Replicating a Long Equity Swap with a Stock Short and Cash Investment
Summary
The document explains how a long equity swap, which pays a floating rate and receives the stock’s return, can be replicated using a short stock position and an investment of the short-sale proceeds. Under the no-dividend assumption, the short is closed at the end of the period by buying back the stock, while the invested proceeds grow at the market rate. The swap’s equity performance receipt and floating-rate payment then offset the financing and stock costs in the replication.
The answer frames fair swap pricing as a no-arbitrage condition and relates it to a forward contract priced using the risk-free rate. The hedge depends on the floating rate matching the assumed market financing rate and on being able to short the stock. It does not fully address dividends, differing borrowing and lending rates, transaction costs, or fees, all of which can change actual cash flows.
Key ideas
- A short stock position produces cash that can be invested at the market rate.
- The short is closed by repurchasing the stock at the end of the period.
- The stock return and floating-rate leg can offset under the stated financing assumptions.
- A fair swap price is linked to a no-arbitrage forward price.
- Dividends, rate differences, costs, and fees complicate the idealized replication.
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Full text
# Hedging a Long Equity Swap by Shorting the Stock
# 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!
## Answer by Richi Wa (score 0, accepted)
https://quant.stackexchange.com/a/12732
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.