# Hedging a Long Equity Swap by Shorting the Stock

Suppose that I enter an Equity Swap, such that I pay a floating rate and I receive the equity return. The payment is every one year for both the rate and the return, and the swap expires in one year. I have been told that shorting the stock should fully hedge the swap. However, I worked out the cash flows as follows:

    Time       |       0       |        1        |        2        |

Pay rate       |       0       |   -S_0 * r_0    |   -S_1 * r_1    |
Receive Equity |       0       |   S_1 - S_0     |   S_2 - S_1     |
Short Stock    |      S_0      |        0        |      -S_2       |
Reinvest       |     -S_0      |        0        |   S_0*(1+r_f)^2 |

Total      |       0       |     not zero    |    not zero     |


r_f is the annual risk free rate, r_0 and r_1 are the annual floating rates.

Am I missing something here? Is shorting one stock enough or do we have to short another stock at time 1? How should the cash flows be?

Thanks!

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What is the floating rate that you receive? The trade should be fair. This means that at the beginning the discounted pay-offs should be equal. This is the no-arbitrage condition and it only holds if you are able to short the stock. Then we can find a replicating strategy. At time zero both sides of the trade could do the following (we assume no dividend for the stock for simplicity):

• enter the contract
• short the stock - you receive $S_0$ cash for this
• invest the cash at the market rate for one year $r_1$
• after one year you close the short on the stock and pay $S_1$ for this. You still have the invested $S_0*(1+r_1)$ from the cash.
• after this year you receive $S_1/S_0-1$ from the counterpart which is the performance of your short and you pay $r_1$ - the sum of all is zero.

The other side can do this:

• The arbitrage free forward price for the stock at time zero is $F_1 = S_0 (1+r_1)$
• The other side of the trade can enter at this forward price and gets $S_1$ at time $1$ from this foward contract and "gives" it to you. The remaining $r_1$ is what you pay.

After all: shorting the stock, closing the short on your side, a forward trade on the other side makes things fair.

Things get difficult if you have to estimate dividends, if we consider the real workd where both paries usually get different rates and other transaction costs and fees enter.

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