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

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 = – Put theta = –

S0 N' (d1 )σe–yt

2√t S0 N' (d1 )σe–yt 2√t

+ yS0N(d1)e–yt – RXe–RtN(d2) – 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 = Binomial trees p* =

∆

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

Portfolio beta × Portfolio value Option delta × option contract value

Known Dividend S* = S0 – (Dividend)e–Rt Continuous dividend yield and binomial trees ∆

p= √∆

u= d=

√∆

= 1/ u

Options on futures √∆

u= d= p=

√∆

= 1/ u

Money Markets Price = Face value 1RBD =

Par-Price

RBEY = RBEY =

×

Days

RBD

360

Par n Par-Price 365

×

Price 365×RBD

n

360- RBD ×n

Equivalent taxable yield = Critical tax rate = 1-

RM

Tax-exempt yield 1-Marginal tax rate

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 ∂P ⎛ ∂R ⎞ = –D ⎜ ⎟ P ⎝1+ R ⎠ ∂P =–D P D=

⎛ ∂R ⎞ ⎜⎜ ⎟⎟ ⎝ 1 + (R/2) ⎠

∑ DCF × t ∑ DCF (price)

(1 + y) + T(c - y) 1+ y y c[(1 + y) T - 1] + y 1+ y Duration of a perpetuity is: y T 1+ y Duration for a level annuity is: y (1 + y) T - 1 ⎡ ∂R ⎤ ] ∂P = P ×[(– D) × ⎢ ⎣1 + R ⎥⎦ ∂P ⎡ ΔR ⎤ 1 = –D ⎢ + CX(ΔR)2 ⎥ P ⎣1 + R ⎦ 2 D=

CX = convexity = Scaling factor [capital loss from one basis point rise 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

+

capital gain from] one basis point drop in R   

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) = w A2 σ A2 + wB2 σ B2 + 2w A wB Cov( A, B ) Portfolio variance for 2 asset portfolio (total risk) = w A2 σ A2 + w B2 σ B2 + 2 w A w B σ Aσ B ρ A ,B 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%