Chapter 9 - Stochastic models of security prices

Question 1

Why may the continuous-time lognormal model be inappropriate for modelling investment returns? Give 7 reasons

Answer 1

1) VOLATILITY parameter σ may not be constant over time (estimates of volatility from past data critically depend on the time period chosen for the data & how often the estimate is re-parameterised)
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2) DRIFT parameter η may not be constant over time (in particular, bond yields will influence the drift)
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3) there’s evidence in real markets of MEAN-REVERTING BEHAVIOUR, which is inconsistent with the independent increments assumption
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4) there’s evidence in real markets of MOMENT effects, which is inconsistent with the independent increments assumption.
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5) distribution of security returns log(Su/St) has TALLER PEAK than implied by normal distribution (because there’s more days of little or no movement in financial market)
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6) distribution of security returns log(Su/St) has FATTER TAILS than implied by normal distribution (because there’s more extreme movements in security prices)
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7) SAMPLE PATHS of security prices are not continuous but instead appear to jump occasionally

Question 2

Are stochastic processes good models for asset prices?

Answer 2

i) STARTING PT: asset prices don’t start at zero as assumed by the Wiener process. They represent capital value and start at a value determined by initial capital.
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ii) CONTINUITY: asset prices are not continuous. Trades are made at discrete points in time, contrary to the continuous nature of stochastic models.
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iii) MARKOVIAN PROPERTY: some evidence suggests that asset prices may exhibit Markovian behaviour in the long run, over shorter timeframes, this assumption seems to fail.
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iv) STATIONARITY & DISTRIBUTION: asset price increments are not stationary and do not follow a normal distribution. Identifying stationary increments would be practically impossible, and asset price increments are not normally distributed.

Question 3

Continuous-time lognormal model

Answer 3

lnSu - lnSt ~ N(η(u-t),σ ^2(u-t))
where St is the share price at time t, η is the drift parameter and σ is the diffusion coefficient/volatility parameter & u>t
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E[Su] = Stexp(η(u-t) +0.5σ ^2(u-t))
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Var(Su) = St^2 * exp(2η(u-t) +σ ^2(u-t))*(exp(σ ^2(u-t))-1)
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mean & variance of log returns are proportional to the internal u-t, lnSt has independent& stationary increments & has continuous sample paths