300 Frequently Asked Questions In Quantitative Finance

I

n the following formulæ

N(x)=

1

√

2 π

∫x

−∞

e−

1

2 φ

2

dφ,

d 1 =

ln(S/K)+(r−D+^12 σ^2 )(T−t)

σ

√

T−t

and

d 2 =

ln(S/K)+(r−D−^12 σ^2 )(T−t)

σ

√

T−t

.

The formulæ are also valid for time-dependentσ,Dand

r, just use the relevant ‘average’ as explained in the

previous chapter.

Warning

The greeks which are ‘greyed out’ in the following can

sometimes be misleading. They are those greeks which

are partial derivatives with respect to a parameter (σ,r

orD) as opposed to a variable (Sandt)andwhich are

not single signed (i.e. always greater than zero or always

less than zero). Differentiating with respect a parameter,

which has been assumed to be constantso that we can

find a closed-form solution, is internally inconsistent.

For example,∂V/∂σis the sensitivity of the option price

to volatility, but if volatility is constant, as assumed in

the formula, why measure sensitivity to it? This may

not matter if the partial derivative with respect to the

parameter is of one sign, such as∂V/∂σfor calls and

puts. But if the partial derivative changes sign then

there may be trouble. For example, the binary call has

a positive vega for low stock prices and negative vega

for high stock prices, in the middle vega is small, and

even zero at a point. However, this does not mean that

the binary call is insensitive to volatility in the middle.

It is precisely in the middle that the binary call value is

very sensitive to volatility, but not the level, rather the

volatility skew.