... life.^1.1

Aristotle, Politics ca330BC

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... options.^1.2

Option is derived from the Latin optio, meaning choice.

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... view.^1.3

The underlying assumptions of Bachelier's treatment which are overwhelmingly rejected nowadays are: the modelling of the stock price as normal and not a lognormal diffusion, and the analysis using supply and demand for contracts and not the modern arbitrage or risk neutrality arguments.

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... 01.^1.4

All prices are as of the 18th December 2000.

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... asset.^3.1

Note that this does not hold for indices such as the GSCI [Goldman Sachs Commodity Index] where the underlying asset is not marketable. It does not hold for the NIKKEI225 index when it viewed as a variable with a dollar value; obviously the NIKKEI225 index is measured in yen. In these cases the arbitrage based relationships of this chapter do not hold.

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... point^3.2

See the webpage http://www.liffe.com/products/equities/specs/100.htm for the FTSE100 futures specifications.

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... itself^4.1

Consumption is thought of here as closely linked [or very correlated] to the value of the underlying asset in consideration.

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...^4.2

Perhaps taking into account the inflation uncertainty since the investor is interested in real returns rather than nominal ones.

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... measures.^4.3

This result follows from the duality concept of the linear programming theory. The proof is not really hard but not important.

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....^4.4

Carefull readers should have noticed that the linear pricing measure was defined as $\pi :\Omega \rightarrow \left[ 0,1\right]$ while the risk neutral one is defined as $Q:\Omega \rightarrow \left( 0,1\right)$. The interval is now open!

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... arbitrarily^5.1

Practitioners usually bear in mind the normal distribution which states that over 65% of the values of a normally distributed variable lie in the $\pm \sigma$ interval while more than 95% of the values lie in the $\pm 2\sigma$ interval. More sophisticated methods use derivative prices to compute a forecast of the future volatilities.

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... be^5.2

For example the value $:1.554\approx 1.5536$ comes from the relationship

$$2.92\frac{1}{2}\frac{1}{1+6.618\%/2}+0.29\frac{1}{2}\frac{1}{1+6.618\%/2}%%$$   .

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... prices^5.3

The question of whether or not implied volatilities are an unbiased estimator of future spot volatilities has to be first resolved, if one wants to utilize this approach.

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... sense^7.1

In the Riemann sense if $g$ is a function one can write $I\left( t\right) =\int_{0}^{t}\eta \left( s\right) dg\left( s\right) =\int_{0}^{t}\eta \left( s\right) g^{\prime }\left( s\right) ds$ using the chain rule. Here this won't work since the integrator is nowhere differentiable.

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...]^7.2

This implies that, for example, the density under $P$ of $B^{Q}\left( t\right)$ is given by

$$P\left[ B^{Q}\left( t\right) \in db\right] =\left[ 2\pi T\right] ^{-1/2}\exp \left\{ -\frac{\left( b-\theta T\right) ^{2}}{2T}\right\} db$$.

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... used.^7.3

One approach is to use historical estimated volatilities, resulting into the problem of choosing the appropriate window for estimation. Alternatively, traders use implied volatilities from other contracts to construct volatility matrices, and then interpolate according to the moneyness level and the maturity of the particular contract to be priced, in a fashion that tries to mimic a term structure of implied volatilities. In fact, many option prices are quoted by their implied volatilities.

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... by^8.1

As before, for an at-the-money contract we will have that $C_{ATM}=S_{0}%% \mathcal{N}\left( d_{1}\right) -S^{\star }\mathcal{N}\left( -d_{1}\right)$, where $d_{1}=\frac{\sigma _{ATM}}{2}\sqrt{T}$. The value of theta will therefore be

$$\Theta _{C}=-\frac{S_{0}\mathcal{N}^{\prime }\left( d_{1}\right) ... ...star }\mathcal{N}%% ^{\prime }\left( -d_{1}\right) \sigma }{4\sqrt{T}}\text{,}$$

which yields the result, since $\mathcal{N}^{\prime }\left( d_{1}\right) =%% \mathcal{N}^{\prime }\left( -d_{1}\right)$, due to the symmetry of the normal distribution.

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... case.^8.2

In fact one could say that the analysis of the previous case is not even correct. The result obtained stemmed from the fact that the portfolio and the FTSE250 index were identical.

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