Present value of forward formula

[bpdulog]

Hello

I tried searching the forum but couldn't find any related thread. In the BT slides, the value of a forward is given as:

F0=(S0-K)e^rT

However, in Hull there is a version presented as:

F0=S0e^-qT-Ke^-rT

It's kind of confusing because in Hull, the top formula is the value of a forward with no income. Yet, the price of the forward in slide 16 shows r-q, implying there is a known yield.

 

[brian.field]

Hi @bpdulog

I find it easiest to think of PREPAID forwards. Just memorize the handful of Prepaid Forward rules and the rest is easy.

The prepaid forward is the time=0 value of a time=t forward.

There a few scenarios:

1) Stock with no dividends.
Prepaid Forward = S0
Forward = S0e^rt (note that this is the Future Value of the Prepaid Forward)

2) Stock with Discrete dividends.
Prepaid Forward = S0 - PV(dividends)
Forward = S0e^rt - FV(dividends) (note that this is the Future Value of the Prepaid Forward once again)

3) Stock with Continuous dividend rate q.
Prepaid Forward = S0e^(r-q)t
Forward = S0e^-qt (note that this is the Future Value of the Prepaid Forward once again)

4) Forward on Currency with domestic currency risk free rd and foreign currency risk free rate rf
The trick here is to treat the foreign risk free rate as a continuous dividend yield, i.e., replace r with rd and q with rf. Also, replace S0 with X0 which is the exchange rate in Domestic / Foreign.
Prepaid Forward = S0e^(rd-rf)t
Forward = S0e^-(rf)t (note that this is the Future Value of the Prepaid Forward once again - the prepaid forward is accumulated at the domestic risk free rate)


Once you get a grip on Prepaid Forwards, it will pay huge dividends in the Black-Scholes material.

 

[bpdulog]

Hi @bpdulog

I find it easiest to think of PREPAID forwards. Just memorize the handful of Prepaid Forward rules and the rest is easy.

The prepaid forward is the time=0 value of a time=t forward.

There a few scenarios:

1) Stock with no dividends.
Prepaid Forward = S0
Forward = S0e^rt (note that this is the Future Value of the Prepaid Forward)

2) Stock with Discrete dividends.
Prepaid Forward = S0 - PV(dividends)
Forward = S0e^rt - FV(dividends) (note that this is the Future Value of the Prepaid Forward once again)

3) Stock with Continuous dividend rate q.
Prepaid Forward = S0e^(r-q)t
Forward = S0e^-qt (note that this is the Future Value of the Prepaid Forward once again)

4) Forward on Currency with domestic currency risk free rd and foreign currency risk free rate rf
The trick here is to treat the foreign risk free rate as a continuous dividend yield, i.e., replace r with rd and q with rf. Also, replace S0 with X0 which is the exchange rate in Domestic / Foreign.
Prepaid Forward = S0e^(rd-rf)t
Forward = S0e^-(rf)t (note that this is the Future Value of the Prepaid Forward once again - the prepaid forward is accumulated at the domestic risk free rate)


Once you get a grip on Prepaid Forwards, it will pay huge dividends in the Black-Scholes material.

Thanks @brian.field for providing these

However, I am still confused as to how the value of a Prepaid Forward = S0. Using the example in Hull, delivery price of a stock is $24 and the current (new S0) price is $25, no dividend, time to expiry in 6 months and r=10%. They then value the forward contract as:

Price of 6 month forward t=0; F0= 25e^(0.1*0.5)=26.28

f = (26.28-24)e^(-0.1*0.5) = 2.17

So I'm struggling to see how the calculated value of 2.17 comes even close to S0, because the value of a forward at t=0 is usually 0 or close to 0.

I guess my real question is why Hull equation 5.5 was presented and not 5.7, given the forward price formula on that slide is Hull 5.3

 

[brian.field]

Consider a forward on a nondividend stock entered into at time 0. This forward provides for transferring of the stock at time T. Let St by the price of the stock at time t. There are two ways you could own the stock at time T:

1) Buy stock at time t=0 and hold it to time T.
2) Buy forward on the stock at time t=0 and hold it to time T.

1) requires a payment of S0 at time t=0 and a payment of 0 at time T.
2) requires a payment of 0 at time t=0 and a payment of F(0,T) at time T - we are representing the Forward Price as F(0,T)

This is where we need to rely on the Law of One Price of the assumption of no arbitrage. If we rely on this, we conclude that the two scenarios above must have the same cost.

In other words, the cost at time t=0 of 1) is S0 which si the price of the stock at time 0. The second scenario costs F(0,T) at time T. Since we can invest at the risk free rate, every dollar we invest at time T=0 grows to E^(rT) dollars at time T.

We can pay F(0,T) at time T by investing F(0,T)*e^(-rT) at time 0.

Therefore, we have the following prepaid forward relationship for a nondividend stock:

Prepaid Forward = F(0,T)*e^(-rT) = S0.

Accumulating both sides to time T, we have the desired Forward relationship, namely,

F(o,T) = S0e^(rT)

 

[brian.field]

It is also important to be very clear on what you are contemplating. I provided "pricing" formulas for the prices of forwards at forward inception. The "value" of a forward can change over time as the stock price and forward price change.

 

[brian.field]

Hello

I tried searching the forum but couldn't find any related thread. In the BT slides, the value of a forward is given as:

F0=(S0-K)e^rT

However, in Hull there is a version presented as:

F0=S0e^-qT-Ke^-rT

It's kind of confusing because in Hull, the top formula is the value of a forward with no income. Yet, the price of the forward in slide 16 shows r-q, implying there is a known yield.

To answer your question, the top expression is the forward price at time t=T and the bottom expression is the forward price at time t=0.

 

[bpdulog]

After looking over slide 16 again, the BT formula uses F0 notation: (F0-K)e^-rT

as opposed to S0 in Hull. Maybe that explains the difference. That is my suspicion