Financial Markets

CML

market expected return

beta

ARA

continuous return from discrete

duration

change in price using duration

convexity using discrete compounding

convexity using continuous compounding

hedge ratio

Pseudo probability, p

price of call option

min. number of ups to get positive payoff

RRA

= ARA*w

amount you would be willing to give up in a fair bet

market variance

covariance

put-call parity

call pricing using binomial approximation

Z₁

SD₁

P^I

upper bound for call option

upper bound for put option

lower bounds of call option

lower bounds for put options

two period binomial option pricing

three period binomial option pricing

risk neutral probabilities, q

forward prices, no income

forward prices, predictable income

forward prices, predictable yield

value of long forward

value of short forward

forward rate on currencies

forward rate on investment commodities

forward rate on consumption commodities

d₁

d₂

u =

d =

1/u

CAPM Assumptions

• no transaction costs

• all assets are infinitely divisible

• no personal taxes

• markets are purely competitive

• investors choose portfolios purely on expected return and standard deviation

• short sales permitted

• unlimited borrowing and lending at the risk free rate

• investors have homogenous expectations

arithmetic returns

• discrete compounding

• multiplicative cumulation

• non-symmetric reversal in return

• not normally distributed

logarithmic returns

• continuous compounding

• additive cumulation

• reversal in return is symmetric

• more normally distributed

fama French model

• extends CAPM to include two additional factors that capture systemic risk that is not explained by the market alone

• factors: market, size, value

• largely empirical

• explains more of the variation in average returns than CAPM

• empirically supported across many markets and long time periods

• lacks strong theoretical foundations

• interpretations of size and value factors still debated

carhart four factor model

• extends on mama French by adding momentum factor

• strong empirical support: captures size, value and momentum effects

• widely used in performance evaluation

• lacks deep theoretical foundation

• momentum may arise from behavioural biases rather than risk premia

CAPM vs APT

• CAPM is one factor only, APT has multiple factors

• CAPM has market risk only, several system risks priced in APT

• CAPM has weak empirical performance but APT has stronger

• core idea of CAPM = excess returns depend on exposure to market portfolio

• core idea of APT = expected returns are determined by multiple sources of systemic risk. condition of no arbitrage prevents mispricing.

effect of a decrease in market interest rate on utility of borrowers and lenders

• market interest rate = -(1+r)

• a fall in r makes the budget line flatter

• can now access a higher utility curve

• consumers consume more this period and next

effect of a decrease in market interest rate on the present wealth of borrowers and lenders

• pv of future income = income*1/1+r

• as r falls, pv of future income rises

• borrower has negative future net income, lower r reduces the pv of loan repayments, increasing the present wealth of borrowers

• lender has positive future income, lower r reduces pv of returns from lending - present wealth of lenders decreases

strategy for arbitrage

• buy the undervalued option

• short the stock if the option is a call

• buy the stock if the option is a put

• do this without borrowing - borrow cash if necessary

credit default swaps

• derivative contract that transfers credit risk of a borrower from one party to another

• protection buyer pays premium

• protection seller compensates buyer if credit event occurs

• settlement may be physical or cash

• mainly used to hedge credit risk, speculate on changes in credit quality, and improve pricing efficiency between bond and credit markets

• CDS involves counterparty risk, may encourage excessive speculation and can contribute to systemic risk if poorly regulated

Explain the nature and properties of Brownian motion(BM); and explain the relevance of the standard form of scaled BM with drift to the pricing of options

stochastic process with following properties:

• W0 = 0

• independent increments: movements over non-overlapping time intervals are independent

• normally distributed increments

• continuous paths

• stationary increments: distribution depends only on the length of the time interval

• in finance it is used calculate stock prices

  

Itos lemma

• applying Ito’s lemma gives the stochastic differential equation for option price

• decomposition separates random part from deterministic part

• by forming a hedged portfolio that eliminates dwt, creating a riskless portfolio

• no arbitrage arguments lead directly to the Black-Scholes partial differential equation

complete vs incomplete market

• complete = every possible payoff across all states of the world can be replicated exactly using traded assets

• incomplete = some contingent payoffs cannot be replicated

redundant assets

• if its payoff can be exactly replicated by a portfolio of other traded assets

• its presence does not expand the set of attainable payoffs

  • removing it does not affect market completeness

Type 1 vs Type 2 arbitrage

type 1 = zero initial cost, non-negative payoff in every state, strictly positive payoff in at least one state

type 2 = negative or zero initial cost, non-negative payoff in every state, weak arbitrage

Arrow-Debreu security

• hypothetical asset that pays 1 unit of consumption in one state of the world, and 0 in all other states

state price vector

• is the vector of prices of all arrow debreu securities

• each element gives the todays price of receiving 1 unit of payoff in a specific state tomorrow