In: Finance

Are the profit and payoff of buying an index and buying a put the same as investing in zero-coupon bonds and buying a call?

Given that a 950-strike call has a premium of $120.405 and a 950-strike put has a premium of $51.777 and the 6-month interest rate is 2%. Suppose you buy the S&R index for $1000 and buy a 950-strike put. What are the profit and payoff for this position. Is this the same profit and payoff as investing $931.37 in a zero-coupon bonds and buying a 950-strike call.


Expert Solution

Option 1: Buy S&R index for 1000 and buy 950-strike put

The payoff at anytime for buying an S&R index is its spot price, denoted \( S_T \). Think of it like buying a stock. Once you've bought the stock, your payoff at anytime is the amount you can sell the stock for which is usually it's current price in the stock market which is also called it's spot price. So payoff = \( S_T \)

The payoff for buying a 950-strike put can be found using the put payoff formula: \( \text{put payoff} = \max(0, \text{strike price - spot price}) \) because remember that a put contract gives you the right to sell but not the obligation to sell some amount of the underlying security. So you can choose when to use or exercise the contract meaning if it looks like you will lose money you can simply choose not to exercise the contract. If you think you can make money, the amount of money you can make is the strike price which you've used to buy the put contract minus the current price of the underlying asset. So the payoff = \( \max(0,950-S_T) \).

Then your total payoff is the combined value of the above two payoffs: total payoff = \( S_T + \max(0,950-S_T) \)

\( \begin{align*} \text{total payoff} &= S_T + \max(0,950-S_T) \\ &= \max(0+S_T,950-S_T+S_T) \\ &= \max(S_T,950) \end{align*} \)

Your profit is payoff minus what you paid to get into this position at this current point in time. You paid $1000 to buy the index at \( t=0 \) and a premium of $51.777 to enter the put contract at \( t=0 \). So altogether you paid 1000 + 51.777 = 1051.777 at \( t=0 \) to enter this position. Now we pull that value forward to this current time when the interest rate is 2% is:  1051.777(1 + 0.02) = 1072.81. So then your profit is 

\( \begin{align*} \text{total profit} &= \max(S_T,950) - 1072.81\\ &= \max(S_T - 1072.81,-122.81) \end{align*} \)

Option 2: Invest 931.37 in zero-coupon bonds and buy a 950-strike call

You invested $931.37 in zero coupon bonds at \( t=0 \). So by now that amount has grown to \( 931.37(1+0.02) \approx 950 \) which is also your current payoff for this investment.

Like the 950-strike put, the 950-strike call formula gives the payoff: \( \begin{align*} \text{call payoff} &=\max(0,\text{spot price} -\text{strike price})\\ &= \max(0,S_T-950) \end{align*} \)

So the total payoff is \( \begin{align*} \text{total call payoff} &= 950 + \max(0,S_T-950) \\ &= \max(0 + 950,S_T-950+950) \\ &= \max(950,S_T) \\ \end{align*} \)

We can see that option 1 and option 2 both have the same payoff.

The amount spent to get into this position is 931.37 which you spent to get the zero-coupon bond and the 120.405 you spent to buy the call contract. So the total amount spent and pulled forward to current time is (931.37 + 120.405)1.02 = 1072.81. Then the profit is:\( \begin{align*} \text{total profit} &= \max(950, S_T) - 1072.81\\ &= \max(-122.81, S_T - 1072.81) \end{align*} \)

We can see that this is the same profit as Option 1 so therefore Option 1 and Option 2 are equivalent.

Both options have:

payoff = \( S_T + \max(0,950) \) 

profit = \( \max(-122.81, S_T - 1072.81) \)

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