CS代考 Lecture 5: Put-call parity, Forward Price, Expected utility, Constraint opt

Lecture 5: Put-call parity, Forward Price, Expected utility, Constraint optimization Economics of Finance
School of Economics, UNSW

Put-Call Parity

Put-Call Parity is a relationship, first identified by Stoll (1969), that must exist between the prices of European Put and Call options that both have:

Put-Call Parity
Put-Call Parity is a relationship, first identified by Stoll (1969), that must exist between the prices of European Put and Call options that both have:
• the same underlying stock;

Put-Call Parity
Put-Call Parity is a relationship, first identified by Stoll (1969), that must exist between the prices of European Put and Call options that both have:
• the same underlying stock; • the same strike price;
• the same expiration date.
The relationship is derived using arbitrage arguments. Consider two portfolios consisting of:

Put-Call Parity: Cash and Call

Put-Call Parity: Underlier and Put

Put-Call Parity
• The two portfolios (call + cash and put + underlier) have identical expiration values.

Put-Call Parity
• The two portfolios (call + cash and put + underlier) have identical expiration values.
• Irrespective of the value of the underlier at expiration, each portfolio will have the same value as the other.
• If the two portfolios are going to have the same value at expiration, then they must have the same value today. Otherwise, an investor could make an arbitrage profit

Put-Call Parity
Accordingly, we have the price equality:
pcall + PV (X) = pput + punderlier (1)
• pcall is the current market value of the call;
• P V (X) is the present value of the strike price, X;
• pput is the current market value of the put;
• punderlier is the current market value of the underlying stock.
Note: “Current” refers to Period 0 since you are evaluating today prices

Put-Call Parity: An example
• We have priced a European Call option that gives the holder a right to Buy the Stock at Period 2 at the Exercise Price, X = 1.10. We found its price to be pCall = 0.0816.
• Consider a European Put option that gives the holder a right to sell the Stock at Period 2 at the Exercise Price, X = 1.10.
The cash flow associated with the Put option: c=􏰀0 0 0 0 0 0.1784􏰁′
The atomic prices are still the same:
g b gg gb bg bb
patom = 􏰀0.2857 0.6666 0.0816 0.1904 0.1904 0.4444􏰁 The value of the Put option is:
pPut = patom · c =0.4444 · 0.1784 = 0.0793

Put-Call Parity: An example (cont’d)
According to the Put-Call parity we have
pcall + PV (X) = pput + punderlier
Notice that PV (X) = df(2) · X, where df(2) is the discount factor for Period 2. df(2) is the present value of one certain dollar received at Period 2. It must equal to the sum of atomic security prices for states: gg, gb, bg and bb.
df (2) = 0.0816 + 0.1904 + 0.1904 + 0.4444 = 0.9070 PV(X)=df(2)·X =0.9070·1.10=0.9977
pcall =pput +punderlier −PV(X)
= 0.0793 + 1 − 0.9977 = 0.0816
This is the same value as the one we found before.

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?

Using Forward Atomic Prices
Forwardvalueofthetreeisftree =63·0.3+48·0.7=52.5 Present value of the tree is ptree = 52.5 · 0.95 = 49.875

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 state.
• 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.

Preference
Economics studies individual choice:
• Preference relation describes ordering of choices, e.g.,
a 􏰛 b, where a and b are not necessarily numbers, apple is preferred or indifferent to banana.
• We use a utility function, u(·) to represent the preference relation,
u (a) 􏰓 u (b) ⇔ a 􏰛 b,
• Utility function u(·) gives us relative numbers such that above holds (can be strict, i.e., > and a ≻ b)

Uncertainty
• Two periods: today (time 0) and future (time 1)
• Today’s state of nature s0 is known.
• Set of possible future events – good weather (G) and bad weather (B): S = {G, B} .
• G occurs with probability π(s1 = G); B with probability π(s1 = B).
• Aim – design optimal state-contingent consumption plan:
• c(s0)− consumption at time 0.
• c(s1 = G)−consumption at time 1 if state is G; • c(s1 = B)−consumption at time 1 if state is B;

Expected Utility: an introduction
• A consumer has a time and state separable utility function over consumption c(s0) and c(s1) – each period-state instantaneous utility function u(c) does not depend on other period-state consumption directly.
• Consumers discount future expected utility with time discount factor β ∈ (0, 1) which represent time preferences. The lower the β, the more impatient are the consumers.
• The period utility function u(c) is assumed to be strictly increasing and concave, i.e., u′ > 0 and u′′ ≤ 0;
• The consumer maximises expected utility, U, given by U =u(c(s0))+β[π(G)·u(c(G))+π(B)·u(c(B))]
expected discounted future utility

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(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)]

Background
Since (1776), “The Wealth of Nations”, economists strive to prove the existence of “invisible hands”.
• Advocates free trade;
• Individual’s selfish decision drives the aggregate economy; Edgeworth (1881), “Mathematical Psychics: An Essay on the
Application of Mathematics to the Moral Sciences”.
• Two people, two goods;
• Free trade is allowed;
• He suspects the outcome is always social desirable;
• Pareto (1906) confirmed this social desirability, now known as “Pareto Efficiency”;

Arrow-Debreu Approach
Arrow, K. J.; Debreu, G. (1954). “Existence of an equilibrium for a competitive economy” filled this important gap.
• Proved ’s and Edgeworth’s congestion in a more general context.
• Two separate Nobel prizes were awarded for this ground breaking contribution;
• Arrow (1972); Debreu (1983)

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,

Constrained Optimization: A refresher
• To deal with the consumer’s problem we have to maximize a function subject to several equality constraints.
• Consider a problem of choosing x, y, z to maximize the objective function f subject to equality constraints g1 and g2 :
max f (x, y, z) x,y,z
s.t. g1(x,y,z)=b1, g2 (x,y,z) = b2,
where b1 and b2 are constants.

Step 1: Set up a Lagrangian
We want to “translate” the constrained maximization problem into a unconditional maximisation question.
The Lagrangian, L(x, y, z, λ1, λ2), contains: • the objective function f
• minus Lagrange multiplier for constraint 1, λ1, times the difference between LHS and RHS of the constraint 1;
• minus Lagrange multiplier for constraint 2, λ2, times the difference between LHS and RHS of the constraint 2:
L (x, y, z, λ1, λ2) = f (x, y, z) − λ1 [g1 (x, y, z) − b1] − λ2 [g2 (x, y, z) − b2]

Step 2: First order conditions
Then we take partial derivatives of the Lagrangian with respect to its every argument and equate them to zero;
∂L = ∂y ∂L = ∂z
∂f − λ ∂g1 − λ ∂g2 ≡ 0; ∂x 1 ∂x 2 ∂x
∂f − λ ∂g1 − λ ∂g2 ≡ 0; ∂y 1 ∂y 2 ∂y
∂f − λ ∂g1 − λ ∂g2 ≡ 0; ∂z 1 ∂z 2 ∂z
∂L =−g1(x,y,z)+b1 ≡0; ∂λ1
∂L =−g2(x,y,z)+b2 ≡0. ∂λ2
Now we have five equations with five unknowns: x, y, z, λ1, λ2 and, hence, can find the solution.

An illustration: two-dimensional optimisation problem

An illustration: contour lines
At the optimum, the tangency point of f(x,y) and g(x,y), ∇f = λ∇g,

What does λ mean?
When we set up Lagrangian, we are looking for x, y, z that satisfies:
∇f = λ1∇g1 + λ2∇g2;
where ∇(·) ≡ 􏰂∂(·), ∂(·), ∂(·)􏰃 is the gradient vector.
λs measure the importance of each constraint, i.e., how much the maximum value will change if we marginally change the value of that constraint.

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