1. Put-call parity

C(K,T ) − P(K,T ) = e−rT (F0,T − K) = FP0 − Ke−rT

Continuous: FP0,T = S 0e−δT , so C(K,T ) − P(K,T ) = S 0e−δT − Ke−rT .

Discrete: FP0,T = S 0 −

∑dte−rt = S 0 − PV(dividends), so C(K,T ) − P(K,T ) = S 0 −

∑dte−rt − Ke−rT .

To create one unit of synthetic stock: S 0 =C(S ,K)−P(S ,K)

e−δT+ Ke−rT

e−δT, so buy 1

e−δTcalls, sell 1

e−δTputs, buy Ke−rT

e−δTbonds.

To create a synthetic treasury with maturity value $B, buy BKeδT

or BK+CV(divs) shares of stock.

2. Comparing options

For a Call: higher strike =⇒ lower price. For a Put: higher strike =⇒ higher price. −1 ≤ ∂C∂K ≤ 0 and 0 ≤ ∂P

∂K ≤ 1.American options are worth more than European.

S ≥ CAmer(S ,K,T ) ≥ CEur(S ,K,T ) ≥ max(0, FP0,T − Ke−rT )

K ≥ PAmer(S ,K,T ) ≥ PEur(S ,K,T ) ≥ max(0,Ke−rT − FP0,T )

The only rational time to exercise an American call option early is just before a dividend. If no dividends, then never exercise early.Early Call exercise may be rational if PVt,T (Div) ≥ K(1 − e−r(T−t)). Early Put exercise may be rational if Call + PVt,T (Div) ≥ K(1 − e−r(T−t)).An American option with exercise time T is ≥ one with exercise time t < T .A European call option on a non-dividend stock with exercise time T is ≥ one with exercise time t < T .A European option on a non-dividend stock with strike price Ker(T−t) and expiry T is ≥ one with strike price K and expiry t.The gain of early exercise of a Put is interest on the strike price. If the strike increases at the risk-free rate, it is never rational to exercise early.

3. Binomial trees— stock, one period

Replicating portfolio: solving S u∆eδh + Berh = Cu and S d∆eδh + Berh = Cd gives

# shares to buy: ∆ =Cu −Cd

S (u − d)e−δh , # bonds to buy: B =

uCd − dCu

u − de−rh , risk-neutral probability: p∗ =

e(r−δ)h − du − d

=1

1 + eσ√

h.

These formulas can also be used to pull back the replicating portfolio (∆, B) at each node. Must have d < e(r−δ)h < u to prevent arbitrage.

For a tree built using forward rates: u/d = e(r−δ)h±σ√

h . Option price: C = S ∆ + B = e−rh(p∗Cu + (1 − p∗)Cd)

Remember: to pull back to previous node, discount with e−rh, not with the e−(r−δ)h that appears in the expression for p∗!

4. Binomial trees— general

For European trees, use binomial theorem shortcut:∑n

k=0

(nk

)pk(1 − p)n−k .

For American options, compare pulled-back value to exercise value (S node − K) ∧ 0 or (K − S node) ∧ 0.For currency options, use δ = rforeign.

Futures: p∗ =1 − du − d

, ∆ =Cu −Cd

Fu − Fd, B = C = e−rh(p∗Cu + (1 − p∗)Cd , u = eσ

√h, d = e−σ

√h .

For computing the forward price, F = S e(r−δ)T where T is the length of the contract (nothing to do with the period h of the tree).

For multinomial trees, set up “expected value=forward price” equations for each asset: p∗1S (1)T + p∗2S (2)

T + (1 − p∗1 − p∗2)S (3)T = S 0e(r−δ)t .

Here S 0 is the spot price, S (i)T is the possible value of S in outcome i (at time t = T ) and δ is the corresponding rate for this asset.

5. Risk-neutral pricing

5.1. Pricing with true probabilities. α is the expected return on a stock, and γ is the corresponding discounting rate for the option α > r.

γ is also the compound annual return for the option. For puts, usually γ < 0. For Cd = 0, use e−γh p = e−rh p∗ .p∗ is not really the probability that the stock increase in value. It is the probability that makes α = r.

p =e(α−δ)h − d

u − deγh =

S ∆

S ∆ + Beαh +

BS ∆ + B

erh Ceγh = S ∆eαh + Berh = pCu + (1 − p)Cd

5.2. Risk-neutral pricing. Let Qu be the price of a security that pays $1 iff the stock moves to state u, and Qd pays $1 iff stock moves to d.

Qi is partial expected value of PV(1), so Qu = pUu =p∗

1 + r, Qd =

1 − p∗

1 + r, where Ui is PV(1), given that the stock is in state i.

Current value of a stock: C0 = QuCu + QdCd = pUuCu + (1 − p)UdCd .

Effective annual rates of return. True: 1 + α =E[C]C0

=pCu + (1 − p)Cd

C0vs. risk-neutral: 1 + r =

E∗[C]C0

=p∗Cu + (1 − p∗)Cd

C0=

1Qu + Qd

Risk-neutral probabilities: p∗ =Qu

Qu + Qd=

pUu

pUu + (1 − p)Ud, 1 − p∗ =

(1 − p)Ud

pUu + (1 − p)Ud← utility-weighted avgs of true probs

In a binomial tree, set up and solve: S uQu + S dQd = S 0, CuQu + CdQd = C0, Qu + Qd = e−r .1

2

6. Binomial trees: miscellany

For early exercise to be optimal: div − int ≥ implicit put, or S (1 − e−δt) − K(1 − e−rt) ≥ Put(S ,K)

Standard tree centered on (r − δ)h, so u/d = e(r−δ)h±σ√

h . Based on forward prices.

Cox-Ross-Rubenstein tree centered on 1, so u/d = e±σ√

h . Allows arbitrage iff eσ√

h < e(r−δ)h.

Lognormal tree centered on (r − δ − σ2

2 )h, so u/d = e(r−δ− σ2

2 )h±σ√

h . Allows arbitrage iff e(r−δ− σ2

2 )h+σ√

h < e(r−δ)h.Also called Jarrow-Rudd. Makes p very close to 0.5.

For all trees, risk-neutral probability comes by looking at forward pricecurrent price , so p∗ =

(FT /S 0)−du−d = e(r−δ)h−d

u−d in each case.

Estimating volatility: given stock prices S ti for i = 0, 1, . . . , n, compute τi := ln(S ti/S ti−1 ), for i = 1, . . . , n. Then

x =1n

n∑i=1

τi =ln(S n/S 0)

nand µ2 =

1n

n∑i=1

τ2i and unbiased sample variance = s2 =

nn − 1

(µ2 − x2

)(1)

Then the estimated annualized unbiased sample volatility is s√

p, where p is number of periods per year (h = 1p ).

7. Modeling stock prices with the lognormal distribution

Weaknesses/assumptions: (i) constant volatility, (ii) stock returns for different periods are independent, (iii) stock prices don’t jump.For S t lognormal, α is the continuously compounded annual rate of return, and σ2t := Var[ln FP

t (S t)] is variance.For the associated arithmetic process, ln(S t/S 0) ∼ N

(mt, v2t

), where m = α − δ − σ2

2 is the expected value of the annual rate of return.

Pr(S t > K) = N(d2) and Pr(S t < K) = N(−d2) , where d2 has α in place of r: d2 =ln(S 0/K) + (α − δ − σ2

2 )t

σ√

t

Median stock price: M = S 0emt = E[S t]e−(σ2/2)t < E[S t] = S 0e(α−δ)t = mean stock price .

Confidence intervals: let ζt = σ√

t N−1( 1+p

2

). Then have Me−ζt = S 0emt−ζt ≤ S t ≤ S 0emt+ζt =Meζt with probability p.

Partial expectation: PE[X|Y] = E[X|Y]Pr(Y), so PE[S t |S t < K] = E[S t |S t < K]Pr(S t < K) = S (α−δ)t0 N(−d1)

Conditional expected stock price: E[S t |S t > K] = S e(α−δ)t N(d1)N(d2)

and E[S t |S t < K] = S e(α−δ)t N(−d1)N(−d2)

; see §14 to compare with AoNs.

Expected payoff Call: E[max0, S t − K] = S e(α−δ)tN(d1) − KN(d2) Put: E[max0,K − S t] = KN(−d2) − S e(α−δ)tN(−d1)

C = e−rt∫ ∞

K (S t − K)g∗(S t) dS t = e−rtE∗[S − K|S > K]P∗[S > K] = S e(α−r−δ)t N(d1)N(d2)

N(d2) − Ke−rt N(−d1)N(−d2)

N(−d2) = B-S, for α = r.

8. Fitting stock prices to a lognormal distribution

Estimate of growth rate α: α =

(x +

s2

2

)p , where x and s and p = 1

h are as in (1). This is for estimating α alone (i.e., dividends removed).

Confidence interval for estimated α: α ± σ√n

N−1( 1+P2 ), where n is number of estimates.

9. The Black-Scholes formula

Assumptions:

• Continuously compounded returns on the stock (δ) are normally distributed and independent over time.• Continuously compounded returns on the strike asset (r) are known and constant.• Volatility is known and constant.• Dividends are known and constant.• There are no transaction costs or taxes.• It is possible to short-sell any amount of stock and to borrow any amount of money at the risk-free rate.

General Black-Scholes C(S ,K, σ,T, r, δ) = FP(S )N(d1) − FP(K)N(d2) , where d1 =ln

(FP(S )/FP(K)

)+ σ2

2 T

σ√

Tand d2 = d1 − σ

√T .

For stocks, FP(S ) = S e−δT and FP(K) = Ke−rT , so d1 = 1σ√

T

(ln

(SK

)+

(r − δ + σ2

2

)T)

and d2 = d1 − σ√

T and

Call(S ,K) = S e−δT N(d1) − Ke−rT N(d2), Put(S ,K) = Ke−rT N(−d2) − S e−δT N(−d1).

For currency, let rf = rforeign and let x be the amount of foreign currency to buy. Then δ = rf and r = rdomestic

Call(x,K) = xe−rfT N(d1) − Ke−rT N(d2), Put(x,K) = Ke−rT N(−d2) − se−rfT N(−d1).

For futures, δ = r, so d1 = 1σ√

T

(ln

(SK

)+ σ2

2 T). When K = S , this becomes d1 = σ

√T/2. The Black formula is:

Call(x,K) = Fe−rT N(d1) − Ke−rT N(d2), Put(x,K) = Ke−rT N(−d2) − Fe−rT N(−d1).

3

10. The Black-Scholes formula: Greeks

∆ ∆put = ∆call − e−δT S-shaped curveΓ Γput = Γcall symmetric hump, peak to left of stock price, further left with higher T

vega vegaput = vegacall asymmetric hump, peak like Γ

θ θput = θcall + 1365 (rKe−rT − δS e−δT ) Sulcus for short lives, gradual decrease for long lives.

ρ ρput = ρcall −T

100 Ke−rT increasing curve (pos for calls, neg for puts)Ψ Ψput = + T

100 S e−δT Ψcall decreasing curve (neg for calls, pos for puts)To convert between Put and Call greeks, differentiate both sides of C − P = S e−δt − Ke−rt with respect to the appropriate variable.θ is almost always negative for calls; negative for puts unless far in the money.

To replicate a call option: buy ∆ shares of stock and borrow Ke−rT N(d2). ∆call = e−δT N(d1) and ∆put = −e−δT N(−d1) = ∆call − e−δT

Elasticity of the option with premium C is Ω =S ∆

C=

ε∆/Cε/S =

%option change%stock change = percentage risk. Dollar risk is ≈ ε∆.

For calls, Ω ≥ 1 because ∆ ≥ 0 and S ∆ = S e−δT N(d1) > C(S ). For puts, Ω ≤ 0 because ∆ ≤ 0.Volatility of the option is σoption = σstock |Ω| .

Sharpe ratio isγ − rσoption

=α − rσstock

. γ − r = Ω(α − r) is true instantaneously, and follows from eγh = erh + Ω(eαh − erh).

For a portfolio consisting of ai of options Ci on the same stock: A greek for the porfolio is computed by ∆portfolio =∑

ai∆Ci .

Elasticity for the porfolio is computed by Ωportfolio =S ∆portfolio

Cportfolio=

S∑

ai∆Ci∑aiCi

, where Cportfolio is the value of the portfolio of options.

11. The Black-Scholes formula: applications, implied volatility

Calendar spread Sell C(S ,K, t) and buy C(S ,K,T ), with t < T (or buy C(S ,K, t) and sell C(S ,K,T )).If Ct << C0, then loss on C(S ,K,T ) outweighs profit on sale of C(S ,K, t).If Ct >> C0, then obligation from sale of C(S ,K, t) outweighs profit when exercising C(S ,K,T ).

To find holding period profit/calendar spread profit for [0, t], where 0 < t < Ti, use BS to compute

(portfolio value at time t)︸ ︷︷ ︸earned from investing

− (portfolio value at time 0)ert︸ ︷︷ ︸could’ve earned from lending

=

n∑i=1

Ci(S ,K,Ti)︸ ︷︷ ︸still extant

−

m∑j=1

C j(S ,K, t)ert

︸ ︷︷ ︸expired at t

+(cash settlements) (2)

Implied volatility (i) Allows pricing other options on the same stock, without market prices. (ii) Is a quick way to describe option prices. (iii)Volatility skew measures accuracy of Black-Scholes model. Volatility skew: implied volatility tends to be lower for high strike prices.

Historical volatility: this is (1) (annualized) with x = 0, so just the second moment σ2H = 1

n−1∑n

i=1 τ2i , τi = ln(S i/S i−1).

12. Delta hedging

If a market-maker sells an option, he buys ∆ of the stock to hedge so there will be no profit or loss if the stock price changes; see (4).

12.1. Overnight profit. Ignoring dividends, profit is change in option value & ∆ (change in stock price) & interest on borrowed money:market-maker profit = − (C(S t) −C(S 0)) + ∆(S t − S 0) −

(e(r−δ)t − 1

)(∆S 0 −C(S 0)), where usually t = 1

365 . This is a special case of (2):

market-maker profit = (∆S t −C(S t)) − e(r−δ)t(∆S 0 −C(S 0)) which is positive iff S − Sσ√

t < S t < S + Sσ√

t . (3)

12.2. Delta-gamma-theta approximation (∆Γθ). C(S t + ε) = C(S ) + ∆ε + 12 Γε2 + θt + Taylor remainder error.

The whole point of ∆-hedging is to separate ε (or Sσ√

h) from ∆; putting ∆Γθ into (3) cancels the ∆ε terms. For ε = Sσ√

h, get

market-maker profit = −

[ε2

2Γ + (r − δ)S ∆h − rC(S ) + θh

]= −

[σ2

2S 2Γ + (r − δ)S ∆ − rC(S ) + θ

]h (4)

Holds for most options (exception: when early exercise is optimal). To get the Black-Scholes equation, set market-maker profit = 0; see §19.

12.3. Greeks for binomial trees. Compute ∆0 = ∆(S , 0) =Cue−δt−Cde−δt

S u−S d as in §3 and approximate Γ0 = Γ(0, S ) with Γ(S , h) =∆(S u,h)−∆(S d,h)

S u−S d .Back θ0 = θ(S , 0) out of the ud node using the ∆Γθ approximation: C(S ud , 2h) = C0 + ∆0(S ud − S ) + 1

2 Γ0(S ud − S )2 + 2hθ0.

12.4. Rehedging. Buy ∆t − ∆0 of the stock, where ∆t = N(d1) is computed at time t, and ∆0 = N(d1) computed at time 0. If negative, sell.Let Rh,i be period-i return to a delta-hedged market maker who has written a call. Then Boyle-Emanuel says

Rh,i =σ2

2S 2(x2

i − 1)Γh, and Var(Rh,i) = 12

(σ2S 2Γh

)2and annual variance = 1

2

(σ2S 2Γ

)2h

Note: variance, not volatility!

4

12.5. Hedging practices. (i) use options to obtain gamma-neutrality, (ii) static option replication (eg, P-C parity), (iii) out-of-the-money optionsas insurance, (iv) sell the hedging error (variance swap). Market-makers cannot be gamma-neutral in the aggregate; besides, this hedge incurs abid-ask spread.

12.6. Hedging multiple greeks. Sell option Csell, then ∆-Γ hedge by buying x1 shares of stock and buying x2 of another option Cbuy:

Delta-gamma hedging

stock Cbuy Csell

∆ : x1 + ∆buy x2 = ∆sell

Γ : Γbuy x2 = Γsell

compare to Delta-hedgingstock Csell

∆ : x1 = ∆sell

13. Asian, barrier, and compound options

13.1. Asian options. Arithmetic average A(S ) = 1T

∑Tt=1 S t and geometric average G(S ) =

T√∏T

t=1 S t . Ignore initial price (exclude S 0)

arithmetic geometric

average price Call = max(0, A(S ) − K), Put = max(0,K − A(S )) Call = max(0,G(S ) − K), Put = max(0,K −G(S ))

average strike Call = max(0, S T − A(S )), Put = max(0, A(S ) − S T ) Call = max(0, S T −G(S )), Put = max(0,G(S ) − S T )

Asian is cheaper than European, since less volatile. Similarly, average over more items is cheaper.Daily average price < monthly average price. Monthly average strike < daily average strike.G(S ) ≤ A(S ) =⇒ Geometric average price call < arithmetic average price call. Reverse inequality for puts, also reverse for average strikes.

For claims on dS t = αS t dt + σS t dZt formulated as geometric means, like G = (S 1S 2S 3)1/3, introduce Qt := S tS t−1

for t = 1, 2, 3. These are

normal with the same µ, σ2 and independent. Then S 1S 2S 3 = S 0Q1S 1Q2S 2Q3 = · · · = S 0Q31Q2

2Q13, so ln G

S 0= ln Q1 + 2

3 ln Q2 + 13 Q3 is normal

with parameters 2µ and (1 + ( 23 )2 + ( 1

3 )2)σ2 = 149 σ

2.

13.2. Barrier options. Rebate options: pay a fixed amount when the barrier is hit.Parity: knock-in option + knock-out option = ordinary option .

13.3. Maxima and minima. maxS ,K = S + max0,K − S = K + max0, S − K, and maxcS , cK = c maxS ,K for c > 0, and minS ,K +maxS ,K = S + K, and maxS ,K = −min−S ,−K.

Given a min, convert it to a max in order to price an option.

13.4. Compound options. For binomial tree models, work out the binomial tree for the underlying first. Then for the compound option, work outa second tree with initial vertices given by the prices of the underlying.

13.4.1. Compound option parity.

CallOnCall(C(S ,K,T ), x, t) − PutOnCall(C(S ,K,T ), x, t) = C(S ,K,T ) − xe−rt

CallOnPut(P(S ,K,T ), x, t) − PutOnPut(P(S ,K,T ), x, t) = P(S ,K,T ) − xe−rt

13.4.2. American options on stocks with one discrete dividend. If a dividend D is paid at time t, then value of an American call at time t is greaterof exercise value and option for remaining period: maxS t + D − K,C(S t ,K,T − t). If S t is cum-dividend and S ′t = S t − D is ex-dividend, then

time t : S ′t + D − K + max0, P(S ′t ,K,T − t) + K(1 − e−r(T−t)) − D =⇒ time 0 : S 0 − Ke−rt + CallOnPut[S ,K,D − K(1 − e−r(T−t))] .

Early exercise is not rational if x = D − K(1 − e−r(T−t)) < P(S ′t ,D − K(1 − e−r(T−t))) . For P, use FP(S ) = S 0 − De−rt and FP(K) = Ke−rT .

14. Gap, exchange, and other options

14.1. All-or-nothing options. Recall from §7 that Pr(S t > K) = N(d2) and Pr(S t < K) = N(−d2).Option S | S > K S | S < K c | S > K c | S < K PE[S t |S t > K] E[S t |S t > K] PE[S t |S t < K] E[S t |S t < K]

Value S e−δT N(d1) S e−δT N(−d1) Ke−rT N(d2) Ke−rT N(−d2) S 0e(α−δ)tN(d1) S 0e(α−δ)t N(d1)N(d2)

S 0e(α−δ)tN(−d1) S 0e(α−δ)t N(−d1)N(−d2)

14.2. Gap options. K splits into Kstrike and Ktrigger. Gap options satisfy parity at Kstrike. Occurrence of payoff is determined by Ktrigger, so use itto determine the probabilities N(di). Amount of payoff is determined by Kstrike, so use it to compute option price:

Call = S e−δtN(d1) − Kstrikee−rtN(d2), Put = Kstrikee−rtN(−d2) − S e−δtN(−d1).

where d1 = 1σ√

T

(ln(S/Ktrigger) +

(r − δ + σ2

2

)T)

and d2 = d1 − σ√

T . Decompose gap in terms of AoNs:

C(S ,Kstrike,Ktrigger) = C(S ,Ktrigger) + (Ktrigger − Kstrike)|S > Ktrigger P(S ,Kstrike,Ktrigger) = P(S ,Ktrigger) + (Ktrigger − Kstrike)|S < Ktrigger

14.3. Exchange options. Let S be the asset you receive, with dividend rate δ1, and K be the asset you may exchange for it, with dividend rate δ2.

The volatility of S − K is σ2 = σ2S + σ2

K − 2ρσSσK

14.4. Chooser options. V = C(S ,K,T ) + e−δ(T−t)P(S ,Ke−(r−δ)(T−t), t)

5

14.5. Forward-start options. Let dT−ti be di computed with time T − t instead of t, and K = cS . Then to buy an option with strike cS t ,

S te−δ(T−t)N(dT−t1 ) − cS te−r(T−t)N(dT−t

2 )discount to t=0

−−−−−−−−−−−−−−−−→ S 0e−δT N(dT−t1 ) − cS 0e−r(T−t)−δtN(dT−t

2 )

14.6. For hedging: differentiate using∂N(di)∂S

=∂

∂S

∫ di(S )

−∞

e−(x)2/2√

2πdx =

e−(di)2/2√

2π

(∂di

∂S

)=

e−(di)2/2√

2π

(1

Sσ√

T

)=

e−(di)2/2

Sσ√

2πT.

15. Monte Carlo valuation

To simulate a lognormal random variable, let u ∼ U(0, 1) be uniform. Then N−1(u) ∼ N(0, 1) and eN−1(u) is lognormal.

If V(S T ,T ) is option payoff at T , the Monte Carlo time-0 price is V(S 0, 0) =e−rT

n

n∑i=1

V(S iT ,T )

For a European call, this would be C =e−rT

n

n∑i=1

max

0, S 0e(r−δ− σ2

2 )T+σ√

TZi − K

. If discounting is not used, replace r with α .

x is sample mean and σC = sn =(

1n∑n

i=1(xi − x)2)1/2

is sample stdev for one draw. Then σn =sn√

nis stdev of the Monte Carlo estimate.

To attain a given target standard error of σn, need n = (sn/σn)2 trials.Control variate method: compute X (unknown) using control C by assuming Xtrue − Xsimulated = Ctrue −Csimulated. (β = 1)

Boyle’s modification: Xtrue − Xsimulated = β (Ctrue −Csimulated), for β =〈X,Y〉Var(Y)

=

∑xiyi − nxy∑y2

i − ny2. Here, Xtrue is the estimate you compute.

Var(Xtrue) = Var(Xsimulated) + Var(Csimulated) − 2β〈Xsimulated,Csimulated〉, and min (Var(Xtrue)) = σ2Xsim

(1 − ρ2Xtrue ,Ctrue

).For discrete dividends: find periodic multipliers first. Then multiply starting price, subtract dividend, multiply again, subtract again, etc.

16. Brownian motion

At is stock price at time t, α is continuous rate of return, δ is continuous dividend rate, σ is volatility, N(x) is the normal (cumulative) distributionfunction. So total drift is α − δ =“capital gains return”=“contin. compounded expected incr.”.

If α or σ is given in terms of a time unit, use this to denominate time (e.g., σ = 2 per quarter =⇒ 1 year would be t = 4).For stock to exceed a continuously compounded annual return (yield) of y means S (t)

S (0) ≥ eyt , or ln(

S (t)S (0)

)≥ yt.

16.1. Arithmetic BM:. X(t) = αs + σZ(t). The increment X(t + h) − X(t) has mean µ = (α − δ)h and var= σ2h, so

P[X(t + h) < At+h |X(t) = At] = N(

At+h − At − µ

σ√

h

), dX(t) = α dt + σdZ(t)

16.2. Geometric BM:. Y(t) = eX(t). The increment Y(t + h) − Y(t) has mean µ = (α − δ − σ2

2 )h and var= σ2h, so

P[Y(t + h) < At+h |Y(t) = At] = P[

Y(t + h)Y(t)

<At+h

AtY(t) = At

]= N

(ln At+h − ln At − µ

σ√

h

), dY(t) = αY(t) dt + σX(t)dZ(t)

To go from geometric to arithmetic, subtract σ2

2 and replace X(t) with ln X(t) .

Covariance. For standard BM: 〈Zs,Zt〉 := Cov(Zs,Zt) = mins, t.

For Xt = X0 + αt + σZt , 〈Xs, Xt〉 = σ2〈Zs,Zt〉. For Xt = X0eµt+σZt , 〈Xs, Xt〉 = X20e(µ+ σ2

2 )(s+t)(eσ2〈Zs ,Zt〉 − 1).

17. Differentials

Watch for variance (σ2) given instead of volatility (σ). Var(ln S (t)|S (0)) = Var(ln F0,T (S )) = Var(ln FP0,T (S )) = σ2t.

ln X(t)|X(0) ∼ N(X(0) + (ξ − σ2

2 )t, σ2t)

X(t) = X(0)e(ξ−σ2

2 )t+σZ(t)

ln X(t) − ln X(0) = (ξ − σ2

2 )t + σZ(t) d (ln X(t)) = (ξ − σ2

2 ) dt + σdZ

18. Ito’s lemma

dC = CS dS + 12 CS S (dS )2 + Ct dt. If dS is arithmetic BM, (dS )2 = σ2dt. Don’t forget σ2!

(dt)2 = 0, dt × dZt = 0, (dZt)2 = dt, dZt × dZ′t = ρdt, where ρ is correlation coefficient.Ornstein-Uhlenbeck process: dXt = λ(α − Xt) dt + σ dZt , where λ is speed of reversion to mean α.

19. The Black-Scholes equation

Set (4) equal to 0 to obtain: σ2

2 CS S S 2 + (r − δ)CS S + Ct = rC or σ2

2 S 2Γ + (r − δ)S ∆ + θ = rC

Use it to price a claim C or to determine the parameters of a derivative security. Remember to annualize θ. Applies only when S t is GBM.

BS tells which parameters ensure C is arbitrage-free. C might only satisfy BS for certain r, δ, and σ, or maybe only if C itself pays dividends.

6

20. Sharpe ratio

Express process: dXX =

(α(t, X) − δ(t, X)

)dt + σ(t, X)dZ. Then φ =

α(t, X) − rσ(t, X)

.

Risk-free portfolios: buy xS of dSS = αS dt + σS dt and buy xQ of dQ

Q = αQ dt + σQ dt, to solve either of:

portfolio yield rate =αS S 0 xS + αQQ0 xQ

S 0 xS + Q0 xQ

= r or coefficient of uncertainty (dZ term) = σS S 0 xS + σQQ0 xQ = 0.

CAPM: φi =αi−rσi

= ρi,mαm−rσm

= φm, and βi = ρi,mσiσm

, so β relates the risk premiums: αi − r = β(αm − r).

21. Risk-neutral pricing and proportional portfolios

Risk-neutral process for stocks (Girsanov’s theorem): let φ = α−rσ be the Sharpe ratio, so dZ = dZ + φ dt is arithmetic BM. Then convert

true process =dSS

= (α − δ) dt − σφ dt + σφ dt + σ dZ = (α − δ − σφ)dt + σ(dZ + φ dt) = (r − δ) dt + σ dZ = risk-neutral process

Blended portfolio:dWW

=(ηα − δW + (1 − η)r

)dt + ησ dZ. Then

Wt

W0=

(S t

S 0

)ηe

(ηδS −δW +(1−η)(r+η σ

22

)t

22. Monomial securities Sa

The process is S at = S a

0ea(α−δ− σ2

2 )t+σaZt and its expected value is E[S a

T

]= S a

0ea(α−δ+(a−1) σ2

2 )T

Forward price is F0,T(S a

T

)= S a

0ea(r−δ+(a−1) σ2

2 )T . Prepaid forward price is FP0,T

(S a

T

)= e−rT S a

0ea(r−δ+(a−1) σ2

2 )T

The Ito process C = S a is given byd(S a)

S a =

(a(α − δ) + a(a − 1)σ

2

2

)dt + aσdZ, and ln S s

tS a

0= a ln S t

S 0∼ N(a(α − δ − σ2

2 ), a2σ2t).

Sharpe ratios γ−raσ = α−r

σ show that C = S a earns γ = a(α − r) + r. Then ln FP0,T (S a) = lnE [S a] − γ relates the above formulas.

Use α for S at and E

[S a

T

]; use r for F0,T

(S a

T

)and FP

0,T

(S a

T

). For options on S a, use σ = aσ in BS .

Suppose the price of euros C (in $) is given by dxx = ξdt + σdZ. Then the forward price of $1 (in C ) is F0,T

(S −1

T

)with r = ξ.

23. Stochastic integration

For differentiating an integral: ∂∂t

(∫ t0 f (s)dZs

)= f (t) dZt . Don’t forget the dZt!

To solve dXt + A(t)Xt dt = B(t) dt, use integrating factor ρ = e∫

A(t) dt .

Quadratic variation: for [0,T ], let ti = in T , so equally spaced increments. Then QV(X, 0,T ) = lim

n→∞

∑n

i=1

(X(ti) − X(ti−1)

)2.

Ornstein-Uhlenbeck: dXt = λ(α − Xt)dt + σ f (r) dZt ↔ Xt = X0e−λt + α(1 − e−λt) + σe−λt∫ t

0 eλs f (r(s)) dZs, for any function f .

24. Binomial trees for interest rates

24.1. Generic (nonBDT) interest rate trees. Let P(t,T ) be the price at time t for a bond maturing at time T .Given a continuously compounded interest rate tree with entries r (or prices P(t, t + 1) = e−r),

P(0,T ) =∑

γ∈Paths

P(γ)∏n∈γ

e−rn , where rn is the rate at node n in the path γ (5)

Prices at the end are just e−r or 11+r ; for intermediate prices, use (5).

This gives forward prices (not rates): P(1, 2) = 12 (Pu(1, 2) + Pd(1, 2)) . To obtain P(0, t), start in column t, walk back by avg & discount.

rates ruu

ru

ggggggggg

UUUUUUUUU

r∅

hhhhhhhhhh

VVVVVVVVVV rudrdu

rdWWWWWWWWW

iiiiiiiii

rdd

=⇒

prices e−ruu

Pu(1, 2) = e−ru2

(e−ruu + e−rud

) ddddddd

YYYYYYYYe−r∅|Paths|

∑∏e−rn

dddddddZZZZZZZ e−rud

e−rdu

Pd(1, 2) = e−rd2

(e−rdu + e−rdd

)ZZZZZZZ

eeeeeeee

e−rdd

To obtain the continuously compounded yield to maturity, solve P(0,T ) = e−rT for r, where T =number of periods (columns).To compute the premium of a call at time t, strike K: compute (P(0,T ) − K) ∧ 0 for column t. Then walk back by average & discount.For American options: immediate exercise at a node with rate r is worth e−r , so exercise a call early if e−r − K > (pulled-back value).

7

24.2. Black-Derman-Toy trees. Use P j(t − 1,T ) =1

1 + R j(t − 1, t)12

(P j(t,T ) + P j+1(t,T )

)to walk back, and P =

11 + R

to convert.

BDT tree R2he4σ√

h

Rhe2σ√

h

fffffff

XXXXXXX

R0

fffffffffXXXXXXXXXXX R2he2σ

√h

RhYYYYYYYYYYY

fffffffff

R2h

Construct with:P(0, 2)P(0, 1)

=12

11 + R1

+1

1 + R1e2σ√

h

Ratio in column t: rate j

rate j+1= e2σ

√h

To compute P(0,T ), start with prices P j(T − 1,T ) = 11+R j(T−1,T ) in column T . Then walk back by average & discount.

To compute F0,t(P(t,T )), start with discounted rates R j(T − 1,T ) = R1+R in column T . Then walk back by average & discount.

To compute a cap on a loan of L with strike K, start with last caplet (discounted) L R−K1+R in column T . Then walk back by average & discount.

To compute volatility for yields, compute (T − t)-year bond prices P j(t,T ) at time t and get rates R j(t,T ) = P j(t,T )1/(T−t).

Then σ = 12 ln

(R j(t,T )

R j+1(t,T )

). Note: R j(t,T ) is expressed in terms of one period and that

√h depends on tree, not on bond duration.

25. The Black formula

25.1. Bond options. Black formula for bond options: Ft = F0,t[P(t,T )] =P(0,T )P(0, t)

is the price at t to purchase a bond maturing at T .

Call : C(Ft ,K) = P(0, t)[Ft · N(d1) − K · N(d2)]

Put : P(Ft ,K) = P(0, t)[K · N(−d2) − Ft · N(−d1)]d1 =

ln(F/K) + σ2

2 t

σ√

t, d2 = d1 − σ

√t (6)

25.2. Caps via the Black formula. Each caplet is (1 + K) puts with strike 11+K . A Put with strike 1

1+K and exercise time t has value P(Ft ,1

1+K ).

Cap price = (1 + K)[P0 + P(F1,

11+K ) + · · · + P(FT ,

11+K )

]Caplet price = (1 + K)P(Ft ,

11+K ) from (6)

where P0 = ( 1bond price at 0 − 1) ∧ 0 = R0 ∧ 0 is the initial payoff, and P(Ft ,

11+K ) is computed using F0,t and σt .

The strike K is constant throughout, but Ft =P(0, t)

P(0, t − 1)and t changes for each caplet . Remember to multiply by (1 + K)P(0, t)!

26. Equilibrium interest rate models

26.1. The impossible model. Use continuously compounded interest P(0,T ) = e−rT .

Hedge ratio for duration-hedge: −N = −t1P(0, t1)t2P(0, t2)

. (< 0 =⇒ sell), N = Nt2 =number of t2-bonds

26.2. Equilibrium models. P = P(r, t,T ) is the price of a zero-coupon bond when the short-term rate is r, and dr = a(r) dt + σ(r) dZ.

dP = αP dt − qP dZ, α = α(r, t,T ) =1P

(a(r)Pr +

σ(r)2

2Prr + Pt

)q = q(r, t,T ) = −

σ(r)Pr

Pand Sharpe ratio φ(r, t) =

α(r, t,T ) − rq(r, t,T )

Risk-neutral process for bonds: dr = a(r) dt + σ(r) dZ 7→ risk-neutral interest rates dr = (a(r) + σ(r)φ(r)) dt + σ(r) dZ

Note that Pr < 0, so q = −σPr

P > 0, for α > r. Then the risk premium is −φσ = −(

1q

)(α − r)σ = P

Pr(α − r) < 0. So subtract −φσ for r-n.

Black-Scholes equation:σ2

2 S 2Γ + rS ∆ + θ = rC(S ) 7→ σ2

2 Prr + (a − σφ)Pr + Pt = rP

26.2.1. Rendelman-Bartter: dr = ar dt + σr dZ . Advantages: r ≥ 0, vol ∼ r. Disadvantages: no mean reversion, r unbounded.

26.2.2. Vasicek: dr = a(b − r) dt + σ dZ . Advantages: mean reversion. Disadvantages: vol is constant, can have r < 0.

P(r, t,T ) = Ae−rB , where A(t,T ) = er[B−(T−t)]−B2 σ24a , B = B(t,T ) = 1

a (1 − e−a(T−t)) , r = b +σ

aφ −

σ2

2a2 . (7)

Also, B(t,T ) = aT−t a and r is the formula for the yield to maturity on an infinitely-lived bond. The price model (7) satisfies

σ2

2Prr + (a(b − r) + σφ)Pr + Pt = rP, for Pr = ∆ = −BP and Prr = Γ = B2P.

For Vasicek, q = σB and α = −a(b − r)B + σ2

2 B2 +PtP .

For the case a = 0: dr = σdz, r is undefined, B = T − t, and A = exp(σφ B2

2 + σ2

2B3

3 ).

26.2.3. Cox-Ingersoll-Ross: dr = a(b − r) dt + σ√

r dZ . Advantages: mean reversion, vol ∼ r, r ≥ 0.

P(r, t,T ) = Ae−rB, where A = A(t,T ) =

[2γe(a−ϕ+γ)(T−t)/2

(a − ϕ + γ)(eγ(T−t) − 1) + 2γ

]2ab/σ2

, B = B(t,T ) =2(eγ(T−t) − 1)

(a − ϕ + γ)(eγ(T−t) − 1) + 2γ,

and γ =√

(a − ϕ)2 + 2σ2 and ϕ = φ(r, t)σ(r, t)/r and r =2ab

a − ϕ + γis yield to maturity on an infinitely-lived bond.

8

26.2.4. Facts for Vasicek and CIR models.

Higher volatility =⇒ lower yield. “Instantaneous rate of change” = drift term of dr.For CIR: higher risk premium ϕ =⇒ lower yield (same for Vasicek when a = 0).A = A(t,T ) = A(T − t) and B = B(t,T ) = B(T − t) depend only on T − t.

∆ = Pr = −BP and Γ = Prr = B2P , so q = −σPr

P =⇒ q(r, t,T ) = σ(r)B .

Vasicek: α(r, t,T ) = −a(b − r)B + σ2

2 B2 +PtP and the Sharpe ratio φ = α−r

−σB does not vary with r or t.

CIR: Sharpe ratio φ = ϕ√

rσ , so φ(r1, t) , φ(r2, t) and φ(r, t)σ(r) =

(ϕ√

rσ

) (σ√

r)

= ϕr, and φ√

rdoes not vary with r or t.

To convert Sharpes for CIR: since α(r)−rσ√

rB=

ϕ√

rσ = φ, can use

α1 − r1

r1= φB =

α2 − r2

r2

26.3. Delta-hedging.

duration-hedge :∑

(bond value) (bond duration) = 0 =⇒ need to sell N =t1P(r, 0, t1)t2P(r, 0, t2)

of bond 2

delta-hedge :∑

(bond value) (bond delta) = 0 =⇒ need to sell N =Pr(r, 0, t1)Pr(r, 0, t2)

of bond 2, Pr =∂P∂r

Direction and Convexity for option premiums as a function of strike price K, as in §2.

30 35 40 45 50 55 60 65 700

5

10

15

20

25

30

Strike price K

Cal

l pre

miu

m

S=40S=45S=50S=55S=60

30 35 40 45 50 55 60 65 700

5

10

15

20

25

Strike price K

Cal

l pre

miu

mt=0.005t=0.08t=0.25t=0.55t=1.0

30 35 40 45 50 55 60 65 700

5

10

15

20

25

Strike price K

Put p

rem

ium

t=0.005t=0.08t=0.25t=0.55t=1.0

30 35 40 45 50 55 60 65 700

5

10

15

20

25

30

Strike price K

Put p

rem

ium

S=40S=45S=50S=55S=60

2040

6080

00.20.40.60.81

Stock

price

0.1 ye

ar0.5

year

1 year

2 year

Delta

for a

call

2040

6080

010203040

0.1 ye

ar0.5

year

1 year

2 year

s

Call p

remium

as a

functi

on of

price

S

Stock

price

D

2040

6080

05101520250.1

year

0.5 ye

ar1.2

year

30 ye

ar

Vega

for a

call o

r a pu

t

00.1

0.20.3

0.40.5

5101520

0.1 ye

ar0.5

year

1 year

2 year

s

volat

ility

Call p

remium

as a

functi

on of

volat

ility s

v

2040

6080

−15

−10−50

0.1 ye

ar0.5

year

1 year

2 year

s

Theta

for a

call

Call p

remium

as a

functi

on of

t

01

23

05101520

Time t

o exp

iry

q

2040

6080

0204060800.1

year

0.5 ye

ar1 y

ear2 y

ears

Stock

price

Rho f

or a c

all

00.0

50.1

0.15

0.2510152025

0.1 ye

ar0.5

year

1 year

2 year

s Risk−

free r

ate r

Call p

remium

as a

functi

on of

rr

2040

6080

−100−8

0

−60

−40

−200

0.1 ye

ar0.5

year

1 year

2 year

s

Psi fo

r a ca

ll

Stock

price

00.0

50.1

0.15

0.25101520253035

0.1 ye

ar0.5

year

1 year

2 year

s

Divid

end y

ield r

ate d

Call p

remium

as a

functi

on of

d

Y

2040

6080

−1−0.8

−0.6

−0.4

−0.20

Delta

for a

put

Stock

price

0.1 ye

ar0.5

year

1 year

2 year

s

2040

6080

051015202530Pu

t prem

ium

0.1 ye

ar0.5

year

1 year

2 year

s

Stock

price

Put p

remium

as a

functi

on of

price

S

D

2040

6080

Delta

for a

call o

r a pu

t Stock

price

0.1 ye

ar0.5

year

1 year

2 year

s

2040

6080

0

0.02

0.04

0.06

0.08

Gamm

a for

a call

or a

put

0.1 ye

ar0.5

year

1.2 ye

ar10

year

Stock

price

G

2040

6080

−15

−10−505

0.1 ye

ar0.5

year

1 year

2 year

s

Stock

price

Theta

for a

put

01

23

051015

Strike

K=4

0Str

ike K

=50

Strike

K=5

5Str

ike K

=58

Strike

K=6

5

Put p

remium

as a

functi

on of

tq

Put p

remium

as a

functi

on of

r

00.0

50.1

0.15

0.20246810

0.1 ye

ar0.5

year

1 year

2 year

s

Risk−

free r

ate r

2040

6080

−100−8

0

−60

−40

−200

0.1 ye

ar0.5

year

1 year

2 year

s

Rho f

or a p

ut

Stock

pricer

2040

6080

0204060800.1

year

0.5 ye

ar1 y

ear2 y

ears

Stock

price

Psi fo

r a pu

t

Put p

remium

as a

functi

on of

d

00.0

50.1

0.15

0.20

0.050.10.150.20.25

0.1 ye

ar0.5

year

1 year

2 year

s

Divid

end y

ield r

ate d

Y

Strike

K=4

0Str

ike K

=50

Strike

K=5

5Str

ike K

=65

Strike

K=8

0

Normal Distribution Table

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.50000 0.50399 0.50798 0.51197 0.51595 0.51994 0.52392 0.52790 0.53188 0.53586

0.1 0.53983 0.54380 0.54776 0.55172 0.55567 0.55962 0.56356 0.56749 0.57142 0.57535

0.2 0.57926 0.58317 0.58706 0.59095 0.59483 0.59871 0.60257 0.60642 0.61026 0.61409

0.3 0.61791 0.62172 0.62552 0.62930 0.63307 0.63683 0.64058 0.64431 0.64803 0.65173

0.4 0.65542 0.65910 0.66276 0.66640 0.67003 0.67364 0.67724 0.68082 0.68439 0.68793

0.5 0.69146 0.69497 0.69847 0.70194 0.70540 0.70884 0.71226 0.71566 0.71904 0.72240

0.6 0.72575 0.72907 0.73237 0.73565 0.73891 0.74215 0.74537 0.74857 0.75175 0.75490

0.7 0.75804 0.76115 0.76424 0.76730 0.77035 0.77337 0.77637 0.77935 0.78230 0.78524

0.8 0.78814 0.79103 0.79389 0.79673 0.79955 0.80234 0.80511 0.80785 0.81057 0.81327

0.9 0.81594 0.81859 0.82121 0.82381 0.82639 0.82894 0.83147 0.83398 0.83646 0.83891

1.0 0.84134 0.84375 0.84614 0.84849 0.85083 0.85314 0.85543 0.85769 0.85993 0.86214

1.1 0.86433 0.86650 0.86864 0.87076 0.87286 0.87493 0.87698 0.87900 0.88100 0.88298

1.2 0.88493 0.88686 0.88877 0.89065 0.89251 0.89435 0.89617 0.89796 0.89973 0.90147

1.3 0.90320 0.90490 0.90658 0.90824 0.90988 0.91149 0.91309 0.91466 0.91621 0.91774

1.4 0.91924 0.92073 0.92220 0.92364 0.92507 0.92647 0.92785 0.92922 0.93056 0.93189

1.5 0.93319 0.93448 0.93574 0.93699 0.93822 0.93943 0.94062 0.94179 0.94295 0.94408

1.6 0.94520 0.94630 0.94738 0.94845 0.94950 0.95053 0.95154 0.95254 0.95352 0.95449

1.7 0.95543 0.95637 0.95728 0.95818 0.95907 0.95994 0.96080 0.96164 0.96246 0.96327

1.8 0.96407 0.96485 0.96562 0.96638 0.96712 0.96784 0.96856 0.96926 0.96995 0.97062

1.9 0.97128 0.97193 0.97257 0.97320 0.97381 0.97441 0.97500 0.97558 0.97615 0.97670

2.0 0.97725 0.97778 0.97831 0.97882 0.97932 0.97982 0.98030 0.98077 0.98124 0.98169

2.1 0.98214 0.98257 0.98300 0.98341 0.98382 0.98422 0.98461 0.98500 0.98537 0.98574

2.2 0.98610 0.98645 0.98679 0.98713 0.98745 0.98778 0.98809 0.98840 0.98870 0.98899

2.3 0.98928 0.98956 0.98983 0.99010 0.99036 0.99061 0.99086 0.99111 0.99134 0.99158

2.4 0.99180 0.99202 0.99224 0.99245 0.99266 0.99286 0.99305 0.99324 0.99343 0.99361

2.5 0.99379 0.99396 0.99413 0.99430 0.99446 0.99461 0.99477 0.99492 0.99506 0.99520

2.6 0.99534 0.99547 0.99560 0.99573 0.99585 0.99598 0.99609 0.99621 0.99632 0.99643

2.7 0.99653 0.99664 0.99674 0.99683 0.99693 0.99702 0.99711 0.99720 0.99728 0.99736

2.8 0.99744 0.99752 0.99760 0.99767 0.99774 0.99781 0.99788 0.99795 0.99801 0.99807

2.9 0.99813 0.99819 0.99825 0.99831 0.99836 0.99841 0.99846 0.99851 0.99856 0.99861

3.0 0.99865 0.99869 0.99874 0.99878 0.99882 0.99886 0.99889 0.99893 0.99896 0.99900

3.1 0.99903 0.99906 0.99910 0.99913 0.99916 0.99918 0.99921 0.99924 0.99926 0.99929

3.2 0.99931 0.99934 0.99936 0.99938 0.99940 0.99942 0.99944 0.99946 0.99948 0.99950

3.3 0.99952 0.99953 0.99955 0.99957 0.99958 0.99960 0.99961 0.99962 0.99964 0.99965

3.4 0.99966 0.99968 0.99969 0.99970 0.99971 0.99972 0.99973 0.99974 0.99975 0.99976

3.5 0.99977 0.99978 0.99978 0.99979 0.99980 0.99981 0.99981 0.99982 0.99983 0.99983

3.6 0.99984 0.99985 0.99985 0.99986 0.99986 0.99987 0.99987 0.99988 0.99988 0.99989

3.7 0.99989 0.99990 0.99990 0.99990 0.99991 0.99991 0.99992 0.99992 0.99992 0.99992

3.8 0.99993 0.99993 0.99993 0.99994 0.99994 0.99994 0.99994 0.99995 0.99995 0.99995

3.9 0.99995 0.99995 0.99996 0.99996 0.99996 0.99996 0.99996 0.99996 0.99997 0.99997

Inverse lookup: P(Z<z) 0.800 0.850 0.900 0.950 0.975 0.990 0.995 0.999z 0.84162 1.03643 1.28155 1.64485 1.95996 2.32635 2.57583 3.09023