Formulas for the MFE Exam

Formulas for MFE 1 Chapter 9 Parity and Other Relationships

1.1 Options on Stock C(K , T ) = P (K , T ) + [So − PV0,T (Div)] − e−r TK C(K , T ) = P (K , T ) + Soe−δ T − PV0,T (K)

1.2 Options on Currencies From Chapter 5, dollar forward price for a euro is F0,T = x0e(r −re u r o )T , where x0 is the current exchange rate denominated as $/euro. C(K , T ) − P (K , T ) = x0e−re u r o T − Ke−rT

1.3 Options on Bonds C(K , T ) = P (K , T ) + [B0 − PV0,T (Coupons)] − PV0,T (K)

1.4 Generalized Parity and Exchange Options C(ST , QT , 0) = max (0, ST − QT ) P (ST , QT , 0) = max (0, QT − St) P P C(St , Qt , T − t) − P (St , Qt , T − t) = Ft,T (S) − Ft,T (Q)

1.5 Currency Options C$ (x0, K , T ) = x0KP f



1 1 , ,T x0 K



1.6 Maximum and Minimum Options Prices S > CAmer (S , K , T ) > CEur (S , K , T ) > max [0, PV0,T (F0,T ) − PV0,T (K)] K > PAm er (S , K , T ) > PEur (S , K , T ) > max [0, PV0,T (K) − PV0,T (F0,T )]

1.7 Early Exercise Exercise on a call is not optimal when: K − PVt,T (K) > PVt,T (Div) 1

2

Section 2

1.8 Different Strike Prices K1 < K2 < K3 1. C(K1) > C(K2) 2. P (K2) > P (K1) 3. C(K1) − C(K2) 6 K2 − K1 4. P (K2) − P (K1) 6 K2 − K1 5.

C(K1) − C(K2) K2 − K1

>

C(K2) − C(K3) K3 − K2

6.

P (K2) − P (K1) K2 − K1

6

P (K3) − P (K2) K3 − K2

2 Binomial Option Pricing: I 2.1 The Binomial Solution ∆ = e−δh

Cu − Cd

S(u − d)

B = e−rh

uCd − d Cu

u−d

u − e(r −δ)h e(r −δ)h − d ∆S + B = e−rh Cu + Cd u−d u−d (r −δ)h No arbitrage ⇒ u > e >d

2.2 Risk-Neutral Pricing p∗ =

e(r −δ)h − d u−d

C = e−r h[p∗Cu + (1 − p∗)Cd] √

u = e(r −δ)h+σ√h d = e(r −δ)h−σ h

2.3 Options on Currencies √

ux = xe(r −rf )h+σ√h dx = xe(r −rf )h−σ h p∗ =

e

(r −r f )h

u−d

−d

2.4 Options on Futures Contracts √

u = eσ√ h d = eσ h p∗ =

1−d u−d



3

The Black-Scholes Formula

3 Binomial Option Pricing: II √ σh = σ h

σ

Example: σmonthly = √

12

4 The Black-Scholes Formula 4.1 Calls and Puts C(S , K , σ, r, T , δ) = Se−δTN (d1) − Ke−rTN (d2) 1

ln(S/K) + (r − δ + 2 σ 2)T √ σ T √ d2 = d1 − σ T d1 =

P (S , K , σ, r, T , δ) = Ke−r TN ( − d2) − Se−δTN ( − d1)

4.2 Options on Stocks with Discrete Dividends P F0,T (S) = S0 − PV0,T (Div)

4.3 Options on Currencies P F0,T (S) = x0e−rfT

C(x, K , σ, r, T , r f ) = xe−rfTN (d1) − Ke−r TN (d2) 1

ln(x/K) + (r − r f + 2 σ 2)T √ d1 = σ T P (x, K , σ, r, T , r f ) = C(x, K , σ, r, T , r f ) + Ke−r T − xe−rfT

4.4 Options on Futures δ =r This is know as the Black Formula.

4.5 Greek Measures for Portfolios ∆p ortfolio =

n X

ωi∆i

i=1

Holds true for other greeks as well.

4

Section 6

4.6 Option Elasticity ǫ is the change in the stocks price. Ω is option elasticity. Ω≡

% change in option price % change in sto ck price

=

ǫ∆ C ǫ S

=

S∆ C

4.7 Volatility of an option σoption = σsto ck × |Ω|

4.8 Risk Premium of an Option γ is the expected return of the option. γ − r = (α − r) × Ω

4.9 Calendar Spreads Calendar spreads: buy and sell options with different expirations.

5 The Standard Normal Distribution 1 − 1 x2 φ(x) ≡ √ e 2 2π

N (x) ≡

Z

x

φ(x)dx −∞

6 Option Greeks Important identities: N (x) = 1 − N ( − x)

Se−δTN ′(d1) = Ke−rT N ′(d2)

6.1 Delta (∆)

∆call =

∂C(S , K , σ, r, T − t, δ) = e−δ(T −t)N (d1) ∂S

∆put =

∂P (S , K , σ, r, T − t, δ) = e−δ(T −t)N ( − d1) ∂S

6.2 Gamma (Γ)

5

Market-Making and Delta-Hedging

Γcall = Γput =

∂ 2C(S , K , σ, r, T − t, δ) e−δ(T −t)N ′(d1) √ = ∂ 2S Sσ T − t

6.3 Theta (θ) Ke−r(T −t)N ′(d2)σ ∂C(S , K , σ, r, T − t, δ) √ = δSe−δ(T −t)N (d1) − rKe−r(T −t)N (d2) − ∂t 2 T −t ∂P (S, K , σ, r, T − t, δ) θput = = θcall + rKe−r(T −t) − δSe−δ(T −t) ∂t

θcall =

6.4 Vega

Vegacall = Vegaput =

√ ∂C(S , K , σ, r, T − t, δ) = Se−δ(T −t)N ′(d1) T − t ∂σ

6.5 Rho (ρ) ∂C(S , K , σ, r, T − t, δ) = (T − t)Ke−r(T −t)N (d2) ∂r ∂P (S , K , σ, r, T − t, δ) ρput = = (T − t)Ke−r(T −t)N ( − d2) ∂r

ρcall =

6.6 Psi (ψ) ∂C(S , K , σ, r, T − t, δ) = − (T − t)Se−δ(T −t)N (d1) ∂δ ∂P (S , K , σ, r, T − t, δ) ψput = = (T − t)Se−δ(T −t)N ( − d1) ∂δ ψcall =

7 Market-Making and Delta-Hedging 7.1 Understanding Market-Makers Profit 1

∆t(St+h − St) − [∆t(St+h − St) + 2 (St+h − St)2Γt + θh] − rh[∆tSt − C(St)] =   1 − 2 ǫ2Γt + θth + rh[∆tSt − C(St)]

One stardard deviation move: ǫ2 = σ 2St2h. In that case we have   1 Market-maker profit= − 2 σ 2St2Γt + θ + r[∆tSt − C(St)] h. Set profit equal to zero and rearrange terms to get:

6

Section 8

1 2 2 σ St Γt + rSt∆t + θ = rC(St) 2

7.2 Re-hedgeing Re-hedge every h, market has moved xi standard deviations. 1

Rh,i = 2 S 2σ 2Γ(x2i − 1)h 1

Var(Rh,i) = 2 (S 2σ 2Γh)2 Example Variance for a day for a daily re-hedger: 1

Var(R1/365,1) = 2 (S 2σ 2Γ/365)2

7.3 Greeks in the Binomial Model ∆(S , 0) = e−δh Γ(Sh , h) =

Cu − Cd uS − dS

∆(uS , h) − ∆(dS , h) uS − dS

ǫ = udS − S 1

θ(S , 0) =

C(udS , 2h) − ǫ∆(S , 0) − 2 ǫ2Γ(S , 0) − C(S , 0) 2h

8 Exotic Options: I 8.1 Asian Options max [0, ± (G(T ) − K)] where G(T ) is some sort of average and the sign depends on it being a call or put.

8.2 Barrier Options 1. Knock-out. Go out of existence when a barrier is crossed. 2. Knock-in. Go into existence when a barrier is crossed. 3. Rebate. Make a fixed payment if barrier is crossed. “Knock-in” option + “Knock-out” option = Oridinary Option

8.3 Compound Options max [C(St , K , T − t1) − x, 0]. Compound Option Parity:

7

The Lognormal Distribution

CallOnCall(S , K , x, σ, r, t1, t2, δ) − PutOnCall(S , K , x, σ, r, t1, t2, δ) + xe−r t1 = BSCall(S , K , σ, r, t2, δ)

8.4 Gap Options K1 strike. K2 trigger. C(S , K1, K2, σ, r, T , δ) = Se−δTN (d1) − K1e−r TN (d2) 1

ln(Se−δT /K2e−rT ) + 2 σ 2T √ d1 = σ T √ d2 = d1 − σ T

8.5 Exchange Options max (0, ST − Kt) C(S , K , σ, r, T , δ) = Se−δSTN (d1) − K1e−δK TN (d2) 1

ln(Se−δST /Ke−δK T ) + 2 σ 2T √ σ T √ d2 = d1 − σ T d1 =

σ=

p 2 σS2 + σK − 2ρσSσK

9 The Lognormal Distribution

9.1 The Normal Distribution 1

− 1 φ(x; µ, σ) ≡ √ e 2 σ 2π



x− µ σ

2

9.2 Sum of Normal Random Variables xi ∼ N (µi , σi2) and Cov(xi , x j ) = σi, j . σij = ρijσiσ j

E

n X

ωi x i =

i=1

Var

!

n X i=1

!

n X

ωiµi

i=1

ωixi =

n X n X

i=1 j =1

ωiω jσij

8

Section 10

9.3 The Lognormal Distribution

Continously compounded return definition: R(0, t) = ln(St/S0) or St = S0eR(0,t) 1

If x ∼ N (m, v 2) then E(ex) = e

m+ 2 v 2

2

2

and Var(ex) = e2m+v (ev − 1)

ln(St/S0) ∼ N [(α − δ − 0.5σ 2)t, σ 2t]

√ ln(St/S0) = (α − δ − 0.5σ 2)t + σ t z

St = S0e(α−δ −0.5σ

2

√ )t+σ t z

E(St) = S0e(α−δ)t

9.4 Lognormal Probablity Calculations Prob(St < K) = N ( − dˆ2) with r → α. N (St |St < K) = Se(α−δ)t

Prob(St > K) = N (dˆ2) where dˆ2 is the standard Black-Schoes argument

N ( − dˆ1) N ( − dˆ2)

N (St |St > K) = Se(α−δ)t

N (dˆ1) N (dˆ2)

10 Monte Carlo Valuation 10.1 Using Sums of Uniformly Distributed Random Variables

Z˜ =

12 X i=1

ui − 6

10.2 Monte Carlo Valuation ST = S0e

√ P 1 (α−δ − 2 σ 2)T +σ h [ n i=1Z(i)]

1

V (S0, 0) = n e−r T

n X

V (STi , T )

i=1

10.3 Control Variate Method A∗ = A¯ + β(G − G¯ ) where β = Cov(A¯ , G¯ )/Var(G¯ )

√ 1 Pn 1 (α−δ − 2 σ 2)T +σ T [ √ i=1Z(i)]

=e

n

9

ˆ ’s Lemma Brownian Motion and Ito

11 Brownian Motion and Ito ˆ’s Lemma 11.1 Black-Scholes Assumptions about Stock Prices dS(t) S(t)

= αdt + σdZ(t)

ln[S(T )] ∼ N (ln[S(0)] + [α − 0.5σ 2]T , σ 2T )

11.2 Brownian Motion

•

Z(0) = 0

•

Z(t + s) − Z(t) ∼ N (0, s)

•

Z(t + s1) − Z(t) is independant of Z(t) − Z(t − s2)

•

Z(t) is continuous

s1, s2 > 0

Above implies that Z(t) is a martingale (i.e. E[Z(t + s)|Z(t)] = Z(t)

small h, Y (t) = { − 1, 1}, E[Y (t)] = 0, Var[Y (t)] = 1, √ Z(t + h) − Z(t) = Y (t + h) h h = T /n " # n √ 1 X Z(T ) − Z(0) = T √ Y (ih) n i=1 √ dZ(t) = Y (t) dt " # Z T n √ 1 X Z(T ) = Z(0) + lim T √ Y (ih) → Z(0) + dZ(t) n→∞ n i=1 0

11.3 Properties of Brownian Motion lim

n→∞

n X i=1

(Z[ih] − Z[(i − 1)/h])2 = lim

n→∞

n  X √ i=1

Thus it has finite quadratic variation so: lim

n→∞

n X i=1

(Z[ih] − Z[(i − 1)/h])n = 0 for n > 2.

h Yih

2

= lim

n→∞

n X i=1

2 hYih =T

10

Section 11

But infinite total variation: lim

n→∞

n X i=1

|Z[ih] − Z[(i − 1)/h]| = ∞

11.4 Arithmetic Brownian Motion

X(T ) − X(0) = αT + σZ(T ) dX(t) = αdt + σdZ(t) X(T ) − X(0) ∼ N (αT , σ 2T )

11.5 The Ornstein-Uhenbeck Process

dX(t) = λ[α − X(t)]dt + σdZ(t)

11.6 Geometric Brownian Motion

dX(t) = α[X(t)]dt + σ[X(t)]dZ(t) dX(t) = αdt + σdZ(t) X(t) ln[X(t)] ∼ N (ln[X(0)] + (α − 0.5σ 2)t, σ 2t) X(t) = X(0)e(α−0.5σ

2

√ )t+σ t Z

E[X(t)] = X(0)eαt

11.7 Multiplication Rules dt × dZ = 0 (dt)2 = 0 (dZ)2 = dt dZ × dZ ′ = ρdt

11.8 The Sharpe Ratio Sharpe ratioi =

αi − r σi

11

The Black-Scholes Equation

11.9 The Risk Neutral Process dS(t) = (α − δ)dt + σdZ(t) S(t) Z˜ (t) generates a martingale in utility terms for a risk-averse investor. dS(t) = (r − δ)dt + σd Z˜ (t) S(t) dZ˜ (t) = dZ(t) + ηdt where η = (α − r)/σ

11.10 Ito ˆ’s Lemma n o dS(t) = αˆ[S(t), t] − δˆ[S(t), t] dt + σˆ[S(t), t]dZ(t) 1

dC(S , t) = CSdS + 2 CS S (dS)2 + Ctdt n o 1 dC(S , t) = [αˆ(S , t) − δˆ(S , t)]CS + 2 σˆ(S , t)2CS S + Ct dt + σˆ(S , t)CSdZ

11.11 Valuing a Claim on S a

P F0,T [S(T )a] = e−rTS(0)ae

1

[a(r −δ)+ 2 a(a−1)σ 2]T 1

F0,T [S(T )a] = S(0)ae

[a(r −δ)+ 2 a(a−1)σ 2]T

12 The Black-Scholes Equation dS = (α − δ)dt + σdZ S Option V [S(t), t]. Invest W in bonds that pay return r. dW = rWdt Total investment in option, stocks (N shares) and bonds should be zero. I = V (S , t) + NS + W = 0 1

dI = dV + N (dS + δSdt) + dW = Vtdt + VSdS + 2 σ 2S 2VSSdt + N (dS + δSdt) + rWdt Delta-hedge so N = − VS . Bonds: W = VSS − V . So, 1

dI = Vt + 2 σ 2S 2VS Sdt − VSδSdt + r(VSS − V )dt

12

Section 13

Zero-investment, zero-risk portfolio so dI = 0. 1

Vt + 2 σ 2S 2VS S + (r − δ)SVS − rV = 0

12.1 Risk Neutral Pricing dS = (r − δ)dt + σdZ˜ S 1 1 E(dV ) = Vt + σ 2S 2VSS + (α − δ)SVS 2 dt E ∗(dS) = (r − δ)dt 1 E ∗(dV dt

1

) = Vt + 2 σ 2S 2VS S + (r − δ)SVS so

1 E ∗(dV dt

) = rV

13 Exotic Options: II

13.1 All-Or-Nothing Options

CashCall(S , K , σ, r, T − t, δ) = e−r(T −t)N (d2) CashPut(S , K , σ, r, T − t, δ) = e−r(T −t)N ( − d2) AssetCall(S , K , σ, r, T − t, δ) = e−δ(T −t)SN (d1) AssetPut(S , K , σ, r, T − t, δ) = e−δ(T −t)SN ( − d1)

13.2 Ordinary Options and Gap Options

BSCall(S , K , σ, r, T − t, δ) = AssetCall(S , K , σ, r, T − t, δ) − K × CashCall(S , K , σ, r, T − t, δ) BSPut(S , K , σ, r, T − t, δ) = K × CashPut(S , K , σ, r, T − t, δ) − AssetPut(S , K , σ, r, T − t, δ) Gap option that pays S − K1 if S > K2. AssetCall(S , K2, σ, r, T − t, δ) − K1 × CashCall(S , K2, σ, r, T − t, δ)

Interest Rate Models

14 Volatility

dSt/St = (α − δ)dt + σ(St , Xt , t)dZ ǫt+h = ln(St+h/St) n

2 σˆH =

X 1 ǫ2i (n − 1) i=1

14.1 ARCH ln(St/St−h) = (α − δ − 0.5σ 2)h + ǫt Var(ǫt) = σ 2h

15 Interest Rate Models

15.1 Behavior of Bonds and Interest Rates dP = α(r, t)dt + q(r, t)dZ P dr = a(r)dt + σ(r)dZ

15.2 Impossible Bond Pricing Model P (t, T ) = e−r(T −t)

15.3 An Equilibrium Equation for Bonds

dP (r, t, T ) =

  ∂P 1 ∂ 2P ∂P ∂P 1 ∂ 2P ∂P ∂P 2 2 dr + (dr) + dt = a(r) dr + σ(r) + dt + σ(r)dZ ∂r 2 ∂r 2 ∂t ∂r 2 ∂r 2 ∂t ∂r

  ∂P 1 ∂ 2P ∂P 1 2 a(r) dr + σ(r) + α(r, t, T ) = ∂r 2 ∂r 2 ∂t P (r, t, T ) q(r, t, T ) = −

∂P 1 σ(r) P (r, t, T ) ∂r

13

14

Section 16

dP (r, t, T ) = α(r, t, T )dt − q(r, t, T )dZ P (r, t, T ) Delta-hedged portfolio

dI = N [α(r, t, T1)dt − q(r, t, T1)dZ]P (r, t, T1) + [α(r, t, T2)dt − q(r, t, T2)dZ]P (r, t, T2) + rWdt Set N = −

P (r, t, T2) P (r, t, T2) q(r, t, T2) =− r Pr(r, t, T2) P (r, t, T1) q(r, t, T1)

and dI = 0

Thus Sharpe ratio for the two bonds is equal α(r, t, T1) − r α(r, t, T2) − r = q(r, t, T2) q(r, t, T1)

φ(r, t) =

α(r, t, T ) − r q(r, t, T )

1 ∂ 2P ∂P ∂P σ(r)2 2 + [a(r) + σ(r)φ(r, t)] + − rP = 0 2 dr ∂r ∂t The risk-neutral process for the interest rate: dr = [a(r) + σ(r)φ(r, t)]dt + σ(r)dZ P [t, T , r(t)] = Et∗[e−R(t,T )] Z T R(t, T ) = r(s)ds t

15.4 Delta-Gamma Approximations for Bonds 1 ∗ E (dP ) = rP dt

16 Equilibrium Short-Rate Bond Price Models

16.1 The Rendelman-Bartter Model

15

Equilibrium Short-Rate Bond Price Models

dr = adt + σdZ

16.2 The Vasicek Model

dr = a(b − r)dt + σdz 1 2 ∂ 2P ∂P ∂P σ + [a(b − r) − σφ] + − rP = 0 2 ∂r 2 ∂r ∂t P [t, T , r(t)] = A(t, T )e−B(t,T )r(t) A(t, T ) = er¯(B(t,T )+t−T )−B

2 2

σ /4a

B(t, T ) = (1 − e−a(T −t))/a r¯ = b + σφ/a − 0.5σ 2/a2 r¯ is the yield to maturity for an infinite length bond.

16.3 The Cox-Ingersoll-Ross Model √ dr = a(b − r)dt + σ r dz

√ Sharpe Ratio: φ(r, t) = φ¯ r /σ 1 2 ∂ 2P ∂P ∂P σ + [a(b − r) − rφ¯ ] + − rP = 0 2 ∂r 2 ∂r ∂t P [t, T , r(t)] = A(t, T )e−B(t,T )r(t) "

¯

2γe(a− φ + γ)(T −t)/2 A(t, T ) = (a − φ¯ + γ)(e γ(T −t) − 1) + 2γ B(t, T ) =

γ=

#

2(e γ(T −t) − 1) ¯ (a − φ + γ)(eγ(T −t) − 1) + 2γ

p (a − φ¯ )2 + 2σ 2

16.4 Bond Options, Caps, and The Black Model

•

Pt(T , T + s) is zero-coupon bond price at time t purchased at time T and paying $1 at time T + s

16

Section 16

•

•

If t = T then P (T , T + s) is the spot price If t < T then Pt(T , T + s) is a forward price Ft,T [(P (T , T + s)]

Call option payoff = max [0, P (T , T + s) − K] Ft,T [P (T , T + s)] = P (t, T + s)/P (t, T )

Volatility = Var(ln(Ft,T [P (T , T + s)]))

C[F , P (0, T ), σ, T ] = P (0, T )[FN (d1) − KN (d2)] d1 =

ln(F /K) + 0.5σ 2T √ σ T

√ d2 = d1 − σ T where F = F0,T [P (T , T + s)]

R0(T , T + s) =

P (0, T ) −1 P (0, T + s)

Foward rate agreement (FRA)

Payoff to FRA = RT (T , T + s) − R0(T , T + s) Call option on FRA is a caplet.

Payoff to caplet = max [0, RT (T , T + s) − KR] If settled at time T , the option pays: 1 max [0, RT (T , T + s) − KR] 1 + RT (T , T + s) Let RT = RT (T , T + s)    1 1 R T − KR = (1 + KR)max 0, − (1 + KR)max 0, 1 + KR 1 + R T (1 + KT )(1 + KR) 

cap is a collection of caplets

17

Jensen’s Inequality

Cap payment at time ti+1 = max [0, Rti(ti , ti+1) − KR]

16.5 A Binomial Interest Rate Model Pi(i, i + 1; j) = e−ri(i,i+1; j)h P0(0, 1; 0) = e−r h P0(0, 2; 0) = e−r h[pe−ruh + (1 − p)e−rdh] = e−r h[pP1(1, 2; 1) + (1 − p)P1(1, 2; 0)]   Pn Using risk neutral E ∗ e− i=0rih Yields= − ln(P (0, T ))/T

16.6 The Black-Derman-Toy Model Distance between up node and down node is

√ h √ −σ h Ae

Aeσ

Yield: y[h, T , r(h)] = P [h, T , r(h)]−1/(T −h) − 1   y(h, T , ru) √ / h Yield volatility = 0.5 × ln y(h, T , rd) P (0, 1) × (0.5 × P (1, 2; Ru) + 0.5 × P (1, 2; Rd)) = P (0, 2)

17 Interest Rates Effective annual rate: r in (1 + r)n Continuously compounded rate: r in er n

18 Jensen’s Inequality If f (x) is convex: E[f (x)] > f [E(x)] If f (x) is concave: E[f (x)] 6 f [E(x)]