Financial Derivatives Formulas SheetFormelblad Finansiella Derivat

Farid Bonawiede

The Black-Scholes formula is assumed to be known!

Lt =dQ

dPis a P -martingale (1)

{dt = φtLtdwP

t

L0 = 1(2)

E[LT ] = L0 = 1 (3)

Lt = eR t0 φsdws− 1

2

R t0 φ2

sds (4)

∫ t

0a(s, T )dWs ∼ N

(0,

∫ t

0a2(s, T )ds

)where W is a wiener-process

(5)

dwpt = dwQ

t + φtdt (6)

Affine Term Structure:

P (t, T ) = F (t, rt; T ) = eA(t,T )−B(t,T )rt (7)

Term Structure Equation:

F Tt + (µ− λσ)F T

r +12σ2F T

rr − F Tr = 0 (8)

1

λ = market price of risk =µ− α

σ(9)

α is the drift of the process. More specifically, it is the same as the shortrate under the measure Q.

λ = −φ (10)

where φ is the girsanov kernel from P → Q

E[etX ] = eµt+σ2t2

2 iff X ∼ N(µ, σ2) (11)

Normalized Gain-process

G′(t) =S(t)B(t)

+∫ t

0

1B(s)

dDs (12)

EQ[G′(T )|Ft] = G′(t) (13)

Zero Coupon Bond

P (t, T ) = e−R T

t f(t,s)ds (14)

p(t, T ) is numeraire under the measure QT

∆V = ∆P when you want to hedge an option (15)

Put-Call parity

p(t, s) = Ke−r(T−t) + c(t, s)− s (16)

Meta-Theorem

M denotes the number of risky assetsR denotes the number of random sources

The model isarbitrage free iff M ≤ R

complete iff M ≥ R

complete and arbitrage free iff M = R

2

Black-Scholes equation

Ft(t, s) + rsFs(t, s) +12s2σ2(t, s)Fss(t, s)− rF (t, s) = 0 (17)

F (T, s) = Φ(s) (18)

Differential of an integral

Y (t, T ) =∫ T

tf(t, s)ds (19)

dY (t, T ) =∂

∂t

(∫ T

tf(t, s)ds

)dt +

∫ T

tdf(t, s)ds (20)

dY (t, T ) = f(t, t)ds +∫ T

tdf(t, s)ds (21)

3