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 oT− 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) C(St, Qt, T− t) − P (St, Qt, T− t) = Ft,TP (S) − Ft,TP (Q)

1.5 Currency Options

C$(x0, K , T) = x0KPf ₁ x0, 1 K, T 

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_{− PV}t,T(K) > PVt,T(Div)

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_{− K}1 >C(K2) − C(K3) K3_{− K}2 6. P(K2) − P (K1) K2_{− K}1 6 P(K3) − P (K2) K3_{− K}2

2 Binomial Option Pricing: I

2.1 The Binomial Solution

∆ = e−δh Cu−Cd S_{(u − d)} B= e−rh u Cd−d Cu u_{− d} ∆S + B = e−rh _C u e(r −δ)h_{− d} u− d + Cd u_{− e}(r −δ)h u− d



No arbitrage ⇒ u > e(r −δ)h_{> d}

2.2 Risk-Neutral Pricing

p∗₌e (r −δ)h_{− d} u_{− d} C= e−r h_[p∗_C u+ (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 √ d x= xe(r −rf)h−σ h √ p∗₌e (r −rf)h_{− d} u− d

2.4 Options on Futures Contracts

u= eσ√h

d= eσ√h

p∗₌1 − d

3 Binomial Option Pricing: II

σh= σ h √ Example: σmonthly= σ 12 √

4 The Black-Scholes Formula

4.1 Calls and Puts

C(S , K , σ, r, T , δ) = Se−δT_N(d 1) − Ke−rTN(d2) d1= ln(S/K) + (r − δ +1₂σ2)T σ T√ d2= d1− σ T √ P(S , K , σ, r, T , δ) = Ke−r T_{N( − d} 2) − Se−δTN( − d1)

4.2 Options on Stocks with Discrete Dividends

F0,TP (S) = S0− PV0,T(Div)

4.3 Options on Currencies

F0,TP (S) = x0e−rfT C(x, K , σ, r, T , rf) = xe−rfTN(d1) − Ke−r TN(d2) d1= ln(x/K) + (r − rf+1₂σ2)T σ T√ P(x, K , σ, r, T , rf) = C(x, K , σ, r, T , rf) + 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=

X

i=1 n

ωi∆i

Holds true for other greeks as well.

4.6 Option Elasticity

ǫis 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

φ(x) ≡ 1 2π √ e− 1 2x 2 N(x) ≡ Z −∞ x φ(x)dx

6 Option Greeks

Important identities: N (x) = 1 − N( − x) Se−δT_N′_(d 1) = Ke−rTN′(d2)

6.1 Delta

(∆)

∆call= ∂C(S , K , σ, r, T − t, δ) ∂S = e−δ(T −t)N(d1) ∆put= ∂P_{(S , K , σ, r, T − t, δ)} ∂S = e−δ(T −t)N( − d1)

6.2 Gamma

(Γ)

Γcall= Γput= ∂2_{C(S , K , σ, r, T − t, δ)} ∂2_S = e−δ(T −t)_N′_(d1) Sσ T√ − t

6.3 Theta

(θ)

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

∂t = θcall+ rKe−r(T −t)− δSe−δ(T −t)

6.4 Vega

Vegacall= Vegaput=

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

6.5 Rho

(ρ)

ρcall= ∂C_{(S , K , σ, r, T − t, δ)} ∂r = (T − t)Ke −r(T −t)_N(d 2) ρput= ∂P(S , K , σ, r, T − t, δ) ∂r = (T − t)Ke−r(T −t)N( − d2)

6.6 Psi

(ψ)

ψcall= ∂C_{(S , K , σ, r, T − t, δ)} ∂δ = − (T − t)Se −δ(T −t)_N(d 1) ψput= ∂P(S , K , σ, r, T − t, δ) ∂δ = (T − t)Se−δ(T −t)N( − d1)

7 Market-Making and Delta-Hedging

7.1 Understanding Market-Makers Profit

∆t(St+h− St) − [∆t(St+h− St) + 1 2(St+h− St) 2_Γ t+ θh] − rh[∆tSt− C(St)] = −1₂ǫ2_Γ t+ θth+ rh[∆tSt− C(St)] 

One stardard deviation move: ǫ2_{= σ}2_S

t2h. In that case we have

Market-maker profit= −1₂σ2St2Γt+ θ + r[∆tSt− C(St)]



h. Set profit equal to zero and rearrange terms to get:

1 2σ

2_S

t2Γt+ rSt∆t+ θ = rC(St)

7.2 Re-hedgeing

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

Rh,i=1₂S2σ2Γ(xi2− 1)h

Var(Rh,i) =1₂(S2σ2Γh)2 Example Variance for a day for a daily re-hedger:

Var(R1/365,1) =1₂(S2σ2Γ/365)2

7.3 Greeks in the Binomial Model

∆(S , 0) = e−δhCu− Cd uS_{− dS} Γ(Sh, h) =∆(uS , h) − ∆(dS, h) uS− dS ǫ_{= udS − S} θ(S , 0) =C(udS , 2h) − ǫ∆(S, 0) − 1 2ǫ 2_{Γ(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].

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

K1strike. K2trigger. C(S , K1, K2, σ, r, T , δ) = Se−δTN(d1) − K1e−r TN(d2) d1= ln(Se−δT_/K2e−rT_{) +}1 2σ 2_T σ T√ d2= d1− σ T √

8.5 Exchange Options

max (0, ST− Kt) C(S , K , σ, r, T , δ) = Se−δST_N(d 1) − K1e−δKTN(d2) d1= ln(Se−δST_/Ke−δK T_{) +}1 2σ 2_T σ T√ d2= d1− σ T √ σ= σS2+ σK2 − 2ρσSσK p

9 The Lognormal Distribution

9.1 The Normal Distribution

φ_{(x; µ, σ) ≡} 1 σ√2πe

−1₂x−µ_σ 2

9.2 Sum of Normal Random Variables

xi∼ N(µi, σi2) and Cov(xi, xj) = σi, j. σi j= ρi jσiσj E X i=1 n ωixi ! =X i=1 n ωiµi Var X i=1 n ωixi ! =X i=1 n X j=1 n ωiωjσi j

9.3 The Lognormal Distribution

Continously compounded return definition: R(0, t) = ln(St/S0) or St= S0eR(0,t) If x ∼ N(m, v2_{) then E(e}x_{) = e}m+ 1 2v 2 and Var(ex_{) = e}2m+v2 (ev2 − 1) ln(St/S0) ∼ N[(α − δ − 0.5σ2)t, σ2t] ln(St/S0) = (α − δ − 0.5σ2)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 ( − dˆ ) Prob(S2 t> K) = N (dˆ ) where d2 ˆ is the standard Black-Schoes argument2

with r → α. N(St|St< K) = Se(α−δ)t N( − dˆ )1 N_{( − d}ˆ )2 N(St|St> K) = Se(α−δ)t N(dˆ )1 N(dˆ )2

10 Monte Carlo Valuation

10.1 Using Sums of Uniformly Distributed Random Variables

Z˜=X

i=1 12

ui− 6

10.2 Monte Carlo Valuation

ST= S0e(α−δ − 1 2σ 2 )T +σ h√ [P i=1 n _Z(i)] = e(α−δ − 1 2σ 2 )T +σ T√ [ 1 n √ P i=1 n Z(i)] V(S0,0) =1_ne−r T X i=1 n V(STi, T)

10.3 Control Variate Method

A∗_{= A¯ + β(G − G¯)}

11 Brownian Motion and Ito

ˆ’s Lemma

11.1 Black-Scholes Assumptions about Stock Prices

d S(t) S(t) = αdt + σdZ(t) ln[S(T )] ∼ N(ln[S(0)] + [α − 0.5σ2_{]T , σ}2_T)

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) s1, s2>0 • Z(t) is continuous

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 Z(T ) − Z(0) = T√ " 1 n √ X i=1 n Y(ih) # d Z(t) = Y (t) dt√ Z(T ) = Z(0) + lim n→∞ T √ " 1 n √ X i=1 n Y(ih) # → Z(0) + Z 0 T d Z(t)

11.3 Properties of Brownian Motion

lim

n→∞

X

i=1 n

(Z[ih] − Z[(i − 1)/h])2_{= lim} n→∞ X i=1 n  h √ Yih 2 = lim n→∞ X i=1 n hYi h2 = T

Thus it has finite quadratic variation so: lim

n→∞

X

i=1 n

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

But infinite total variation: lim n→∞ X i=1 n |Z[ih] − Z[(i − 1)/h]| = ∞

11.4 Arithmetic Brownian Motion

X(T ) − X(0) = αT + σZ(T ) d X(t) = αdt + σdZ(t) X_{(T ) − X(0) ∼ N(αT , σ}2T)

11.5 The Ornstein-Uhenbeck Process

d X_{(t) = λ[α − X(t)]dt + σdZ(t)}

11.6 Geometric Brownian Motion

d X(t) = α[X(t)]dt + σ[X(t)]dZ(t) dX(t) X(t) = αdt + σdZ(t) ln[X(t)] ∼ N(ln[X(0)] + (α − 0.5σ2_{)t, σ}2_t) X(t) = X(0)e(α−0.5σ2 )t+σ t√Z E[X(t)] = X(0)eαt

11.7 Multiplication Rules

d t× dZ = 0 (dt)2_{= 0} (dZ)2_{= dt} d Z_{× dZ}′_{= ρdt}

11.8 The Sharpe Ratio

Sharpe ratioi=αi− r

11.9 The Risk Neutral Process

dS(t)

S(t) = (α − δ)dt + σdZ(t)

Z˜(t) generates a martingale in utility terms for a risk-averse investor. dS(t) S(t) = (r − δ)dt + σd Z˜(t) dZ˜_{(t) = dZ(t) + ηdt where η = (α − r)/σ}

11.10 Ito

ˆ’s Lemma

d S(t) =nαˆ[S(t), t] − δˆ[S(t), t]odt+ σˆ[S(t), t]dZ(t) d C(S , t) = CSd S+1₂CS S(dS)2+ Ctdt d C(S , t) =n[αˆ(S , t) − δˆ(S, t)]CS+ 1 2σˆ(S , t) 2_C S S+ Ct o dt+ σˆ(S , t)CSdZ

11.11 Valuing a Claim on

S

a F0,TP [S(T )a] = e−rTS(0)ae[a(r −δ)+ 1 2a(a−1)σ 2 ]T F0,T[S(T )a] = S(0)ae[a(r −δ)+ 1 2a(a−1)σ 2 ]T

12 The Black-Scholes Equation

dS

S = (α − δ)dt + σdZ

Option V [S(t), t]. Invest W in bonds that pay return r. d W= rWdt

Total investment in option, stocks (N shares) and bonds should be zero. I= V (S , t) + NS + W = 0 d I= dV + N (dS + δSdt) + dW = Vtd t+ VSdS+ 1 2σ 2_S2_V SSd t+ N (dS + δSdt) + rWdt

Delta-hedge so N = − VS. Bonds: W = VSS− V . So,

d I= Vt+1₂σ2S2VS Sdt− VSδSdt+ r(VSS− V )dt

Zero-investment, zero-risk portfolio so dI = 0. Vt+ 1 2σ 2_S2_V S S+ (r − δ)SVS− rV = 0

12.1 Risk Neutral Pricing

dS S = (r − δ)dt + σdZ˜ 1 dtE(dV ) = Vt+ 1 2σ 2_S2_V SS+ (α − δ)SVS E∗_{(dS) = (r − δ)dt} 1 d tE∗(dV ) = Vt+ 1 2σ 2_S2_V S S+ (r − δ)SVS so _{d t}1E∗(dV ) = rV

13 Exotic Options: II

13.1 All-Or-Nothing Options

CashCall(S , K , σ, r, T − t, δ) = e−r(T −t)_N(d 2) CashPut(S , K , σ, r, T − t, δ) = e−r(T −t)_{N( − d} 2) AssetCall(S , K , σ, r, T − t, δ) = e−δ(T −t)_SN(d 1) AssetPut(S , K , σ, r, T − t, δ) = e−δ(T −t)_{SN( − d} 1)

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 − K1if S > K2.

14 Volatility

d St/St= (α − δ)dt + σ(St, Xt, t)dZ ǫt+h= ln(St+h/St) σˆ_H2 = 1 (n − 1) X i=1 n ǫi2

14.1 ARCH

ln(St/St−h) = (α − δ − 0.5σ2)h + ǫt Var(ǫt) = σ2h

15 Interest Rate Models

15.1 Behavior of Bonds and Interest Rates

dP

P = α(r, t)dt + q(r, t)dZ d r= 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

d P(r, t, T ) =∂P ∂rd r+ 1 2 ∂2P ∂r2(dr) 2₊∂P ∂td t=  a(r)∂P ∂rd r+ 1 2 ∂2P ∂r2σ(r) 2₊∂P ∂t  dt+∂P ∂rσ(r)dZ α(r, t, T ) = 1 P(r, t, T )  a(r)∂P ∂rdr+ 1 2 ∂2_P ∂r2σ(r) 2₊∂P ∂t  q(r, t, T ) = −_{P(r, t, T )}1 ∂P_∂rσ(r)

dP(r, t, T ) P(r, t, T ) = α(r, t, T )dt − q(r, t, T )dZ Delta-hedged portfolio d I= 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_P(r, t, T_{(r, t, T}2) 1) q(r, t, T2) q(r, t, T1)= − Pr(r, t, T2) Pr(r, t, T2) and dI = 0

Thus Sharpe ratio for the two bonds is equal α(r, t, T1) − r q(r, t, T1) = α(r, t, T2) − r q(r, t, T2) φ(r, t) =α(r, t, T ) − r q(r, t, T ) 1 2σ(r) 2∂2P dr2+ [a(r) + σ(r)φ(r, t)] ∂P ∂r + ∂P ∂t − rP = 0 The risk-neutral process for the interest rate: d r= [a(r) + σ(r)φ(r, t)]dt + σ(r)dZ P[t, T , r(t)] = Et∗[e−R(t,T )] R(t, T ) = Z t T r(s)ds

15.4 Delta-Gamma Approximations for Bonds

1

dtE∗(dP ) = rP

16 Equilibrium Short-Rate Bond Price Models

d r= adt + σdZ

16.2 The Vasicek Model

d r= a(b − r)dt + σdz 1 2σ 2∂2P ∂r2 + [a(b − r) − σφ] ∂P ∂r + ∂P ∂t − rP = 0 P[t, T , r(t)] = A(t, T )e−B(t,T )r(t) A(t, T ) = er¯(B(t,T )+t−T )−B2 σ2 /4a B(t, T ) = (1 − e−a(T −t)_)/a r¯ = b + σφ/a − 0.5σ2_/a2

r¯ is the yield to maturity for an infinite length bond.

16.3 The Cox-Ingersoll-Ross Model

d r_{= a(b − r)dt + σ r}√ d z Sharpe Ratio: φ(r, t) = φ¯ r√ /σ 1 2σ 2∂2P ∂r2 + [a(b − r) − rφ¯] ∂P ∂r + ∂P ∂t − rP = 0 P[t, T , r(t)] = A(t, T )e−B(t,T )r(t) A(t, T ) = " 2γe(a−φ¯+γ)(T −t)/2 (a − φ¯ + γ)(eγ(T −t)− 1) + 2γ # B(t, T ) = 2(e γ(T −t)_{− 1)} (a − φ¯ + γ)(eγ(T −t)− 1) + 2γ γ=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

• 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 2_T σ T√ d2= d1− σ T √ where F = F0,T[P (T , T + s)] R0(T , T + s) = P(0, T ) P(0, T + s)− 1 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 1 + RT(T , T + s)max [0, RT(T , T + s) − KR] Let RT= RT(T , T + s) (1 + KR)max  0, RT− KR (1 + KT)(1 + KR)  = (1 + KR)max  0, 1 1 + KR− 1 1 + RT 

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)]

Using risk neutral E∗_e−P

i=0 n

rih

Yields= − ln(P (0, T ))/T

16.6 The Black-Derman-Toy Model

Distance between up node and down node is Ae

σ√h

Ae−σ√h

Yield: y[h, T , r(h)] = P [h, T , r(h)]−1/(T −h)_{− 1}

Yield volatility = 0.5 × ln y(h, T , r_{y(h, T , r}u)

d)

 / h√

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)]