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Prices, Returns & Trading Costs

r0,1 = ( P1 – P0 ) / P0 = P0,1 / P0

r1,2 = ( P2 – P1 ) / P1 = P1,2 / P1

rT-1,T = ( PT – PT-1 ) / PT-1 = PT-1,T / PT-1

PR0,1 = P1 / P0 = ( P0 + P0,1 ) / P0 =1 + r0,1

PR1,2 = P2 / P1 = ( P1 + P1,2 ) / P1 = 1 + r1,2

PRT-1,T = PT / PT-1 = ( PT-1 + PT-1,T ) / PT-1 = 1 + rT-1,T

PR0,T = PT / P0 = (P1 / P0) * (P2 / P1) *…* (PT / PT-1)

R = ln(1+r) = ln(price relative)

R0,1 = ln(P1/P0)= ln(P1) – ln(P0) = ln(1+r0,1)

R1,2 = ln(P2/P1) = ln(P2) – ln(P1) = ln(1+r1,2)

RT-1,T = ln(PT) – ln(PT-1) = ln(1+rT-1,T)

PR0,T = PT / P0 = (P1 / P0) * (P2 / P1) *…* (PT / PT-1)

R0,T = R0,1 + R1,2 +…+ RT-1,T =

Ri-1,i = ln(PT) – ln(P0)

P2 = P0 ( 1 + r0,2 )

P2 = P0 ( 1 + r0,1 ) ( 1 + r1,2 )

1 + r0,2 = P2 / P0 = ( P1 / P0 ) * ( P2 / P1 ) =

= ( 1 + r0,1 ) ( 1 + r1,2 )

R0,2 = R0,1 + R1,2

- When P* follows a random walk, P* returns are generated by draws from two distributions:
- Poisson distribution (when does P* jump)
- A lognormal distribution (how big is the jump)

- Ln(P*t ) = Ln(P*t-1 ) + Rt

the jump

Assume a constant Mean:

E(R0,1) = E(R1,2)

E(R0,2) = E(R0,1) + E(R1,2)

E(R0,2) = 2E(R0,1)

Log Returns: Two Period Variance

Var(R0,2)=Var(R0,1)+Var(R1,2)+2Cov(R0,1,R1,2)

Assume a constant Variance:

Var(R0,1) = Var(R1,2)

For Cov(R0,1,R1,2) = 0

Var (R0,2) = 2 Var(R0,1)

What if Cov(R0,1,R1,2) < 0 ?

Trading Costs

1.Explicit costs

- commissions
- taxes
- etc.
2.Execution Costs(the implicit costs of trading)

- Bid-ask spread
- Market impact
- Delay/opportunity cost
- Implementation shortfall

From Trading Costs to Volatility

- The bid-ask spread
- Market impact
- Momentum trading
- Imperfect price discovery

Trading costs

cause prices to bounce between higher and lower values

Time

Trading Costs & Volatility

C

= Implicit transaction cost of buy or sell

= Transaction price (triggered by buy order)

= Transaction price (triggered by sell order)

= Magnitude of C

= Unobserved, costless trading price

P*

P*

C

= Implicit transaction cost of buy or sell

= Observed price of buy-triggered trade

= Observed price of sell-triggered trade

= C

= Unobserved, costless trading price

P*

Price

P*

Time

P* or

the transaction price that we observe?

Price

P*

Observed

Transaction

Price

Time

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