Continuously compounded forwards

[johntavv]

The continuous forward contract price on an equity index =
Spot x e^[(cont. compo. risk free - cont. compo. div yield) xT
The continuous forward contract value on an equity index =
Spot / e^(cont. compo. div yield x T-t) - FP / e^(cont. compo. risk free) x T-t)
What is the intuition for the contract price being: risk free - div yield and then the contract value being div yield - risk free?

 

[naren]

At t = 0, contract value is zero, so
forward contract value = spot / e^(divyld)(T-0) - forward price / e^(CCR)(T-0) = 0 which means
spot / e^(divyld)(T) = forward price / e^(CCR)(T) which means
forward price = spot * e^(CCR)(T) / e^(DY)(T)

 

[johntavv]

Is there an easy way to memorize this, or do we just have to know the order or div yield and cont. comp. risk free?

 

[JSobes]

I like to think of the simpler formula here:
[St - PV(div)] - FP/(1+rf)^(t-T) = Vt
If we think of things this way we can think of st/e^rdx(T-t) as simply being the continuously compounded form of [St - PV(div)]
the reduction of FP/(1+rf)^(T-t) is simple enough to understand now too
hope it helps

 

[S2000magician]

johntavv wrote:The continuous forward contract price on an equity index =
Spot x e^[(cont. compo. risk free - cont. compo. div yield) xT
The continuous forward contract value on an equity index =
Spot / e^(cont. compo. div yield x T-t) - FP / e^(cont. compo. risk free) x T-t)
What is the intuition for the contract price being: risk free - div yield and then the contract value being div yield - risk free?

They pulled a fast one on you: the first formula:
Spot × e^[(cont. compo. risk free – cont. compo. div yield) × T]
is equivalent to:
Spot × e^[(cont. compo. risk free) × T] / e^[(cont. compo. div yield) × T]
So both formulae are discounting the spot rate by the dividend yield.
The second formula (for valuing the forward) also discounts the forward price by the risk-free rate to get the present value.