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Volatility
where P (t, T ) = exp − tT r(u)du is the zero coupon bond price and What matters is the average (or integrated) (squared) volatility during
the life of the option. “Future volatility, not past volatility”.
1 T
2
σAV = σ 2 (u)du.
T −t t If volatility is stochastic, σ(t, ω), then the random variable tT σ 2(u, ω)du
still plays an important role for option pricing, although things are
This is (sometimes) called the Black/Scholes/Merton formula. quite as nice as in the r-case.
Not too hard to prove with martingale methods; hard to guess with If σ is exogenous to the stock price (driven by another BM) then by
PDEs. conditioning:
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call(t) = callBSM (. . . , σAV = x, . . .)φσ 2 (x)dx,
Deterministic changes may not be what we ultimately want, but it’s AV
a first step. “Shades of the real world” & gives useful information. where φσ 2 denotes the density of T 1 tT σ 2(u, ω)du. Expansions lead
−t
AV
to (approximate) formulas. (Hull & White 1987.)
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The Black/Scholes/Merton formula & Some Consequences When r and σ are constant original B/S formula. Time-varying
dividend yields: DIY (relevant in currency models).
Exercise 4 on WN 9 extends base-case B/S to time-varying (but
deterministic) interest rate r(t) and volatility σ(t). Interest rates
If there is a term structure of interest rates, then in place of r in
The arbitrage-free call-option price is original B/S formula you should plug in the zero coupon rate with
maturity equal to the time to expiry of the option:
S(t)
ln( P (t,T )K ) + 1 σAV (T − t)
2
2
call(t) = S(t)Φ √ r y(t, T ) (since by definitionP (t, T ) = exp(−y(t, T )(T − t)))
σAV T − t
With stochastic interest rates, the ZC yield y(t, T ) can also be plugged
S(t)
ln( P (t,T )K ) − 1 σAV (T − t)
2
2
in. A good approximation for stocks. Can be theoretically supported
−KP (t, T )Φ √ ,
σAV T − t when interest rates are Gaussian & independent of the stock.
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The LHS can be estimated from data by summing squared log-returns
over the observation period [Ti; Ti]. This is called realized volatility
If the volatility path is known to market participants, then the BSM-
analysis tells us that the (Black/Scholes) implied volatility of an op-
T
tion (after scaling) is T i+1 σ 2 (u, ω)du. This can be recorded/observed
i
already at time Ti
This leads to the empirical/econometric question: Is implied volatil-
ity a good forecast for realized volatility? In particular, does implied
volatility contain information that can’t be extracted from past/historical
data?
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Other Spin-Off: Implied vs. Realized Volatility
Empirical analysis: Record data time series of realized volatilities,
If σ is stochastic and
σAV (tk ) and corresponding implied volatilities σIM P (tk ).
dS(t) = S(t)µ(t)dt + σ(t, ω)S(t)dW P ,
then with Y = ln S we have Run regression (possibly in logs)
dY = (µ − σ 2/2)dt + σ(t, ω)dW P σAV (tk ) = α0 + α1σIM P (tk ) + (tk+1)
Now look at some [Ti ; Ti+1] and suppose it has been split into n pieces If α1 = 0, then implied volatility contains some information.
(intermediate points tj ). If (α0, α1) = (0, 1) then implied volatility is an unbiased forecast.
Include eg. α2σAV (tk−1 ); test if α2 = 0.
Then from our analysis of quadratic variation we know that
n→∞ Ti+1 Results: There is some information content in implied volatility that
(Y (tj+1 ) − Y (tj ))2 −→ σ 2 (u, ω)du
j Ti historical volatility does not capture. Beware of overlapping data!
Reference: Christensen & Prabala (JFE 1998).
(All objects are random variables; convergence is in L2-sense.)
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