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Understanding Prices, Returns, and Trading Costs in Financial Markets

Discover the concepts of prices, returns, and trading costs in financial markets and how they affect volatility. Learn about arithmetic and percentage returns, price relatives, log returns, and the impact of trading costs on prices and returns.

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Understanding Prices, Returns, and Trading Costs in Financial Markets

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  1. Topic 3 Prices, Returns & Trading Costs

  2. Returns Measurement

  3. Prices(P) P0 , P1 , P2 , … , PT Time P0 P1 P2 PT

  4. Arithmetic Returns (P) P0,1 = P1 – P0 P1,2 = P2 – P1 PT-1,T = PT – PT-1

  5. Percentage Returns (r) 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

  6. Price Relatives (PR) 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)

  7. Log Returns (R) 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)

  8. Question: Which of the following may be normally distributed? P r PR R

  9. Two Period Log Returns 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

  10. P* Returns in TraderEx • 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

  11. Means and Variances

  12. Log Returns: Two Period Mean 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)

  13. 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 ?

  14. Costs

  15. 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

  16. 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

  17. Price 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*

  18. Trading Costs & Volatility 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

  19. Trading Costs & Returns Price P T Time

  20. Which is More Volatile? P* or the transaction price that we observe? Price P* Observed Transaction Price Time

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