7v-4 - Forward Price Definition: Forward price, f (t ), is...

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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 years-two 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 years-two 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. Risk-free 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 good-state 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 Risk-neutral 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 risk-neutral (value certain payoff as much as expected (average) payoff ) we can think about forward atomic prices as risk-neutral probabilities. Expected value of discrete random variable X : E (X ) = ∑ xi P ( X = xi ) Forward value of the tree is expected payoff under risk-neural probabilities ftree = Erisk-neutral (c ) = 63 · 0.3 + 48 · 0.7 = 52.5 Note: investors are typically risk-averse and therefore there is a difference between physical and risk-neutral 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.

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