Nobody22 May 11, 2010

Variables

S = Stock price F = Forward price K = Strike price C = Call option P = Put option r = Continuous risk-free interest rate δ = Continuous dividend rate t = Time σ = Volatility (Normal distribution) ∆ = Shares of stock to replicate option B = Amount to borrow to replicate option p∗ = % Chance stock will increase (using r) p = % Chance stock will increase (using α) q = % Chance stock will decrease u = Ratio increase in the price d = Ratio decrease in the price α = Expected rate of return on a stock γ = Expected rate of return on an option C0 = Current value of a stock Ci = The value of the stock if it ( H=increases, L=decreases) Ui = The utility value of a dollar if the stock (H=increases, L=decreases) s = Sample volatility x = Sample average ratio of price movement ¯ µ2 = Second raw empirical moment n = Number of stock movements m = Mean of lognormal model v = Volatility of lognormal model = Change in stock price φ = Sharp Ratio ρ = Correlation Coeﬃcient X(t) = Arithmetric Brownian Motion Z(t) = Geometric Brownian Motion

Formulas

Building Binomial Trees

Put-Call Parity C − P = S0 e−δt − Ke−rt Replicating Portfolio C = S∆ + B = e−rt [p∗ Cu + (1 − p∗ )Cd ] ∆= Cu − Cd e−δt S(u − d) uCd − dCu e−rt B= u−d Rate of return relationship Utility Value QH = pUH QL = qUL 1 = QH + QL 1+r C0 = QH CH + QL CL pUH QH = p∗ = QH + QL pUH + qUL pCH + qCL −1 α= C0 Estimating volatility xi = ln n

Using forward rates u = e(r−δ)t+σ d = e(r−δ)t−σ

√ √ t t

Risk Neutral Pricing e(r−δ)t − d u−d 1 √ Using forward rates p∗ = 1 + eσ t Cox-Ross-Rubinstein p∗ = u = eσ √ t t

Ceγt = S∆eαt + Bert

d = e−σ

√

Lognormal (Jarrow-Rudd) u = e(r−δ−0.5σ d = e(r−δ−0.5σ 2 )t+σ 2 )t−σ

√ √

t t

For futures option 1−d p∗ = u−d u = eσ

√ t

St St−1 x2 i n

Sn S0

µ2 =

i=1

d = e−σ t Cu − Cd ∆= F (u − d) B = Option Price √ u All trees = e2σ t d

√

ln x= ¯

2

n n s = (µ2 − x2 ) ¯ n−1 Must annualize from interval period

Lognormal model m = µt = (α − δ − 0.5σ 2 )t √ v=σ t E[St |S0 ] = S0 e(µ+0.5σ Median = S0 eµt Mode = S0 e (µ−σ 2 )t

2 )t

Elasticity S∆ Ω= C σoption = σstock |Ω| Risk Premium γ − r = (α − r)Ω Sharpe Ratio γ−r α−r φ= = σoption σstock Overnight Proﬁt Proﬁt = C0 − C1 + ∆ − (er/365 − 1)(∆S0 − C0 ) ‘∆ − Γ − θ approximation’ 2

Conﬁdence Int. = S0 em±N (...)v

Note: The following are not set to PV

P r(St < K) = N (−d2 ) P r(St > K) = N (d2 ) P E[St |St < K] = S0 e(α−δ)t N (−d1 ) P E[St |St > K] = S0 e (α−δ)t

Ch = C0 + ∆ + Γ

N (d1 )

+ θh 2 Maket-maker Proﬁt

2

E[Cpayof f ] = S0 e(α−δ)t N (d1 ) − KN (d2 ) 1 Sn = ‘Data’ µ = ln ˆ n S0 St into ‘Data’ σ = Put each ln ˆ St−1 n n n St 2 St 2 σ = ˆ ln − ln n−1 St−1 St−1 1 1

Proﬁt = − rh(∆S − C) + Γ

2

+ θh 2 Black-Scholes Equation 1 rC = S 2 σ 2 Γ + rS∆ + θyear 2 Boyle-Emanuel (Re-hedging h-times) Where h is a year:

Black-Scholes d1 = ln ln

S0 K

+ (r − δ + 0.5σ 2 )t √ σ t

+ 0.5σ 2 t √ = σ t √ d2 = d1 − σ t + (r − δ − 0.5σ 2 )t √ = σ t C = S0 e−δt N (d1 ) − Ke−rt N (d2 ) ln P = Ke−rt N (−d2 ) − S0 e−δt N (−d1 ) ∆call = e−δt N (d1 ) ∆put = −e−δt N (−d1 ) = ∆call − e−δt Note: Future prices disregard δ and r S0 K

S0 e−δt Ke−rt

1 V ar(Rh,i ) = (S 2 σ 2 Γh)2 2 1 Annual Var = (S 2 σ 2 Γ)2 h 2 Exchange option volatility 2 2 2 σoption = σS + σK − 2ρσS σK

Calculating ∆ of exotic options ∂N (di ) e−di /2 √ = ∂S Sσ 2πT 2

Chooser-Option Formula V = C(S, K, T ) + e−δ(T −t) P S, Ke−(r−δ)(T −t) , t American Call option with one discrete dividend: = S0 − Ke−rt + CallonPut S, K, D − K 1 − e−r(T −t) If D − K(1 − e−r(T −t) ) < implicit put then it’s not rational to excercise early Forward-Start Options V = Se−δt NT −t (d1 ) − cSe−r(T −t)−δt NT −t (d2 ) Geometric Brownian Motion dSt = (α − δ)dt + σdZ(t) St K ln S0 − (α − δ − 0.5σ 2 )t √ P (St > K) = 1 − N σ t Note this is NOT B.S.

Arithmetic Brownian Motion X(t) = αt + σZ(t) P (St > K) = 1 − N αarithmetric (K − S0 ) − αt √ σ t...