Put-Call-Forward Parity

Hey guys! I have a little confusion regarding the put-call-forward parity. In the CFAI curriculum, it says that one can create a synthetic protective put by going long a forward contract and long a risk-free bond (which will be equal to going long the asset). But in the formula, we only have a derivative (the forward contract) and the put, but where is the risk-free bond? They only say that the formula is going long a forward (derivative) and long a put.

If someone could clarify this, I would be much appreciated.


If you’re talking about the formula:

p0c0 = [XF0(T)] / (1 + r)T

then X is the risk-free bond.

Yes, that formula.

But in the side of the call, the X is the risk-free bond, but on the side of the put, there is only the derivative. Shouldn’t be another risk-free bond? (because it is going long with a forward and long a risk-free bond in order to have S0

The put-call parity is

C + PV(X) = S + P

where C = call prem / P = put prem / S = spot price of underlying asset / X = a risk-free bond with a value equal to strike price of the put and call

The put-call-forward parity is just a modification of put-call parity where S (the stock or underlying asset) is bought in a deferred manner (i.e. a forward on that asset). So:

C + PV(X) = PV(F) + P

A synthetic protective put can be created being long a call option and long a risk-free bond:

PV(F) + P = C + PV(X)

Remember that PV(F) is being long in a forward to purchase (obviously) the underlying asset, say, a stock.

There’s no point in having a bond on both sides of the equation; they’ll just net to one bond.

The point of the bond is to have cash available to be able to exercise the call option, or to pay for the underlying stock when the futures contract expires.

Correct, but the curriculum just mentions two strategies (long a forward and a bond), and later they keep just the derivative in the formula. It seems counterintuitive.

The forward contract is associated with buying a risk free bond.

So, instead of buying the asset, you buy a forward contract and a risk free bond in which the face value is the forward price, in addition to long put. (this will create a protective put)

This formula is a little odd.

I’d rather see is as:

c0p0 = [F0(T) − X] / (1 + r)T = F0(T) / (1 + r)TX / (1 + r)T

The left side – long a call option and short a put option – is the same as buying the underlying (stock) today at a price that is the present value of the strike.

The right side is the present value of the futures price of the underlying (stock) – which is the spot price of the underlying (stock) – less the present value of the strike price; it’s the same as owning the underlying (stock) today at a price that is the present value of the strike.

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Ah ok, perfect, that makes sense now.

Thank you! And thanks everyone that helped with the question.

In case someone lands here having the same doubt. How I understood this! So instead of buying an asset, we buy a forward and a risk-free bond with the long put. I am buying forward (‘0’ value at the inception, hence I don’t pay anything now, therefore it doesn’t reflect in the equation) to purchase the asset in the future, but I simultaneously will buy a risk-free bond having the same face value as my forward price contract to cover the asset purchase. So the risk-free bond (same as forward price) and a long put will create a synthetic protective put which is similar to fiduciary call…