(Ff1(u)u0 and that the quadratic variation is hZiu = Z f1 (u) 0 f 0
(s)ds = f(f 1 (u) f(0) = u so by Levys characterisation (Zu)u0 is a
Brownian motion. Define the stopping time by = infcfw_u 0, Zu = K.
Since (Zu)u0 is a Brownian motion, we have < almost
we need to be able to compute the moment generating function for
some interesting models. We first consider a general stochastic volatility
model: dBt = Btrdt dSt = St(rdt + vtdWS t ) dvt = A(vt)dt + B(vt)dWv t
Here WS and Wv are assumed to be correlated
formula to get the right price. The BlackScholes formula says that in
the context of a BlackScholes model the call price is given by (*) Ct(T,
K) = C BS(t, T, K, St , r, ) for an explicit function C BS written out the
previous section. But in reality we d
a.s., the conclusion is that if you are willing to wait a while, investing in
this strategy will result in an almost sure gain X = 1. But the amount of
time you have to wait is very long: one can show that E() = +. One
can improve upon the above idea by t
2 t ( 2 t 2 ) 2V S2 dt Subtracting this equation from equation (*)
and solving yields 1 2 Z T 0 e r(T t) ( 2 2 t )S 2 t 2V S2 (t, St ,
)dt = XT V (T, ST , ) e rT (X0 V (0, S0, ) = XT g(ST ) since X0
= 0 by assumption and 0 = V (0, S0, ) by the definition
constant Y0 > 0 or in equivalent differential form dYt = Yt(rtdt t
dWt). Then Y is a state price density. Furthermore, if the filtration is
generated by the m-dimensional Brownian motion W, all state price
densities have this form. Remark. The m-dimensio
0). Note that H0 P0 = 1 = K0 P0 = 1 but HT PT = 1 > KT PT = 0. The
point of this example is that the asset with price S seems like a good
deal - it costs nothing at time 0 but pays a positive amount at time 1.
However, holding one share of the asset, corr
t = ( r)/(t, St). Recall that by Girsanovs theorem dW t = dWt
tdt defines a Q-Brownian motion. The next theorem in the present
context is usually attributed to Dupires 1994 paper. Theorem. Suppose
that C0(T, K) = E Q [e rT (ST K) +] Then C0 T (T, K) + rK
martingale, then there is no arbitrage relative to K. In particular, there is
no absolute arbitrage. The proof of this fact is based on an important
lemma: Lemma. Suppose H is a self-financing pure investment strategy
and let Xt = Ht Pt = X0 + Z t 0 Hs dP
this model in terms of the calendar time t, the current stock price St ,
spot interest rate r, the option maturity T and strike K, and a volatility
parameter . 85 However, since the implied volatility surface t(T, K) of
real-world option prices is usually
), the previous theorem says we should solve the BlackScholes PDE V
t + rS V S + 1 2 2S 2 2V S2 = rV V (T, S) = g(S) Now, lets
specialise to the case of the call option where g(S) = (S K) +. From last
section we have V (t, S) =S log(K/S) T t + (r/ + /2) T
admissible, then XY is a non-negative local martingale. Non-negative
local martingales are supermartingales by Fatous lemma. Proof that
existence of a local martingale deflator implies no arbitrage. Let Y be a
local martingale deflator, and let H and K be
continuous-time arbitrage theory. Definition. An admissible strategy H is
called an absolute arbitrage iff there is a nonrandom time T such that
H0 P0 = 0 HT PT a.s. and P (HT PT > 0) > 0. An admissible strategy H
is called an arbitrage relative to an adm
converse implication. See the recent book of Delbaen and
Schachermayer The Mathematics of Arbitrage for an account of the
modern theory. Remark. Here is an example of a market with a relative
arbitrage and no absolute arbitrage. Fix T > 0 and let (t)0tT b
volatility smiles, the BlackScholes model cannot be considered an
adequate description of how stock prices fluctuate. However, it should
be considered an approximation of reality, and we will now do a
calculation to see how to quantify how good this appro
7 t(T, K) often resembles a convex parabola4 at least for strikes K
close to the money, i.e. such that Ker(T t)/St 1. That is why
practictioner refer to the function K 7 t(T, K) as the implied volatility
smile or smirk. One could either conclude BlackScho
martingale. Therefore, we must ceck that BY is a true martingale in
order to claim the density above defines an equivalent probability
measure. In discrete time, there is no problems since positive local
martingales are true martingales. However, in conti
into debt in order to 71 secure the K winning. Indeed, if such strategies
were a good model for investor behaviour, we all could be much richer
by just spending some time trading over the internet. The above
discussion shows that the integrability necessa
Since X is integrable, the previous corollary implies X is a martingale.
Theorem. Suppose that Mt = M0 + X t s=1 Ks(Xs Xs1) where K is
predictable, X is a martingale and M0 is a constant. If MT 0 a.s. for
some non-random T > 0, then (Mt)0tT is a true mart
price was a geometric Brownian motion, that is, of the form St = S0e
at+Wt , then one could insert the value 2 into the BlackScholes
formula to obtain the price of a call option. Notice that we have done
the statistics under the objective measure P, not t
t for 0 t T and that the augmented market with (n+1)- dimensional
price process (P, ) has no arbitrage. Then there exists a martingale
deflator Y of the original market such that t = 1 Yt E(T YT |Ft) for all 0
t T. Proof. This is just the first fundament
BlackScholes theory, he puts the initial wealth of X0 = 0 in 4 .but be
careful: for large K, the graph can grow no faster than p 2 log K/(T t).
See example sheet 4. 84 his account and holds a portfolio of t = V S
(t, St , ) shares of the stock at all time
random variables Yi = log Sti Sti1 are independent with distribution Yi
= ( 2 /2)(ti ti1) + (Wti Wti1 ) N(aT/n, 2T/n) where a =
2/2. The maximum likelihood estimator of a is a = 1 T Xn i=1 Yi and of
2 is 2 = 1 T Xn i=1 (Yi aT/n ) 2 . Notice that this e
assumption that V solves a certain PDE to go from the second to third
line above. Now letting and be as in the statement of the theorem
we have that V (t, St) = tBt + t St dV (t, St) = tdBt + t dSt . Hence
H = (, ) is a self-financing strategy with associ
see that this condition isnt strong enough to make our economic
analysis interesting. Example. Consider a discrete-time market model
with two assets P = (1, S) where S is a simple symmetric random walk: St
= 1 + . . . + t where the random variables 1, 2,
time by speeding up the clock. Consider the market with prices P = (1,
W) where W is a Brownian motion. We will now construct a pure
investment trading strategy such that the corresponding wealth process
has X0 = 0 and XT = K a.s. where T > 0 is an arbitr
tdt to mean Xt = X0 + Z t 0 sdWs + Z t 0 sds. Recall that the sample
paths of the Brownian motion are nowhere differentiable, so the
notation dWt is only formal, and can only be interpreted via the
stochastic integration theory. But in this differential n
another application of Itos formula. The significance of this result is
that if we can prove the local martingale is a true martingale, then E[e
log ST ] = e log S0F(0, v0; ) and hence we have found the moment
generating function. To use this result, we
process with respect to the martingale, and hence is a martingale. The
next theorem gives a sufficient condition that a local martingale is a
true martingale. Theorem. Let X be a local martingale in either discrete
or continuous time. Let Yt be a process
the current market prices. Furthermore, the pricing function V can be
found by solving a certain linear parabolic partial differential equation2
with terminal data to match the payout of the claim. Solving this
equation may be difficult to do by hand, but