Download Notes on Stochastic Finance

Chapter 5
The Black-Scholes PDE
In this chapter we review the notions of assets, self-financing portfolios, riskneutral measures, and arbitrage in continuous time. We also derive the BlackScholes PDE for self-financing portfolios, and we solve this equation using the
heat kernel method.
5.1 Continuous-Time Market Model
Let (At )t∈R+ be the riskless asset given by
dAt
= rdt,
At
t ∈ R+ ,
i.e. At = A0 e rt ,
t ∈ R+ ,
(5.1)
where r > 0 is the risk-free interest rate.† We will model the risky asset
price process (St )t∈R+ using a geometric Brownian motion defined from the
equation
dSt
= µdt + σdBt , t ∈ R+ ,
(5.2)
St
see Section 4.6. By Proposition 4.12 we have
1
St = S0 exp σBt + µ − σ 2 t ,
2
t ∈ R+ .
“Anyone who believes exponential growth can go on forever in a finite world is either
a madman or an economist”, Kenneth E. Boulding, in: Energy Reorganization Act of
1973: Hearings, Ninety-third Congress, First Session, on H.R. 11510, page 248, United
States Congress, U.S. Government Printing Office, 1973.
†
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install.packages("quantmod")
library(quantmod)
getSymbols("0005.HK",from="2016-02-15",to=Sys.Date(),src="yahoo")
stock=Ad(`0005.HK`)
write(stock, file = "data_exp", sep="\n")
chartSeries(stock,up.col="blue",theme="white")
The next Figure 5.1 presents a graph of underlying market data that can be
compared to the geometric Brownian motion of Figure 4.10.
stock
70
[2016−02−15/2017−05−09]
Last 67.15
Feb 15
2016
May 02
2016
Jul 01
2016
Sep 01
2016
Nov 01
2016
Jan 02
2017
Mar 01
2017
May 01
2017
65
60
60
55
55
St
65
50
50
45
45
40
0005.HK
eµt
Feb 16 May 16
Jul 16
Sep 16 Nov 16
Jan 17
Mar 17 May 17
t
Fig. 5.1: Graphs of underlying prices.
5.2 Self-Financing Portfolio Strategies
Let ξt and ηt denote the (possibly fractional) quantities invested at time t
over the time period [t, t + dt), respectively in the assets St and At , and let
ξ¯t = (ηt , ξt ),
S̄t = (At , St ),
t ∈ R+ ,
denote the associated portfolio and asset price processes. The portfolio value
Vt at time t is given by
Vt = ξ¯t · S̄t = ηt At + ξt St ,
t ∈ R+ .
(5.3)
Our description of portfolio strategies proceeds in four steps which correspond
to different interpretations of the self-financing condition.
Portfolio update
The portfolio strategy (ηt , ξt )t∈R+ is self-financing if the portfolio value remains constant after updating the portfolio from (ηt , ξt ) to (ηt+dt , ξt+dt ), i.e.
ξ¯t · S̄t+dt = At+dt ηt + St+dt ξt = At+dt ηt+dt + St+dt ξt+dt = ξ¯t+dt · S̄t+dt , (5.4)
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The Black-Scholes PDE
which is the continuous-time equivalent of the self-financing condition already
encountered in the discrete setting of Chapter 2, see Definition 2.1. A major difference with the discrete-time case of Definition 2.1, however, is that
the continuous-time differentials dSt and dξt do not make pathwise sense
as continuous-time stochastic integrals are defined by L2 limits, cf. Proposition 4.9, or by convergence in probability.
Portfolio value
ξ¯t · S̄t
- ξ¯t · S̄t+dt = ξ¯t+dt · S̄t+dt
- ξ¯t+dt · S̄t+2dt
Asset value
St
St+dt
St+dt
St+2dt
Time scale
t
ξt
t + dt
ξt
t + dt
ξt+dt
t + 2dt
ξt+dt
Portfolio allocation
Fig. 5.2: Illustration of the self-financing condition (5.4).
Portfolio update
Equivalently, Condition (5.4) can be rewritten as
At+dt dηt + St+dt dξt = 0,
where
dηt := ηt+dt − ηt
(5.5)
and dηt := ξt+dt − ξt
denote the respective changes in portfolio allocations. Equivalently, we have
At+dt (ηt − ηt+dt ) = St+dt (ξt+dt − ξt ).
(5.6)
In other words, when one sells a (possibly fractional) quantity ηt − ηt+dt > 0
of the riskless asset priced At+dt at the end of the time period [t, t + dt] for
the total amount At+dt (ηt − ηt+dt ), one should entirely spend this income to
buy the corresponding quantity ξt+dt − ξt > 0 of the risky asset for the same
amount St+dt (ξt+dt − ξt ) > 0.
Similarly, if one sells a quantity −dξt > 0 of the risky asset St+dt between
the time periods [t, t + dt] and [t + dt, t + 2dt] for a total amount −St+dt dξt ,
one should entirely use this income to buy a quantity dηt > 0 of the riskless
asset for an amount At+dt dηt > 0, i.e.
At+dt dηt = −St+dt dξt .
Condition (5.6) can be rewritten as
St (ξt+dt − ξt ) + At+dt (ηt+dt − ηt ) + (St+dt − St )(ξt+dt − ξt ) = 0,
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i.e.
St dξt + At dηt + dSt · dξt = 0
(5.7)
in differential notation.
Portfolio differential
In practice, the self-financing portfolio property will be characterized by the
following proposition.
Proposition 5.1. A portfolio allocation (ξt , ηt )t∈R+ with price
Vt = ηt At + ξt St ,
t ∈ R+ ,
is self-financing according to (5.4) if and only if the relation
dVt = ηt dAt + ξt dSt
(5.8)
holds.
Proof. We check that by Itô’s calculus we have
dVt = ηt dAt + ξt dSt + At dηt + St dξt + dηt · dAt + dξt · dSt
= ηt dAt + ξt dSt + At dηt + St dξt + dξt · dSt ,
since
(At+dt − At ) · (ηt+dt − ηt ) = dAt · dηt = rAt (dt · dηt ) ' 0
in the sense of the Itô calculus by the Itô Table 4.1. Hence, Condition (5.7)
rewrites as (5.8), which is equivalent to (5.4) and (5.5).
Let
Vet = e −rt Vt
and
Xt = e −rt St
respectively denote the discounted portfolio value and discounted risky asset
prices at time t > 0. We have
dXt = d( e −rt St )
= −r e −rt St dt + e −rt dSt
= −r e −rt St dt + µ e −rt St dt + σ e −rt St dBt
= Xt ((µ − r)dt + σdBt ).
In the next lemma we show that when a portfolio is self-financing, its discounted value is a gain process given by the sum over time of discounted
profits and losses (number of risky assets ξt times discounted price variation
dXt ).
The following lemma is the continuous-time analog of Lemma 3.2.
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The Black-Scholes PDE
Lemma 5.2. Let (ηt , ξt )t∈R+ be a portfolio strategy with value
Vt = ηt At + ξt St ,
t ∈ R+ .
The following statements are equivalent:
i) the portfolio strategy (ηt , ξt )t∈R+ is self-financing,
ii) we have
Vet = Ve0 +
wt
0
ξu dXu ,
(5.9)
t ∈ R+ .
Proof. Assuming that (i) holds, the self-financing condition and (5.1)-(5.2)
show that
dVt = ηt dAt + ξt dSt
= rηt At dt + µξt St dt + σξt St dBt
= rVt dt + (µ − r)ξt St dt + σξt St dBt ,
t ∈ R+ ,
hence
e −rt dVt = r e −rt Vt dt + (µ − r) e −rt ξt St dt + σ e −rt ξt St dBt ,
t ∈ R+ ,
and
dVet = d e −rt Vt
= −r e −rt Vt dt + e −rt dVt
= (µ − r)ξt e −rt St dt + σξt e −rt St dBt
= (µ − r)ξt Xt dt + σξt Xt dBt
= ξt dXt ,
t ∈ R+ ,
i.e. (5.9) holds by integrating on both sides as
Vet − Ve0 =
wt
0
dVeu =
wt
0
ξu dXu ,
t ∈ R+ .
(ii) Conversely, if (5.9) is satisfied we have
dVt = d( e rt Vet )
= r e rt Vet dt + e rt dVet
= r e rt Vet dt + e rt ξt dXt
= rVt dt + e rt ξt dXt
= rVt dt + e rt ξt Xt ((µ − r)dt + σdBt )
= rVt dt + ξt St ((µ − r)dt + σdBt )
= rηt At dt + µξt St dt + σξt St dBt
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= ηt dAt + ξt dSt ,
hence the portfolio is self-financing according to Definition 5.1.
As a consequence of (5.9), the hedging problem of a claim C with maturity
T is reduced to that of finding the representation of the discounted claim
C̃ = e −rT C as a stochastic integral:
C̃ = Ve0 +
wT
0
ξu dXu .
Note also that (5.9) shows that the value of a self-financing portfolio can be
written as
wt
wt
Vt = e rt V0 + (µ − r)
e r(t−u) ξu Su du + σ
e r(t−u) ξu Su dBu , t ∈ R+ .
0
0
(5.10)
5.3 Arbitrage and Risk-Neutral Measures
In continuous-time, the definition of arbitrage follows the lines of its analogs
in the discrete and two-step models. In the sequel we will only consider admissible portfolio strategies whose total value Vt remains nonnegative for all
times t ∈ [0, T ].
Definition 5.3. A portfolio strategy (ξt , ηt )t∈[0,T ] with price Vt = ξt St +ηt At ,
t ∈ R+ , constitutes an arbitrage opportunity if all three following conditions
are satisfied:
i) V0 6 0,
ii) VT > 0,
iii) P(VT > 0) > 0.
Roughly speaking, (ii) means that the investor wants no loss, (iii) means
that he wishes to sometimes make a strictly positive gain, and (i) means that
he starts with zero capital or even with a debt.
Next, we turn to the definition of risk-neutral measures in continuous time.
Recall that the filtration (Ft )t∈R+ is generated by Brownian motion (Bt )t∈R+ ,
i.e.
Ft = σ(Bu : 0 6 u 6 t),
t ∈ R+ .
Definition 5.4. A probability measure P∗ on Ω is called a risk-neutral measure if it satisfies
IE∗ [St |Fu ] = e r(t−u) Su ,
0 6 u 6 t,
(5.11)
where IE∗ denotes the expectation under P∗ .
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The Black-Scholes PDE
From the relation
At = e r(t−u) Au ,
0 6 u 6 t,
we interpret (5.11) by saying that the expected return of the risky asset St
under P∗ equals the return of the riskless asset At . The discounted price Xt
of the risky asset in $ at time 0 is defined by
Xt = e −rt St =
St
,
At /A0
t ∈ R+ ,
i.e. At /A0 plays the role of a numéraire in the sense of Chapter 12.
Definition 5.5. A continuous-time process (Zt )t∈R+ of integrable random
variables is a martingale with respect to the filtration (Ft )t∈R+ if
IE[Zt |Fs ] = Zs ,
0 6 s 6 t.
Note that when (Zt )t∈R+ is a martingale, Zt is in particular Ft -measurable
for all t ∈ R+ .
In continuous-time finance, the martingale property can be used to characterize risk-neutral probability measures, for the derivation of pricing partial
differential equations (PDEs), and for the computation of conditional expectations.
As in the discrete-time case, the notion of martingale can be used to characterize risk-neutral measures as in the next proposition.
Proposition 5.6. The measure P∗ is risk-neutral if and only if the discounted
price process (Xt )t∈R+ is a martingale under P∗ .
Proof. If P∗ is a risk-neutral measure we have
IE∗ [Xt |Fu ] = IE∗ [ e −rt St |Fu ]
= e −rt IE∗ [St |Fu ]
= e −rt e r(t−u) Su
= e −ru Su
= Xu ,
0 6 u 6 t,
hence (Xt )t∈R+ is a martingale. Conversely, if (Xt )t∈R+ is a martingale then
IE∗ [St |Fu ] = e rt IE∗ [Xt |Fu ]
= e rt Xu
= e r(t−u) Su ,
0 6 u 6 t,
hence the measure P is risk-neutral according to Definition 5.4.
∗
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As in the discrete-time case, P] would be called a risk premium measure if it
satisfied
IE] [St |Fu ] > e r(t−u) Su ,
0 6 u 6 t,
meaning that by taking risks in buying St , one could make an expected return
higher than that of
At = e r(t−u) Au ,
0 6 u 6 t.
Similarly, a negative risk premium measure P[ satisfies
IE[ [St |Fu ] < e r(t−u) Su ,
0 6 u 6 t.
Next, we note that the first fundamental theorem of asset pricing also holds
in continuous time, and can be used to check for the existence of arbitrage
opportunities.
Theorem 5.7. A market is without arbitrage opportunity if and only if it
admits at least one (equivalent) risk-neutral measure P∗ .
Proof. cf. [HP81] and Chapter VII-4a of [Shi99].
5.4 Market Completeness
Definition 5.8. A contingent claim with payoff C is said to be attainable if
there exists a (self-financing) portfolio strategy (ηt , ξt )t∈[0,T ] such that
C = VT .
In this case the price of the claim at time t will be equal to the value Vt of
any self-financing portfolio hedging C.
Definition 5.9. A market model is said to be complete if every contingent
claim C is attainable.
The next result is a continuous-time restatement of the second fundamental
theorem of asset pricing.
Theorem 5.10. A market model without arbitrage opportunities is complete
if and only if it admits only one (equivalent) risk-neutral measure P∗ .
Proof. cf. [HP81] and Chapter VII-4a of [Shi99].
In the Black-Scholes model one can show the existence of a unique risk-neutral
measure, hence the model is without arbitrage and complete.
5.5 The Black-Scholes Formula
We start by deriving the Black-Scholes Partial Differential Equation (PDE)
for the price of a self-financing portfolio. Note that the drift parameter µ in
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The Black-Scholes PDE
(5.2) is absent of the PDE (5.12) and is not involved in the Black-Scholes
formula (5.17).
Proposition 5.11. Let (ηt , ξt )t∈R+ be a portfolio strategy such that
(i) (ηt , ξt )t∈R+ is self-financing,
(ii) the value Vt := ηt At + ξt St , t ∈ R+ , takes the form
Vt = g(t, St ),
for some function g ∈ C
1,2
t ∈ R+ ,
((0, ∞) × (0, ∞)).
Then the function g(t, x) satisfies the Black-Scholes PDE
rg(t, x) =
∂g
1
∂2g
∂g
(t, x) + rx (t, x) + σ 2 x2 2 (t, x),
∂t
∂x
2
∂x
x > 0, (5.12)
t ∈ [0, T ], and ξt is given by
ξt =
∂g
(t, St ),
∂x
(5.13)
t ∈ R+ .
Proof. First, note that the self-financing condition (5.8) in Proposition 5.1
implies
dVt = ηt dAt + ξt dSt
= rηt At dt + µξt St dt + σξt St dBt
(5.14)
= rVt dt + (µ − r)ξt St dt + σξt St dBt
= rg(t, St )dt + (µ − r)ξt St dt + σξt St dBt ,
t ∈ R+ . We now rewrite (4.26) under the form of an Itô process
St = S0 +
wt
0
vs ds +
wt
0
us dBs ,
t ∈ R+ ,
as in (4.20), by taking
ut = σSt ,
and vt = µSt ,
t ∈ R+ .
By (4.22), the application of Itô’s formula Theorem 4.11 to Vt = g(t, St ) leads
to
∂g
∂g
(t, St )dt + ut (t, St )dBt
∂x
∂x
∂g
1
∂2g
+ (t, St )dt + |ut |2 2 (t, St )dt
∂t
2
∂x
dVt = dg(t, St ) = vt
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=
∂g
∂g
∂g
1
∂2g
(t, St )dt + µSt (t, St )dt + σ 2 St2 2 (t, St )dt + σSt (t, St )dBt .
∂t
∂x
2
∂x
∂x
(5.15)
By respective identification of the terms in dBt and dt in (5.14) and (5.15)
we get

2

 g(t, St ) + (µ − r)ξt St dt = ∂g (t, St )dt + µSt ∂g (t, St )dt + 1 σ 2 S 2 ∂ g (t, St )dt,

t

∂t
∂x
2
∂x2



 ξt St σdBt = St σ ∂g (t, St )dBt ,
∂x
hence

∂g
∂g
1 2 2 ∂2g



 rg(t, St ) = ∂t (t, St ) + rSt ∂x (t, St ) + 2 σ St ∂x2 (t, St ),



 ξt = ∂g (t, St ).
∂x
(5.16)
The derivative giving ξt in (5.13) is called the Delta of the option price, cf.
Proposition 5.13 below.
The amount invested on the riskless asset is
ηt At = Vt − ξt St = g(t, St ) − St
∂g
(t, St ),
∂x
and ηt is given by
ηt =
=
=
Vt − ξt St
At
g(t, St ) − St
∂g
(t, St )
∂x
At
g(t, St ) − St
∂g
(t, St )
∂x
.
A0 e rt
In the next proposition we add a terminal condition g(T, x) = f (x) to the
Black-Scholes PDE in order to price a claim C of the form C = h(ST ). As
in the discrete-time case, the arbitrage price πt (C) at time t ∈ [0, T ] of the
claim C is defined to be the price Vt of the self-financing portfolio hedging
C.
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The Black-Scholes PDE
Proposition 5.12. The arbitrage price πt (C) at time t ∈ [0, T ] of the option
with payoff C = h(ST ) is given by πt (C) = g(t, St ), where the function g(t, x)
is solution of the following Black-Scholes PDE:

∂g
1
∂g
∂2g

 rg(t, x) =
(t, x) + rx (t, x) + σ 2 x2 2 (t, x),
∂t
∂x
2
∂x


g(T, x) = h(x), x > 0.
Market terms and data
The gearing at time t ∈ [0, T ] of the option with payoff C = h(ST ) is defined
as the ratio
St
St
Gt :=
=
,
t ∈ [0, T ].
πt (C)
g(t, St )
The effective gearing at time t ∈ [0, T ] of the option with payoff C = h(ST )
is defined as the ratio
EGt := Gt ξt
∂g
= Gt (t, St )
∂x
St ∂g
=
(t, St )
πt (C) ∂x
St ∂g
=
(t, St )
g(t, St ) ∂x
∂ log g
= St
(t, St ),
∂x
t ∈ [0, T ].
The break-even price BEPt of the underlying is the value of S for which the
intrinsic option payoff h(S) equals the option price πt (C) at time t ∈ [0, T ].
For European call options it is given by
BEPt := K + πt (C) = K + g(t, St ),
t = 0, 1, . . . , N.
whereas for European put options it is given by
BEPt := K − πt (C) = K − g(t, St ),
t = 0, 1, . . . , N.
The option premium OPt can be defined as the variation required from the
underlying in order to reach the break-even price, i.e. we have
OPt :=
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BEPt − St
K + g(t, St ) − St
,=
,
St
St
t = 0, 1, . . . , N,
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for European call options, and
OPt :=
St − BEPt
St + g(t, St ) − K
=
,
St
St
t = 0, 1, . . . , N,
for European put options. The term “premium” is sometimes also used to
denote the arbitrage price g(t, St ) of the option.
Forward contracts
When C = ST − K is the (linear) payoff of a forward contract, i.e. f (x) =
x − K, the Black-Scholes PDE admits the easy solution
g(t, x) = x − K e −r(T −t) ,
x > 0,
t ∈ [0, T ],
and the Delta of the option price is given by
ξt =
∂g
(t, St ) = 1,
∂x
t ∈ [0, T ],
cf. Exercise 5.4. The forward contract can be realized by the option issuer as
follows:
a) At time t, receive the option premium St − e −r(T −t) K from the option
buyer.
b) Borrow e −r(T −t) K from the bank, to be refunded at maturity.
c) Buy the risky asset using the amount St − e −r(T −t) K + e −r(T −t) K = St .
d) Hold the risky asset until maturity (do nothing, constant portfolio strategy).
e) At maturity T , hand in the asset to the option holder, who gives the price
K in exchange.
f) Use the amount K = e r(T −t) e −r(T −t) K to refund the lender of e −r(T −t) K
borrowed at time t.
Forward contracts can be used for physical delivery, e.g. live cattle.
For a future contract expiring at time T we take K = S0 e rT and the contract
is usually quoted at time t using the forward price
e r(T −t) (St − K e −r(T −t) ) = e r(T −t) St − K = e r(T −t) St − S0 e rT ,
or simply using e r(T −t) St . Future contracts are non-deliverable forward contracts which are “marked to market” at each time step via a cash flow between
both parties, ensuring that the absolute difference | e r(T −t) St − K| has been
credited to the buyer’s account if e r(T −t) St > K, or to the seller’s account if
e r(T −t) St < K.
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The Black-Scholes PDE
Black-Scholes formula for European call options
Recall that in the case of a European call option with strike price K the
payoff function is given by f (x) = (x − K)+ and the Black-Scholes PDE
reads

∂g
∂g
1
∂2g

 rgc (t, x) = c (t, x) + rx c (t, x) + σ 2 x2 2c (t, x)
∂t
∂x
2
∂x


gc (T, x) = (x − K)+ .
In Sections 5.6 and 5.7 we will prove that the solution of this PDE is given
by the Black-Scholes formula
gc (t, x) = Bl(K, x, σ, r, T − t) = xΦ d+ (T − t) − K e −r(T −t) Φ d− (T − t) ,
(5.17)
with
2
log(x/K) + (r + σ /2)(T − t)
√
d+ (T − t) :=
,
(5.18)
σ T −t
d− (T − t) :=
log(x/K) + (r − σ 2 /2)(T − t)
√
,
σ T −t
(5.19)
cf. Proposition 5.16 below. Here, “log” denotes the neperian logarithm “ln”,
and
2
1 wx
Φ(x) := √
e −y /2 dy,
x ∈ R,
2π −∞
denotes the standard Gaussian distribution function, with
√
d+ (T − t) = d− (T − t) + σ T − t.
In other words, a European call option with strike price K and maturity T
is priced at time t ∈ [0, T ] as
gc (t, St ) = Bl(K, St , σ, r, T −t) = St Φ d+ (T − t) − K e −r(T −t) Φ d− (T − t) ,
|
{z
} |
{z
}
risky investment
riskless investment
t ∈ [0, T ]. The following script is an implementation of the Black-Scholes
formula for European call options in R.∗
One can easily check that

 +∞, x > K,
lim d+ (T − t) = lim d− (T − t) =

t%T
t%T
−∞, x < K,
∗
Download the corresponding IPython notebook that can be run here.
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BSCall <- function(S, K, r, T, sigma)
{d1 <- (log(S/K)+(r+sigma^2/2)*T)/(sigma*sqrt(T))
d2 <- d1 - sigma * sqrt(T)
BSCall = S*pnorm(d1) - K*exp(-r*T)*pnorm(d2)
BSCall}
Table 5.1: The Black-Scholes call function in R.
which allows us to recover the boundary condition


x>K
 xΦ(+∞) − KΦ(+∞) = x − K,
gc (T, x) =
= (x − K)+


xΦ(−∞) − KΦ(−∞) = 0,
x<K
at t = T . Similarly we can check that

 +∞,
lim d− (T − t) =
T →∞

−∞,
r > σ 2 /2,
r < σ 2 /2,
and limT →∞ d+ (T − t) = +∞, hence
lim Bl(K, St , σ, r, T − t) = St ,
T →∞
t ∈ R+ .
Figure 5.3 presents an interactive graph of the Black call price function, i.e.
the solution
(t, x) 7−→ gc (t, x) = xΦ d+ (T − t) − K e −r(T −t) Φ d− (T − t)
of the Black-Scholes PDE for a call option.
Fig. 5.3: Graph of the Black-Scholes call price function with strike price
K = 100.∗
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The Black-Scholes PDE
The next proposition is proved by a direct differentiation of the Black-Scholes
function, and will be recovered later using a probabilistic argument in Proposition 6.11 below.
Proposition 5.13. The Black-Scholes Delta of the European call option is
given by
∂gc
ξt =
(t, St ) = Φ d+ (T − t) ∈ [0, 1],
(5.20)
∂x
where d+ (T − t) is given by (5.18).
Proof. By (5.17) we have
∂
log(x/K) + (r + σ 2 /2)(T − t)
∂gc
√
(t, x) =
xΦ
(5.21)
∂x
∂x
σ T −t
2
∂
log(x/K) + (r − σ /2)(T − t)
√
− K e −r(T −t)
Φ
∂x
σ T −t
log(x/K) + (r + σ 2 /2)(T − t)
√
=Φ
σ T −t
∂
log(x/K) + (r + σ 2 /2)(T − t)
√
+x Φ
∂x
σ T −t
log(x/K) + (r − σ 2 /2)(T − t)
−r(T −t) ∂
√
−Ke
Φ
∂x
σ T −t
log(x/K) + (r + σ 2 /2)(T − t)
√
=Φ
σ T −t
2 !
1
1 log(x/K) + (r + σ 2 /2)(T − t)
√
+ p
exp −
2
σ T −t
σ 2π(T − t)
2 !
−r(T −t)
1 log(x/K) + (r − σ 2 /2)(T − t)
Ke
p
√
exp −
−
2
σ T −t
σx 2π(T − t)
2
log(x/K) + (r + σ /2)(T − t)
√
=Φ
σ T −t
2 !
1
1 log(x/K) + (r + σ 2 /2)(T − t)
√
+ p
exp −
2
σ T −t
σ 2π(T − t)
!
2
K e −r(T −t)
1 log(x/K) + (r + σ 2 /2)(T − t)
x
p
√
−
exp −
+ r(T − t) − log
2
K
σ T −t
σx 2π(T − t)
log(x/K) + (r + σ 2 /2)(T − t)
√
=Φ
.
σ T −t
∗
Right click on the figure for interaction and “Full Screen Multimedia” view.
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In Figure 5.4 we plot the the Delta of the European call option as a function
of the underlying and of time to maturity.
2
1.5
1
0.5
020
15
200
10
150
Time to maturity T-t
100
5
50
underlying
0 0
0198
Fig. 5.4: Delta of a European call option with strike price K = 100.
The Gamma of the European call option is defined as the second derivative
of the option price with respect to the underlying, which gives
γt =
=
1
√
Φ0 d+ (T − t)
St σ T − t
1
St σ
p
2π(T − t)
exp −
1
2
log(St /K) + (r + σ 2 /2)(T − t)
√
σ T −t
2 !
.
In Figure 5.5 we plot the (truncated) value of the Gamma of a European call
option as a function of the underlying and of time to maturity.
4
3.5
3
2.5
2
1.5
1
0.5
0
101
100.5
0
0.005
100
0.01
underlying
0.015
99.5
0.02
0.025
Time to maturity T-t
99 0.03
Fig. 5.5: Gamma of a European call option with strike price K = 100.
0198
Since Gamma is always nonnegative, the Black-Scholes hedging strategy is
to keep buying the underlying risky asset when its price increases, and to sell
it when its price decreases, as can be checked from Figure 5.5.
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The Black-Scholes PDE
Option price
Delta
Gamma
Vega
Theta
Rho
g(t, St )
∂g
(t, St )
∂x
∂2g
(t, St )
2
∂x
∂g
(t, St )
∂σ
∂g
(t, St )
∂t
∂g
(t, St )
∂t
Call
St Φ(d+ (T − t)) − K e −r(T −t) Φ(d− (T − t))
Put
K e −r(T −t) Φ(−d− (T − t)) − St Φ(−d+ (T − t))
Φ(d+ (T − t))
−Φ(−d+ (T − t))
Φ0 (d+ (T − t))
√
St σ T − t
√
St T − tΦ0 (d+ (T − t))
−
St σΦ0 (d+ (T − t))
St σΦ0 (d+ (T − t))
√
√
− rK e −r(T −t) Φ(d− (T − t)) −
+ rK e −r(T −t) Φ(−d− (T − t))
2 T −t
2 T −t
K(T − t) e −r(T −t) Φ(d− (T − t))
−K(T − t) e −r(T −t) Φ(−d− (T − t))
Table 5.2: Black-Scholes Greeks (Wikipedia).
T=
K=
gc
St ∂ log
∂x
(t,St ,σ,T )=
= StS−K
t
=σ
c (t,S ,σ,T )
= ∂g
t
∂σ
K +gc =
gc (t,St ,σ,T )
=
6000
c (t,S ,σ,T )
= ∂g
t
∂t
T −t=
c (t,S ,σ,T )
= ∂g
t
∂x
= g (t,SSt,σ,T )
c
t
=
St +gc (t,St ,σ,T )−K
K +gc (t,St ,σ,T )
=St
Fig. 5.6: Warrant terms and data.
The R package bizdays can be used to compute times to maturity.
Black-Scholes formula for European put options
Similarly, in the case of a European put option with strike price K the payoff
function is given by f (x) = (K − x)+ and the Black-Scholes PDE reads
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
∂g
∂g
1
∂ 2 gp

 rgp (t, x) = p (t, x) + rx p (t, x) + σ 2 x2
(t, x),
∂t
∂x
2
∂x2


gp (T, x) = (K − x)+ ,
with explicit solution
gp (t, x) = K e −r(T −t) Φ − d− (T − t) − xΦ − d+ (T − t) ,
with
(5.22)
d+ (T − t) =
log(x/K) + (r + σ 2 /2)(T − t)
√
,
σ T −t
(5.23)
d− (T − t) =
log(x/K) + (r − σ 2 /2)(T − t)
√
,
σ T −t
(5.24)
as illustrated in Figure 5.7.
In other words, a European put option with strike price K and maturity
T is priced at time t ∈ [0, T ] as
gp (t, St ) = K e −r(T −t) Φ − d− (T − t) − St Φ − d+ (T − t) ,
|
{z
} |
{z
}
riskless investment
risky investment
t ∈ [0, T ]. The following script is an implementation of the Black-Scholes
formula for European put options in R.
BSPut <- function(S, K, r, T, sigma)
{d1 = (log(S/K)+(r-sigma^2/2)*T)/(sigma*sqrt(T))
d2 = d1 - sigma * sqrt(T)
BSPut = K*exp(-r*T) * pnorm(-d2) - S*pnorm(-d1)
BSPut}
Table 5.3: The Black-Scholes put function in R.
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The Black-Scholes PDE
14
12
10
8
6
4
2
0
7
90
95
100 105
underlying HK$
110
115
120
0
1
2
3
8
9
10
6
5
4 time to maturity T-t
Fig. 5.7: Graph of the Black-Scholes put price function with strike price K = 100.
Note that the call-put parity relation
g(t, St ) = x − K e −r(T −t) = gc (t, St ) − gp (t, St ),
0 6 t 6 T,
(5.25)
is also satisfied from (5.17) and (5.22).
Numerical examples
In Figure 5.8 we consider the historical stock price of HSBC Holdings
(0005.HK) over one year:
Fig. 5.8: Graph of the stock price of HSBC Holdings.
Consider a call option issued by Societe Generale on 31 December 2008 with
strike price K=$63.704, maturity T = October 05, 2009, and an entitlement
ratio of 100, meaning that one option contract is divided into 100 warrants, cf.
page 7. The next graph gives the time evolution of the Black-Scholes portfolio
price
t 7−→ gc (t, St )
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driven by the market price t 7−→ St of the underlying risky asset as given in
Figure 5.8, in which the number of days is counted from the origin and not
from maturity.
40
35
30
25
20
15
10
5
0
100
90
80
underlying HK$
70
60
50
40
150
200
100
0
50
time in days
Fig. 5.9: Path of the Black-Scholes price for a call option on HSBC.
As a consequence of Proposition 5.13, in the Black-Scholes model the amount
invested in the risky asset is
log(St /K) + (r + σ 2 /2)(T − t)
√
St ξt = St Φ d+ (T − t) = St Φ
> 0,
σ T −t
which is always positive, i.e. there is no short selling, and the amount invested
on the riskless asset is
log(St /K) + (r − σ 2 /2)(T − t)
√
ηt At = −K e −r(T −t) Φ
6 0,
σ T −t
which is always negative, i.e. we are constantly borrowing money, as noted
in Figure 5.10.
Black-Scholes price
Risky investment
Riskless investment
Underlying
100
80
K
60
HK$
40
20
0
-20
-40
-60
0
50
100
150
200
Fig. 5.10: Time evolution of a hedging portfolio for a call option on HSBC.
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The Black-Scholes PDE
Cash settlement. In the case of a cash settlement, the option issuer will satisfy the option contract by selling ξT = 1 stock at the price ST = $83,
refund the K = $63 riskless investment, and hand in the remaining amount
C = (ST − K)+ = 83 − 63 = $20 to the option holder.
Physical delivery. In the case of physical delivery of the underlying asset, the
option issuer will deliver ξT = 1 stock to the option holder in exchange for
K = $63, which will be used to refund the $2 riskless investment.
For one more example, we consider a put option issued by BNP Paribas on
04 November 2008 with strike price K=$77.667, maturity T = October 05,
2009, and entitlement ratio 92.593, cf. page 7. In the next Figure 5.11, the
number of days is counted from the origin and not from maturity.
45
40
35
30
25
20
15
10
5
0
0
50
100
time in days
150
200
100
90
80
70
40
50
60
underlying HK$
Fig. 5.11: Path of the Black-Scholes price for a put option on HSBC.
The Delta of the Black-Scholes put option is given by
ξt = −Φ − d+ (T − t) ∈ [−1, 0],
and the amount invested on the risky asset is
log(St /K) + (r + σ 2 /2)(T − t)
√
−St Φ d+ (T − t) = −St Φ −
6 0,
σ T −t
i.e. there is always short selling, and the amount invested on the riskless asset
is
log(St /K) + (r − σ 2 /2)(T − t)
√
> 0,
K e −r(T −t) Φ −
σ T −t
which is always positive, i.e. we are constantly investing on the riskless asset.
In the above example the put option finished out of the money (OTM), so
that no cash settlement or physical delivery occurs.
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Black-Scholes price
Risky investment
Riskless investment
Underlying
100
80
K
60
HK$
40
20
0
-20
-40
-60
0
50
100
150
200
Fig. 5.12: Time evolution of the hedging portfolio for a put option on HSBC.
5.6 The Heat Equation
In this section we study the heat equation
∂g
1 ∂2g
(t, y) =
(t, y)
∂t
2 ∂y 2
(5.26)
which is used to model the diffusion of heat over time through solids. Here,
the data of g(x, t) represents the temperature measured at time t and point
x. We refer the reader to [Wid75] for a complete treatment of this topic.
We can check by a direct calculation
that the Gaussian probability density
√
2
function g(t, y) := e −y /(2t) / 2πt solves the heat equation (5.26), as follows:
!
2
∂g
∂
e −y /(2t)
√
(t, y) =
∂t
∂t
2πt
2
2
y 2 e −y /(2t)
e −y /(2t)
√ + 2 √
=−
3/2
2t
2π 2t
2πt
1
y2
= − + 2 g(t, y),
2t 2t
and
1 ∂
1 ∂2g
(t, y) = −
2 ∂y 2
2 ∂y
2
y e −y /(2t)
√
t
2πt
!
2
2
1 e −y /(2t)
y 2 e −y /(2t)
√
=−
+ 2 √
2t
2t
2πt
2πt
1
y2
= − + 2 g(t, y),
t ∈ R+ ,
2t 2t
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The Black-Scholes PDE
Fig. 5.13: Time-dependent solution of the heat equation.∗
In Section 5.7 this equation will be shown to be equivalent to the BlackScholes PDE after a change of variables. In particular this will lead to the
explicit solution of the Black-Scholes PDE.
Proposition 5.14. The heat equation

2

 ∂g (t, y) = 1 ∂ g (t, y)

∂t
2 ∂y 2



g(0, y) = ψ(y)
(5.27)
with initial condition
g(0, y) = ψ(y)
has the solution
g(t, y) =
w∞
−∞
2
ψ(z) e −(y−z)
/(2t)
dz
√
,
2πt
t > 0.
(5.28)
Proof. We have
2
∂g
∂ w∞
dz
(t, y) =
ψ(z) e −(y−z) /(2t) √
∂t
∂t −∞
2πt
!
2
w∞
∂
e −(y−z) /(2t)
√
=
ψ(z)
dz
−∞
∂t
2πt
2
1w∞
(y − z)2
1
dz
=
ψ(z)
−
e −(y−z) /(2t) √
2
2 −∞
t
t
2πt
∂ 2 −(y−z)2 /(2t) dz
1w∞
√
ψ(z) 2 e
=
2 −∞
∂z
2πt
w
2
2
1 ∞
∂
dz
=
ψ(z) 2 e −(y−z) /(2t) √
2 −∞
∂y
2πt
∗
The animation works in Acrobat Reader on the entire pdf file.
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2
1 ∂2 w ∞
dz
ψ(z) e −(y−z) /(2t) √
2 ∂y 2 −∞
2πt
1 ∂2g
=
(t, y).
2 ∂y 2
=
On the other hand it can be checked that at time t = 0,
lim
t→0
w∞
−∞
ψ(z) e −(y−z)
2
/(2t)
w∞
2
dz
dz
√
= lim
ψ(y + z) e −z /(2t) √
= ψ(y),
2πt t→0 −∞
2πt
y ∈ R.
Let us provide a second proof of Proposition 5.14 using stochastic calculus
and Brownian motion. Note that under the change of variable x = z − y we
have
w∞
2
dz
g(t, y) =
ψ(z) e −(y−z) /(2t) √
−∞
2πt
w∞
−x2 /(2t) dx
√
=
ψ(y + x) e
−∞
2πt
= IE[ψ(y − Bt )],
where (Bt )t∈R+ is a standard Brownian motion. Applying Itô’s formula we
have
w
w
t
t
1
IE[ψ(y − Bt )] = ψ(y) − IE
ψ 0 (y − Bs )dBs + IE
ψ 00 (y − Bs )ds
0
0
2
1wt
00
= ψ(y) +
IE [ψ (y − Bs )] ds
2 0
1 w t ∂2
= ψ(y) +
IE [ψ(y − Bs )] ds,
2 0 ∂y 2
since the expectation of the stochastic integral is zero. Hence
∂g
∂
(t, y) =
IE[ψ(y − Bt )]
∂t
∂t
1 ∂2
=
IE [ψ(y − Bt )]
2 ∂y 2
2
1∂ g
=
(t, y).
2 ∂y 2
Concerning the initial condition we check that
g(0, y) = IE[ψ(y − B0 )] = IE[ψ(y)] = ψ(y).
The expression g(t, y) = IE[ψ(y − Bt )] provides a probabilistic interpretation
of the heat diffusion phenomenon based on Brownian motion. Namely, when
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The Black-Scholes PDE
ψ(y) = 1[−ε,ε] (y), we find that
g(t, y) = IE[ψ(y − Bt )]
= IE[1[−ε,ε] (y − Bt )]
= P y − Bt ∈ [−ε, ε]
= P y − ε 6 Bt 6 y + ε
represents the probability of finding Bt within a neighborhood of the point
y ∈ R.
5.7 Solution of the Black-Scholes PDE
In this section we will solve the Black-Scholes PDE by the kernel method
of Section 5.6 and a change of variables. This solution method uses a transformation of variables (5.30) which involves the time inversion t 7−→ T − t
on the interval [0, T ], so that the terminal condition at time T in the BlackScholes equation (5.29) becomes an initial condition at time t = 0 in the heat
equation (5.32).
Proposition 5.15. Assume that f (t, x) solves the Black-Scholes PDE

∂f
∂f
1
∂2f

 rf (t, x) =
(t, x) + rx (t, x) + σ 2 x2 2 (t, x),
∂t
∂x
2
∂x


+
f (T, x) = (x − K) ,
(5.29)
with terminal condition h(x) = (x − K)+ . x > 0. Then the function g(t, y)
defined by
2
g(t, y) = e rt f T − t, e σy+(σ /2−r)t
(5.30)
solves the heat equation (5.27) with initial condition
g(0, y) = h( e σy ),
i.e.
y ∈ R,

2

 ∂g (t, y) = 1 ∂ g (t, y)

∂t
2 ∂y 2



g(0, y) = h( e σy ).
Proof. Letting s = T − t and x = e σy+(σ
2
/2−r)t
(5.31)
(5.32)
we have
2
2
∂g
∂f
(t, y) = r e rt f (T − t, e σy+(σ /2−r)t ) − e rt
T − t, e σy+(σ /2−r)t
∂t
∂s
2
2
2
σ
∂f
+
− r e rt e σy+(σ /2−r)t
T − t, e σy+(σ /2−r)t
2
∂x
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2
∂f
σ
∂f
(T − t, x) +
− r e rt x (T − t, x)
∂s
2
∂x
σ 2 rt ∂f
1 rt 2 2 ∂ 2 f
(T − t, x) +
e x (T − t, x),
(5.33)
= e x σ
2
∂x2
2
∂x
= r e rt f (T − t, x) − e rt
where on the last step we used the Black-Scholes PDE. On the other hand
we have
2
2
∂g
∂f
(t, y) = σ e rt e σy+(σ /2−r)t
T − t, e σy+(σ /2−r)t
∂y
∂x
and
2
1 ∂g 2
σ 2 rt σy+(σ2 /2−r)t ∂f
(t, y) =
e e
T − t, e σy+(σ /2−r)t
2
2 ∂y
2
∂x
2
2
σ2
∂2f
+ e rt e 2σy+2(σ /2−r)t 2 T − t, e σy+(σ /2−r)t
2
∂x
σ 2 rt 2 ∂ 2 f
σ 2 rt ∂f
e x (T − t, x) +
e x
(T − t, x).
(5.34)
=
2
∂x
2
∂x2
We conclude by comparing (5.33) with (5.34), which shows that g(t, x) solves
the heat equation (5.27) with initial condition
g(0, y) = f (T, e σy ) = h( e σy ).
In the next proposition we recover the Black-Scholes formula (5.17) by solving
the PDE (5.29). The Black-Scholes will also be recovered by probabilistic
arguments and the computation of an expectation in Proposition 6.4.
Proposition 5.16. When h(x) = (x−K)+ , the solution of the Black-Scholes
PDE (5.29) is given by
f (t, x) = xΦ d+ (T − t) − K e −r(T −t) Φ d− (T − t) ,
where
2
1 wx
Φ(x) = √
e −y /2 dy,
2π −∞
and
x ∈ R,
d+ (T − t) :=
log(x/K) + (r + σ 2 /2)(T − t)
√
,
σ T −t
d− (T − t) :=
log(x/K) + (r − σ 2 /2)(T − t)
√
.
σ T −t
Proof. By inversion of (5.30) with s = T − t and x = e σy+(σ
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2
/2−r)t
we get
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The Black-Scholes PDE
−(σ 2 /2 − r)(T − s) + log x
f (s, x) = e −r(T −s) g T − s,
.
σ
Hence using the solution (5.28) and Relation (5.31) we get
−(σ 2 /2 − r)(T − t) + log x
f (t, x) = e −r(T −t) g T − t,
σ
w ∞ −(σ 2 /2 − r)(T − t) + log x
2
dz
−r(T −t)
ψ
+ z e −z /(2(T −t)) p
= e
−∞
σ
2π(T − t)
w∞ 2
2
dz
= e −r(T −t)
h x e σz−(σ /2−r)(T −t) e −z /(2(T −t)) p
−∞
2π(T − t)
+
w∞ 2
2
dz
= e −r(T −t)
x e σz−(σ /2−r)(T −t) − K
e −z /(2(T −t)) p
−∞
2π(T − t)
w∞
2
2
dz
= e −r(T −t) (−r+σ2 /2)(T −t)+log(K/x) x e σz−(σ /2−r)(T −t) − K e −z /(2(T −t)) p
2π(T − t)
σ
w∞
2
2
dz
e σz−(σ /2−r)(T −t) e −z /(2(T −t)) p
= x e −r(T −t)
√
−d− (T −t) T −t
2π(T − t)
w∞
2
dz
e −z /(2(T −t)) p
−K e −r(T −t)
√
−d− (T −t) T −t
2π(T − t)
w∞
2
2
dz
e σz−σ (T −t)/2−z /(2(T −t)) p
=x
√
−d− (T −t) T −t
2π(T − t)
w∞
2
dz
e −z /(2(T −t)) p
−K e −r(T −t)
√
−d− (T −t) T −t
2π(T − t)
w∞
2
dz
e −(z−σ(T −t)) /(2(T −t)) p
=x
√
−d− (T −t) T −t
2π(T − t)
w∞
2
dz
−K e −r(T −t)
e −z /(2(T −t)) p
√
−d− (T −t) T −t
2π(T − t)
w∞
2
dz
=x
e −z /(2(T −t)) p
√
−d− (T −t) T −t−σ(T −t)
2π(T − t)
w∞
2
dz
−K e −r(T −t)
e −z /(2(T −t)) p
√
−d− (T −t) T −t
2π(T − t)
w∞
w∞
2
2
dz
dz
=x
e −z /2 √ − K e −r(T −t)
e −z /2 √
√
−d− (T −t)−σ T −t
d− (T −t)
2π
2π
= x 1 − Φ − d+ (T − t) − K e −r(T −t) 1 − Φ − d− (T − t)
= xΦ d+ (T − t) − K e −r(T −t) Φ d− (T − t) ,
where we used the relation
1 − Φ(a) = Φ(−a),
"
a ∈ R.
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Exercises
Exercise 5.1 Black-Scholes PDE with dividends. Consider an underlying
asset price process (St )t∈R+ modeled as
dSt = (µ − δ)St dt + σSt dBt ,
where (Bt )t∈R+ is a standard Brownian motion and δ > 0 is a continuoustime dividend rate. By absence of arbitrage, the payment of a dividend entails
a drop in the stock price by the same amount occuring generally on the exdividend date, on which the purchase of the security no longer entitles the
investor to the dividend amount. The list of investors entitled to dividend
payment is consolidated on the date of record, and payment is made on the
payable date.
library(quantmod)
getSymbols("0005.HK",from="2010-01-01",to=Sys.Date(),src="yahoo")
getDividends("0005.HK",from="2010-01-01",to=Sys.Date(),src="yahoo")
a) Assuming that dividends are reinvested into the portfolio with value Vt
at time t, write down the portfolio change dVt .
b) Assuming that the portfolio value Vt takes the form Vt = g(t, St ) at time
t, derive the Black-Scholes PDE for the function g(t, x) with its terminal
condition.
c) Compute the price at time t ∈ [0, T ] of a European call option with strike
price K by solving the corresponding Black-Scholes PDE.
Exercise 5.2
Power option. (Exercise 3.7 continued).
a) Solve the Black-Scholes PDE
rg(x, t) =
∂g
∂g
σ2 2 ∂ 2 g
(x, t) + rx (x, t) +
x
(x, t)
∂t
∂x
2
∂x2
(5.35)
with terminal condition g(x, T ) = x2 , x > 0.
Hint: Try a solution of the form g(x, t) = x2 f (t), and find f (t).
b) Find the respective quantities ξt and ηt of the risky asset St and riskless
asset At = e rt in the portfolio with value
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"
The Black-Scholes PDE
Vt = g(St , t) = ξt St + ηt At
hedging the contract with payoff ST2 at maturity.
Exercise 5.3
On December 18, 2007, a call warrant has been issued by
Fortis Bank on the stock price S of the MTR Corporation with maturity
T = 23/12/2008, strike price K = HK$ 36.08 and entitlement ratio=10.
Recall that in the Black-Scholes model, the price at time t of a European
claim on the underlying asset St , with strike price K, maturity T , interest
rate r and volatility σ is given by the Black-Scholes formula as
f (t, St ) = St Φ d+ (T − t) − K e −r(T −t) Φ d− (T − t) ,
where
d− (T − t) =
and
Recall that
(r − σ 2 /2)(T − t) + log(St /K)
√
σ T −t
√
d+ (T − t) = d− (T − t) + σ T − t.
∂f
(t, St ) = Φ d+ (T − t) ,
∂x
see Proposition 5.13.
a) Using the values of the Gaussian cumulative distribution function, compute the Black-Scholes price of the corresponding call option at time
t =November 07, 2008 with St = HK$ 17.200, assuming a volatility σ =
90% = 0.90 and an annual risk-free interest rate r = 4.377% = 0.04377,
b) Still using the values of the Gaussian cumulative distribution function,
compute the quantity of the risky asset required in your portfolio at time
t =November 07, 2008 in order to hedge one such option at maturity
T = 23/12/2008.
c) Figure 1 represents the Black-Scholes price of the call option as a function
of σ ∈ [0.5, 1.5] = [50%, 150%].
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N. Privault
0.6
Black-Scholes price
0.5
HK$
0.4
0.3
0.2
0.1
0
0.5
0.6
0.7
0.8
0.9
1
sigma
1.1
1.2
1.3
1.4
1.5
Fig. 5.14: Option price as a function of the volatility σ.
Knowing that the closing price of the warrant on November 07, 2008 was
HK$ 0.023, which value can you infer for the implied volatility σ at this
date ?∗
Exercise 5.4 Forward contracts. Recall that the price πt (C) of a claim
C = h(ST ) of maturity T can be written as πt (C) = g(t, St ), where the
function g(t, x) satisfies the Black-Scholes PDE

∂g
∂g
1
∂2g

 rg(t, x) =
(t, x) + rx (t, x) + x2 σ 2 2 (t, x),
∂t
∂x
2
∂x


g(T, x) = h(x),
(1)
with terminal condition g(T, x) = h(x), x > 0.
a) Assume that C is a forward contract with payoff
C = ST − K,
at time T . Find the function h(x) in (1).
b) Find the solution g(t, x) of the above PDE and compute the price πt (C)
at time t ∈ [0, T ].
Hint: search for a solution of the form g(t, x) = x − α(t) where α(t) is a
function of t to be determined.
c) Compute the quantity
∂g
(t, St )
ξt =
∂x
of risky assets in a self-financing portfolio hedging C.
Exercise 5.5
∗
Download the corresponding R code or the IPython notebook that can be run here.
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The Black-Scholes PDE
a) Solve the Black-Scholes PDE
rg(t, x) =
∂g
∂g
σ2 2 ∂ 2 g
(t, x)
(t, x) + rx (t, x) +
x
∂t
∂x
2
∂x2
(5.36)
with terminal condition g(T, x) = 1, x > 0.
Hint: Try a solution of the form g(t, x) = f (t) and find f (t).
b) Find the respective quantities ξt and ηt of the risky asset St and riskless
asset At = e rt in the portfolio with value
Vt = g(t, St ) = ξt St + ηt At
hedging the contract with payoff $1 at maturity.
Similar exercise: Repeat the above questions with the terminal condition
g(T, x) = x.
Exercise 5.6
Binary options. Consider a price process (St )t∈R+ given by
dSt
= rdt + σdBt ,
St
S0 = 1,
under the risk-neutral measure P∗ . A binary (or digital) call option is a
contract with maturity T , strike price K, and payoff

 $1 if ST > K,
Cd := 1[K,∞) (ST ) =

0 if ST < K.
a) Derive the Black-Schole PDE satisfied by the pricing function Cd (t, St ) of
the binary call option, together with its terminal condition.
b) Show that we have
(r − σ 2 /2)(T − t) + log(x/K)
√
Cd (t, x) = e −r(T −t) Φ
σ T −t
= e −r(T −t) Φ d− (T − t) ,
where
d− (T − t) :=
(r − σ 2 /2)(T − t) + log(St /K)
√
,
σ T −t
0 6 t < T.
Exercise 5.7
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N. Privault
a) Solve the stochastic differential equation
dSt = αSt dt + σdBt
(5.37)
in terms of α, σ > 0, and the initial condition S0 .
b) Write down the Black-Scholes PDE satisfied by the function C(t, x), where
C(t, St ) is the price at time t ∈ [0, T ] of the contingent claim with payoff
φ(ST ) = exp(ST ), and identify the process Delta (ξt )t∈[0,T ] that hedges
this claim.
c) Solve the Black-Scholes PDE of Question (b) with the terminal condition
φ(x) = e x , x > 0.
Hint: Search for a solution of the form
σ2 2
(h (t) − 1) ,
C(t, x) = exp −r(T − t) + xh(t) +
4r
(5.38)
where h(t) is a function to be determined, with h(T ) = 1.
d) Compute the strategy (ξt , ηt )t∈[0,T ] that hedges the contingent claim with
payoff exp(ST ).
Exercise 5.8
Consider the backward induction relation (3.13), i.e.
ve(t, x) = (1 − p∗N )e
v (t + 1, x(1 + aN )) + p∗N ve (t + 1, x(1 + bN )) ,
using the renormalizations rN := rT /N and
√
√
aN := (1 + rN ) e −σ T /N − 1, bN := (1 + rN ) e σ T /N − 1,
N > 1,
of Section 3.6, with
p∗N =
rN − aN
bN − aN
and p∗N =
bN − rN
.
bN − aN
a) Show that the Black-Scholes PDE of Proposition 5.11 can be recovered
from the induction relation (3.13) when the number N of time steps tends
to infinity.
∂gc
b) Show that the expression of the Delta ξt =
(t, St ) can be similarly
∂x
recovered from the finite difference relation (3.18), i.e.
(1)
ξt (St−1 ) =
v (t, (1 + bN )St−1 ) − v (t, (1 + aN )St−1 )
St−1 (bN − aN )
as N tends to infinity.
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The Black-Scholes PDE
Problem 5.9 Stop-loss start-gain strategy [Lip01] § 8.3.3. (Exercise 4.18
continued). Let (Bt )t∈R+ be a standard Brownian motion started at B0 ∈ R.
a) We consider a simplified foreign exchange model in which the AUD is a
risky asset and the AUD/SGD exchange rate at time t is modeled by Bt ,
i.e. AU$1 equals SG$Bt at time t. A foreign exchange (FX) European
call option gives to its holder the right (but not the obligation) to receive
AU$1 in exchange for K = SG$1 at maturity T . Give the option payoff
at maturity, quoted in SGD.
In the sequel, for simplicity we assume no time value of money (r = 0),
i.e. the (riskless) SGD account is priced At = A0 = 1, t ∈ [0, T ].
b) Consider the following hedging strategy for the European call option of
Question (a):
i) If B0 > 1, charge the premium B0 − 1 at time 0, borrow SG$1 and
purchase AU$1.
ii) If B0 < 1, issue the option for free.
iii) From time 0 to time T , purchase∗ AU$1 every time Bt crosses K = 1
from below, and sell† AU$1 each time Bt crosses K = 1 from above.
Show that this strategy effectively hedges the foreign exchange European
call option at maturity T .
Hint. Note that it suffices to consider four scenarios based on B0 < 1 vs
B0 < 1 and BT > 1 vs BT < 1.
c) Determine the quantities ηt of SGD cash and ξt of AUD to be held in the
portfolio and express the portfolio value
V t = η t + ξt Bt
at all times t ∈ [0, T ].
d) Compute the integral summation
wt
0
ηs dAs +
wt
0
ξs dBs
of portfolio profits and losses at any time t ∈ [0, T ].
Hint. Apply the result of Question (e).
e) Is the portfolio (ηt , ξt )t∈[0,T ] self-financing? How to interpret the answer
in practice?
∗
†
We need to borrow SG$1 if this is the first AUD purchase.
We use the SG$1 product of the sale to refund the loan.
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