1. Derivatives Pricing a Forward / Futures Contract
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
The objective for this session is to pave to way for the understanding of derivative products.
The most important result will the fundamental valuation model for forward contracts. This model is based on the absence of arbitrage opportunities (no free lunch). The value of a (long) forward contract is simply the difference between the current spot price (adjusted, if necessary, for income and costs of storage) and the present value of delivery price.
The important insight from this model is that the forward price does not depend on expectations regarding the future spot price.
We will show that this valuation principle breaks down if the underlying asset can not be stored or if short position are not possible.
We will first analyze forward contracts when the underlying asset does not pay any dividend and is costless to store. One important application will be the analysis of forward contracts on zero-coupons. We then extend the model to incorporate income. We apply the more general model to forward contracts on bonds, stock indices and currencies. We conclude the analysis with commodity contracts.
Although the timing of cash flows is different for futures and forward contracts, it will turn out that their prices should be equal is interest rates are non stochastic.

2. Forward price and value of forward contract: review
Remember: the forward price is the delivery price which sets the value of a forward contract equal to zero.
Value of forward contract with delivery price K
You can check that f = 0 for K = S0 e r T
The valuation formula
f = S0 – K e –rT
gives the recipe to create a synthetic long forward contract: buy the asset spot and borrow. A long position on a forward contract can be thus viewed as a portfolio composed of:
A long position on the underlying asset
A short position on a zero-coupon with face value K
The valuation formula can also be written as:
f = (F0 – K) e-rT
This formula expresses the value of a forward contract as the difference between the forward price (namely the delivery price for a new contract) and the delivery price.
Note that the value of a forward contract can be positive, null or negative. A negative value means that you would have to be paid to take a long position in the contract.
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3. Forward on a zero-coupon : example
Consider a:
6- month forward contract
on a 1- year zero-coupon with face value A = 100
Current interest rates (with continuous compounding)
6-month spot rate: 4.00%
1-year spot rate: 4.30%
Step 1: Calculate current price of the 1-year zero-coupon
 use 1-year spot rate
S0 = 100 e –(0.043)(1) = 95.79
Step 2: Forward price = future value of current price
 use 6-month spot rate
F0 = e(0.04)(0.50) = 97.73
A zero-coupon bond is a bond that pays no coupon but only the principal at maturity. Zero-coupons are the “elementary particles of finance”. Many complex bonds can be decomposed into packages of zero-coupons. Being able to value a forward contract on a zero-coupon is a first step toward a full understanding of interest rate derivatives.
As a first step, we can simply apply the Forward Valuation Equation derived previously. The only subtlety is that the spot price of the zero-coupon has to be calculated based on the spot rate. Remember that the spot interest rate of maturity t is defined as the yield to maturity on a zero-coupon maturing in year t:
S0 = A e-rt where A is the face value.
Be attentive to the fact that, in our example, the term structure is not flat: spot interest rates vary with maturity. The interest rate to use to calculate the spot price of the zero-coupon is different from the interest rate to use to calculate the forward price.
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4. Forward on zero coupon A r* t T T* R r Ft Value of underlying asset:
Notations (continuous rates):
A  face value of ZC with maturity T*
r*  interest rate from 0 to T*
r  interest rate from 0 to T
R  forward rate from T to T*
Value of underlying asset:
S0 = A e-r* T*
Forward price:
F0 = S0 er T
= A e (rT - r* T*)
= A e-R(T*-T) A r* t T T*
The analysis of forward contracts on zero coupons is the most straightforward application of the basic principles of forward contract valuation on securities that provides no income. As we shall discover later, these contracts are widely used under different names (term deposit and term borrowing). They are also traded over the counter as Forward Rate Agreements (FRAs) and on organized exchange markets as Interest Rate Futures (IRF).
Remember that any long forward contract can be viewed as buying the underlying asset and borrowing. As a consequence, a long forward contract can be decomposed into:
a long position on a zero-coupon maturing at time T* (the underlying asset)
a short position on a zero-coupon maturing at time T (borrowing)
Note that buying forward a zero coupon is equivalent to investing the forward price F0 at time T and receiving the face value A at time T*. The forward interest rate is the interest rate locked in today from T to T*. R r Ft
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5. Forward interest rate
Rate R set at time 0 for a transaction (borrowing or lending) from T to T*
With continuous compounding, R is the solution of:
A = F0 eR(T* - T)
The forward interest rate R is the interest rate that you earn from T to T* if you buy forward the zero-coupon with face value A for a forward price F0 to be paid at time T.
Replacing A and F0 by their values:
When buying a zero-coupon maturing at time T*, you know exactly the interest that you will earn from now until T*. You can decompose this interest into two parts: the interest from now until time T and the interest locked in today from T until T*. The forward interest rate is by definition locked in today for a future period. The forward rate for a future time period is implied by current spot rates.
For instance, in our example, when buying a one-year zero-coupon you earn a continuously compounded interest rate of 4.30% for one year. If, on the other hand, you invest in a six-month zero-coupon, the interest you receive is 4% for the next 6-month. So, buying a one-year zero-coupon can be decomposed into:
Buying a six-month zero-coupon (with a rate of 4%)
Reinvesting forward the future value for an additional 6-month period at the current forward interest for a 6-month period beginning in 6-month.
The forward rate would then be calculated as the solution of:
e(0.04)(0.50) eR(0.50) = e(0.043)(1)
With a little bit of algebra that you check that:
F0 = A e-R(T*-T) ↔ erT eR (T*-T) = er* T*
In previous example:
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6. Forward rate: Example
Current term structure of interest rates (continuous):
6 months: 4.00%  d = exp(  0.50) =
12 months: 4.30%  d* = exp(  1) =
Consider a 12-month zero coupon with A = 100
The spot price is S0 = 100 x = 95.79
The forward price for a 6-month contract would be:
F0 =  exp(0.04  0.50) = 97.73
The continuous forward rate is the solution of:
97.73 e0.50 R = 100
 R = 4.60%
Here is useful trick to compute the forward interest rate (as I hate memorizing formulas, I find the trick useful)
The value of the zero coupon can be written as: S0 = A d* [where d* is the T* year discount factor].
Similarly, the forward price can be written as: F0 = S0 / d
(dividing S0 by the discount factor gives the future value)
As a consequence A = F0 d / d*
As the face value of the zero-coupon is the future value of the forward price based on the forward interest rate:
A = F0 e R (T*-T)
the forward rate R can be calculated as:
R = ln(d / d*) / (T* - T)
In our example, we have d = and d* =
So d / d* = / = 1.023
The forward rate is R = ln(1.023)/(0.50) = 4.60%
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7. Forward borrowing View forward borrowing as a forward contract on a ZC
You plan to borrow M for  years from T to T*
The simple interest rate set today is RS
You will repay M(1+RS) at maturity
In fact, you sell forward a ZC
The face value is M(1+RS)
The maturity is is T*
The delivery price set today is M
The interest will set the value of this contract to zero
There is a subtle difference between buying forward a zero-coupon (with a fixed face value) and borrowing forward.
When you buy forward a zero-coupon, you first set the face value of the zero-coupon. In other words, you determine how much you will receive at the maturity of the zero-coupon. The forward price that you will pay for this zero-coupon will then be determined by current market interest rates.
But when you borrow forward, you start by determining how much you will want to borrow. Current market interest rates then let you know how much you will have to repay at the maturity of the loan. In a way, you first decide on the forward price of the zero-coupon. Then market rates let you calculate the face value of this zero-coupon.
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8. Forward borrowing: Gain/loss
At time T* :
Difference between the interest paid RS and the interest on a loan made at the spot interest rate at time T : rs
M  rs- Rs  
At time T:
T = [M ( rs- Rs )  ] / (1+rS)
To comply with market practices, RS and rS are simple interest
Another calculation of T
We have seen previously that the value of a forward contract at maturity is :
fT= ST – F0
For forward borrowing:
ST = M(1+RS )/(1+rS) (the face value discounted from T* to T)
Ft = M
As a consequence:
T =fT = M(1+RS )/(1+rS) -M =[ M(rS - RS) ]/(1+rS)
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9. FRA (Forward rate agreement)
Example: 3/9 FRA
Buyer pays fixed interest rate Rfra 5%
Seller pays variable interest rate rs 6-m LIBOR
on notional amount M $ 100 m
for a given time period (contract period)  6 months
at a future date (settlement date or reference date) T ( in 3 months, end of accrual period)
Cash flow for buyer (long) at time T:
Inflow (100 x LIBOR x 6/12)/(1 + LIBOR x 6/12)
Outflow (100 x 5% x 6/12)/(1 + LIBOR x 6/12)
Cash settlement of the difference between present values
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10. FRA: Cash flows 3/9 FRA (buyer)
General formula:CF = M[(rS - Rfra)]/(1+rS)
Same as for forward borrowing - long FRA equivalent to cash settlement of result on forward borrowing
(100 rS 0.50)/(1+rS 0.50)
T=0,25
T*=0,75
(100 x 5% x 0.50)/(1+rS)
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11. Basis: definition DEFINITION : SPOT PRICE - FUTURES PRICE bt = St - Ft
Depends on:
- level of interest rate
- time to maturity ( as maturity )
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12. Extension : Known cash income
F0 = (S0 – I )erT
where I is the present value of the income
Ex: forward contract to purchase a coupon-bearing bond
0.25
T =0.50
r = 5%
S0 =
C = 6
We now extend the analysis to take into account cash income provided by the underlying asset between now and the maturity of the forward contract. Buying forward means that you do not receive the income paid buy the underlying asset before the final maturity. This opportunity loss has an impact of the forward price.
Let I = Present value of C = PV(C). Consider the two following portfolios:
Portfolio A: One long forward contract on the security plus an amount of cash equal to PV(F)
Portoflio B One unit of the security plus borrowing of amount I at the risk-free rate
t t1 T
PORTFOLIO A Long fwd 0 +ST - F Cash - PV(F) + F Total A -PV(F) +ST
PORTFOLIO B Security -S +C +ST Borrowing +I -C Total B -S + I 0 +ST
Conclusion : No arbitrage condition PV(F) = S-I
F = FV(S-I)
f = (S-I) - PV(K)
Forward price : f = 0
Note : as before
I = 6 e –(0.05)(0.25) = 5.85
F0 = ( – 5.85) e(0.05)(0.50) =
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13. F0 =  e-qT S0  erT = S0 e(r-q)T
Known dividend yield
q : dividend yield p.a. paid continuously
Examples:
Forward contract on a Stock Index
r = interest rate
q = dividend yield
Foreign exchange forward contract:
r = domestic interest rate (continuously compounded)
q = foreign interest rate (continuously compounded)
F0 =  e-qT S0  erT = S0 e(r-q)T
Consider the two following strategies
Strategy A:Buy forward contract
Strategy B:Buy e-q(T-t) units of underlying asset and reinvest all dividend received in this asset (the number of units of the underlying asset will grow at a continuous rate q - after (T-t) years, you will own e-q(T-t)  eq(T-t) = 1 unit of the underlying asset)
Borrow the present value of the Forward Price
Arbitrage table (with future profit)
CFt CFT
Strategy A
Buy forward (ST - F)
Strategy B
Buy spot -e-q(T-t) St + ST
Borrow + PV(F) -F
Total -e-q(T-t) St + PV(F) + (ST - F)
Both strategies lead to the same future cash flow, their costs should be identical
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14. Commodities I = - PV of storage cost (negative income)
q = - storage cost per annum as a proportion of commodity price
The cost of carry:
Interest costs + Storage cost – income earned c=r-q
For consumption assets, short sales problematic. So:
The convenience yield on a consumption asset y defined so that:
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15. Value of forward contract
Summary
Value of forward contract
Forward price
No income
Known income
I =PV(Income)
F=(S – I)erT
Known yield q
f =S e-qT – K e-rT
F = S e(r-q)T
Commodities
f=Se(u-y)T- Ke-rT
F=Se(r+u-y)T
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16. Valuation of futures contracts
If the interest rate is non stochastic, futures prices and forward prices are identical
NOT INTUITIVELY OBVIOUS:
Total gain or loss equal for forward and futures
but timing is different
Forward : at maturity
Futures : daily
PROOF: S Spot price
F Futures price
G Forward price
r Daily interest rate
AT T: ST = GT = FT
AT T-1:
T-1 T
LONG 1 FWD 0 ST - GT-1
SHORT 1 FUTURE 0 -(FT - FT-1)
TOTAL 0 FT-1 - GT-1
 FT-1 = GT-1
AT T-2:
T-2 T-1
LONG (1+r) FWD 0 (1+r)[GT-1 -GT-2]/(1+r)
SHORT 1 FUTURE 0 -(FT-1 - FT-2)
TOTAL FT-2 - GT-2
 FT-2 = GT-2
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17. Forward price & expected future price
Is F0 an unbiased estimate of E(ST) ?
F < E(ST) Normal backwardation
F > E(ST) Contango
F = E(ST) e(r-k) (T-t)
If k = r F = E(ST)
If k > r F < E(ST)
If k < r F > E(ST)
To understant the relation between F and E(ST), consider the following strategy :
t T
Invest - F e-r(T-t) +F
Long forward 0 ST - F
Total - F e-r(T-t) ST
PV = - F e-r(T-t) + E(ST) e-k(T-t) = 0
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