Exam Mfe Formulars
aMFE Reference
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
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