Some notes on continuous time finance
Basics:
• Instantaneous total return:
dpt
Dt
+
dt
pt
pt
where pt is price and Dt is instantaneous rate of dividend.
• Model price as a diffusion:
dpt
= µ (·) dt + σ (·) dz
pt
• Risk free security can be modeled as a security with constant price and
dividend:
p
=
1
= rtf
Dt
or a security with no dividend whose price grows at the deterministic rate:
dpt
= rtf dt
pt
Pricing equation
• Utility flows
Z∞
U ({ct }) = E
e−δt u(ct )dt
t=0
• Arbitrage:
0
Z∞
pt u (ct ) = E
e−δs u0 (ct+s )Dt+s dt
s=0
• Since
0
u (c+∆c)
u0 (c)
not well behaved, define
Λt = e−δt u0 (ct )
then
Z∞
pt Λt = Et
s=0
1
Λt+s Dt+s ds
• Write this as
s+∆
Z
pt Λt
Λt+s Dt+s ds + Et pt+∆ Λt+∆
= Et
s=0
+
Λt Dt ∆ + Et pt+∆ Λt+∆
• For small ∆ approximate integral as Λt Dt ∆ so that
0 = Λt Dt ∆ + Et (pt+∆ Λt+∆ − pt Λt )
Now take ∆ → 0
0 = Λt Dt + Et [d (Λt pt )]
• This is continuous time equivalent to p = E (mx) for return x and stochastic discount factor m.
Risk premium
• Ito’s Lemma implies
d (Λp) = pdΛ + Λdp + dpdΛ
which for diffusions we can write as
dΛ dp dΛ dp
D
+
+
0 = dt + Et
P
Λ
p
Λ p
• If p = 1 and D = rtf we have the risk-free rate equation
rtf dt = −Et
dΛt
Λt
• We can then get the risk-premium equation as
dpt
Dt
dΛt dpt
f
Et
+
dt − rt dt = −Et
pt
Pt
Λt pt
CRRA:
• Taylor series expansion:
1
dΛ = −δe−δt dt + e−δt u00 (c)dc + e−δt u000 (c)dc2
2
• Local curvature:
so that
cu00 (c)
u0 c)
γ
= −
η
= γ(γ + 1) = −
c2 u000 (c)
u0 (c)
dΛ
dc 1 dc2
= −δdt + γ + η 2
Λ
c
2 c
2
Risk premium with CRRA:
• Risk free rate
rtf dt
1
dct
− ηEt
= δdt + γEt
ct
2
dc2t
c2t
• Risk premium
Et
dpt
pt
+
Dt
dct dpt
dt − rtf dt = γEt
Pt
ct pt
• Let
dc = µc cdt + σc cdz
then Sharpe-ratio satisfies
µp +
t
where µp = Et dp
pt , σp = Et
Dt
Pt dt
− rtf dt
σp
2
≤ γσc
dpt
pt
Stochastic discount factor for stock with geometric brownian motion:
• Assume stock value follows
dS = µSdt + σSdz
• Market completeness: all stochastic discount factors that price the stock
S satisfy
dΛ
(µ − r)
= −rdt −
dz − σw dw
Λ
σ
where
E (dw dz) = 0
Since dw uncorrelated we can set it to zero to price the asset.
• Ito’s lemma implies
d ln S
d ln Λ
µ − 0.5σ 2 dt + σdz
µ−r
µ−r
= − r + 0.5
dt −
dz
σ
σ
=
Integrating these expression implies
ln ST
=
ln Λt
=
√
ln So + µ − 0.5σ 2 T + σ T ε
µ − r√
µ−r
T−
ln Λo − r + 0.5
Tε
σ
σ
where
ε=
zt − zo
√
∼ N (0, 1)
T
3
European call option:
• Let X denote strike price and ST denote stock value on expiration day.
Payoff is
C = max (ST − X, 0)
• Option value:
Z
Co
T
ΛT
max (ST − X, 0)
Λt
Z0 ∞
ΛT
(ST − X) df (Λt , ST )
Et
ST =X Λt
Z ∞
ΛT
Et
(ST (ε) − X) df (ε)
ST =X Λt
=
Et
=
=
Deriving the Black-Scholes Formula:
• Guess C = C (S, t) ie. function of price and time to expiration.
• Ito’s Lemma:
dC
=
=
1
Ct dt + Cs dS + Css dS 2
2
1
Ct + Cs Sµ + Css S 2 σ 2 + Cs Sσdz
2
• Asset pricing equation:
0 = Et (dΛC) = CEt dΛ + ΛEt dC + Et dΛdC
Since
Et
dΛ
= −rdt
Λ
we have
0 = −rC + Ct + Cs Sµ + 0.5Css S 2 σ 2 − S (µ − r) Cs
which gives the Black-Scholes Formula
1
0 = −rC + Ct + SrCS + CSS S 2 σ 2
2
Solution:
• We are looking for a solution to
1
0 = −rC + Ct + SrCS + CSS S 2 σ 2
2
with boundary condition
C = max [ST − X, 0]
4
• Guess the solution satisfies:
Co = So Φ
!
ln So /X + r + σ 2 /2 T
√
−Xe−rT Φ
σ T
5
!
ln So /X + r − σ 2 /2 T
√
σ T