Statistics and FinanceLiving on the Hedge

Vaibhav GuptaStatistics for Management

October2011

History on the Explosion of Statistics in Finance

“Quote” (optional)– Quote source

“ Employing data bases and statistical skills, academics compute with precision the beta of a stock … then build arcane investment and capital-allocation theories around this calculation. In their hunger for a single statistic to measure risk they forget a fundamental principle: It is better to be approximately right than precisely wrong. ”

– Warren Buffet

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The economic crisis of Europe between 1875 to 1895 made way for application of academic statistical concepts.

New statistical data such as CPI of workers, family budgets and unemployment days was now organized and recorded.

Survey techniques and mechanical processing saw wider use now to solve economic problems.

Rise of the planet of the Statisticians

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• 1900- Louis Bachelier: Movement of stock prices as limits of random walks.

• Connected it to diffusion process: Prices diffuses.• Stock prices follow a movement which is now known as

Brownian Motion.• Allowed prices to be negative• 1960-Samuelson proposed Geomotric B.M. [ log(price)].• 1965- Fama’s Efficient market theory and theory of

Random Walk of a stock price.• PRICE IS A RANDOM VARIABLE

Emergence of Theory of Random Walk

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Black –Scholes Formula• Black and Scholes (1973) and Merton (1973) derive option prices

under the following assumption on the stock price dynamics,

• The binomial model: Discrete states and discrete time (The number of possible stock prices and time steps are both finite).

• The BSM model: Continuous states (stock price can be anything between 0 and infinite ) and continuous time (time goes continuously).

• Scholes and Merton won Nobel price. Black passed away.

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Primer on Continuous Time process• The Driver of the process is Wt , a Brownian Motion, or a Weiner

Process.

• The sample paths of a Brownian Motion are continuous over time, but nowhere differentiable.

• It is idealization of the trajectory of a single particle being constantly bombarded by an infinite number of infinitesimally small random forces.

• Like a shark, a Brownian motion must always be moving, or else it dies.

• If you sum the absolute values of price changes over a day (or any time horizon) implied by the model, you get an infinite number.

• If you tried to accurately draw a Brownian motion sample path, your pen would run out of ink before one second had elapsed.

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Properties of Brownian Motion

• The process Wt generates a random variable that is normally distributed with mean 0 and variance t, ɸ(0, t). (Also referred to as Gaussian.)

• The process is made of independent normal increments

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Properties of Normally Distributed Random Variable

Under the BSM Model, µ is the annualized mean of the instantaneous return. Here, σ 2 is the annualized variance of the instantaneous return – instantaneous

return variance . And, σ is the annualized standard deviation of the instantaneous return - –

instantaneous return volitality.

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Geometric Brownian Motion

• The stock price is said to follow , a geometric Brownian Motion.• µ is often referred as the drift and σ the diffusion of the process.

• Instantaneously, the stock price is normally distributed.

• Over long horizons, the price change is lognormally distributed.

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Normal v/s Lognormal Distribution

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The Key Idea Behind BSM

• The option price and the stock price depend on the same underlying source of uncertainty.

• The Brownian motion dynamics imply that if we slice the time thin enough(dt), it behaves like a binominal tree.

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The Key Idea Behind BSM

• Reversely, if we cut t small enough and add enough time steps, the binomial tree converges to the distribution behaviour of the geometric Brownian motion.

– Under this thin slice of time interval, we can combine the option with the stock to form a riskfree portfolio.

– The portfolio is riskless (under this thin slice of time interval) and must earn the riskfree rate.

– Magic: μ does not matter for this portfolio and hence does not matter for the option valuation. Only σ matters.

– We do not need to worry about risk and risk premium if we can hedge away the risk completely.

ASSET VALUATION AND PROBABILITY

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ASSET VALUATION

• Asset: An asset can been seen as the promise of receiving a set of future payments, called cash-flows (CF).

• The asset value (P) could be approached by the sum of all CF associated with that asset.

• CF should be weighted inversely to the time to fulfilment.

• The weights are obtained from a discount function dependent on time to maturity and a parameter called interest rate (i) that can be seen as a price of time.

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where is a continuous discount function.• Once we know all the CFj (and its correspondent tj) and

the appropriate i, P is uniquely determined. Let us suppose we have an US coupon-zero Treasury Bond that promise to pay $100 in one year. Given an i=0.04, then

P=100·e-0.04·1=96.08• Unluckily, most of the CFs present in financial assets are

not known in advance. This is the case with shares, where CFs are called dividends and depends on firm performance year by year. Even in case of bonds you can not be sure.

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CF as Random Variable

CF can be seen as a random variable.

If we knew the probability distribution of each CF, then it is possible to reconstruct a mean value of an asset from the expected CF.

Everyday millions of people make such valuation (consciously or not), using their own subjective probabilities.

Buying and selling is a consequence , the movements on prices , produce a so-called Price Discovery Process (PDP).

PDP allows agents to improve their knowledge of the hidden probabilities by the joint opinion of the other investors.

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Efficient Market Hypothesis, Fama(1965)

• Fama (1965) proposed the Efficient Market Hypothesis • (EMH) states that if all information relevant to the pricing of

an asset is known by investors, this information will be incorporated into the price via PDP,

• And no available information in the market can be used to improve such evaluation.• Eg. Bond that promise to pay $100 in one year and the market price it

at $93 .• The difference comes from the possibility Treasury Bond that is

supposed to be risk-free.

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EMH Example continued…

• The $100 CF can be modeled as governed by a binomial variable B(1,π), where π has been implicitly established by the market,

• E[P]= (1−π )·100·e−0.04 = 93

• So π= 0.032 is the subjective probability given by the market to the company to default in one year.

• Above example is the simplest possible one to see binomial distributions. One can pick up a newspaper to do some more exercises at home.

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Markowitz’s Portfolio Theory (1991)

• Where to invest was a problem of Pareto optimal decision between risk and return.

• For the same level of return we would prefer assets with less risk, and for the same level of risk we would prefer more returns.• Eg. tracking 32 world indexes during 2003.• For each one of these markets we can proxy the expected daily

returns from,

• and risk by standard deviation of rt, also called volatility.

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Scatter-plot of risk and expected returns…

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Scatter-plot of risk and expected returns

• From the figure in the last slide you could see , that it is not wise to invest in some stocks, but it is not so true.

• Markowitz showed that sometimes it is possible to obtain the same amount of return of a single asset but with less risk via diversification with an appropriate portfolio including “not so good” assets.

• We can use Montecarlo Simulation to generate a set of portfolios, to obtain something similar to the figure on the next slide.

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Montecarlo Simulation

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Interpretations from last exercise

• It shows, how it is possible to find portfolios with better performance (measured in terms of returns and risk) than individual assets.

These portfolios define an efficient frontier as it is called in Portfolio Theory.

PRICE MODELLING AND STOCHASTIC PROCESSES

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Concept of Prices

• Asset prices evolve as some kind of random walk.

• This conception of prices can be tracked back to Bachelier (1900) seminal work.

• Improved by Samuelson (1960) that defined it in continuous time as a geometric Brownian motion.

• Samuelson’s improvement consisted of the appearance of P in the denominator of price changes, which avoid the theoretical appearance of negative prices.

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Stochastic process

t represents time,

r is the long term expected return,

σ is a measure of volatility,

so price movements distribute as N(r,σ t1/2 ), in the presence of Weiner Process (dW).

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Madelbrot(1963) and Brock et al., (1996)

Efficiency will say that returns would be independent and identically distributed (iid). But results show that

• Returns diverge from Gaussian distribution by having more observations close to the mean and the extremes of the distribution (heavy tails).

• Returns are skewed to the left of the distribution, as result of heavier reaction to bad news(losses) than to good news, coherent to the expected risk-aversion of economic agents.

• Extreme movements tend to cluster in some periods of time, what suggest that volatility changes over time.

(a) Histogram of daily returns on British FTSE-100 against normal distribution.

(a)

(b) Histogram ofdaily returns of FTSE-100 and normal distribution after subtracting 2.3% of largest movements.

(b)

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Jarque-Bera test (Jarque and Bera, 1980)

• We can see , How divergence from normality is due to the presence of extreme values.

• If we take a time series of returns and evaluate its normality with a standard test of Gaussianity we will reject the Gaussian hypothesis.

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Recursive Jarque-Bera Test

In spite of this, if we order the sample by the magnitude of absolute returns and recursively test normality subtracting the largest movements, it can be seen that we just need to erase a small part of

the sample to obtain a gaussian distribution, where there is no trackof the leptokurtosis or skewness.

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Value at Risk

Value at Risk measures the potential loss in value of a risky asset or portfolio over a defined period for a given confidence interval

Example if the VaR on an asset is $ 100 million at a one-week, 95% confidence level, there is a only a 5% chance that the value of the asset will drop more than $ 100 million over any given week

The focus in VaR is on downside risk and potential losses

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Acceptance of VaR

Its use in banks reflects their fear of a liquidity crisis, where a low-probability catastrophic occurrence creates a loss that wipes out the capital and creates a client exodus

The demise of Long Term Capital Management, the investment fund with top Wall Street traders was a trigger in the widespread acceptance of VaR

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Elements of VaR

There are three key elements of VaR – a specified level of loss in value a fixed time period over which risk is assessed a confidence interval

The VaR can be specified for an individual asset, a portfolio of assets or for an entire firm

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Application of Value At Risk

It is used most often by commercial and investment banks to capture the potential loss in value of their traded portfolios from adverse market movements over a specified period, this can then be compared to their available capital and cash reserves to ensure that the losses can be covered without putting the firms at risk

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Further Study

• RISK MANAGEMENT

• Use of Autoregressive Conditional Heterokedasticity, or ARCH Models

• Use of Generalized Autoregressive Conditional Heterokedasticity, or GARCH Models

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