MFE Notes - Spring 2010 Sitting

Lesson 1 - Put-Call Parity

• Bull Spread: pays off if stock moves up in price

– with Calls: buy CK2 and sell CK1 ; K1 > K2

– with Puts: buy PK2 and sell PK1 ; K1 > K2

• Bear Spread: pays off if the stock moves down in price

– with Calls: buy CK1 and sell CK2 ; K1 > K2

– with Puts: buy PK1 and sell PK2 ; K1 > K2

• Straddle: buy a Call and a Put

– same K ⇒ Payoff = |ST − S0|– bet on volatility

– Strangle - buy PK2 and CK1 ; K1 > K2

• Synthetic Stock: Solve for S0: S0 = eδt (C − P +Ke−rt)

• Synthetic Treasury: Solve for Ke−rt: Ke−rt = S0e−δt − C + P

• Synthetic Options: Solve for C or P

• Conversion

– Synthetically buy a T-bill

– Lend dollars

• Reverse Conversion

– Synthetically sell a T-bill

– Borrow dollars

1

• Converting between domestic and foreign currency options:

– A put denominated in the base currency is equivalent to somenumber of calls denominated in the foreign currency

– KPd

(1x0, 1K, T)

= Cd(x0, K, T )

– Kx0Pf

(1x0, 1K, T)

= Cd(x0, K, T )

• Bid-Ask Prices

– The verb applie to the market-maker, not the retail customer

∗ The market-maker bids the bid price when buying a share ofstock

∗ The market-maker asks the ask price when selling a share ofstock

∗ Bid Price < Ask Price

Lesson 2 - Comparing Options

• American Options:

– Calls: S ≥ CA ≥ CE ≥ max(0, F P

0,T (S)−Ke−rT , S0 −K)

– Puts: K ≥ PA ≥ PE ≥ max(0, Ke−rT − F P

0,T (S), K − S0

)• Early exercise of American Options

– Calls:

∗ lose the implicit Put

∗ if non-dividend then CA = CE

∗ not rational if PVt,T (Div) < K(1− e−r(T−t)) + P

· b/c you get stock and Divs. but pay K and lose the im-plicit Put

– Puts:

∗ lose the implicit Call

2

∗ may be rationa even if no dividends

∗ earn interest on K

• Different Strike Prices

– Direction:

∗ C1 ≤ C2 and P1 ≤ P2

∗ ∂C∂K≤ 0 and ∂P

∂K≥ 0

– Slope:

∗ C1 − C2 ≥ K2 −K1 and P1 − P2 ≤ K1 −K2

∗ ∂C∂K≥ −1 and ∂P

∂K≤ 1

– Convexity:

∗ C1−C2

K1−K2≥ C2−C3

K2−K3and P1−P2

K1−K2≥ P2−P3

K2−K3

∗ ∂2C∂K2 ≥ 0 and ∂2P

∂K2 ≤ 0

· K1 > K2 > K3

• Strike Price Increases Over Time on a Call - suppose that a stock doesnot pay dividends and the strike price increases at a rate that is less thanor equal to r:

KT ≤ Kter(T−t)

The longer the call option, the more valuable it is:C(S0, KT , T ) ≥ C(S0, Kt, t) for T > t

If the inequality above is violated, then arbitrage is available.

That is if:KT ≤ Kte

r(T−t) and C(S0, KT , T ) < C(S0, K, T )

then arbitrage can be obtained with the following steps:

1. Buy the longer option and sell the shorter one

2. At time t, the shorter option is in the money, sell stock short andlend Kt at the risk-free rate

Lesson 3 - Binomial Trees - Stock, One Period

3

• Replicating Portfolio: B stands for bond, not borrowing; amount welend

• ∆ = e−δh(Cu−CdS(u−d)

)• B = e−rh

(uCd−dCuu−d

)• p∗ = e(r−δ)h−d

u−d

• C = S∆ +B

• Multinomial Trees: Set up equations as:

n+1 times

Price1∆1+ · · ·+ Pricen∆n+ Berh = Payoff1

... · · · ......

...Price1∆1+ · · · Pricen∆n+ Berh = Payoffn

• Volatility: d < e(r−δ)h < u

• If tree is based on forward prices:

– p∗ = 1

1+eσ√h

Lesson 4 - Binomial Trees - General

• Pricing Options on Futures

– p∗ =Ft,t+h/St−d

u−d = 1−du−d

– ∆ = Cu−CdF (u−d)

– B = C = e−rh(uCd−dCuu−d + Cu−Cd

u−d

)∗ B = C because there is no initial cost

Lesson 5 - Risk-Neutral Pricing

4

• Seαh = puSeδh + (1− p)dSeδh

– ⇒ p = e(α−δ)h−du−d

• eγh = S∆S∆+B

eαh + BS∆+B

erh

– ⇒ taking E(.)⇒ C = e−γh(pCu + (1− p)Cd)

• Ceγh = S∆eαh +Berh

– γ ∼ discount rate for an option

– γCall > α > r > γPut

• These results are equivalent to using p∗

– ⇒ C = e−γh(pCu + (1− p)Cd) = e−rh(p∗Cu + (1− p∗)Cd)

∗ If Cd = 0⇒ e−γhp = e−rhp∗

• Risk-Neutral Pricing and Utility(annual not cont. rates)

– Ui ∼ current value of $1 paid at the end of one year when theprice of the stock is in state i

∗ UH ≤ UL because of decling MU

∗ if risk-neutral: UH = UL = 11+r

– Ci ∼ cash flow of the stock at the end of one year in state i

– Qi ∼ the current value of $1 paid at the end of one year only ifthe price of the stock is Ci

• Important Formulas:

– QH = pUH QL = (1− p)UL– QH +QL = 1

1+r

– C0 = pUHCH + (1− p)CLQL = QHCH +QLCL

– 1 + α = pCH+(1−p)CLC0

= pCH+(1−p)CLpUHCH+(1−p)ULCL

= pCH+(1−p)CLQHCH+QLCL

– p∗ = pUHpUH+(1−p)CL

= QHQH+QL

5

∗ ⇒ solve for p⇒ p = p∗ULp∗UL+(1−p∗)UH

Lesson 6 - Binomial Trees: Misc. Topics

• Understanding early exercise of Options

– Compare S(1− e−δt) vs. K(1− e−rt)

∗ depends on Call vs. Put

• Lognormality and Alternative Trees

– if annual volatility is σ ⇒ monthly is σ√12

– Cox-Ross-Rubinstein Tree

∗ centered on 1

∗ u = eσ√h d = e−σ

√h

– Lognormal / Jarrow-Rudd Tree

∗ centered on er−δ−12σ2

• Estimating Volatility

– σ =√p

√nn−1

(∑x2i

n− x2

)– use ln

(StSt−1

)for data points

Lesson 7 - Modeling Stock Price with the Lognormal Distribution

• If X ∼ N(µ, σ2) then Y = eX ∼ LogNormal(µ, σ)

• Properties

– E(Y ) = em+ 12v2

– V (Y ) = e2m+v2(ev

2 − 1)

6

– mode = em−v2

∗ m = µt =(α− δ − 1

2σ2)t

∗ v = σ√t

• Lognormal Confidence Intervals: Assume that the stock prices are log-normally distributed:

ln(STSt

)∼ N

[(α− δ − 1

2σ2)(T − t), σ2(T − t)

]T > t

The (1− p) confidence interval is:Pr(SLT < ST < SUT

)= 1− p

The lower and upper stock prices defining the confidence interval are:

– SLT = Ste(α−δ− 1

2σ2)(T−t)+|σLZ |

√T−t

– SUT = Ste(α−δ− 1

2σ2)(T−t)+|σUZ |

√T−t

∗ where Pr(z < zL) = p2

and Pr(z > zU) = p2

∗ σ can be given as negative, that’s why abs. value signs arethere

• Jensen’s Inequality: E(g(X)) ≥ g(E(X))

– E(X2) ≥ (E(X))2

• Pricing European Options using the Lognormal Model

– Pr(ST < K) = N(−d2) Pr(ST > K) = N(d2)

– E(X|Y ) = PE(X|Y )Pr(Y )

– PE[ST |ST > K] = S0e(α−δ)tN(d1)

– PE[ST |ST < K] = S0e(α−δ)tN(−d1)

– PE[K|ST < K] = KN(−d2)

– PE[K|ST > K] = KN(d2)

– E[K − ST |ST < K] = PE[K−ST |ST<K]Pr(ST<K)

= KN(−d2)−S0e(α−δ)tN(−d1)

N(−d2)

– E[ST −K|ST > K] = PE[ST−K|ST>K]Pr(ST>K)

= S0e(α−δ)tN(d1)−KN(d2)

N(d2)

7

• Expected Payoff

– Call: E[max(0, ST −K)] = S0e(α−δ)tN(d1)−KN(d2)

– Put: E[max(0, K − ST )] = KN(−d2)− S0e(α−δ)tN(−d1)

• Expected Value

– E[ST |S0] = S0e(µ+ 1

2σ2)t

Lesson 8 - Fitting Stock Prices to a Lognormal Distribution

• Estimate using ln(

StSt−1

)as data points

• Annual Return: α = µ+ 12σ2

• Drawing a Normal Probability Plot in 5 Easy Steps

1. Sort the data into order statistics, from smallest to largest

2. Convert the order statistics into quantiles by matching them withthe appropriate cumulative probabilities

3. Match each cumulative probability with its corresponding z-value

4. Graph the points with the quantiles on the horizontal axis andthe z-values on the vertical axis

5. Draw a straight line through the 25% and 75% quantiles

Lesson 9 - The Black-Scholes Formula

• Black-Scholes Formula for Options on Futures

– C = Fe−rtN(d1)−Ke−rtN(d2)

– P = Ke−rtN(−d2)− Fe−rtN(−d1)

– d1 =ln( FK )+ 1

2σ2

σ√t

– d2 = d1 − σ√t

8

• The futures period affects the forward price of the stock but does notaffect the option price in any other way

Lesson 10 - The Black-Scholes Formula: Greeks

• ∆C −∆P = e−δt

– ∆C = e−δtN(d1)

– S-shaped

• ΓC = ΓP

– Symmetric hump, peak to the left of K (further with higher t)

• V egaC = V egaP

– Asymmetric hump; peak similar to Γ

• Ct − Pt = −δSe−δt + rKe−rt

– Θ = − 1365Ct ⇒ ΘC −ΘP = δSe−δt−rKe−rt

365

– Upside-down hump; almost always < 0 unless far in the money

• ρC − ρP = .01tKe−rt

– Assuming ρ expressed in terms of % points

– Increasing curve; positive for C, negative for P

• ΨC −ΨP = −.01tSe−δt

– Assuming Ψ expressed in terms of % point

– Decreasing curve; negative for C, positive for P

• Elasticity and Related Concepts

– Ω = ε∆/Cε/S

= S∆C

– σoption = σstock|Ω|

9

– γ − r = Ω(α− r)⇒ γ−rσoption

= Ω(α−r)Ωσstock

= α−rσstock

• Greek for Portfolio: Σ of the greeks

• Elasticity for Portfolio: Wtd. Average of the Ω’s

Lesson 11 - The Black-Scholes Formula: Applications and Volatility

• Purchase a t-day call and hold it for 1 day. Profit =

1. Change in call premium (Ct−1 − Ct)2. Lost interest (er/365Ct − Ct)

– Ct−1 − Cter/365 ∼ Difference of 1. and 2.

• Volatility

– Black-Scholes assumes σ is constant

– Implied Volatility: volatility that reproduces the price of an optionin a pricing model.

∗ Common patterns for implied equity volatilities:

1. Decreases with strike price

2. Flatter curve for longer time until expiration

– Volatility Skew: refers to the fact that the implied volatility is notconstant across strike prices

∗ implied volatility declined as time to expiry increased

∗ implied volatility decreased as K increased

∗ in-the-money call has higher volatility than an out-of-the-money call

∗ in-the-money put has lower volatility than an out-of-the-moneyput

• ∆ = change in Cchange in S

Lesson 12 - Delta Hedging

10

• Overnight Profit on a Delta-Hedged Portfolio

– Profit = −(C1 − C0) + ∆(S1 − S0)− (er

365 − 1)(∆S0 − C0)

• Break even for Market Maker

– S ± Sσ√h

• Delta-gamma-theta approximation

– C1 = C0 + ∆ε+ 12Γε2 + θh

∗ ε = Sh − S0

• Black-Scholes Equation

– rC = S∆(r − δ) + 12ΓS2σ2 + θh

• Greeks for Binomial Trees

– ∆(S, 0) = e−δh(Cu−CdS(u−d)

)– Γ(S, h) ≈ Γ(S, 0) = ∆(Su,h)−∆(Sd,h)

S(u−d)

– C(Sud, 2h) = C(S, 0) + ∆(S, 0)ε+ 12Γ(S, 0)ε2 + 2hθ(S, 0)

∗ ⇒ θ(S, 0) =C(Sud,2h)−C(S,0)−∆(S,0)ε− 1

2Γ(S,0)ε2

2h

· ε = Sud− S

• Reheding

– Variance of the return for a single period

∗ V ar[Rh,i] = 12(S2σ2Γh)2

– If we re-hedge every h (measured per year)

∗ Annual Variance of Return = 1hV ar[Rh,i] = 1

2(S2σ2Γ)2h

• Misc. Notes

– Sell Call ⇒ Buy Stock

11

– Sell Put ⇒ Sell Stock

Lesson 13 - Asian, Barrier and Compound Options

• Maxima and Minima

– max(S,K) = S +max(0, K − S) = K +max(0, S −K)

– max(cS, cK) = c ·max(S,K)

– max(S,K) +min(S,K) = S +K

• Compound Options

– CoC − PoC = C − x0e−rt1

– CoP − PoP = P − x0e−rt1

• American options on Stock with 1 discrete dividend

– CA = S0 −Ke−rt1 + CoP (S,K,D −K(1− e−r(T−t1)), t1, T )

• Asian Options - ignore initial price

Lesson 14 - Gap, Exchange and Other Options

• All-or-nothing Options

– S|S > K = S0e(r−δ)TN(d1)

– S|S < K = S0e(r−δ)TN(−d1)

– c|S > K = ce−rTN(d2)

– c|S < K = ce−rTN(−d2)

– Delta for all-or-nothing options

∗ ∂N(di)∂S

= e−d2i2

Sσ√

2πT

• Gap Options

12

– Remember that ST > trigger for Calls and ST < trigger for Puts

– Put-Call Parity applies

– If two otherwise identical gap options have different strike prices,then use linear interpolation to find the price of a third otherwiseidentical gap option with a different strike price.

• Exchange Options

– volatility measures the variance of rate of return (not the dollarreturn)i.e. 2 shares have the volatility as 1 share

• Chooser Options

– Derivation

Vt = max(C(S,K, T − t), P (S,K, T − t)) (1)

= C(S,K, T − t) +max(0, P (S,K, T − t)− C(S,K, T − t) (2)

= C(S,K, T − t) +max(0, Ke−r(T−t) − Se−δ(T−t)) (3)

= C(S,K, T − t) + e−δ(T−t) ·max(0, Ke−(r−δ)(T−t) − S) (4)

@ t0 ⇒ V0 = C(S,K, T ) + e−δ(T−t) · P (S,Ke−(r−δ)(T−t), t)

• Forward Start Options

– Purchase a call @ t with K = cSt expiring @ T , then the value ofthe forward start option is:

∗ V = Se−δTN(d1)− cSe−r(T−t)−δtN(d2)

· di are computed using T − t as time to expiry

Lesson 15 - Monte Carlo Valuation

• Generating LogNormal random numbers

1. Let zj =∑12

i=1 ui − 6 where ui ∈ U [0, 1]

2. Let zj = N−1(uj)

13

• Use r to discount when pricing options

• Use α for true expected payoffs

• Control Variate Method

– Let X∗ = X + (E(Y )− Y ), Y ∼ control variate

∗ ⇒ V (X∗) = V (X) + V (Y )− 2Cov(X, Y )

∗ Always use sample variance / covariance formula

– Boyle modification:

∗ X∗ = X + β(E(Y )− Y )

· ⇒ V (X∗) = V (X) + β2V (Y )− 2βCov(X, Y )

· Optimal value for β = Cov(X,Y )

V (Y )

· Variance becomes: V (X∗) = V (X)(

1− ρ2X,Y

)• Other Variance Reduction Techniques

– Antithetic Variates: for every ui, use 1− ui

– Stratified Sampling: break sampling space into strata and thenscale uniform #s to be in these strata

∗ If you had 4 strata: [0, .25), . . . , [.75, 1) then generate sets of4 ui on [0, 1), multiply all 4 by .25, put the first in [0, .25),add .25 to 2nd number, etc.

– Latin Hypercube Sampling

– Importance Sampling

– Low Discrepancy Sequences

Lesson 16 - Brownian Motion

• Random Walk

14

1. X(0) = 0

2. For t > 0, if X(t− 1) = k, then X(t) =

k + 1, with p = 1

2

k − 1, with p = 12

3. Memoryless.

– Pr(X(t+ u) = l|X(t) = k) = Pr(X(u) = l − k)

4. X(t) is random, distance traversed is not.

– Sum of the squares of the movement is t

5. X(t) ∼ Bin(t, 1

2

)• Brownian Motion

– Move√h per h units of time and take limh→0

∗ ⇒ Cont. Random Walk and Binomial → Normal

– Properties

1. Z(0) = 0

2. Z(t+ s)|Z(t) ∼ N(Z(t), s)

3. Z(t+ s1)− Z(t) is independent of Z(t)− Z(t− s2)

4. Z(t) is cont. in t

– Expected Values Under Pure Brownian Motion

∗ E[Z(t)] = 0

∗ E[Z(t+ h)|Z(t)] = Z(t)

∗ E[Z(t+ h)− Z(t)] = 0

∗ E[dZ(t)] = 0

∗ E[dZ(t)|Z(t)] = 0

∗ E [(Z(t))2] = t

15

∗ E [(dZ(t))2] = dt

∗ E[Z(t)Z(s)] = Min(t, s)

– Variances under Pure Brownian Motion

∗ V [Z(t)] = t

∗ V [Z(t+ h)|Z(t)] = h

∗ V [Z(t+ h)− Z(t)] = h

∗ V [dZ(t)] = dt

∗ V [dZ(t)|Z(t)] = dt

– is a diffusion process - cont. process in which the absolute valueof the R.V. tends to get larger

– is a martingale - process X(t) for which E[X(t+ s)|X(t)] = X(t)

∗ ABM and GBM are martingales iff they have zero drift

• Arithmetic Brownian Motion

– X(t) = αt+ σZ(t)

– X(t+ s)−X(t) ∼ N(µs, σ2s)

– X(t+ s)|X(t) ∼ N(X(t) + µs, σ2s)

• Geometric Brownian Motion

– If ln(X(t)X(0)

)∼ N(µt, σ2t) then X(t)−X(0) ∼ LogNormal

∗ Mean = e(µ+ 12σ2)t

∗ Variance = e(2µ+σ2)t(eσ2t − 1)

• To go from GBM to ABM, you must subtract 12σ2

– When dealing with probabilities, you must convert to ABM

16

• V ar(ln(S(t))|S(0)) = V ar(ln(F0,T (S))) = V ar(ln(F P0,T (S)))

• Forms of BM

– GBM: dSS

= (α− δ)dt+ σdZ

– ABM: d(ln(S)) = (α− δ − 12σ2)dt+ σdZ

• When you add δ to total return (for Sharpe Ratio), only add to S, notC

• Portfolio Returns: Suppose that a portfolio P consists of 2 assets, Aand B. If x is the percentage of the portfolio is invested in A and(1− x) is the percentage invested in B, then the instantaneous changein the price of the portfolio is:

dP (t)P (t)

= xdA(t)A(t)

+ (1− x)dB(t)B(t)

To find the instantaneous return on the portfolio, include the dividends.

– Instantaneous Return on Portfolio

∗ dP (t)P (t)

+(xδA+(1−x)δB)dt = x[dA(t)A(t)

+ δA

]+(1−x)

[dB(t)B(t)

+ δB

]Lesson 17 - Ito’s Lemma

• dC = CSdS + 12CSS(dS)2 + Ctdt

• Multiplication rules: All → 0 except (dZ)2 = dt

• The Black-Scholes Equation

– rC = S∆(r − δ) + 12ΓS2σ2 + θh

• Sharpe Ratio(Only works for GBM)

– φ = α−rσ

∗ α is total return (includes δ)

17

– For 2 Ito processes with the same dZ, the Sharpe Ratios are equal

• Problems which give 2 processes, Prices and ask how much should beallocated to each process. Such as:

1. dS1

S1= α1dt+ σ1dZ and dS2

S2= α2dt+ σ2dZ

2. x shares of S1 and y shares of S2, r = r

(a) Solve S1 · x · α1 + S2 · y · α2 = (S1 · x+ S2 · y)r

(b) If you know x and need y, look @ σ1

σ2, that’s ratio of value of

S2 you need to buy/sell. Since S1 costs S1 · x then you need

to buy/sell S1 · x(σ1

σ2

)= y

• CAPM: αi−rσi

= ρi,M

(αM−rσM

)– φi = ρi,MφM

• Risk-Neutral Processes

– True Ito Process: dS = (α− δ)dt+ σdZ

– Risk-Neutral Ito Process: dS = (r − δ)dt+ σdZ

∗ dZ = dZ + ηdt

· where η = α−rσ

∗ E∗[Z(T )] = 0

∗ E∗[Z(T )] =(r−ασ

)T

∗ E[Z(T )] = 0

∗ E[Z(T )] =(α−rσ

)T

• Valuing a Forward on Sa

– E[S(T )a] = Sa0e[a(α−δ)+ 1

2σ2a(a−1)]T

18

– F0,T (Sa) = Sa0e[a(r−δ)+ 1

2σ2a(a−1)]T

– F P0,T (Sa) = e−rT · F0,T (Sa)

• Ito Process for Sa

– If C = Sa and dSS

= (α− δ)dt+ σdZ then

∗ dCC

= (a(α− δ) +1

2σ2a(a− 1) + δ∗)︸ ︷︷ ︸γ

dt+ σadZ

· δ∗ ∼ derivative’s dividend yield

∗ ⇒ Sharpe Ratios = γ−raσ

= α−rσ

· ⇒ γ = a(α− r) + r

• Stochastic Integration

– Regular Calculus Rules Apply (e.g. FTC)

1.∫ T

0dZ(t) = Z(T )− Z(0) ∼ N(0, T )

2.∫ T

0(dZ(t))2 =

∫ T0dt = T − 0 = T

3.∫ T

0(dZ(t))n = 0, n > 2

4. S(t) =∫ T

0sZ(s)ds⇒ dS = t · Z(t)dt

5. S(t) =∫ T

0tdZ(s)⇒ dS = dt

(∫ T0dZ(s)

)+ t(∫ T

0dZ(s)

)′=

Z(t)dt+ tdZ(s)

• Ornstein-Uhlenbeck Process

– DE: dX = λ(α−X(t))dt+ σdZ

– Integral: X(t) = X0e−λt + α(1− e−λt) + σ

∫ t0eλ(s−t)dZ(s)

• Misc. Notes

19

– volatility of Sn is n · σ (remember when working with Black-Scholes)

Lesson 18 - Binomial Tree Models for Interest Rates

• Ft,T (P (T, T + s)) ∼ forward price @ t for an agreement to buy a bond@ T maturing @ T + s

– Ft,T (P (T, T + s)) = P (t,T+s)P (t,T )

• Binomial Trees

– don’t necessarily recombine

– risk-neutral probs. are given

– list out all paths and discount by that factor

• The Black-Derman-Toy model

– Bond Price = 1(1+R)n

– Ratio between interest rates @ successive nodes is constant

∗ it is e2σt√h

– σ =12ln(RuRd

)√h

– 2 year Bond Price1 year Bond Price

= 12

(1

1+R1+ 1

1+R1e2σ

)• Pricing Forwards using BDT

– use annual not cont. compouding

• Pricing Caps using BDT

20

– Discount difference due to cap by the discount rate appropriateto the beginning of the year

– Cap pays max(

0, RT−KR1+RT

)– For multiple year trees, start @ end and calculate the value then

weigh the results and add in the additional cap values as youmove to t0

Lesson 19 - The Black Formula for Bond Options

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

• P (F, P (0, T ), σ, T ) = P (0, T )(KN(−d2)− FN(−d1))

– where d1 =ln( FK )+ 1

2σ2T

σ√T

and d2 = d1 − σ√T

• Pricing Caps with the Black Formula

– (1 +KR) Puts with strike price 11+KR

– Calculate @ each node and then add together and multiply thesum by (1 +KR)

Lesson 20 - Eq. Interest Rate Models: Vasicek and Cox-Ingersoll-Ross

• Eq. Models - Theory

– dr = a(r)dt+ σ(r)dZ and dPP

= α(r, t, T )dt− q(r, t, T )dZ

– by Ito’s, dP = Prdr + 12Prr(dr)

2 + Ptdt

∗ ⇒ dP = α(r, t, T )dt− q(r, t, T )dZ, where

· α(r, t, T ) = 1P

(a(r)Pr + 1

2σ2(r)Prr + Pt

)· q(r, t, T ) = 1

PPrσ(r)

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• Black-Scholes equation for Bonds

– rP = (a(r) + σ(r)φ)Pr + 12σ2(r)Prr + Pt

• Risk Premium = σ(r)φ

• To go to Risk Neutral, add σ(r)φ

– dr = (a(r) + σ(r)φ)dt+ σ(r)dZ

– Z(t) = Z(t)− φ

• The Rendelman-Barter Model(GBM)

– dr = ardt+ σrdZ

– Interest rates cannot go negative (+)

– Volatility is proportional to interest rate (+)

– Interest rates can get arbitrarily high, no mean reversion (-)

– Determine probabilities like with any GBM problem

• The Vasicek Model

– dr = a(b− r)dt+ σdZ

– There is mean reversion (+)

– Volatility is constant (-)

– Interest rates can go negative (-)

– DE: (a(b− r) + σφ)Pr + 12σ2Prr + Pt = rP

– P (r, t, T ) = A(t, T )e−B(t,T )r

– a 6= 0

22

∗ A(t, T ) = er[B−(T−t)]−B2 σ2

4a

∗ B(t, T ) = 1−e−a(T−t)

a

∗ r = b+ σ φa− 1

2

(σa

)2

– a = 0

∗ A(t, T ) = e12σφ(T−t)2+σ2 (T−t)3

6

∗ B(t, T ) = T − t

– ∆ = Pr = −BP

– Γ = Prr = B2P

• The Cox-Ingersoll-Ross Model

– dr = a(b− r)dt+ σ√rdZ

– Interest rates cannot go negative (+)

– Volatility varies with interest rate (+)

– There is mean reversion (+)

– φσ = φr

– DE: [a(b− r) + φr]Pr + 12σ2Prr + Pt = rP

– P (r, t, T ) = A(t, T )e−B(t,T )r

– A(t, T ) =[

2γe(a−φ+γ)(T−t)/2

(a−φ+γ)(eγ(T−t)−1)+2γ

] 2abσ2

– B(t, T ) = 2(eγ(T−t)−1)

(a−φ+γ)(eγ(T−t)−1)+2γ

∗ where γ =√

(a− φ)2 + 2σ2

• Misc. Notes

– q(r, t, T ) = −PrPσ(r) = Bσ(r)

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– Vasicek

∗ α(r, t, T ) = −a(b− r)B + 12σ2B2 + Pt

P

• Delta Hedging

– Duration Hedge: N = −T1P (r,0,T1)T2P (r,0,T2)

– Delta Hedge: N = −Pr(r,0,T1)Pr(r,0,T2)

∗ where numerator is what you are hedging

• Delta-Gamma-Theta Approximation

– P (r + ε, 0, t+ h) = P (r, 0, t) + ∆ε+ 12Γε2 + θh

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