Replicating a Stock Position by Shorting a Synthetic Long

Summary

The document presents an options parity question involving a call, a put, a strike price, a risk-free rate, and the market price of the underlying stock. It calculates a synthetic long position as long one call, short one put, and holding a discounted amount equal to the strike price in a risk-free bond. Given the stated inputs, the calculated synthetic price exceeds the quoted stock price, prompting a question about how to short the synthetic position and match the stock exposure.

Reversing the synthetic long means shorting the call, buying the put, and borrowing the present value of the strike amount. The stock market price does not determine the number of option contracts or the bond quantity in the parity replication; the quantities are set by the contract multiplier and the present value of the strike for each matched share. The example assumes one option contract represents 100 shares and a zero risk-free rate. It does not discuss dividends, transaction costs, early exercise, or whether the quoted premiums and prices are contemporaneous, each of which can affect practical parity comparisons.

Key ideas

• A synthetic long combines a long call, a short put, and a bond position equal to the discounted strike.
• Shorting the synthetic reverses each leg: short the call, long the put, and borrow the present value of the strike.
• Option and bond quantities must be scaled to match the underlying share exposure and contract multiplier.
• The example assumes a zero risk-free rate and does not account for dividends or trading frictions.

Tags

# Shorting a Synthetic Long


# Shorting a Synthetic Long












I have the following information:

```
Call Premium: 0.30
Put Premium: 40.4
Strike: 130
1-Month Risk-Free Rate: 0%
Market Price: $85.00
```

If I use the Synthetic Long formula I get a price of: `$89.90`

```
Synthetic-Long = call - put + X/(1+Rf)^t
Synthetic-Long = 0.30 - 40.4 + 130/[(1+0.00)^(30/360)]
Synthetic-Long = $89.90
```

This is significantly higher than the market price. If the Investor buys the stock at the market price `$85.00` how many options and bonds must they buy/sell?

Shorting the Synthetic would make the formula:

```
-(Synthetic-Long) = -call + put - X/(1+Rf)^t
```

Since 1 option contract = 100 shares,

then sell 1 call & buy 1 put, but how much of the bond must they sell? (assuming US treasury, & assuming the Risk-Free Rate is 0%)

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.