Put-Call Parity with Known Dividend C – P = S – (Div)e–Rt – Xe–Rt

Put-Call Parity with Continuous Dividends P = C + Xe–Rt – S0e–yt

Black-Scholes-Merton Model C0 = S0e–ytN(d1) – Xe–RtN(d2) P0 = Xe–RtN(–d2) – S0e–ytN(–d1)

d1 = –

√

d2 = d1 – σ√ Delta of a call = e–ytN(d1) Delta of a put = –e–ytN(–d1) Eta of a call = e–ytN(d1)(S/C) Eta of a put = –e–ytN(–d1)(S/P) < 0 Vega = S0e–ytN′(d1) √

N′(x) = √

Gamma = N

′

S σ√

Call theta = – S0N'(d1)σe–yt

2√t + yS0N(d1)e–yt – RXe–RtN(d2)

Put theta = – S0N'(d1)σe–yt

2√t – yS0N(–d1)e–yt + RXe–RtN(–d2)

Call rho = Xte–RtN(d2) Put rho = –Xte–RtN(–d2) Hedging with index options Number of option contracts = Portfolio beta × Portfolio value

Option delta × option contract value

Binomial trees p* =

∆

u = √∆ d = √∆ = 1/ u f = e–RΔt[pfu + (1 – p)fd]

Known Dividend S* = S0 – (Dividend)e–Rt

Continuous dividend yield and binomial trees p =

∆

u = √∆ d = √∆ = 1/ u Options on futures u = √∆ d = √∆ = 1/ u p = Money Markets

Price = Face value 1- Days RBD

RBD = Par-Price

Par× 360

n

RBEY = Par-Price

Price× 365

n

RBEY = 365×RBD

360- RBD×n

Equivalent taxable yield = Tax-exempt yield

1-Marginal tax rate

Critical tax rate = 1- RM

R

Accrued interest 30/360 If D1 = 31, change to 30 If D2 = 31 and D1 = 30 or 31, change D2 to 30, otherwise leave D2 at 31 # of days (Y2 – Y1)×360 + (M2 – M1)×30 + (D2 – D1) 30E/360 – Assumes a 30-day month If D1 = 31, change to 30 If D2 = 31 Change to 30 # of days (Y2 – Y1)×360 + (M2 – M1)×30 + (D2 – D1)

w = # of days between settlement and next coupon payments

# of days in coupon period

Accrued interest = C # of days since last coupon# of days in period

Duration and Convexity

PP∂

= –D ⎟⎠⎞

⎜⎝⎛+∂

R 1R

PP∂

= – D ⎟⎟⎠

⎞⎜⎜⎝

⎛+∂(R/2) 1R

D = ∑∑ ×

(price) DCF t DCF

D = y

y 1+-

y 1] - y) c[(1y) - T(c y) (1

T ++++

Duration of a perpetuity is: y

y 1+

Duration for a level annuity is: y

y 1+ -

1 - y) (1T

T+

∂P = P ×[(– D) × ⎥⎦⎤

⎢⎣⎡+∂

R 1R

]

PP∂

= – D ⎥⎦⎤

⎢⎣⎡+Δ

R 1R

+ 21

CX(ΔR)2

CX = convexity = Scaling factor [capital loss from capital gain from] one basis point + one basis point rise in R drop in R    DM = D

1+ y

%Δ in bond price = –DM(ΔR)

DE = V–- V+

2V0(∆R)

V0 = initial price V– = price if YTM decreases by R V+ = price if YTM increases by R

CXE = V–+V+ – 2V0

2V0(∆R)2

Futures FT = S(1+ R – d)T Stock hedging with futures # of contracts = Bond hedging with futures # of contracts =

Cross Hedging h = ρS,F   Value at Risk Portfolio variance for 2 asset portfolio (total risk) = )B,A(Covwwww BABBAA 22222 ++ σσ Portfolio variance for 2 asset portfolio (total risk) = B,ABABABBAA wwww ρσσσσ 22222 ++ E(RP,T) = E(RP) × T σP,T = σP × √ Prob[RP,T ≤ E(Rp) × T – 2.326σP√ ] = 1%  Prob[RP,T ≤ E(Rp) × T – 1.96σP√ ] = 2.5% Prob[RP,T ≤ E(Rp) × T – 1.645σP√ ] = 5%