Stochastic Calculas

Question 1

State the five defining properties that apply to standard Brownian motion.

Answer 1

• Independent increments
• Stationary increments
• Gaussian increments
• Continuous sample paths
• B0 = 0.

Question 2

Explain why the standard Brownian motion is less suitable than the geometric Brownian motion as a model of stock prices.

Answer 2

However successful the Brownian motion model may be for describing the
movement of market indices in the short run, it is useless in the long run, if
only for the reason that a standard Brownian motion is certain to become
negative eventually.
It could also be pointed out that the Brownian model predicts that daily
movements of size 100 or more would occur just as frequently when the
process is at level 100 as when it is at level 10,000.

Question 3

Describe the empirical evidence relating to the continuous-time lognormal model for
security prices.

Answer 3

Share prices are always positive, which is consistent with this model.
The increments of share prices are proportional to the share price itself.
However, estimates of σ vary widely according to what time period is considered.
Examination of historic option prices suggests that volatility expectations fluctuate
markedly over time.
One way of modelling this behaviour is to take volatility as a process in its own right.
This can explain why we have periods of high volatility and periods of low volatility.
A more contentious area relates to whether the drift parameter μ is constant over time.
There are good theoretical reasons to suppose that μ should vary over time.
One unsettled empirical question is whether markets are mean reverting, or not.
There appears to be some evidence for this…
… but the evidence rests heavily on the aftermath of a small number of dramatic
crashes.
Furthermore, there also appears to be some evidence of momentum effects.
A further strand of empirical research questions the use of the normality assumptions
in market returns.
Actual returns tend to have many more extreme events, both on the upside and
downside, than is consistent with such a model.
While the random walk produces continuous price paths, jumps or discontinuities
seem to be an important feature of real markets.
Furthermore, days with no change, or very small change, also happen more often than
the normal distribution suggests.
However, whilst a non-normal distribution can provide an improved description of the
actual returns observed, the improved fit to empirical data comes at the cost of losing
the tractability of working with normal (and lognormal) distributions.
Market jumps are consistent with the arrival of information in packets rather than
continuously.
After a crash, many investors may have lost a significant proportion of their total
wealth; it is not irrational for them to be more averse to the risk of losing what
remains. [½]
Many orthodox statistical tests are based around assumptions of normal distributions.
If we reject normality, we will also have to re-test various hypotheses. In particular,
the evidence for time-varying mean and volatility is greatly weakened.

Question 4

Assumptions of Black-Scholes

Answer 4

• The price of the underlying share follows a geometric Brownian motion.
• There are no risk-free arbitrage opportunities.
• The risk-free rate of interest is constant, the same for all maturities and the same for borrowing or lending.
• Unlimited short selling (that is, negative holdings) is allowed.
• There are no taxes or transaction costs.
• The underlying asset can be traded continuously and in infinitesimally small numbers of units

Question 5

Limitations of the Black-Scholes model

Answer 5

• Share prices can jump. This invalidates assumption that the stock price evolves as geometric Brownian motion, as this process has continuous sample paths. However, hedging strategies can still be constructed which substantially reduce the level of risk.
• The risk-free rate of interest does vary and in an unpredictable way. However, over the short term of a typical derivative, the assumption of a constant risk-free rate of interest is not far from reality. (More specifically the model can be adapted in a simple way to allow for a stochastic risk-free rate, provided this is a predictable process.)
• Unlimited short selling may not be allowed, except perhaps at penal rates of interest. These problems can be mitigated by holding mixtures of derivatives which reduce the need for short selling. This is part of a suitable risk management strategy.
• Shares can normally only be dealt in integer multiples of one unit, not continuously, and dealings attract transaction costs. Again we are still able to construct suitable hedging strategies which substantially reduce risk
• Distributions of share returns tend to have fatter tails than suggested by the lognormal model.

Question 6

Define a complete market

Answer 6

The market is complete if for any contingent claim X there is a replicating strategy (Φ𝑡 ,Ψ𝑡 )
i.e. is a self-financing strategy, defined for 0 ≤ t <u></u>