LINKÖPING UNIVERSITY
Department of Mathematics
Mathematical Statistics
John Karlsson
TAMS29
Stochastic Processes with
Applications in Finance
12. More on Black-Scholes pricing
Notation:
ln
d1 (t) ≡ d1 (x, T − t, K) :=
x
K
2
+ ρ + σ2 (T − t)
√
σ T −t
√
d2 (t) ≡ d2 (x, T − t, K) := d1 (x, T − t, K) − σ T − t =
ln
x
K
2
+ ρ − σ2 (T − t)
√
.
σ T −t
If X ∼ N (0, 1) we have
1
Φ(t) := P (X ≤ t) = √
2π
Price for European call is given by
Z
t
e−
x2
2
dx.
−∞
C(S, t) = St · Φ(d1 (t)) − Ke−ρ(T −t) · Φ(d2 (t)).
Definition 12.1. A portfolio is said to be self-financing if there is no exogenous
infusion or withdrawal of money so that the purchase of a new asset must be
financed by the sale of an old one. In mathematical terms let hi (t) denote the
number of stocks of type i in the portfolio and let the stock have value Si (t). Then
the value of the portfolio (h1 (t), h2 (t), . . . , hk (t)) is given by
V (t) =
k
X
hi (t)Si (t).
i=1
The portfolio is self-financing if
dV (t) =
k
X
hi (t)dSi (t).
i=1
Model: In a portfolio let ξt denote the quantity of a risky asset with price process
St , where
dSt = St (µdt + σdWt ),
i.e.
1
St = S0 exp σWt + µ − σ 2 t .
2
Let ηt denote the quantity of a risk free asset with price process At = A0 eρt . The
value of the portfolio at any given t is given by
Vt = ηt At + ξt St .
Proposition 12.2. Let a portfolio strategy (ηt , ξt ) be self financing and let Vt take
the form
Vt = ηt At + ξt St = g(t, St ).
Then the function g(t, x) satisfies the Black–Scholes PDE
∂
1
∂2
∂
g(t, x) + σ 2 x2 ·
g(t, x) + ρx ·
g(t, x) − ρg(t, x) = 0,
∂t
2
∂x2
∂x
and
ξt =
∂
g(t, St ).
∂x
1/1