Lesson 5: Forward Prices, Physical and Risk-neutral Probabilities, Utility-based Models
Economics of Finance
School of Economics, UNSW
Copyright By PowCoder代写 加微信 powcoder
Forward Price
Definition: Forward price, f(t), is the value of the payment at the time t.
Relation with present (spot) price:
p = df(t)f(t) ⇒ f(t) = p/df(t) = p(1 + i(t))t
Forward Atomic Prices
Note: Forward Atomic prices are positive and sum to 1. Why?
Forward Atomic Prices as Risk-neutral 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) = xiP(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.
Physical 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 = 0.113
Risk premium
Expected return of the risky tree:
Ephysical(rtree) = E(c)/p − 1 = 55.5/49.875 − 1 = 0.113
Return of the riskless asset:
rriskless =1/df−1=i=1/0.95−1=0.053
Risk premium: difference between expected risky return and riskless return Ephysical(rtree) − rriskless = 0.113 − 0.053 = 0.06
Atomic risk premia
• Risk premium of the GW atomic security is positive, 0.7, because the forward price of 1 GW apple is lower than the physical probability of GW state. We value GAs not that much because they are more abundant.
• Risk premium of the BW atomic security is negative (risk discount), -0.3, because the forward price of 1 BW apple is higher than the physical probability of BW state. This is like buying an insurance to cover your consumption in BW.
• Remember that the whole tree still carried risk premium.
Two different perspectives on asset pricing
• Relative Pricing – covered up until now
• assuming arbitrage-free environment and a competitive
market which eliminates any arbitrage;
• pricing using the Law-of-One-Price and replicating
portfolios;
• relying on existing securities for market completeness;
• atomic (state) prices used to price any future
state-contingent payoffs patom = pS × Q−1
• Pricing from microfoundations – from now on
• expected utility optimisation
• assumptions on preference, i.e., functional form of the
utility function;
• market is completed by introducing securities;
• market clearing: matching aggregate demand/supply;
• explains how we arrive at the equilibrium.
Risk Aversion
u(c) is assumed to be strictly increasing and concave, e.g., u(c) = ln(c).
u(c)′ > 0, u(c)′′ ≤ 0
Risk aversion ⇔ u[E(c)] > E[u(c)]
Risk Neutrality
u(c) is assumed linear, i.e., u(c) = a + b c.
u(c) = b, a constant; u(c)′′ = 0
Risk neutrality ⇔ u[E(c)] = E[u(c)]
Assumptions:
• Everyone is self-interested and optimises own utility;
U =u(c(s0))+β[π(G)·u(c(G))+π(B)·u(c(B))]
expected discounted future utility
• subject to budget constraints;
• free trade is allowed;
• everyone take price as given;
• market clears (demand=supply);
• Not only such outcome is Pareto efficient (First
Fundamental Theory of Welfare);
• Any Pareto efficient outcome can be produced by such
economic environment (Second Fundamental Theory of Welfare).
Endowments:
• There is an (exogenously given) supply or endowment of a non-storable consumption good at each time and state;
• At t = 0, the consumer does not know which state will realise in the future.
• Notation (Endowments):
• e(s0) – the initial endowment of consumption good;
• e (s1 = G) – the quantity of the consumption good consumer
receives (say apples from a tree) at time 1 if the realized
state is Good Weather;
• e (s1 = B) – the endowment available at time 1 in the Bad
Weather state;
Market structure:
• The consumer can freely borrow or lend in a complete set of atomic (Arrow-Debreu) securities.
• We assume the existence of two securities: Bad Weather security and Good Weather security.
• One unit of ’G security’ sells at time 0 at a price
q(s0, s1 = G) and pays one unit of consumption at time 1 if state ’G’ occurs and nothing otherwise.
• One unit of ’B security’ sells at time 0 at a price
q(s0, s1 = B) and pays one unit of consumption in state ’B’ only.
• In this notation: s0 refers to the state when securities are traded; s1 = G refers to a particular realization of the state s1 when the security pays off.
Flow budget constraints: Time 0
• In the first period the consumer has initial endowment e(s0). They can consume or buy Arrow-Debreu securities:
c(s0)+q(s0,s1 =G)·a(s0,s1 =G) +q(s0,s1 =B)·a(s0,s1 =B)=e(s0)
• a(s0 , s1 = G) – quantity G securities acquired in state s0 ; • a(s0, s1 = B) – quantity B securities acquired in state s0;
• In our two-period model all trades occur in state s0. The only uncertainty is about the realization of the state s1. Therefore, we can use simplified notation:
• for atomic security prices: qG , qB
• for quantities of the atomic security purchased (sold):
c0 +qG ·aG +qB ·aB =e0
Flow budget constraints: Time 1
• If the realized state at time 1 is Good Weather:
• Each of aG G atomic securities pays off 1 unit of consumption;
• Bad Weather atomic securities do not pay off at all;
• Consumer receives an endowment corresponding to G state:
eG and consumes every unit of consumption they have got: cG = 1 · aG + 0 · aB + eG,
• If the realized state at time 1 is Bad Weather:
• Each of aB B atomic securities pays off 1 unit of consumption;
• G atomic securities do not pay off at all;
• Consumer receives an endowment corresponding to B state:
eB and consumes every unit of consumption they have got: cB = 0 · aG + 1 · aB + eB .
Market Equilibrium:
• A Market Equilibrium in this economy is defined as an allocation c0,cG,cB,aG,aB and prices qG,qB such that:
• Given the prices, the allocation solves the consumer’s problem of maximizing expected utility
u(c0)+β[πG ·u(cG)+πB ·u(cB)] subject to a sequence of budget constraints
c0 +qG ·aG +qB ·aB =e0, cG = aG + eG,
cB =aB +eB.
• Prices are such that markets clear in every period and state:
c0 =e0;cG =eG;cB =eB,
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com