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Unformatted text preview: Forward Price
Deﬁnition: Forward price, f (t ), is the agreed amount determined
at the present time that have to be paid at the speciﬁed future
time, t . Lecture 7:
Forward Prices, Probabilities and Risk Premia
Valentyn Panchenko Relation with present (spot) price:
p = df (t )f (t ) ⇒ f (t ) = p /df (t ) = p (1 + i (t ))t
Example: Two yearstwo states: good and bad. Contract:
Party A might agree to deliver 1 dollar next year if the state is
good; Party B agrees to deliver 0.30 dollars next year no matter
what the state has been. In this case we would say that the forward price of a good state
dollar is 0.30 dollars delivered next year: party B bought one good
state dollar for 0.30 dollars to be delivered (with certainty) in one
year. Relation between forward and present (spot) price: Arbitrage Example cont.: Two yearstwo states: good and bad. Contact:
Party A might agree to deliver 1 dollar next year if the state is
good; Party B agrees to deliver 0.30 dollars next year no matter
what the state has been. Suppose there is a ﬁrm willing to pay 0.29 dollars now for 1 dollar
next year if the state is good. Would it be proﬁtable to transact
with this ﬁrm?
Discount factor: df = 0.95
p = 0.95 · 0.30 = 0.285 year 0: enter into agreement with the ﬁrm to deliver 1 dollar
next year if the state is good, receive 0.29 dollars, deposit to
the bank 0.285 dollars year 0: enter into agreement with the party A as party B: no
current payment necessary year 1: receive 0.285/0.95 = 0.30 from the bank and use the
proceeds to pay party A. Riskfree proﬁt: 0.005 current dollars per contact.
If there are many traders they will be competing to transact with
the ﬁrm lowering the price of the 1 goodstate dollar and
eliminating arbitrage. Forward Atomic Prices Using Forward Atomic Prices Note: Forward Atomic prices are positive (normally) and sum to 1.
Why? Forward Atomic Prices as Riskneutral probabilities Forward value of the tree is ftree = 63 · 0.3 + 48 · 0.7 = 52.5
Present value of the tree is ptree = 52.5 · 0.95 = 49.875 Physical probabilities If we assume that
all investors agree on the same probabilities all investors are riskneutral (value certain payoﬀ as much as
expected (average) payoﬀ ) we can think about forward atomic prices as riskneutral
probabilities.
Expected value of discrete random variable X :
E (X ) = ∑ xi P ( X = xi ) Forward value of the tree is expected payoﬀ under riskneural
probabilities ftree = Eriskneutral (c ) = 63 · 0.3 + 48 · 0.7 = 52.5
Note: investors are typically riskaverse and therefore there is a
diﬀerence between physical and riskneutral probabilities. Expected payoﬀ (wrt physical probability):
Ephysical (ctree ) = 63 · 0.5 + 48 · 0.5 = 55.5
Expected return (wrt physical probability):
Ephysical (rtree ) = E (ctree )/p − 1 = 55.5/49.875 − 1 = 0.1128 Risk premium Expected return of the risky tree:
Ephysical (rtree ) = E (c )/p − 1 = 55.5/49.875 − 1 = 0.1128
Return of the riskless asset:
rriskless = 1/df − 1 = i = 1/0.95 − 1 = 0.0526
Risk premium: diﬀerence between expected risky return and
riskless return Ephysical (rtree ) − rriskless 0.1128 = −0.0526 = 0.0602 Atomic risk premia ...
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This note was uploaded on 07/13/2011 for the course ECON 3107 at University of New South Wales.
 '11
 valentyn
 Economics

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