Some notes on continuous time finance Basics: Instantaneous total - - PDF document

Some notes on continuous time finance Basics:

• Instantaneous total return:

dpt pt + Dt pt dt where pt is price and Dt is instantaneous rate of dividend.

• Model price as a diffusion:

dpt pt = µ (·) dt + σ (·) dz

• Risk free security can be modeled as a security with constant price and

dividend: p = 1 Dt = rf

t

• r a security with no dividend whose price grows at the deterministic rate:

dpt pt = rf

t dt

Pricing equation

• Utility flows

U ({ct}) = E

∞

• t=0

e−δtu(ct)dt

• Arbitrage:

ptu′ (ct) = E

∞

• s=0

e−δsu′(ct+s)Dt+sdt

• Since u′(c+∆c)

u′(c)

not well behaved, define Λt = e−δtu′ (ct) then ptΛt = Et

∞

• s=0

Λt+sDt+sds 1

• Write this as

ptΛt = Et

s+∆

• s=0

Λt+sDt+sds + Etpt+∆Λt+∆ + ΛtDt∆ + Etpt+∆Λt+∆

• For small ∆ approximate integral as ΛtDt∆ so that

0 = ΛtDt∆ + Et (pt+∆Λt+∆ − ptΛt) Now take ∆ → 0 0 = ΛtDt + Et [d (Λtpt)]

• This is continuous time equivalent to p = E (mx) for return x and stochas-

tic discount factor m. Risk premium

• Ito’s Lemma implies

d (Λp) = pdΛ + Λdp + dpdΛ which for diffusions we can write as 0 = D P dt + Et dΛ Λ + dp p + dΛ Λ dp p

• If p = 1 and D = rf

t we have the risk-free rate equation

rf

t dt = −Et

dΛt Λt

• We can then get the risk-premium equation as

Et dpt pt

• + Dt

Pt dt − rf

t dt = −Et

dΛt Λt dpt pt

• CRRA:
• Taylor series expansion:

dΛ = −δe−δtdt + e−δtu′′(c)dc + 1 2e−δtu′′′(c)dc2

• Local curvature:

γ = −cu′′(c) u′c) η = γ(γ + 1) = −c2u′′′(c) u′(c) so that dΛ Λ = −δdt + γ dc c + 1 2η dc2 c2 2

Risk premium with CRRA:

• Risk free rate

rf

t dt = δdt + γEt

dct ct − 1 2ηEt dc2

t

c2

t

• Risk premium

Et dpt pt

• + Dt

Pt dt − rf

t dt = γEt

dct ct dpt pt

• Let

dc = µccdt + σccdz then Sharpe-ratio satisfies µp + Dt

Pt dt − rf t dt

σp ≤ γσc where µp = Et

dpt pt , σp = Et

• dpt

pt

2 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 Λ = −rdt − (µ − r) σ dz − σwdw 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 =

• µ − 0.5σ2

dt + σdz d ln Λ = −

• r + 0.5µ − r

σ

• dt − µ − r

σ dz Integrating these expression implies ln ST = ln So +

• µ − 0.5σ2

T + σ √ Tε ln Λt = ln Λo −

• r + 0.5µ − r

σ

• T − µ − r

σ √ Tε where ε = zt − zo √ T ∼ N(0, 1) 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:

Co = Et T ΛT Λt max (ST − X, 0) = Et ∞

ST =X

ΛT Λt (ST − X) d f (Λt, ST ) = Et ∞

ST =X

ΛT Λt (ST (ε) − X) d f (ε) Deriving the Black-Scholes Formula:

• Guess C = C (S, t) ie. function of price and time to expiration.
• Ito’s Lemma:

dC = Ctdt + CsdS + 1 2CssdS2 =

• Ct + CsSµ + 1

2CssS2σ2

• + CsSσdz
• Asset pricing equation:

0 = Et (dΛC) = CEtdΛ + ΛEtdC + EtdΛdC Since Et dΛ Λ = −rdt we have 0 = −rC + Ct + CsSµ + 0.5CssS2σ2 − S (µ − r) Cs which gives the Black-Scholes Formula 0 = −rC + Ct + SrCS + 1 2CSSS2σ2 Solution:

• We are looking for a solution to

0 = −rC + Ct + SrCS + 1 2CSSS2σ2 with boundary condition C = max [ST − X, 0] 4

• Guess the solution satisfies:

Co = SoΦ

• ln So/X +
• r + σ2/2
• T

σ √ T

• −Xe−rT Φ
• ln So/X +
• r − σ2/2
• T

σ √ T

• 5