividend in exactly 5 months after the announcement of the dividend. When will you break even and when the strategy will be profitable if you hold on to it until the end of the year?

Solutions

Expert Solution

The call is currently $2 out-of-the-money and the stock is currently trading at $50.

.That means that the strike price for the options is $52.

The stock is expected to declare dividend in 8 months, and go ex-dividend in exactly 5 months after the announcement of the dividend. It means that the stock price will go up by the amount of the dividend declared from the end of month 8, when it announces the dividend for 5 months and as soon as the dividend is paid out, it falls back by an equivalent amount of the dividend.

Now, we do not know the price at which the options are purchased.

Assuming that the options are purchased at $2 per option, considering the prices of both options being equal, we spent $4 today.

Since the strike price is $52 and we do not know whether the stock will go up or down in the coming 8 months, we assume that it will go up by the risk-free rate on an average.

We can exercise the put option right away and make a profit of (52-50) = $2 per share.

In the given timeline, we can invest $2 in risk-free deposit, which, by continuous compounding, will become 2*ert, where r = annual rate of return = 10% and t = time in years.

Now, $2 will become 2*e(0.1*2/3) = $2.14 at the end of 2/3 years or 8 months.

And, $2 will become 2*e(0.1*1) = $2.21 at the end of 1 year.

At the end of eight months, the stock price will become 50*e(0.1*2/3) = $53.45.

If we exercise the call option now, we get (53.45-52) = $1.45.

However, with both the options together, (2.14 + 1.45 - purchase price) = (3.59 - 4) = - $0.41. That means we have not reached break-even.

The stock price will now go up faster after 8 months, but it wil depend upon the amount of dividend announced.

Let us assume that a 10% dividend is announced on the stock. So the amount of dividend will be 10% of the stock price at the end of (8+5) = 13 months.

So after 13 months, the stock price will be 50*e(0.1*13/12) = $55.72.

Dividend (D) = 10% of this amount. Thus, D = $5.57.

After 8 months, let the stock price be S = $53.45.

At the end of 12 months, the stock price will become 50*e(0.1*1) = $55.26 + D*(4/5) = $59.71.

If we exercise the call option at the end of year 1, we get (59.71 - 52) = $7.71.

Our total expected profit at the end of year 1 will be (2.21 + 7.71 - purchase price) = (9.92 - 4) = $5.92.

That means we will break even before year 1, but after 8 months.

Let the break even time be x years after month 8.

To break-even, we should be able to cover our purchase cost.

2.14*e(0.1*x/12) + 53.45*e(0.1*x/12) - 52 + D*(x*12/5) - 4 = 0

We solve for x in a scientific calculator or an excel sheet. On computing, we get

x = 0.029 or 0.03 years or 0.36 month.

That means, we expect to break-even at 8.36 months from today under the of assumptions that we will deposit $2 in risk-free deposit, dividend is 10% and the stock will move up at the risk-free rate continuously. At the end of year 1, our expected profit will be $5.92.


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