Synthetic Forward Positions

Options can be combined in all sorts of interesting ways in order to create synthetic assets. It all starts with the parity conditions.

What does "parity" really mean to you?

Exactly. Many parity conditions are assumed to hold. If they don't, arbitrage opportunities exist because parity conditions tell you what should be equal.

Not necessarily. It's a common market assumption.

That's not it. Parity relates to value.

For example, suppose you held a long stock position, and you wanted to limit the downside at a strike price $$X$$. What option contract might you purchase in order to do that?

Of course. That will limit the downside to just the strike price, $$X$$. Now look at the combined payoff:

No, a call option's value is focused on the upside. To limit downside risk, you would want to have a long put position. That will limit the downside to just the strike price, $$X$$. Now look at the combined payoff:

 What sort of profit profile would approximate parity with this combination?

That's not it. The long put is already there. But when combined with a long stock, notice that the downside is eliminated.

Absolutely. The downside nets out, and you're just left with the upside. So the long stock ($$S_0$$) and long put ($$p_0$$) together are on parity with a long call ($$c_0$$), once you adjust for one more little thing: the long stock position requires cash now for $$X$$, so adding this in present value provides the parity condition you have seen before: > $$ S_0 + p_0 = c_0 + \frac{X}{(1+r)^T}$$.

Not really. A short call shows a loss at higher prices of the underlying. Neither of these provides such a loss.

Now you're in a good position to consider a forward contract, and even build one synthetically. What adjusted spot price would show parity with a forward price?

Not quite. This is the future spot price, but it is unknown. You'll need to use the current spot price.

Yes. Simply compound the spot price to time $$T$$, and you have the forward price, $$F_0(T)$$. That means you can also substitute out the spot price using this relationship for put–call-forward parity: > $$\displaystyle \frac{F_0(T)}{(1+r)^T} + p_0 = c_0 + \frac{X}{(1+r)^T}$$.

That's not it. There's no need to discount a spot price that's already in time 0.

Now, create the synthetic forward. You want to essentially have the gain or loss from price $$X$$ of this stock at time $$T$$. What option positions will you need?

That's right. Just as the equation suggests, you can rearrange the parity condition to get a synthetic forward for just a small premium difference: > $$\displaystyle \frac{F_0(T)}{(1+r)^T} - \frac{X}{(1+r)^T} = c_0 - p_0$$.

Not really; a short call will give you a loss if the stock gains. You want the same payoff as having the stock.

No, that will give you a short straddle, where you gain only if the price doesn't change much.

Logically, you get the upside from the long call, and the downside from the short put. How would you expect your payoffs to change if instead you added a short call and a long put?

No, there would certainly be opportunities for both gains and losses with this combination. Consider the payoff diagrams of each, or just rearrange the parity condition equation.

No, there wouldn't be. You could gain or lose with this combination. Consider the payoff diagrams of each, or just rearrange the parity condition equation.

Exactly! This long/short switch of each option contract just switches the long–short nature of the synthetic forward. You can see this by just moving everything in the parity equation to the other side. Synthetic forwards are convenient. You might want to put such an exposure in place without worrying about the cash layout. Market makers create these to hedge exposures when actual forward contracts need to be purchased or sold to satisfy clients.

To summarize: [[summary]]

Purchased together

Equal

Positive

A put option

A call option

A long call

A long put

A short call

$$ S_T $$

$$ S_0(1+r)^T $$

$$\displaystyle\frac{S_0}{(1+r)^T} $$

A long call and a short put

A short call and a long put

A short call and a short put

All payoffs would net to zero

It would create a synthetic short forward

There would be a guaranteed loss

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