Download THE BLACK SCHOLES FORMULA `If options are correctly priced in

THE BLACK SCHOLES FORMULA
MARK H.A. DAVIS
‘If options are correctly priced in the market, it should not be possible to make sure profits by
creating portfolios of long and short positions in options and their underlying stocks. Using this
principle, a theoretical valuation formula for options is derived.’ These are the first two sentences
of the abstract of the great paper [2] by Fischer Black and Myron Scholes on option pricing, and
encapsulate the basic idea, which is that—with the asset price model they employ—insisting on
absence of arbitrage is enough to obtain a unique value for a call option on that asset. The resulting
formula, (1.4) below, is the most famous formula in financial economics, and in fact that whole subject
splits decisively into the pre-Black-Scholes and post-Black-Scholes eras.
This article aims to give a self-contained derivation of the formula, some discussion of the hedge
parameters, and some extensions of the formula, and to indicate why a formula based on a stylized
mathematical model which is known not to be a particularly accurate representation of real asset
prices has nevertheless proved so effective in the world of option trading. Section 1 formulates the
model and states and proves the formula. As is well known, the formula can equally well be stated
in the form of a partial differential equation (PDE); this is equation (1.5) below. Section 2 discusses
the PDE aspects of Black-Scholes. Section 3 summarizes information about the option ‘Greeks’, while
Sections 4 and 5 introduce what is actually a more useful form of Black-Scholes, usually known as the
‘Black formula’. Finally, Section 6 discusses the applications of the formula in market trading. We
define the implied volatility and demonstrate a ‘robustness’ property of Black-Scholes which implies
that effective hedging can be achieved even if the ‘true’ price process is substantially different from
Black and Scholes’ stylized model.
1. The model and formula. Let (Ω, F , (Ft )t∈R+ , P) be a probability space with a given filtration (Ft ) representing the flow of information in the market. Traded asset prices are Ft -adapted
stochastic processes on (Ω, F , P). We assume that the market is frictionless: assets may be held in
arbitrary amount, positive and negative, the interest rate for borrowing and lending is the same, and
there are no transaction costs (i.e. the bid-ask spread is zero). While there may be many traded assets
in the market, we fix attention on two of them. Firstly, there is a ‘risky’ asset whose price process
(St , t ∈ R+ ) is assumed to satisfy the stochastic differential equation
dSt = μSt dt + σSt dwt
(1.1)
with given drift μ and volatility σ. Here (wt , t ∈ R+ ) is an (Ft )-Brownian motion. Equation (1.1) has
a unique solution: if St satisfies (1.1) then by the Itô formula
1
d log St = (μ − σ 2 )dt + σdwt ,
2
so that St satisfies (1.1) if and only if
1 2
St = S0 exp (μ − σ )t + σwt .
2
(1.2)
Asset St is assumed to have a constant dividend yield q, i.e. the holder receives a dividend payment
qSt dt in the time interval [t, t + dt[. Secondly, there is a riskless asset paying interest at a fixed
continuously-compounding rate r. The exact form of this asset is unimportant—it could be a moneymarket account in which $1 deposited at time s grows to $er(t−s) at time t, or it could be a zero-coupon
bond maturing with a value of $1 at some time T , so that its value at t ≤ T is
Bt = exp(−r(T − t)).
1
This grows, as required, at rate r:
dBt = rBt dt
(1.3)
Note that (1.3) does not depend on the final maturity T (the same growth rate is obtained from any
zero-coupon bond) and the choice of T is a matter of convenience.
A European call option on St is a contract, entered at time 0 and specified by two parameters
(K, T ), which gives the holder the right, but not the obligation, to purchase one unit of the risky
asset at price K at time T > 0. (In the frictionless market setting, an option to buy N units of stock
is equivalent to N options on a single unit, so we do not need to include quantity as a parameter.)
If ST ≤ K the option is worthless and will not be exercised. If ST > K the holder can exercise his
option, buying the asset at price K, and then immediately selling it at the prevailing market price ST ,
realizing a profit of ST − K. Thus the exercise value of the option is [ST − K]+ = max(ST − K, 0).
Similarly, the exercise value of a European put option, conferring on the holder the right to sell at a
fixed price K, is [K − ST ]+ . In either case the exercise value is non-negative and, in the above model,
is strictly positive with positive probability, so the option buyer should pay the writer a premium to
acquire it. Black and Scholes [2] showed that there is a unique arbitrage-free value for this premium.
Theorem 1.1. (a) In the above model, the unique arbitrage-free value at time t < T when St = S
of the call option maturing at time T with strike K is
C(t, S) = e−q(T −t) SN (d1 ) − e−r(T −t) KN (d2 )
(1.4)
where N (∙) denotes the cumulative standard normal distribution function
Z x
1 2
1
e− 2 y dy
N (x) = √
2π −∞
and
log(S/K) + (r + σ 2 /2)(T − t)
√
,
σ T −t
√
d2 = d1 − σ T − t.
d1 =
(b) The function C(t, S) may be characterized as the unique C 1,2 solution1 of the Black-Scholes partial
differential equation (PDE)
∂C
1
∂2C
∂C
+ rS
+ σ 2 St2 2 − rC =0
∂t
∂S
2
∂S
solved backwards in time with the terminal boundary condition
C(T, S) = [S − K]+ .
(1.5)
(1.6)
(c) The value of the put option with exercise time T and strike K is
P (t, S) = e−r(T −t) KN (−d2 ) − e−q(T −t) SN (−d1 ).
(1.7)
To prove the theorem, we are going to show that the call option value can be replicated by a
dynamic trading strategy investing in the asset St and in the zero-coupon bond Bt = e−r(T −t) . A
trading strategy is specified by an initial capital x and a pair of adapted processes αt , βt representing
the number of units of S, B respectively held at time t; the portfolio value at time t is then Xt =
αt St + βt Bt , and by definition x = α0 S0 + β0 B0 . The trading strategy (x, α, β) is admissible if
RT
(i) 0 αt2 St2 dt < ∞ a.s,
RT
(1.8)
(ii) 0 |βt |dt < ∞ a.s.
(iii) There exists a constant L ≥ 0 such that Xt ≥ −L for all t, a.s.
1A
two-parameter function is C 1,2 if it is once [twice] continuously differentiable in the first[second] argument.
2
The gain from trade in [s, t] is
Z
t
αu dSu +
s
Z
t
βu dBu +
s
Z
t
qαu Su du,
s
where the first integral is an Itô stochastic integral. This is the sum of the accumulated capital
gains/losses in the two assets plus the total dividend received. The trading strategy is self-financing
if
Z t
Z t
Z t
α t St + βt B t − α s S s − β s B s =
αu dSu +
qαu Su du +
βu dBu ,
s
s
s
implying that the change in value over any interval in portfolio value is entirely due to gains from
trade (the accumulated increments in the value of the assets in the portfolio plus the total dividend
received).
We can always create self-financing strategies by fixing α, the investment in the risky asset, and
investing all residual wealth in the bond. Indeed, the value of the risky asset holding at time t is αt St ,
so if the total portfolio value is Xt we take βt = (Xt − αt St )/B(t). The portfolio value process is then
defined implicitly as the solution of the SDE
dXt = αt dSt + qαt St dt + βt dBt
= αt dSt + qαt St dt + (Xt − αt St )r dt
= rXt dt + αt St σ(θdt + dwt ),
(1.9)
where θ = (μ − r + q)/σ. This strategy is always self-financing since Xt is by definition the gains from
trade process, while the value is αS + βB = X.
Proof of Theorem (1.1). The key step is to put the ‘wealth equation’ (1.9) into a more convenient
form by change of measure. Define a measure Q, the so-called risk-neutral measure on (Ω, FT ) by the
Radon-Nikodým derivative
dQ
1
= exp(−θwT − θ2 T ).
dP
2
(The right-hand side has expectation 1, since wT ∼ N (0, T ).) Expectation with respect to Q will be
denoted EQ . By the Girsanov theorem, w̌ = wt + θt is a Q-Brownian motion, so that from (1.1) the
SDE satisfied by St under Q is
dSt = (r − q)St dt + σSt dw̌t
(1.10)
1
ST = St exp (r − q − σ 2 )(T − t) + σ(w̌T − w̌t ) .
2
(1.11)
so that for t < T
Applying the Itô formula and equation (1.9) we find that, with X̃t = e−rt Xt and S̃t = e−rt St ,
dX̃t = αt S̃t σdw̌t ,
(1.12)
Thus e−rt Xt is a Q-local martingale under condition (1.8)(i). Let h(S) = [S − K]+ and suppose there
exists a replicating strategy, i.e. a strategy (x, α, β) with value process Xt constructed as in (1.9) such
that XT = h(ST ) a.s. Suppose also that αt satisfies the stronger condition
Z T
αt2 St2 dt < ∞.
(1.13)
EQ
0
Then X̃t is a Q-martingale, and hence for t < T
Xt = e−r(T −t) EQ [h(ST )|Ft ]
3
(1.14)
and in particular
x = e−rT EQ [h(ST )].
(1.15)
Now St is a Markov process, so the conditional expectation in (1.14) is a function of St , and indeed
we see from (1.11) that ST is a function of St and the increment (w̌T − w̌t ) which is independent of Ft .
√
Writing (w̌T − w̌t ) = Z T − t where Z ∼ N (0, 1), the expectation is simply a 1-dimensional integral
with respect to the normal distribution. Hence Xt = C(t, St ) where
Z
√
1 2
e−r(T −t) ∞
√
C(t, S) =
h(S exp((r − q − σ 2 /2)(T − t) − σx T − t))e− 2 x dx.
(1.16)
2π
−∞
Straightforward calculations show that this integral is equal to the closed-form expression at (1.4).
The argument so far shows that if there is a replicating strategy the initial capital required must be
x = C(0, S0 ) where C is defined by (1.16). It remains to identify the strategy (x, α, β) and to show that
it is admissible. Let us temporarily take for granted the assertions of part (b) of the theorem; these
will be proved in Proposition 2.1 below, where we will also show that (∂C/∂S)(t, S) = e−q(T −t) N (d1 ),
so that in particular 0 < ∂C/∂S < 1.
The replicating strategy is A = (x, α, β) defined by
∂C
∂C
1
1
∂2C
∂C
x = C(0, S0 ), αt =
.
(t, St ), βt =
+ σ 2 St2 2 − qSt
∂S
rBt ∂t
2
∂S
∂S
Indeed, using the PDE (1.5) we find that Xt = αt St + βt Bt = C(t, St ), so that A is replicating and
also Xt ≥ 0, so that condition (1.8)(iii) is satisfied. From (1.11)
St2 = S02 exp((2r − 2q − σ 2 )t + 2σ w̌t ),
RT
so that EQ [St2 ] = exp((2r−2q +σ 2 )t). Since |e−r(T −t) ∂C/∂S| < 1, this shows that EQ 0 αt2 St2 dt < ∞,
i.e. condition (1.13) is satisfied. Since βt is, almost surely, a continuous function of t it satisfies (1.8)(ii).
Thus A is admissible. Finally, the gain from trade in an interval [s, t] is
Z t
Z t
Z t
Z t
Z t
∂C
∂C
1 2 2 ∂2C
du
(1.17)
αu dSu +
qαu Su du +
βu dBu =
dS +
+ σ St
∂t
2
∂S 2
s
s
s
s ∂S
s
Z t
=
dC
s
= C(t, St ) − C(s, Ss ).
(We obtain the right-hand side of (1.17) from the definition of α, β, and it turns out to be just the
Itô formula applied to the function C.) This confirms the self-financing property and completes the
proof.
Finally, part (c) of the theorem follows from the model-free put-call parity relation C − P =
−q(T −t)
e
S − e−r(T −t) K and symmetry of the normal distribution: N (−x) = 1 − N (x).
The replicating strategy derived above is known as delta hedging: the number of units of the risky
asset held in the portfolio is equal to the Black-Scholes delta Δ = ∂C/∂S.
So far, we have concentrated entirely on the hedging of call options. We conclude this section by
showing that, with the class of trading strategies we have defined, there are no arbitrage opportunities
in the Black-Scholes model.
Theorem 1.2. There is no admissible trading strategy in a single asset and the zero-coupon bond
that generates an arbitrage opportunity, in the Black-Scholes model.
Proof. Suppose Xt is the portfolio value process corresponding to an admissible trading strategy
(x, α, β). There is an arbitrage opportunity if x = 0 and, for some t, Xt ≥ 0 a.s. and P[Xt > 0] > 0,
or equivalently E[Xt ] > 0. This is the P-expectation, but E[Xt ] > 0 ⇔ EQ [X̃t ] > 0 since P and Q are
equivalent measures and e−rt > 0. From (1.12), X̃t is a Q-local martingale which, by the definition of
admissibility, is bounded below by a constant −L. It follows that X̃t is a supermartingale, so if x = 0
then EQ [X̃t ] ≤ 0 for any t. So no arbitrage can arise from the strategy (0, α, β).
4
2. The Black-Scholes partial differential equation.
Proposition 2.1. (a) The Black-Scholes PDE (1.5) with boundary condition (1.6) has a unique
1,2
C
solution, given by (1.4).
(b) The Black-Scholes ‘delta’ Δ(t, S) is given by
Δ(t, S) =
∂
C(t, S) = e−q(T −t) N (d1 ).
∂S
(2.1)
Proof. It can—with some pain—be directly checked that C(t, S) defined by (1.4) does satisfy the
Black-Scholes PDE (1.5), (1.6), and a further calculation (not quite as simple as it appears at first
sight!) gives the formula (2.1) for the Black-Scholes delta. It is however enlightening to take the
orginal route of Black and Scholes and relate the (1.5) to a simpler equation, the heat equation. Note
from the explicit expression (1.11) for the price process under the risk neutral measure that, given the
starting point St , there is a 1-1 relation between ST and the Brownian increment w̌T − w̌t . We can
therefore always express things interchangeably in ‘S coordinates’ or in ‘w̌ coordinates’. In fact we
already made use of this in deriving the integral price expression (1.16). Here we proceed as follows.
For fixed parameters S0 , r, q, σ, define the functions φ : R+ × R → R+ and u : [0, T [×R → R+ by
1
φ(t, x) = S0 exp (r − q − σ 2 )t + σ x
2
and
u(t, x) = C(t, φ(t, x)).
Note that the inverse function ψ(t, s) = φ−1 (t, s) (i.e. the solution for x of the equation s = φ(t, x)) is
1
s
1
log
ψ(t, s) =
− (r − q − σ 2 )t .
σ
S0
2
A direct calculation shows that C satisfies (1.5) if and only if u satisfies the heat equation
∂u 1 ∂ 2 u
+
− r u = 0.
∂t
2 ∂x2
(2.2)
If Wt is Brownian motion on some probability space and u is a C 1,2 function then an application of
the Itô formula shows that
∂u 1 ∂ 2 u
∂u
−rt
−rt
− ru dt + e−rt dWt .
d(e u(t, Wt )) = e
+
∂t
2 ∂x2
∂x
If u satisfies (2.2) with boundary condition u(T, x) = g(x) and
2
Z T
∂u
E
(t, Wt ) dt < ∞
∂x
0
(2.3)
then the process t 7→ e−rt u(t, Wt ) is a martingale so that, with Et,x denoting the conditional expectation given Wt = x,
e−rt u(t, x) = Et,x [e−rT u(T, WT )] = Et,x [e−rT g(WT )].
Since WT ∼ N (x, T − t), this shows that u is given by
Z ∞
e−r(T −t)
1
2
g(y) exp −
u(t, x) = p
(y − x) dy.
2(T − t)
2π(T − t) −∞
A sufficient condition for (2.3) is
√
1
2πT
Z
∞
−∞
g 2 (y)e−y
5
2
/2T
dy < ∞.
(2.4)
Delta
Δ
Gamma
Γ
Theta
Θ
Rho
Vega
P
Υ
∂C
∂S
∂2C
∂S 2
− ∂C
∂τ
∂C
∂r
∂C
∂σ
e−qτ N (d1 )
−e
−qτ
SN 0 (d1 )σ
√
2 τ
e−qτ N 0 (d1 )
√
Sσ τ
−qτ
+ qe
SN (d1 ) − rKe−rτ N (d2 )
Kτ e−rτ N (d2 )
√
e−qτ S τ N 0 (d1 )
Table 3.1
Black-Scholes risk parameters
In our case the boundary condition is g(x) = [φ(t, x) − K]+ < φ(t, x) and this condition is easily
checked. Hence (2.2) with this boundary condition has unique C 1,2 solution (2.4), implying that the
inverse function C(t, S) = u(t, ψ(t, S)) given by (1.16) is the unique C 1,2 , solution of (1.5) as claimed.
3. Hedge parameters. Bringing in all the parameters, the Black-Scholes formula (1.4) is a
5-parameter function C(t, S) = C(τ, S, K, r, σ), where τ = T − t is the time to maturity. For riskmanagement purposes it is important to know the sensitivities of the option value to changes in the
parameters. The conventional hedge parameters or ‘Greeks’ are given in Table 3.1. There are slight
notational problems in that ‘vega’ is not the name of a Greek letter (here we have used upper-case
upsilon, but this is not necessarily a conventional choice) and upper-case rho coincides with Latin P,
so this parameter is usually written ρ, risking confusion with correlation parameters. The expressions
in the right-hand column are readily obtained from the sensitivity parameters (5.3) and (5.4) of the
‘universal’ Black Formula introduced in Section 5 below.
Delta is, of course, the Black-Scholes hedge ratio. Gamma measures the convexity of C and is
at its maximum when the option is close to being at-the-money. Since gamma is the rate of change
of delta, frequent rebalancing of the hedge portfolio will be required in areas of high gamma. Theta
is defined as −∂C/∂τ and is generally negative (as can be seen from the table, it is always negative
for a call option on an asset with no dividends). It represents the ‘time decay’ in the option value as
the maturity time is reduced, i.e. real time advances. As regards rho, it is not immediately obvious,
without doing the calculation, what its sign will be: on the one hand, increasing r increases the forward
price, pushing a call option further into the money, while on the other hand increased r implies heavier
discounting, reducing option value. As can be seen from the table, the first effect wins: rho is always
positive. Vega is in some ways the most important parameter, since a key risk in managing books of
traded options is ‘vega risk’, and in Black-Scholes this is completely ‘outside the model’. Bringing it
back inside the model is the subject of stochastic volatility.
An extensive discussion of the risk parameters and their uses can be found in Hull [6].
4. The Black ‘forward’ option formula. The 5-parameter representation C(τ, S, K, r, σ) is not
the best parametrization of Black-Scholes. For the asset St with dividend yield q the forward price at
time t for delivery at time T is F (t, T ) = St e(r−q)(T −t) (this is a model-free result, not related to the
Black-Scholes model). We can trivially re-express the price formula (1.4) as
C(t, St ) = B(t, T )(F (t, T )N (d1 ) − KN (d2 ))
(4.1)
with
d1 =
log(F (t, T )/K) + 12 σ 2 (T − t)
√
,
σ T −t
√
d2 = d1 − σ T − t,
where B(t, T ) = e−r(T −t) is the zero-coupon bond value or ‘discount factor’ from T to t. There is,
however, far more to this than just a change of notation. Firstly, the continuously-compounding rate
r is not market data. What is market data at time t is the set of discount factors B(t, t0 ) for t0 > t. We
see from (4.1) that ‘r’ plays two distinct roles in Black-Scholes: it appears in the computation of the
6
forward price F and the discount factor B. But both of these are more fundamental than r itself and
are in fact market data which, as (4.1) shows, can be used directly. A further advantage is that the
exact mechanism of dividend payment is not important, as long as there is an unambiguously-defined
forward price.
Formula (4.1) is known as the Black formula and is the most useful version of Black-Scholes,
being widely applied in connection with FX (foreign exchange) and interest-rate options as well as
dividend-paying equities. Fundamentally, it relates to a price model in which the price is expressed in
the risk-neutral measure as St = F (0, t)Mt where Mt is the exponential martingale
1 2
Mt = exp σ w̌t − σ t ,
(4.2)
2
which is equivalent to (1.11). This model accords with the general fact that, in a world of deterministic
interest rates, the forward price is the expected price in the risk-neutral measure, i.e. the ratio
St /F (0, t) is a positive martingale with expectation 1. The exponential martingale (4.2) is the simplest
continuous-path process with these properties.
5. A universal Black formula. The parametrization of Black-Scholes can be further compressed, as follows. First, note that σ and τ = (T − t) do not appear separately, but only in the
√
combination a = σ T − t, where a2 is sometimes known as the ‘operational time’. Next, define the
‘moneyness’ m as m(t, T ) = K/F (t, T ), and define
d(a, m) =
a log m
−
2
a
√
(so that d1 = d(σ T − t, K/F (t, T )).) Then the Black formula (4.1) becomes
C = BF f (a, m),
(5.1)
f (a, m) = N (d(a, m)) − mN (d(a, m) − a).
(5.2)
where
Now BF is the price of a zero-strike call, or equivalently the price to be paid at time t for delivery
of the asset at time T . Formula (5.1) says that the price of the K-strike call is the (model-free) price
of the zero-strike call modified by a factor f that depends only on the moneyness and operational
time. We call f the ‘universal Black-Scholes function’, and a graph of it is shown in Figure 5. With
N 0 = dN/dx and d = d(a, m) we find that mN 0 (d − a) = N 0 (d) and hence obtain the following very
simple expressions for the first-order derivatives:
∂f
(a, m) = N 0 (d),
∂a
∂f
(a, m) = −N (d − a).
∂m
(5.3)
(5.4)
In particular, ∂f /∂a > 0 and ∂f /∂m < 0 for all a, m.
This minimal parametrization of Black-Scholes is is used in studies of stochastic volatility, see for
example Gatheral [5].
6. Implied volatility and market trading. So far, our discussion has been entirely within
the Black-Scholes model. What happens if we attempt to use Black-Scholes delta hedging in real
market trading? This question has been considered by several authors, including El Karoui et al [3]
and Fouque et al [4], though neither of these discusses the effect of jumps in the price process.
In the universal price formula (5.1) the parameters B, F, m are market data, so we can regard
the formula as a mapping a 7→ p = BF f (a, m) from a to price p ∈ [B[F − K]+ , BF ). In a traded
options market, p is market data (but must lie in the stated interval, else there is a static arbitrage
7
1.0
0.9
0.8
0.7
0.6
factor f 0.5
0.4
0.3
0.2
1
0.8
0.1
0.6
0.4
a
0.00
2
1.8
1.4
moneyness, m
1.6
1
0.2
1.2
0.8
0.4
0.6
0.0
0.2
0.0
Fig. 5.1. The universal Black-Scholes function
opportunity). In view of (5.3), f (a, m) is strictly increasing in a and hence there is a unique value
√
a = â(p) such that p = BF f (â(p), m). The implied volatility is σ̂(p) = â(p)/ T − t. If the underlying
price process St actually was geometric Brownian motion (1.1) then σ̂ would be the same, and equal
to the volatility σ, for call options of all strikes and maturities. Of course, this is never the case in
practice—see The Volatility Surface for a discussion. Here we restrict ourselves to examining what
happens if we naı̈vely apply the Black-Scholes delta-hedge when in reality the underlying process is
not geometric Brownian motion. Specifically, we assume that the ‘true’ price model, under measure
P, is
St = S0 +
Z
t
ηt St− dt +
0
Z
t
κt St− dWt +
0
Z
St− vt (z)μ(dt, dz)
(6.1)
[0,t]×R
where μ is a finite-activity Poisson random measure, so that there is a finite measure ν on R such that
μ([0, t] × A) − ν(A)t ≡ (μ − π)([0, t] × A) is a martingale for each A ∈ B(R). η, κ, v are predictable
processes. Assume that η, κ and v are such that the solution to the SDE (6.1) is well-defined and
moreover that almost surely vt (z) > −1 so St > 0 almost surely. This is a very general model including
path-dependent coefficients, stochastic volatility and jumps. Readers unfamiliar with jump diffusion
models can set μ = ν = π = 0 below, and refer to the last paragraph of this section for comments on
the effect of jumps.
Consider the scenario of selling at time 0 a European call option at implied volatility σ̂, i.e.
for the price p = C(T, S0 , K, r, σ̂) and then following a Black-Scholes delta-hedging trading strategy
based on constant volatility σ̂ until the option expires at time T . As usual, we shall denote C(t, s) =
C(T − t, s, K, r, σ̂), so that the hedge portfolio, with value process Xt , is constructed by holding
αt := ∂S C(t, St− ) units of the risky asset S, and the remainder βt := B1t (Xt− − αt St− ) units in the
riskless asset B (a unit notional zero coupon bond). This portfolio, initially funded by the option sale
(so X0 = p), defines a self-financing trading strategy. Hence the portfolio value process X satisfies
the SDE
Xt = p +
Z
+
Z
t
∂S C(u, Su− )ηu Su− du +
0
[0,t]×R
Z
t
∂S C(u, Su− )κu Su− dWu
Z t
∂S C(u, Su− )Su− vu (z)μ(du, dz) +
(Xu − ∂S C(u, Su− )Su )rdu.
0
0
Now define Yt = C(t, St ), so that in particular Y0 = p. Applying the Itô formula (Lemma 4.4.6 of [1])
8
gives
Z
Z
t
t
Yt = p +
∂t C(u, Su− )du +
∂S C(u, Su− )ηu Su− du
0
0
Z
Z t
1 t 2
2
∂S C(u, Su− )κu Su− dWu +
∂ C(u, Su− )κ2u Su−
du
+
2 0 SS
0
Z
C(u, Su− (1 + vu (z))) − C(u, Su− ) μ(dt, dz).
+
[0,t]×R
Thus the ‘hedging error’ process defined by Zt := Xt − Yt satisfies the SDE
Z t
Z t
1
2
2
rXu du −
∂SS
C(u, Su− ) du
rSu− ∂S C(u, Su− ) + ∂t C(u, Su− ) + κ2u Su−
Zt =
2
0
0
Z
C(u, Su− (1 + vu (z))) − C(u, Su− ) − ∂S C(u, Su− )Su− vu (z) μ(du, dz)
−
=
Z
[0,t]×R
t
0
−
Z
1
rZu du +
2
[0,t]×R
Z
t
0
2
Γ(u, Su− )Su−
(σ̂ 2
−
(6.2)
κ2u )du
C(u, Su− (1 + vu (z))) − C(u, Su− ) − ∂S C(u, Su− )Su− vu (z) μ(du, dz),
2
where Γ(t, St ) = ∂SS
C(t, St ), and the last equality follows from the Black-Scholes PDE. Therefore the
final difference between the hedging strategy and the required option payout is given by
ZT = XT − [ST − K]+
Z
1 T r(T −t) 2
=
e
St− Γ(t, St− )(σ̂ 2 − κ2t )dt
2 0
Z 1Z Z
2
er(T −t)
Γ(t, St− (1 + 0 vt (z)))vt2 (z)Su−
d0 d π(dt, dz) − MT
−
[0,T ]×R
0
where MT is the terminal value of the martingale
Z 1Z Z
2
er(T −t)
Γ(t, St− (1 + 0 vt (z)))vt2 (z)Su−
d0 d (μ − π)(dt, dz).
Mt =
[0,T ]×R
0
(6.3)
0
0
Equation (6.3) is a key formula, as it shows that successful hedging is quite possible even under
significant model error. Without some ‘robustness’ property of this kind, it is hard to imagine that the
derivatives industry could exist at all, since hedging under realistic conditions would be impossible.
Consider first the case μ ≡ 0, where St has continuous sample paths and the last two terms in (6.3)
vanish. Then successful hedging depends entirely on the relationship between the implied volatility σ̂
and the true ‘local volatility’ κt . Note from Table 3.1 that Γt > 0. If we, as option writers, are lucky
and σ̂ 2 ≥ βt2 a.s. for all t then the hedging strategy makes a profit with probability one even though
the true price model is substantially different from the assumed model (1.1). On the other hand if we
underestimate the volatility, we will consistently make a loss. The magnitude of the the profit or loss
depends on the option convexity Γ. If Γ is small then hedging error is small even if the volatility has
been grossly mis-estimated.
For the option writer, jumps in either direction are unambiguously bad news. Since C is convex,
ΔC > (∂C/∂S)ΔS, so the last term in (6.2) is monotone decreasing: the hedge profit takes a hit every
time there is a jump, either upwards or downwards, in the underlying price. However, there is some
recourse: in (6.3), MT has expectation zero while the penultimate term is negative. By increasing σ̂
we increase E[ZT ], so we could arrive at a situation where E[ZT ] > 0, although in this case there is
no possibility of with probability one profit because of the martingale term. All of this reinforces the
trader’s intuition that one can offset additional hedge costs by charging more upfront (i.e. increasing
σ̂) and hedging at the higher level of implied volatility.
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REFERENCES
[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004.
[2] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Economy 81 (1973) 637-654
[3] N. El Karoui, M. Jeanblanc-Picqué and S.E. Shreve, Robustness of the Black and Scholes formula, Mathematical
Finance 8 (1998) 93-126
[4] J.-P. Fouque, G. Papanicolaou and K.R. Sircar, Derivatives in financial markets with stochastic volatility, Cambridge University Press 2000
[5] J. Gatheral, The Volatility Surface, Wiley 2006
[6] J.C. Hull, Options, Futures and Other Derivatives, 6th ed. Prentice Hall 2005
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