Here is my master degree thesis on Bitcoin. The topic is :
THE RISK PERCEPTIONS AND TECHNOLOGY ADOPTIONS OF E-COMMERCE USERS
TOWARDS DIGITAL CURRENCY
It is on google drive at the link below :
click at thesis link
Tuesday, December 5, 2017
Wednesday, August 2, 2017
Game Theory
Basic Concepts of Game Theory
Motivating Example
Location Game: setting shop on a beach
On a linear beach, there are two vendors, they charge the same price. Where should the vendors locate their shops?
- in the center, near each other
what if there are three vendors?
- 3 vendors at the same spot, each get 1/3, one vendor moves, it gets more profit
- 3 vendors different positions, one vendor moves to center, it gets more profit
- no equilibrium
Information
Mutual Knowledge vs Common Knowledge
Mutual Knowledge: all players know A
Common Knowledge: everyone knows that everyone knows A
Perfect Information vs Imperfect Information
Perfect Information: the player knows the full history of the game so far
Imperfect Information: the player does not know parts of the history of the game, such as sealed bid auction
Complete Information vs Incomplete Information
Complete Informatio: the player knows the type of other players and rules of the game
Incomplete Information: the incumbent does not know the true type of entrants
Perfect but Incomplete Information Game
- Price negotiation over used car at a dealer shop
Action vs Strategy
Bill has 5 actions and 6 strategies
Normal Form Game (Strategic Form Game)
- simultaneous game
- static setting
- represented by game matrix
Prisoner's Dilemma Game
| C D
---------------------------
C | -8, -8 -2, -15
D | -15, -2 -3, -3
conditions
- each player has dominant strategy
- dominant strategy equilibrium (-8,-8) worse than optimal choice, dominant strategy equilibrium should be pareto inefficient to at least some other outcome (-3,-3)
how to escape from prisoner's dilemma
- price leadership
- price signaling
- focal points
- info agglomeration: online price agglomeration could intensify or mitigate price wars. However, you lower the price, competitor can see it immediately, and copy the rpice, so it is not worth it to lower the price.
- commitment
strictly dominant strategy
u(si, s-i) > u(si, s-i) for all si
weakly dominant strategy
u(si, s-i) >= u(si, s-i) for all si and
u(si, s-i) > u(si, s-i) for some si
Iterated Dominance Equilibrium
- dominated strategy, strategies that will not be played
- eliminate strictly dominated strategy
- eliminate weakly dominated strategy, iterated weak dominance is not robust
Maximin Strategy Equilibrium
- choose the strategy that gives you a max payoff among the min payoff from each strategy
| L R payoff
---------------------------
T | 10, 4 8, 15 8
B | -100, 5 20, 10 -100
payoff 4 10
Nash Equilibrium
(x*, y*) is a NE if
- x* is best choice given 2's choice of y*
- y* is best choice given 1's choice of x*
coordination game
| S R
---------------------------
S | 5, 5 0, 1
R | 1, 0 1, 1
anti-coordination game
| S R
---------------------------
S | -5,- 5 10, 20
R | 20, 10 -3, -3
if multiple NEs:
1. use focal points
- cultural convention
- social convention
- common perception
2. use risk dominance
if (v>=1),(v>=1) at least one of them is strict inequality, then (S,S) payoff dominate (R,R)
| S R
---------------------------
S | v,v 0, 1
R | 1, 0 1, 1
(v-1) > (1-0)
(S,S) risk dominate (R,R)
Mixed Strategy Nash Equilibrium
- assign probabilities to pure strategies
F
| DL (q) CC (1-q) if Nadal choose
-------------------------------
(p) DL | 50,50 80, 20 DL with prob p
N ------------------------------
(1-p) CC | 90,10 . 20,80 CC with prob 1-p
so Federer payoff is
if Federer chooses 50p + 10(1-p) = 20p +80(1-p) => p=0.7
DL with prob q if p> 0.7, q = 1
CC with prob 1-q p < 0.7 , q =0
so Nadal's payoff is p=0.7 , indiff over q
50q + 80(1-q) = 90q + 20(1-q)
q = 0.6
if q<0.6, p =1
q > 0.6, p = 0
q = 0.6, indifference over p
NE (p*, 1-p*) = (0.7, 0.3)
(q*, 1-q*) = (0.6, 0.4)
implications of mixed strategy NE
- each player should mix his pure strategy so that the other player is indifferent among all his pure strategy
choose (p,1-p) so that UDL = UCC
choose (q,1-q) so that UDL = UCC
- assign zero prob to dominated pure strategy
- randomise just right, avoid outguessed by opponent
- each player mix his pure strategies so that the other player is indifferent among all his pure strategies
Oligopoly Games
| compete on quantity compete on price
------------------------------------------------------------------------
simultaneous | Cournot Bertrand
(static) |
| Cournot- Bertrand-
sequential | Stackelberg Stackelberg
substitute: q1 up -> q2 down
- backward induction: for finite dynamic games, start from last stage of the game, not for infinite game
- subgame perfect NE: rule out NE of non credible threat, for finite and infinite game
Sequential Bargaining
Subgame Perfect Equilibrium
Example 1
Example 2
In game theory, having fewer option may be better, because you can manipulate the other player's choices so that outcome is better for you.
Strategic Moves
to influence opponents expectation about your action
to get around prisoner' dilemma
introduced by Thomas Schelling
- cross shareholding
- MFC clause (mutual adoption, 2 period model)
- price matching guarantee policy (mutual adoption)
- entry deterrence:
-- side payment, merge, build a reputation, invest in extra capacity
chicken game
- two gangsters race their cars toward each other , the first one to chicken out loses.
G Gang B
a | Straight Avoid
n --------------------------------
g S | -100,- 100 10, -2
A A | -2, 10 0, 0
- you don't know how to secure the equilibrium that is favorable to you, because they are two NE
- so you use strategic moves to gain advantage
- commitment, play aggressively, scare your opponents
Entry deterrence
- incumbent facing a potential entrant
- entrant moves first, incumbent moves later
- the latter can behave strategically to deter the entry
I Entrant
n | Enter Stay out
c --------------------------------
u E | 100,20 200, 0
m S | 70, -10 130, 0
bent
the NE is (100,20), it is incumbent dominant strategy, but potential entrant will enter
a few options:
- side payment, illegal?
- marge, anti-competitive ?
- build a reputation for being irrational, manipulate rivals choices to your advantage
(incumbent can increase 70 to > 100, or decrease 100 to be < 70)
the SPNE is (100,20)
in chicken game, sequential game, use strategic moves to show commitment and gain advantage
Fudenberg Tirole Taxanomy
How to apply
step 1: calculate your profit as a function of what the other players might do
πyou = f(others actions)
step 2: guess your competitor's profits as a function of what you might do
πothers = f(your actions)
step 3: can legally cooperate?
if yes, use cooperative game theory
if no, use non-cooperative game theory
step 4: create the game's payoff
step 5: pick the game strategies
step 6:strategic moves
step 7: make the moves
(to be continued.)
Motivating Example
Location Game: setting shop on a beach
On a linear beach, there are two vendors, they charge the same price. Where should the vendors locate their shops?
- in the center, near each other
what if there are three vendors?
- 3 vendors at the same spot, each get 1/3, one vendor moves, it gets more profit
- 3 vendors different positions, one vendor moves to center, it gets more profit
- no equilibrium
Information
Mutual Knowledge vs Common Knowledge
Mutual Knowledge: all players know A
Common Knowledge: everyone knows that everyone knows A
Perfect Information vs Imperfect Information
Perfect Information: the player knows the full history of the game so far
Imperfect Information: the player does not know parts of the history of the game, such as sealed bid auction
Complete Information vs Incomplete Information
Complete Informatio: the player knows the type of other players and rules of the game
Incomplete Information: the incumbent does not know the true type of entrants
Perfect but Incomplete Information Game
- Price negotiation over used car at a dealer shop
Action vs Strategy
Normal Form Game (Strategic Form Game)
- simultaneous game
- static setting
- represented by game matrix
Prisoner's Dilemma Game
| C D
---------------------------
C | -8, -8 -2, -15
D | -15, -2 -3, -3
conditions
- each player has dominant strategy
- dominant strategy equilibrium (-8,-8) worse than optimal choice, dominant strategy equilibrium should be pareto inefficient to at least some other outcome (-3,-3)
how to escape from prisoner's dilemma
- price leadership
- price signaling
- focal points
- info agglomeration: online price agglomeration could intensify or mitigate price wars. However, you lower the price, competitor can see it immediately, and copy the rpice, so it is not worth it to lower the price.
- commitment
strictly dominant strategy
u(si, s-i) > u(si, s-i) for all si
weakly dominant strategy
u(si, s-i) >= u(si, s-i) for all si and
u(si, s-i) > u(si, s-i) for some si
Iterated Dominance Equilibrium
- dominated strategy, strategies that will not be played
- eliminate strictly dominated strategy
- eliminate weakly dominated strategy, iterated weak dominance is not robust
Maximin Strategy Equilibrium
- choose the strategy that gives you a max payoff among the min payoff from each strategy
| L R payoff
---------------------------
T | 10, 4 8, 15 8
B | -100, 5 20, 10 -100
payoff 4 10
Nash Equilibrium
(x*, y*) is a NE if
- x* is best choice given 2's choice of y*
- y* is best choice given 1's choice of x*
coordination game
| S R
---------------------------
S | 5, 5 0, 1
R | 1, 0 1, 1
anti-coordination game
| S R
---------------------------
S | -5,- 5 10, 20
R | 20, 10 -3, -3
if multiple NEs:
1. use focal points
- cultural convention
- social convention
- common perception
2. use risk dominance
if (v>=1),(v>=1) at least one of them is strict inequality, then (S,S) payoff dominate (R,R)
| S R
---------------------------
S | v,v 0, 1
R | 1, 0 1, 1
(v-1) > (1-0)
(S,S) risk dominate (R,R)
Mixed Strategy Nash Equilibrium
- assign probabilities to pure strategies
F
| DL (q) CC (1-q) if Nadal choose
-------------------------------
(p) DL | 50,50 80, 20 DL with prob p
N ------------------------------
(1-p) CC | 90,10 . 20,80 CC with prob 1-p
so Federer payoff is
if Federer chooses 50p + 10(1-p) = 20p +80(1-p) => p=0.7
DL with prob q if p> 0.7, q = 1
CC with prob 1-q p < 0.7 , q =0
so Nadal's payoff is p=0.7 , indiff over q
50q + 80(1-q) = 90q + 20(1-q)
q = 0.6
if q<0.6, p =1
q > 0.6, p = 0
q = 0.6, indifference over p
NE (p*, 1-p*) = (0.7, 0.3)
(q*, 1-q*) = (0.6, 0.4)
- each player should mix his pure strategy so that the other player is indifferent among all his pure strategy
choose (p,1-p) so that UDL = UCC
choose (q,1-q) so that UDL = UCC
- assign zero prob to dominated pure strategy
- randomise just right, avoid outguessed by opponent
- each player mix his pure strategies so that the other player is indifferent among all his pure strategies
Oligopoly Games
| compete on quantity compete on price
------------------------------------------------------------------------
simultaneous | Cournot Bertrand
(static) |
| Cournot- Bertrand-
sequential | Stackelberg Stackelberg
substitute: q1 up -> q2 down
collusion outcome less than NE outcome
Application of simultaneous games
Tragedy of the commons
horizontal axis: % of car commuters
vertical axis: payoff for commuters
NE : (q, 1-q)
- q% commute in cars and (1-q)% in busses
still not socially efficient
- all commuting by busses is still Pareto efficient
Sequential Games (Extensive Form Game)
- dynamic setting- backward induction: for finite dynamic games, start from last stage of the game, not for infinite game
- subgame perfect NE: rule out NE of non credible threat, for finite and infinite game
Sequential Bargaining
𝛿 > 50% , first mover adv ; agreement reached in first round of bargaining
𝛿 < 50%, second mover adv
0< 𝛿 <100%, 𝛿 is time value
Example 1
SPNE outcome (0, 4)
SPNE strategy:
If B choose R, A will choose (5, -1), so B will choose L, but A will not choose R, A choose L, B choose R, so (0,4) is SPNE outcome.
Strategic Moves
To solve empty promise problem, make 5 worse than 4, cut (5,-1) branch, make 4 better than 5Example 2
Strategic Moves
to influence opponents expectation about your action
to get around prisoner' dilemma
introduced by Thomas Schelling
- cross shareholding
- MFC clause (mutual adoption, 2 period model)
- price matching guarantee policy (mutual adoption)
- entry deterrence:
-- side payment, merge, build a reputation, invest in extra capacity
chicken game
- two gangsters race their cars toward each other , the first one to chicken out loses.
G Gang B
a | Straight Avoid
n --------------------------------
g S | -100,- 100 10, -2
A A | -2, 10 0, 0
- you don't know how to secure the equilibrium that is favorable to you, because they are two NE
- so you use strategic moves to gain advantage
- commitment, play aggressively, scare your opponents
Entry deterrence
- incumbent facing a potential entrant
- entrant moves first, incumbent moves later
- the latter can behave strategically to deter the entry
I Entrant
n | Enter Stay out
c --------------------------------
u E | 100,20 200, 0
m S | 70, -10 130, 0
bent
the NE is (100,20), it is incumbent dominant strategy, but potential entrant will enter
a few options:
- side payment, illegal?
- marge, anti-competitive ?
- build a reputation for being irrational, manipulate rivals choices to your advantage
(incumbent can increase 70 to > 100, or decrease 100 to be < 70)
the SPNE is (100,20)
in chicken game, sequential game, use strategic moves to show commitment and gain advantage
Fudenberg Tirole Taxanomy
| Your
rival's actions |
Your firm commitment | ||
| Tough | soft | ||
| Strategic complement | You
commit tough, your rival tough too (puppy dog) |
You
commit soft your rival soft too (fat cat) |
|
| Strategic substitutes | You
commit tough, your rival soft
(top dog) |
You
commit soft your rival tough (lean & hungry look) |
|
step 1: calculate your profit as a function of what the other players might do
πyou = f(others actions)
step 2: guess your competitor's profits as a function of what you might do
πothers = f(your actions)
step 3: can legally cooperate?
if yes, use cooperative game theory
if no, use non-cooperative game theory
step 4: create the game's payoff
step 5: pick the game strategies
step 6:strategic moves
step 7: make the moves
(to be continued.)
Monday, July 24, 2017
Leverage and Cost of Capital
Leverage
- the effects that fixed costs have on the returns that shareholders earn
- magnify returns and risks
Operating leverage
- relationship between sales revenue and EBIT
Financial leverage
- relationship between EBIT and EPS
Total leverage
- relationship between sales revenue and EPS
operating leverage and financial leverage influence a firm's beta
breakeven point = fixed costs / contribution margin
= fixed costs / (price - variable costs)
When contribution margin (CM) is higher, profit rises faster
The higher the fixed costs, and low variable cost, the higher the beta
Operating leverage
- comes from mix of fixed and variable cost
- the effects that fixed costs have on the returns that shareholders earn
- magnify returns and risks
Operating leverage
- relationship between sales revenue and EBIT
Financial leverage
- relationship between EBIT and EPS
Total leverage
- relationship between sales revenue and EPS
operating leverage and financial leverage influence a firm's beta
breakeven point = fixed costs / contribution margin
= fixed costs / (price - variable costs)
When contribution margin (CM) is higher, profit rises faster
The higher the fixed costs, and low variable cost, the higher the beta
Operating leverage
- comes from mix of fixed and variable cost
| if sales up by 10% | ||||
| Lite | heavy | Lite | heavy | |
| sales volume | 10000 | 10000 | 11000 | 11000 |
| price | 1000 | 1000 | 1000 | 1000 |
| total revenue | 10000000 | 10000000 | 11000000 | 11000000 |
| fixed cost | 5000000 | 2000000 | 5000000 | 2000000 |
| variable cost (per unit) |
400 | 700 | 400 | 700 |
| total cost | 9000000 | 9000000 | 9400000 | 9700000 |
| EBIT | 1000000 | 1000000 | 1600000 | 1300000 |
Degree of Operating leverage = %ΔEBIT / %ΔSales
if sales up by 10%,
DOL of Lite = 60%/10% = 6
DOL of heavy = 30%/10% = 3
- more fixed cost , DOL increases
if DOL > 1, the firm has operating leverage
Financial leverage
- comes from use of debt
- more EBIT goes to investors in levered firm (as in NI + interest)
if sales up by 10%,
DOL of Lite = 60%/10% = 6
DOL of heavy = 30%/10% = 3
- more fixed cost , DOL increases
if DOL > 1, the firm has operating leverage
Financial leverage
- comes from use of debt
| Unlevered | levered | |
| debt | 0 | 10000 |
| equity | 20000 | 10000 |
| asset | 20000 | 20000 |
| tax rate | 0.4 | 0.4 |
| interest rate | 0.12 | 0.12 |
| EBIT | 3000 | 3000 |
| interest(12%) | 0 | 1200 |
| EBT | 3000 | 1800 |
| tax | 1200 | 720 |
| NI | 1800 | 1080 |
| ROE | 9% | 11% |
- for financially leveraged firm, NI is lower, but equity base is lower too, so ROE is higher
Basic Earning Power
BEP = EBIT/total assets
- BEP is not affected by financial leverage, because EBIT is the same whether you borrow or not
- BEP is affected by operating leverage, because change in EBIT is affected by DOL
Implications
- for leverage to be positive (increase ROE), BEP must be > rd
- for firms with high profit , use more debt, to shield the profit using debt (tax shield)
Basic Earning Power
BEP = EBIT/total assets
- BEP is not affected by financial leverage, because EBIT is the same whether you borrow or not
- BEP is affected by operating leverage, because change in EBIT is affected by DOL
Implications
- for leverage to be positive (increase ROE), BEP must be > rd
- for firms with high profit , use more debt, to shield the profit using debt (tax shield)
Sunday, July 23, 2017
Financial Derivatives
Forward and Futures
pricing:
F(t, T) = S(t) e ^ (r + u - d -y) (T-t)
where T: expiration date
r : risk free rate
u : storage cost
d : dividend yield
y : convenience yield
OTC central clearing to lower counterparty risk, such as London clearing house (LCH)
futures contracts are settled by cash settlement
futures contracts are closed by entering into an offsetting position relative to your original position
Interest Rate Forward
notation:
2f1 - 1 year from now, 6 month rate
14f6 - 7 year from now, 3 year rate
By the principle of no arbitrage,
(1+r1/2)(1+1f1 /2) = (1+r2/2)2, solve for 1f1
r1: 6 month spot
r2: 1 year spot
Interest Rate Swap
interest rate is implied in swap
USD/THB spot
USD LIBOR
# of days
swap point = fwd - spot
Σ PV (fixed) = Σ PV(floating)
money market rate, and bond rate affect swap rate
bank use PV01 to calculate risk
if gap > threshold, bank charge more
DV01 or PVBP
- 30 yr bond with 5.5% coupon
at yield of 5.5%, price = 100
at yield of 5.51%, price = 99.8540
DV01 = 0.146% of par
or DV01(per $1 mm par) = $1460
** take note **
- PV01 is change in market value from bumping the coupon rate by 1 bp
- DV01 is the change in market value for a 1 bp parallel shift of the yield curve
**hedge bond investment with bond futures
- use DV01
DV01 of 6 year bond with coupon of 5.5%: 712.5 per $1million par value
DV01 of 6 year bond with coupon of 5.0%: 613.1 per $1million par value
- the hedge ratio (futures contract to sell) of 6 year bond investment against interest rate risk is
hedge ratio = 712.5/613.1
Cross Currency Swap
- agreement between two parties to exchange principal and interest payments in two currencies over specified period
- may have or my not have initial principal exchange
- interest payments are usually not netted
basis swap (floating vs floating)
FX Forward
EUR/USD ; ieur < iusd
EUR/USD forward, EUR appreciate
=> swap point+ve
=> EUR/USD > spot
AUD/USD forward, AUD depreciate
=> swap point -ve
=> AUD /USD < spot
Case study:
Delox imports machine from Japan. Its revenue is in EUR, while its expense is in JPY. It expects to pay JPY 1000 million in 6 months. How to manage its FX risk?
Ans: one way is to buy JPY forward
EURJPY spot rate 129.45
6 month EURJPY forward rate 128.26
forward: bank charge bid offer spread
swap: bank will not charge bid offer spread
pricing:
F(t, T) = S(t) e ^ (r + u - d -y) (T-t)
where T: expiration date
r : risk free rate
u : storage cost
d : dividend yield
y : convenience yield
OTC central clearing to lower counterparty risk, such as London clearing house (LCH)
futures contracts are settled by cash settlement
futures contracts are closed by entering into an offsetting position relative to your original position
Interest Rate Forward
notation:
2f1 - 1 year from now, 6 month rate
14f6 - 7 year from now, 3 year rate
By the principle of no arbitrage,
(1+r1/2)(1+1f1 /2) = (1+r2/2)2, solve for 1f1
r1: 6 month spot
r2: 1 year spot
Interest Rate Swap
interest rate is implied in swap
USD/THB spot
USD LIBOR
# of days
swap point = fwd - spot
Σ PV (fixed) = Σ PV(floating)
money market rate, and bond rate affect swap rate
bank use PV01 to calculate risk
if gap > threshold, bank charge more
DV01 or PVBP
- 30 yr bond with 5.5% coupon
at yield of 5.5%, price = 100
at yield of 5.51%, price = 99.8540
DV01 = 0.146% of par
or DV01(per $1 mm par) = $1460
** take note **
- PV01 is change in market value from bumping the coupon rate by 1 bp
- DV01 is the change in market value for a 1 bp parallel shift of the yield curve
**hedge bond investment with bond futures
- use DV01
DV01 of 6 year bond with coupon of 5.5%: 712.5 per $1million par value
DV01 of 6 year bond with coupon of 5.0%: 613.1 per $1million par value
- the hedge ratio (futures contract to sell) of 6 year bond investment against interest rate risk is
hedge ratio = 712.5/613.1
Cross Currency Swap
- agreement between two parties to exchange principal and interest payments in two currencies over specified period
- may have or my not have initial principal exchange
- interest payments are usually not netted
basis swap (floating vs floating)
FX Forward
EUR/USD ; ieur < iusd
EUR/USD forward, EUR appreciate
=> swap point
AUD/USD forward, AUD depreciate
=> swap point -
Case study:
Delox imports machine from Japan. Its revenue is in EUR, while its expense is in JPY. It expects to pay JPY 1000 million in 6 months. How to manage its FX risk?
Ans: one way is to buy JPY forward
EURJPY spot rate 129.45
6 month EURJPY forward rate 128.26
swap: bank will not charge bid offer spread
Friday, July 21, 2017
Financial Options
Options
components of option price
m.v of option = time premium + intrinsic value
intrinsic value: the difference of m.v of underlying and strike price of call option
Minimum value of call
American call : Ca(S0, T, X) >= max(0, S0 - X)
European call : Ce(S0, T, X) >= max(0, S0 - X(1+r)-T)
Minimum value of put
American put : max(0, X - S0 )
European put : max(0, X(1+r)-T - S0 )
Put call parity
S0 = C(S0, T, X) - P(S0, T, X) + X/(1 + r )
ATM european call or ATM european put, which one has higher price?
S = c - p + pv(x)
c - p = S - pv(x) ; S=X for ATM option
c - p = S - pv(S) ; S - pv(S) > 0
So : c > p
Put call forward parity
S0 = C(S0, T, X) - P(S0, T, X) + X/(1 + r )
F = S0(1+r)
S0 = F/(1+r)
So : F/(1+r) = C(S0, T, X) - P(S0, T, X) + X/(1 + r )
P(S0, T, X) = C(S0, T, X) + (X-F)/(1+r)
Binomial Model
- discrete time model
- if infinite samples, converges to BSM model
- if interval time getting smaller, converges to BSM model
p = (1+r-d) / (u -d)
c = [ pCu + (1-p)Cd ] / (1+r)
if stock price up by 20%, and down by 10%, then: u = 1.2, d = 0.9
S+ = Su; S- = Sd, find Cu, Cd, then find c
Black Scholes Model
- continuous time model
- assume rf and vol are constant
- no taxes and transaction fee
- assume options are european
- assume stock price is normally distributed
- S, T, X, rf, vol -> find c
calculate implied vol
- work backwards to find it
- C, S, X, T, rf ->BSM model -> find implied vol
volatility smile
- shows implied vol is not consistent
- implied vol depends on exercise price
- violates the constant vol assumption of BSM
Implied vol > forecast vol
- option overvalued
- sell option
selling options, the trade is short volatility. if actual vol is lower than what he priced it at, he makes money
Interest rate Cap
- series of interest rate call options
Interest rate Floor
- series of interest rate put options
components of option price
intrinsic value: the difference of m.v of underlying and strike price of call option
Minimum value of call
American call : Ca(S0, T, X) >= max(0, S0 - X)
European call : Ce(S0, T, X) >= max(0, S0 - X(1+r)-T)
Minimum value of put
American put : max(0, X - S0 )
European put : max(0, X(1+r)-T - S0 )
Ca(S0, T, X) > Ce(S0, T, X)
but prior to expiration, S0 - X(1+r)-T > S0 - X
early exercise?
- for call , if dividend > time premium, then you exercise the call option
- for put , if interest rat is large enough
Maximum value of put
American put : X
European put: X(1+r)-T
S0 = C(S0, T, X) - P(S0, T, X) + X/(1 + r )
ATM european call or ATM european put, which one has higher price?
S = c - p + pv(x)
c - p = S - pv(x) ; S=X for ATM option
c - p = S - pv(S) ; S - pv(S) > 0
So : c > p
Put call forward parity
S0 = C(S0, T, X) - P(S0, T, X) + X/(1 + r )
F = S0(1+r)
S0 = F/(1+r)
So : F/(1+r) = C(S0, T, X) - P(S0, T, X) + X/(1 + r )
P(S0, T, X) = C(S0, T, X) + (X-F)/(1+r)
Binomial Model
- discrete time model
- if infinite samples, converges to BSM model
- if interval time getting smaller, converges to BSM model
p = (1+r-d) / (u -d)
c = [ pCu + (1-p)Cd ] / (1+r)
if stock price up by 20%, and down by 10%, then: u = 1.2, d = 0.9
S+ = Su; S- = Sd, find Cu, Cd, then find c
Black Scholes Model
- continuous time model
- assume rf and vol are constant
- no taxes and transaction fee
- assume options are european
- assume stock price is normally distributed
- S, T, X, rf, vol -> find c
calculate implied vol
- work backwards to find it
- C, S, X, T, rf ->BSM model -> find implied vol
volatility smile
- shows implied vol is not consistent
- implied vol depends on exercise price
- violates the constant vol assumption of BSM
- option overvalued
- sell option
selling options, the trade is short volatility. if actual vol is lower than what he priced it at, he makes money
Interest rate Cap
- series of interest rate call options
Interest rate Floor
- series of interest rate put options
Wednesday, July 12, 2017
Macroeconomics (Part 2)
Marshall-Lerner condition
A depreciation has two good effects (substitution of imports , more demand for exports) and one bad effects (imports are more expensive). Marshall-Lerner condition is real depreciation leads to an increase in NX if sum of export and import elasticity is > 1. Initially, the condition is not met, devaluation worsens the trade balance. In the long run, trade balance will improve, it is called the J curve effect.
PED (x-m) > 1, devaluation improves current account balance.
PED: price elasticity of demand
Mundell-Fleming model
An extension of IS-LM model, the Mundell–Fleming model describes the short-run relationship between an economy's nominal exchange rate, interest rate, and output. It is used in open economy. This model has been used to argue that an economy cannot simultaneously maintain a fixed exchange rate, free capital movement, and an independent monetary policy. This principle is frequently called the "impossible trinity," "unholy trinity," or the "Mundell–Fleming trilemma."
It is also known is IS-LM-BoP model. BoP is (Balance of Payments).
In perfect capital mobility, the BoP curve is horizontal.
In perfect capital mobility and fixed exchange rate, the increase in gov spending (expansionary fiscal policy) forces the central bank to supply more local currency, to keep the interest rate unchanged. At the same interest rate, the output has increased.
In perfect capital mobility and fixed exchange rate, the increase in local currency (expansionary monetary policy) will drop the exchange rate. It forces the government to buy local currency to maintain fixed exchange rate. This reduces the money supply. This means monetary policy has no effect.
So it is impossible to have fixed exchange rate, perfect capital movement, and independent monetary policy.
Minimum wage policy
Why minimum wage is not good for the country?
If wage > productivity,
1) πe ↑ => P↑
w ↑ => C ↑, Pe ↑
Pe => πe ↑ => P ↑
2) P ↑ => X ↓ & IM ↑ => NX ↓
P ↑ => ε = EP/P* => ε↑ => NX ↓
Purchasing power of currency and Purchasing power parity
The purchasing power of a currency refers to the quantity of the currency required to purchase a unit of a good, or basket of goods and services. Purchasing power is determined by the relative cost of living and inflation rates in different countries.
Purchasing power parity means equalising the purchasing power of two currencies by taking into account these cost of living and inflation differences. The basis of PPP is law of one price. In the absence of transportation and other transaction costs, competitive markets will make the price of an identical good the same in two countries, expressed in the same currency.
For example, a particular TV set that sells for 400Rp in India should cost 800Rp in Pakistan if the exchange rate between India and Pakistan is 2 Pak/India. If the price of the TV in Pakistan was only 700 Rp, consumers in India would prefer buying the TV set in Pakistan. If this process (called "arbitrage") is carried out at a large scale, the India consumers buying Pakistan goods will bid up the value of the Pakistan Dollar, thus making Pak goods more costly to them. This process continues until the goods are again the same price.
For example again, if we convert GDP in China to US dollars using market exchange rates, relative purchasing power is not taken into account. By adjusting rates to take into account local purchasing power differences, known as PPP adjusted exchange rates, international comparisons are more valid.
Money illusions and Money neutrality
The illusions of money : people think of money in nominal terms, the nominal value of money is mistaken for its purchasing power.
Neutrality of money : change in stock of money affects nominal variables, such as price, wages, exchange rate, doe not affect real variables, such as employment, GDP, consumption.
Microfoundations in Macroeconomics
It is a macro model that is built up from a formal analysis of the behaviour of individual agents in a consistent way. There may be just a single representative agent, or increasingly heterogeneous agents. So this way of macro analysis will involve lots of optimisation by individual agents, to derive aggregate relationships.
Differences between Micro and Macroeconomics
The difference between micro and macro economics is easy to tell. Microeconomics is the study of economics at an individual, group or company level. Microeconomics focuses on issues that affect individuals and companies. Micro looks at supply and demand and the setting of price level.
Macroeconomics, on the other hand, is the study of a national economy as a whole. Macro studies the GDP and how it is affected by national income, unemployment, growth rate.
A depreciation has two good effects (substitution of imports , more demand for exports) and one bad effects (imports are more expensive). Marshall-Lerner condition is real depreciation leads to an increase in NX if sum of export and import elasticity is > 1. Initially, the condition is not met, devaluation worsens the trade balance. In the long run, trade balance will improve, it is called the J curve effect.
PED: price elasticity of demand
Mundell-Fleming model
An extension of IS-LM model, the Mundell–Fleming model describes the short-run relationship between an economy's nominal exchange rate, interest rate, and output. It is used in open economy. This model has been used to argue that an economy cannot simultaneously maintain a fixed exchange rate, free capital movement, and an independent monetary policy. This principle is frequently called the "impossible trinity," "unholy trinity," or the "Mundell–Fleming trilemma."
It is also known is IS-LM-BoP model. BoP is (Balance of Payments).
In perfect capital mobility, the BoP curve is horizontal.
So it is impossible to have fixed exchange rate, perfect capital movement, and independent monetary policy.
Why minimum wage is not good for the country?
If wage > productivity,
1) πe ↑ => P↑
w ↑ => C ↑, Pe ↑
Pe => πe ↑ => P ↑
2) P ↑ => X ↓ & IM ↑ => NX ↓
P ↑ => ε = EP/P* => ε↑ => NX ↓
Purchasing power of currency and Purchasing power parity
The purchasing power of a currency refers to the quantity of the currency required to purchase a unit of a good, or basket of goods and services. Purchasing power is determined by the relative cost of living and inflation rates in different countries.
Purchasing power parity means equalising the purchasing power of two currencies by taking into account these cost of living and inflation differences. The basis of PPP is law of one price. In the absence of transportation and other transaction costs, competitive markets will make the price of an identical good the same in two countries, expressed in the same currency.
For example, a particular TV set that sells for 400Rp in India should cost 800Rp in Pakistan if the exchange rate between India and Pakistan is 2 Pak/India. If the price of the TV in Pakistan was only 700 Rp, consumers in India would prefer buying the TV set in Pakistan. If this process (called "arbitrage") is carried out at a large scale, the India consumers buying Pakistan goods will bid up the value of the Pakistan Dollar, thus making Pak goods more costly to them. This process continues until the goods are again the same price.
For example again, if we convert GDP in China to US dollars using market exchange rates, relative purchasing power is not taken into account. By adjusting rates to take into account local purchasing power differences, known as PPP adjusted exchange rates, international comparisons are more valid.
Money illusions and Money neutrality
The illusions of money : people think of money in nominal terms, the nominal value of money is mistaken for its purchasing power.
Neutrality of money : change in stock of money affects nominal variables, such as price, wages, exchange rate, doe not affect real variables, such as employment, GDP, consumption.
Microfoundations in Macroeconomics
It is a macro model that is built up from a formal analysis of the behaviour of individual agents in a consistent way. There may be just a single representative agent, or increasingly heterogeneous agents. So this way of macro analysis will involve lots of optimisation by individual agents, to derive aggregate relationships.
Differences between Micro and Macroeconomics
The difference between micro and macro economics is easy to tell. Microeconomics is the study of economics at an individual, group or company level. Microeconomics focuses on issues that affect individuals and companies. Micro looks at supply and demand and the setting of price level.
Macroeconomics, on the other hand, is the study of a national economy as a whole. Macro studies the GDP and how it is affected by national income, unemployment, growth rate.
Monday, July 10, 2017
Fiscal Policy
Fiscal policy
- automatic stabilizer: tax revenue, unemployment benefit
- discretionary policy
tax revenue is endogenous, sensitive to the state of economy than spending
fiscal policy is planned with horizon larger than monetary policy
fiscal consolidation may increase AD, eg. expansionry fiscal contraction
- people expect lower debt, lower taxes in future
- spending cut, wage cut -> labor cost down, Investment up , profit up
- structure reform complement fiscal contraction
Interaction of fiscal and monetary policy
use stackelberg game to model the interaction of policy makers
Government Budget Deficit
deficitt = rBt-1 + Gt -Tt
where r: real interest rate
Bt-1: government debt at the end of year t-1
let deficitt = Bt - Bt-1
therefore:
Bt - Bt-1 = = rBt-1 + Gt -Tt
Bt = (1+r)Bt-1 + (Gt -Tt)
-if government spending is unchanged, a decrease in taxes today will have to be offset by an increase in taxes in the future.
-the longer the government waits to increase taxes, the higher the real interest rate, and higher the increase in future taxes
Debt to GDP Ratio
Bt/Yt = (1+r)Bt-1/Yt + (Gt -Tt)/Yt
after mathematics munipulation:
Bt/Yt - Bt-1/Yt-1 = (r-g)Bt-1/Yt-1 + (Gt -Tt)/Yt
where g: GDP growth rate
How countries reduced their debt ratios
- run budget surplus, (Gt -Tt) < 0
- have low real interest rate and high GDP growth, (r - g) > 0, real interest low can be low or even negative when inflation is high.
- a large part of the decrease in debt ratios was achieved by paying bond holders a negative real interest rate on the bonds
Balanced Budget
- It means Gt = Tt
If economy is good, T up -> G up -> cause overheating
If economy is not good, T down -> G down -> econ could not recover
Therefore, balanced budget is not practical
Ricardian Equivalence
David Ricardo developed a theory about government spending and private spending. When government stimulate demand bu debt financed spending, the people will save money to pat for future tax increases (to be used to pay off the debt).
- So overall demand is unchanged
Cyclically Adjusted Deficits
It is used to indicate whether tax/revenue system is going to create deficit at Yn (output at natural full employment). If it is negative at Yn, deficit is *not* going to go down.
- We never know exact Yn, and Yn changes, so Cyclically Adjusted Deficits is not useful
Money Finance
1. Debt Monetization
Fiscal dominance of monetary policy: Central bank must do what the government tells it to do. Government issues bonds and forces CB to buy. The central bank then pays the government with the money it creates, and the government uses that money to finance its deficit. This process is called debt monetization.
2. Seignorage
The amount of good and services that government can obtain by printing money. The revenue from money creation is called seignorage.
seignorage = ΔH/P = ΔH/H * H/P
seignorage/ Y = (ΔH/H * H/P ) / Y
If government uses seinorage to finance budget deficit of 10% of GDP, seignorage/ Y = 10%, so ΔH/H = 10% and (H/P)/Y = 1, the growth rate of nominal money must be 10%.
- automatic stabilizer: tax revenue, unemployment benefit
- discretionary policy
tax revenue is endogenous, sensitive to the state of economy than spending
fiscal policy is planned with horizon larger than monetary policy
fiscal consolidation may increase AD, eg. expansionry fiscal contraction
- people expect lower debt, lower taxes in future
- spending cut, wage cut -> labor cost down, Investment up , profit up
- structure reform complement fiscal contraction
Interaction of fiscal and monetary policy
use stackelberg game to model the interaction of policy makers
C- cooperation , Pareto efficient equilibrium
OM -monetary leadership, OF -fiscal leadership
Government Budget Deficit
deficitt = rBt-1 + Gt -Tt
where r: real interest rate
Bt-1: government debt at the end of year t-1
let deficitt = Bt - Bt-1
therefore:
Bt - Bt-1 = = rBt-1 + Gt -Tt
Bt = (1+r)Bt-1 + (Gt -Tt)
-if government spending is unchanged, a decrease in taxes today will have to be offset by an increase in taxes in the future.
-the longer the government waits to increase taxes, the higher the real interest rate, and higher the increase in future taxes
Debt to GDP Ratio
Bt/Yt = (1+r)Bt-1/Yt + (Gt -Tt)/Yt
after mathematics munipulation:
Bt/Yt - Bt-1/Yt-1 = (r-g)Bt-1/Yt-1 + (Gt -Tt)/Yt
where g: GDP growth rate
How countries reduced their debt ratios
- run budget surplus, (Gt -Tt) < 0
- have low real interest rate and high GDP growth, (r - g) > 0, real interest low can be low or even negative when inflation is high.
- a large part of the decrease in debt ratios was achieved by paying bond holders a negative real interest rate on the bonds
Balanced Budget
- It means Gt = Tt
If economy is good, T up -> G up -> cause overheating
If economy is not good, T down -> G down -> econ could not recover
Therefore, balanced budget is not practical
Ricardian Equivalence
David Ricardo developed a theory about government spending and private spending. When government stimulate demand bu debt financed spending, the people will save money to pat for future tax increases (to be used to pay off the debt).
- So overall demand is unchanged
Cyclically Adjusted Deficits
It is used to indicate whether tax/revenue system is going to create deficit at Yn (output at natural full employment). If it is negative at Yn, deficit is *not* going to go down.
- We never know exact Yn, and Yn changes, so Cyclically Adjusted Deficits is not useful
Money Finance
1. Debt Monetization
Fiscal dominance of monetary policy: Central bank must do what the government tells it to do. Government issues bonds and forces CB to buy. The central bank then pays the government with the money it creates, and the government uses that money to finance its deficit. This process is called debt monetization.
2. Seignorage
The amount of good and services that government can obtain by printing money. The revenue from money creation is called seignorage.
seignorage = ΔH/P = ΔH/H * H/P
seignorage/ Y = (ΔH/H * H/P ) / Y
If government uses seinorage to finance budget deficit of 10% of GDP, seignorage/ Y = 10%, so ΔH/H = 10% and (H/P)/Y = 1, the growth rate of nominal money must be 10%.
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