1. Put-call parity
C(K, T ) − P(K, T ) = e−rT (F0,T − K) = F0P − Ke−rT
P = S e−δT , so C(K, T ) − P(K, T ) = S e−δT − Ke−rT .
Continuous: F0,T
0
0
P
P −rt
P
Discrete: F0,T = S 0 − dt e = S 0 − PV(dividends), so C(K, T ) − P(K, T ) = S 0 − dt e−rt − Ke−rT .
,K)
1
1
To create one unit of synthetic stock: S 0 = C(S ,K)−P(S
+ Ke
, so buy e−δT
calls, sell e−δT
puts, buy
e−δT
e−δT
B
B
To create a synthetic treasury with maturity value $B, buy KeδT or K+CV(divs) shares of stock.
−rT
Ke−rT
e−δT
bonds.
2. Comparing options
For a Call: higher strike =⇒ lower price. For a Put: higher strike =⇒ higher price. −1 ≤
American options are worth more than European.
∂C
∂K
≤ 0 and 0 ≤
∂P
∂K
≤ 1.
P
− Ke−rT )
S ≥ C Amer (S , K, T ) ≥ C Eur (S , K, T ) ≥ max(0, F0,T
P
K ≥ PAmer (S , K, T ) ≥ PEur (S , K, T ) ≥ max(0, Ke−rT − F0,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 −δh
e
,
S (u − d)
B=
# bonds to buy:
uCd − dCu −rh
e
,
u−d
p∗ =
risk-neutral probability:
e(r−δ)h − d
1
=
√ .
u−d
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
Remember: to pull back to previous node, discount with
.
Option price: C = S ∆ + B = e−rh (p∗ Cu + (1 − p∗ )Cd )
e−rh ,
not with the e−(r−δ)h that appears in the expression for p∗ !
4. Binomial trees — general
P For European trees, use binomial theorem shortcut: nk=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−d
,
u−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∗1 S T(1) + p∗2 S T(2) + (1 − p∗1 − p∗2 )S T(3) = S 0 e(r−δ)t .
Here S 0 is the spot price, S T(i) 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−d
eγh =
S ∆ αh
B
e +
erh
S∆ + B
S∆ + B
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.
p∗
1 − p∗
Qi is partial expected value of PV(1), so Qu = pUu =
, Qd =
, where Ui is PV(1), given that the stock is in state i.
1+r
1+r
Current value of a stock: C0 = Qu Cu + Qd Cd = pUu Cu + (1 − p)Ud Cd .
Effective annual rates of return. True: 1 + α =
Risk-neutral probabilities: p∗ =
E[C]
pCu + (1 − p)Cd
E∗ [C]
p∗ Cu + (1 − p∗ )Cd
1
=
vs. risk-neutral: 1 + r =
=
=
C0
C0
C0
C0
Qu + Qd
Qu
pUu
=
,
Qu + Qd
pUu + (1 − p)Ud
In a binomial tree, set up and solve:
S u Qu + S d Qd = S 0 ,
1 − p∗ =
(1 − p)Ud
pUu + (1 − p)Ud
C u Qu + C d Qd = C 0 ,
1
Qu + Qd =
← utility-weighted avgs of true probs
e−r .
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.
√
±σ h
Cox-Ross-Rubenstein tree centered on 1, so u/d = e
. Allows arbitrage iff eσ
σ2
√
h
√
< e(r−δ)h .
√
σ2
Lognormal tree centered on (r − δ − σ2 )h, so u/d = e(r−δ− 2 )h±σ h . Allows arbitrage iff e(r−δ− 2 )h+σ h < e(r−δ)h .
Also called Jarrow-Rudd. Makes p very close to 0.5.
(r−δ)h
forward price
(F /S )−d
= e u−d−d in each case.
For all trees, risk-neutral probability comes by looking at current price , so p∗ = T u−d0
2
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
x̄ =
n
ln(S n /S 0 )
1X
τi =
n i=1
n
and
µ̄2 =
n
1X 2
τ
n i=1 i
and
unbiased sample variance = s2 =
n µ̄2 − x̄2
n−1
√
Then the estimated annualized unbiased sample volatility is s p, where p is number of periods per year (h =
1
p ).
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 σ2 t := Var[ln FtP (S t )] is variance.
2
For the associated arithmetic process, ln(S t /S 0 ) ∼ N mt, v2 t , where m = α − δ − σ2 is the expected value of the annual rate of return.
Pr(S t > K) = N(d̂2 ) and Pr(S t < K) = N(−d̂2 ) , where d̂2 has α in place of r: d̂2 =
ln(S 0 /K) + (α − δ −
√
σ t
σ2
2 )t
Median stock price: M = S 0 emt = E[S t ]e−(σ /2)t < E[S t ] = S 0 e(α−δ)t = mean stock price .
√
−ζt
Confidence intervals: let ζt = σ t N −1 1+p
= S 0 emt−ζt ≤ S t ≤ S 0 emt+ζt = Meζt with probability p.
2 . Then have Me
2
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 0(α−δ)t N(−d̂1 )
d̂1 )
d̂1 )
Conditional expected stock price: E[S t |S t > K] = S e(α−δ)t N(
and E[S t |S t < K] = S e(α−δ)t N(−
; see §14 to compare with AoNs.
N(d̂ )
N(−d̂ )
2
2
Expected payoff Call: E[max{0, S t − K}] = S e
N(d̂1 ) − KN(d̂2 ) Put: E[max{0, K − S t }] = KN(−d̂2 ) − S e(α−δ)t N(−d̂1 )
R∞
d̂1 )
d̂1 )
C = e−rt K (S t − K)g∗ (S t ) dS t = e−rt E∗ [S − K|S > K]P∗ [S > K] = S e(α−r−δ)t N(
N(d2 ) − Ke−rt N(−
N(−d2 ) = B-S, for α = r.
N(d̂ )
N(−d̂ )
(α−δ)t
2
8. Fitting stock prices to a lognormal distribution
!
s2
p , where x̄ and s and p =
Estimate of growth rate α: ᾱ = x̄ +
2
Confidence interval for estimated α: ᾱ ±
σ
√
N −1 ( 1+P
2 ),
n
1
h
2
are as in (1). This is for estimating α alone (i.e., dividends removed).
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.
2
ln F P (S )/F P (K) + σ2 T
√
P
P
and d2 = d1 − σ T .
General Black-Scholes C(S , K, σ, T, r, δ) = F (S )N(d1 ) − F (K)N(d2 ) , where d1 =
√
σ T
√
2
1
For stocks, F P (S ) = S e−δT and F P (K) = Ke−rT , so d1 = √
ln KS + r − δ + σ2 T and d2 = d1 − σ T and
σ T
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−rf T N(d1 ) − Ke−rT N(d2 ),
Put(x, K) = Ke−rT N(−d2 ) − se−rf T N(−d1 ).
√
2
1
For futures, δ = r, so d1 = √
ln KS + σ2 T . When K = S , this becomes d1 = σ T /2. The Black formula is:
σ T
Call(x, K) = Fe−rT N(d1 ) − Ke−rT N(d2 ),
Put(x, K) = Ke−rT N(−d2 ) − Fe−rT N(−d1 ).
(1)
3
10. The Black-Scholes formula: Greeks
∆
Γ
vega
θ
ρ
∆ put = ∆call − e−δT
Γ put = Γcall
vega put = vegacall
1
θ put = θcall + 365
(rKe−rT − δS e−δT )
T
ρ put = ρcall − 100 Ke−rT
S-shaped curve
symmetric hump, peak to left of stock price, further left with higher T
asymmetric hump, peak like Γ
Sulcus for short lives, gradual decrease for long lives.
increasing curve (pos for calls, neg for puts)
T
Ψ
Ψ put = + 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
S∆
%option change
= ε∆/C
ε/S = %stock change = percentage risk. Dollar risk is ≈ ε∆.
C
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| .
Elasticity of the option with premium C is Ω =
Sharpe ratio is
γ−r
α−r
.
=
σoption
σstock
γ − r = Ω(α − r) is true instantaneously, and follows from eγh = erh + Ω(eαh − erh ).
X
ai ∆Ci .
For a portfolio consisting of ai of options Ci on the same stock: A greek for the porfolio is computed by ∆portfolio =
P
S ∆portfolio
S ai ∆Ci
Elasticity for the porfolio is computed by Ωportfolio =
= P
, where Cportfolio is the value of the portfolio of options.
Cportfolio
ai Ci
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 < T i , use BS to compute
(portfolio value at time t) − (portfolio value at time 0)ert =
|
{z
} |
{z
}
earned from investing
could’ve earned from lending
n
X
Ci (S , K, T i ) −
m
X
C j (S , K, t)ert +(cash settlements)
(2)
j=1
i=1
|
{z
still extant
} |
{z
}
expired at t
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.
1 Pn
2
Historical volatility: this is (1) (annualized) with x̄ = 0, so just the second moment σ̂2H = n−1
i=1 τi , 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:
1
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 = 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 < St < S + Sσ t .
12.2. Delta-gamma-theta approximation (∆Γθ). C(S t + ε) = C(S ) + ∆ε + 21 Γε2 + θt + Taylor remainder error.
√
√
The whole point of ∆-hedging is to separate ε (or S σ h) from ∆; putting ∆Γθ into (3) cancels the ∆ε terms. For ε = S σ h, get
" 2
#
" 2
#
ε
σ 2
market-maker profit = −
Γ + (r − δ)S ∆h − rC(S ) + θh = −
S Γ + (r − δ)S ∆ − rC(S ) + θ h
2
2
(3)
(4)
Holds for most options (exception: when early exercise is optimal). To get the Black-Scholes equation, set market-maker profit = 0; see §19.
C e−δt −C e−δt
d,h)
12.3. Greeks for binomial trees. Compute ∆0 = ∆(S , 0) = u S u−S dd
as in §3 and approximate Γ0 = Γ(0, S ) with Γ(S , h) = ∆(S u,h)−∆(S
.
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 ) + 21 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 2 2
S (xi − 1)Γh,
2
Note: variance, not volatility!
and
Var(Rh,i ) =
1
2
2
σ2 S 2 Γh
and
annual variance =
1
2
2
σ2 S 2 Γ h
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 options
as insurance, (iv) sell the hedging error (variance swap). Market-makers cannot be gamma-neutral in the aggregate; besides, this hedge incurs a
bid-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 :
stock
Cbuy
Csell
stock
∆ : x1 + ∆buy x2 = ∆ sell
Delta-gamma hedging
Γ:
compare to Delta-hedging
∆:
Γbuy x2 = Γ sell
Csell
x1 = ∆ sell
13. Asian, barrier, and compound options
13.1. Asian options. Arithmetic average A(S ) =
1
T
PT
t=1
S t and geometric average G(S ) =
arithmetic
q
T QT
t=1
S t . Ignore initial price (exclude S 0 )
geometric
average price
Call = max(0, A(S ) − K),
average strike
Call = max(0, S T − A(S )),
Put = max(0, K − A(S ))
Call = max(0, G(S ) − K),
Put = max(0, A(S ) − S T )
Put = max(0, K − G(S ))
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.
t
for t = 1, 2, 3. These are
For claims on dS t = αS t dt + σS t dZt formulated as geometric means, like G = (S 1 S 2 S 3 )1/3 , introduce Qt := SSt−1
normal with the same µ, σ2 and independent. Then S 1 S 2 S 3 = S 0 Q1 S 1 Q2 S 2 Q3 = · · · = S 0 Q31 Q22 Q13 , so ln
with parameters 2µ and
(1 + ( 32 )2
+ ( 13 )2 )σ2
=
14 2
9 σ .
G
S0
= ln Q1 +
2
3
ln Q2 + 13 Q3 is normal
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. max{S , K} = S + max{0, K − S } = K + max{0, S − K}, and max{cS , cK} = c max{S , K} for c > 0, and min{S , K} +
max{S , K} = S + K, and max{S , 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 out
a 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 greater
of exercise value and option for remaining period: max{S t + D − K, C(S t , K, T − t)}. If S t is cum-dividend and S t0 = S t − D is ex-dividend, then
time t : S t0 + D − K + max{0, P(S t0 , 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 t0 , D − K(1 − e−r(T −t) )) . For P, use F P (S ) = S 0 − De−rt and F P (K) = Ke−rT .
14. Gap, exchange, and other options
14.1. All-or-nothing options. Recall from §7 that Pr(S t > K) = N(d̂2 ) and Pr(S t < K) = N(−d̂2 ).
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]
Value
S e−δT N(d1 )
S e−δT N(−d1 )
Ke−rT N(d2 )
Ke−rT N(−d2 )
S 0 e(α−δ)t N(d̂1 )
N(d̂1 )
S 0 e(α−δ)t N(
d̂ )
2
PE[S t |S t < K]
E[S t |S t < K]
S 0 e(α−δ)t N(−d̂1 )
d̂1 )
S 0 e(α−δ)t N(−
N(−d̂ )
2
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 it
to determine the probabilities N(di ). Amount of payoff is determined by Kstrike , so use it to compute option price:
where d1 =
1
√
σ T
Call = S e−δt N(d1 ) − Kstrike e−rt N(d2 ),
Put = Kstrike e−rt N(−d2 ) − S e−δt N(−d1 ).
√
2
ln(S /Ktrigger ) + r − δ + σ2 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 + σ2K − 2ρσS σK
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 diT −t be di computed with time T − t instead of t, and K = cS . Then to buy an option with strike cS t ,
discount to t=0
S t e−δ(T −t) N(d1T −t ) − cS t e−r(T −t) N(d2T −t ) −−−−−−−−−−−−−−−−→ S 0 e−δT N(d1T −t ) − cS 0 e−r(T −t)−δt N(d2T −t )
14.6. For hedging: differentiate using
∂
∂N(di )
=
∂S
∂S
Z
di (S )
−∞
!
!
2
2
2
2
e−(x) /2
e−(di ) /2
e−(di ) /2
e−(di ) /2 ∂di
1
= √
dx = √
.
√
√ =
√
∂S
2π
2π
2π S σ T
S σ 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
n
e−rT X
V(S Ti , T )
If V(S T , T ) is option payoff at T , the Monte Carlo time-0 price is V(S 0 , 0) =
n i=1
For a European call, this would be C =
x̄ is sample mean and σC = sn =
is lognormal.
(
)
n
√
σ2
e−rT X
max 0, S 0 e(r−δ− 2 )T +σ T Zi − K . If discounting is not used, replace r with α .
n i=1
P
1 n
n
−1 (u)
i=1 (xi
− x̄)2
1/2
sn
is sample stdev for one draw. Then σn = √ is stdev of the Monte Carlo estimate.
n
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)
P
hX, Yi
xi yi − n x̄ȳ
Boyle’s modification: Xtrue − Xsimulated = β (Ctrue − Csimulated ), for β =
. Here, Xtrue is the estimate you compute.
= P 2
Var(Y)
yi − nȳ2
Var(Xtrue ) = Var(Xsimulated ) + Var(Csimulated ) − 2βhXsimulated , Csimulated i, and min (Var(Xtrue )) = σ2X (1 − ρ2Xtrue ,Ctrue ).
sim
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) distribution
function. 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).
(t) (t)
For stock to exceed a continuously compounded annual return (yield) of y means SS (0)
≥ eyt , or ln SS (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= σ2 h, so
!
At+h − At − µ
P[X(t + h) < At+h |X(t) = At ] = N
,
dX(t) = α dt + σdZ(t)
√
σ h
16.2. Geometric BM:. Y(t) = eX(t) . The increment Y(t + h) − Y(t) has mean µ = (α − δ − σ2 )h and var= σ2 h, so
"
!
#
Y(t + h)
At+h
ln At+h − ln At − µ
P[Y(t + h) < At+h | Y(t) = At ] = P
<
Y(t) = At = N
,
dY(t) = αY(t) dt + σX(t)dZ(t)
√
Y(t)
At
σ h
2
To go from geometric to arithmetic, subtract
σ2
2
and replace X(t) with ln X(t) .
Covariance. For standard BM: hZ s , Zt i := Cov(Z s , Zt ) = min{s, t}.
For Xt = X0 + αt + σZt ,
For Xt = X0 eµt+σZt ,
hX s , Xt i = σ2 hZ s , Zt i.
hX s , Xt i = X02 e(µ+
σ2
2
)(s+t)
(eσ
2 hZ
s ,Zt i
− 1).
17. Differentials
Watch for variance (σ2 ) given instead of volatility (σ).
ln X(t)|X(0) ∼ N X(0) + (ξ −
ln X(t) − ln X(0) = (ξ −
σ2
2 )t
2
σ2
2 )t, σ t
P (S )) = σ2 t.
Var(ln S (t)|S (0)) = Var(ln F0,T (S )) = Var(ln F0,T
X(t) = X(0)e(ξ−
+ σZ(t)
σ2
2 )t+σZ(t)
d (ln X(t)) = (ξ −
σ2
2 ) dt
+ σdZ
18. Itô’s lemma
dC = CS dS + 12 CS S (dS )2 + Ct dt.
If dS is arithmetic BM, (dS )2 = σ2 dt. Don’t forget σ2 !
= 0, dt × dZt = 0, (dZt = dt, dZt × dZt0 = 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.
(dt)2
)2
19. The Black-Scholes equation
Set (4) equal to 0 to obtain:
2
σ2
2 CS S S
+ (r − δ)CS S + Ct = rC or
σ2 2
2 S Γ + (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:
dX
X
α(t, X) − r
.
= α(t, X) − δ(t, X) dt + σ(t, X)dZ. Then φ =
σ(t, X)
portfolio yield rate =
CAPM: φi =
αi −r
σi
= αS dt + σS dt and buy xQ of
dS
S
Risk-free portfolios: buy xS of
αS S 0 xS + αQ Q0 xQ
S 0 xS + Q0 xQ
=r
dQ
Q
= αQ dt + σQ dt, to solve either of:
coefficient of uncertainty (dZ term) = σS S 0 xS + σQ Q0 xQ = 0.
or
= ρi,m ασmm−r = φm , and βi = ρi,m σσmi , so β relates the risk premiums: αi − r = β(αm − r).
21. Risk-neutral pricing and proportional portfolios
α−r
σ
Risk-neutral process for stocks (Girsanov’s theorem): let φ =
true process =
Blended portfolio:
dS
S
be the Sharpe ratio, so dZ̃ = dZ + φ dt is arithmetic BM. Then convert
= (α − δ) dt − σφ dt + σφ dt + σ dZ = (α − δ − σφ)dt + σ(dZ + φ dt) = (r − δ) dt + σ dZ̃ = risk-neutral process
dW Wt
St
= ηα − δW + (1 − η)r dt + ησ dZ. Then
=
W
W0
S0
!η
e
2
ηδS −δW +(1−η)(r+η σ2
t
22. Monomial securities Sa
The process is S ta = S 0a ea(α−δ−
σ2
2
)t+σaZt
h i
σ2
and its expected value is E S Ta = S 0a ea(α−δ+(a−1) 2 )T
σ2
σ2
P
Forward price is F0,T S Ta = S 0a ea(r−δ+(a−1) 2 )T . Prepaid forward price is F0,T
S Ta = e−rT S 0a ea(r−δ+(a−1) 2 )T
d(S a ) 2
2
Ss
The Itô process C = S a is given by
= a(α − δ) + a(a − 1) σ2 dt + aσdZ, and ln S at = a ln SS 0t ∼ N(a(α − δ − σ2 ), a2 σ2 t).
a
S
0
α−r
a
P
a
a
Sharpe ratios γ−r
aσ = σ show that C = S earns γ = a(α − r) + r. Then ln F 0,T (S ) = ln E [S ] − γ relates the above formulas.
h i
P
Use α for S ta and E S Ta ; use r for F0,T S Ta and F0,T
S Ta . For options on S a , use σ̂ = aσ in BS .
−1
Suppose the price of euros C (in $) is given by dx
x = ξdt + σdZ. Then the forward price of $1 (in C ) is F 0,T S T with r = ξ.
23. Stochastic integration
For differentiating an integral:
∂
∂t
R t
0
f (s)dZ s = f (t) dZt . Don’t forget the dZt !
To solve dXt + A(t)Xt dt = B(t) dt, use integrating factor ρ = e
R
A(t) dt
.
Xn 2
so equally spaced increments. Then QV(X, 0, T ) = lim
X(ti ) − X(ti−1 ) .
i=1
n→∞
Rt
Ornstein-Uhlenbeck: dXt = λ(α − Xt )dt + σ f (r) dZt ↔ Xt = X0 e−λt + α(1 − e−λt ) + σe−λt 0 eλs f (r(s)) dZ s , for any function f .
Quadratic variation: for [0, T ], let ti =
i
nT,
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 ),
X
Y
P(0, T ) =
P(γ)
e−rn ,
where rn is the rate at node n in the path γ
γ∈Paths
Prices at the end are just e−r or
1
1+r ;
(5)
n∈γ
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.
gggg ruu
gUgggg
U
UUUU
hhhh
UUU r
hhhh
ud
r∅ V
VVVV
iiii rdu
VVVV
V r Wiiiii
d WWW
WWWWW
prices
rates
hh ru
rdd
=⇒
e−r∅
|Paths|
PQ
e−rn
ddddddd
ZZZZZZZ
Pu (1, 2) =
e−ru
2
e−ruu
Pd (1, 2) =
e−rd
2
e−rdu + e−rdd
+ e−rud
−ruu
ddddddd e
YYYYYY
YY e−rud
ee −r
eeeeee e du
ZZZZZZZ
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
1
to convert.
P j (t, T ) + P j+1 (t, T ) to walk back, and P =
1 + R j (t − 1, t) 2
1+R
√
f R2h e4σ h
ffffff
XXXXXX
ff Rh
X
√
fffff
f
f
R0 XXXXX
f R2h e2σ h
f
XXXXXX
f
f
f
f
ffYfYY
Rh Y
YYYYYYY
BDT tree
Construct with:
√
e2σ h
P(0, 2) 1
=
P(0, 1) 2
Ratio in column t:
1
1+R j (T −1,T )


1
1

+
√ 
1 + R1 1 + R1 e2σ h
√
rate j
= e2σ h
rate j+1
R2h
To compute P(0, T ), start with prices P j (T − 1, T ) =



in column T . Then walk back by average & discount.
R
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 ) = 1+R
To compute a cap on a loan of L with strike K, start with last caplet (discounted) L R−K
1+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) .
√
R j (t,T )
Then σ = 21 ln R j+1
h depends on tree, not on bond duration.
(t,T ) . Note: R j (t, T ) is expressed in terms of one period and that
25. The Black formula
25.1. Bond options. Black formula for bond options: Ft = F0,t [P(t, T )] =
P(0, T )
is the price at t to purchase a bond maturing at T .
P(0, t)
Call : C(Ft , K) = P(0, t)[Ft · N(d1 ) − K · N(d2 )]
d1 =
Put : P(Ft , K) = P(0, t)[K · N(−d2 ) − Ft · N(−d1 )]
25.2. Caps via the Black formula. Each caplet is (1 + K) puts with strike
h
i
1
1
Cap price = (1 + K) P0 + P(F1 , 1+K
) + · · · + P(FT , 1+K
)
where P0 =
1
( bond price
at 0
− 1) ∧ 0 = R0 ∧ 0 is the initial payoff, and
The strike K is constant throughout, but Ft =
1
1+K .
ln(F/K) +
√
σ t
σ2
2 t
A Put with strike
,
1
1+K
√
d2 = d1 − σ t
and exercise time t has value P(Ft ,
Caplet price = (1 + K)P(Ft ,
1
P(Ft , 1+K
)
(6)
1
1+K )
1
1+K ).
from (6)
is computed using F0,t and σt .
P(0, t)
and t changes for each caplet . Remember to multiply by (1 + K)P(0, t)!
P(0, t − 1)
26. Equilibrium interest rate models
26.1. The impossible model. Use continuously compounded interest P(0, T ) = e−rT .
t1 P(0, t1 )
Hedge ratio for duration-hedge: −N = −
. (< 0 =⇒ sell), N = Nt2 =number of t2 -bonds
t2 P(0, t2 )
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.
!
σ(r)2
1
σ(r)Pr
α(r, t, T ) − r
a(r)Pr +
dP = αP dt − qP dZ, α = α(r, t, T ) =
Prr + Pt
q = q(r, t, T ) = −
and Sharpe ratio φ(r, t) =
P
2
P
q(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) dZ̃
1
P
r
Note that Pr < 0, so q = − σP
P > 0, for α > r. Then the risk premium is −φσ = − q (α − r)σ = Pr (α − r) < 0. So subtract −φσ for r-n.
Black-Scholes equation: σ2 S 2 Γ + rS ∆ + θ = rC(S )
2
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 ,
2 σ2
4a
where A(t, T ) = er[B−(T −t)]−B
,
B = B(t, T ) = a1 (1 − e−a(T −t) ) ,
r =b+
σ
σ2
φ− 2 .
a
2a
Also, B(t, T ) = āT −t a and r is the formula for the yield to maturity on an infinitely-lived bond. The price model (7) satisfies
σ2
Prr + (a(b − r) + σφ)Pr + Pt = rP,
2
For Vasicek, q = σB and α = −a(b − r)B +
σ2
2
B2 +
Pt
P.
for Pr = ∆ = −BP and Prr = Γ = B2 P.
2
For the case a = 0: dr = σdz, r̄ is undefined, B = T − t, and A = exp(σφ B2 +
σ2 B3
2 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 ,
and γ =
where A = A(t, T ) =
2γe(a−ϕ+γ)(T −t)/2
(a − ϕ + γ)(eγ(T −t) − 1) + 2γ
p
(a − ϕ)2 + 2σ2 and ϕ = φ(r, t)σ(r, t)/r and r =
#2ab/σ2
,
B = B(t, T ) =
2(eγ(T −t) − 1)
,
(a − ϕ + γ)(eγ(T −t) − 1) + 2γ
2ab
is yield to maturity on an infinitely-lived bond.
a−ϕ+γ
(7)
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.
r
∆ = Pr = −BP and Γ = Prr = B2 P , so q = − σP
P =⇒ q(r, t, T ) = σ(r)B .
α−r
and the Sharpe ratio φ = −σB
does not vary with r or t.
√ √ r
, so φ(r1 , t) , φ(r2 , t) and φ(r, t)σ(r) = ϕ σ σ r = ϕr, and √φr does not vary with r or t.
Vasicek: α(r, t, T ) = −a(b − r)B +
CIR: Sharpe ratio φ = ϕ
√
r
σ
To convert Sharpes for CIR: since
σ2
2
B2 +
α(r)−r
√
σ̄ rB
=
Pt
P
√
ϕ r
σ̄
= φ, can use
α1 − r1
α2 − r2
= φB =
r1
r2
26.3. Delta-hedging.
duration-hedge :
X
(bond value) (bond duration) = 0
=⇒
need to sell
N=
t1 P(r, 0, t1 )
t2 P(r, 0, t2 )
of bond 2
delta-hedge :
X
(bond value) (bond delta) = 0
=⇒
need to sell
N=
Pr (r, 0, t1 )
Pr (r, 0, t2 )
of bond 2,
Direction and Convexity for option premiums as a function of strike price K, as in §2.
30
20
15
10
35
40
45
50
55
Strike price K
60
65
35
40
45
50
55
Strike price K
60
65
70
45
50
55
Strike price K
60
65
70
25
S=40
S=45
S=50
S=55
S=60
20
15
t=0.005
t=0.08
t=0.25
t=0.55
t=1.0
20
Put premium
25
Put premium
10
0
30
70
30
10
15
10
5
5
0
30
15
5
5
0
30
t=0.005
t=0.08
t=0.25
t=0.55
t=1.0
20
Call premium
25
Call premium
25
S=40
S=45
S=50
S=55
S=60
35
40
45
50
55
Strike price K
60
65
70
0
30
35
40
Pr =
∂P
∂r
Stock price
60
40
Stock price
D
60
40
Stock price
60
Put premium
Put premium as a function
of price S
0
20
5
10
15
20
25
30
−1
20
−0.8
−0.6
−0.4
−0.2
0
Delta for a put
40 Stock price 60
0.1 year
0.5 year
1 year
2 years
Call premium as a function of price S
0
20
10
20
30
40
0
20
0.2
0.4
80
0.1 year
0.5 year
1 year
2 years
0.1 year
0.5 year
1 year
2 years
80
80
80
v
60
0.1
G
0.2
0.3
volatility
Stock price
20
40
Stock price
Delta for a call or a put
40
Gamma for a call or a put
0
0.1 year
0.5 year
1 year
2 years
60
60
Call premium as a function of volatility s
0
20
0.02
0.04
0.06
0.08
5
10
15
20
40
Vega for a call or a put
0
20
5
10
15
0.1 year
0.5 year
1 year
2 year
0.6
25
20
40
D
0.8
1
Delta for a call
0.1 year
0.5 year
1 year
2 years
0.1 year
0.5 year
1.2 year
10 year
0.4
0.1 year
0.5 year
1.2 year
30 year
80
80
0.5
80
0
5
10
15
40
Stock price
q
Time to expiry
0
1
Put premium as a function of t
−15
20
1
0.1 year
0.5 year
1 year
2 years
Theta for a put
0
−10
40
0.1 year
0.5 year
1 year
2 years
q
Call premium as a function of t
−5
0
5
0
5
10
15
20
−15
20
−10
−5
0
Theta for a call
2
60
2
60
Strike K=40
Strike K=50
Strike K=55
Strike K=58
Strike K=65
Strike K=40
Strike K=50
Strike K=55
Strike K=65
Strike K=80
3
80
3
80
Stock price
Stock price
0.15
0.2
0
0.1
Risk−free rate r
0.15
0.2
0.25
0
0.05
Psi for a put
0
Stock price
0.05
60
0.1
0.15
Dividend yield rate d
Put premium as a function of d
40
Y
0.1 year
0.5 year
1 year
2 years
0.1 year
0.5 year
1 year
2 years
0.1 year
0.5 year
1 year
2 years
0.1
0.15
Dividend yield rate d
Call premium as a function of d
0
20
0
0.05
80
20
40
60
80
5
0.05
0.1 year
0.5 year
1 year
2 years
0.1 year
0.5 year
1 year
2 years
0.2
10
15
20
25
30
35
Y
40 Stock price 60
0.1 year
0.5 year
1 year
2 years
Psi for a call
−100
20
2
60
0.15
80
0.1
0
Put premium as a function of r
40
r
0.1
Risk−free rate r
60
−80
−60
−40
−20
0
4
6
8
10
0.05
0.1 year
0.5 year
1 year
2 years
Rho for a put
−100
20
−80
−60
−40
−20
40
0.1 year
0.5 year
1 year
2 years
r
Call premium as a function of r
0
0
5
10
15
20
25
0
20
20
40
60
80
Rho for a call
0.2
80
0.2
80
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
0.1
0.2
0.3
0.4
0.50000
0.53983
0.57926
0.61791
0.65542
0.50399
0.54380
0.58317
0.62172
0.65910
0.50798
0.54776
0.58706
0.62552
0.66276
0.51197
0.55172
0.59095
0.62930
0.66640
0.51595
0.55567
0.59483
0.63307
0.67003
0.51994
0.55962
0.59871
0.63683
0.67364
0.52392
0.56356
0.60257
0.64058
0.67724
0.52790
0.56749
0.60642
0.64431
0.68082
0.53188
0.57142
0.61026
0.64803
0.68439
0.53586
0.57535
0.61409
0.65173
0.68793
0.5
0.6
0.7
0.8
0.9
0.69146
0.72575
0.75804
0.78814
0.81594
0.69497
0.72907
0.76115
0.79103
0.81859
0.69847
0.73237
0.76424
0.79389
0.82121
0.70194
0.73565
0.76730
0.79673
0.82381
0.70540
0.73891
0.77035
0.79955
0.82639
0.70884
0.74215
0.77337
0.80234
0.82894
0.71226
0.74537
0.77637
0.80511
0.83147
0.71566
0.74857
0.77935
0.80785
0.83398
0.71904
0.75175
0.78230
0.81057
0.83646
0.72240
0.75490
0.78524
0.81327
0.83891
1.0
1.1
1.2
1.3
1.4
0.84134
0.86433
0.88493
0.90320
0.91924
0.84375
0.86650
0.88686
0.90490
0.92073
0.84614
0.86864
0.88877
0.90658
0.92220
0.84849
0.87076
0.89065
0.90824
0.92364
0.85083
0.87286
0.89251
0.90988
0.92507
0.85314
0.87493
0.89435
0.91149
0.92647
0.85543
0.87698
0.89617
0.91309
0.92785
0.85769
0.87900
0.89796
0.91466
0.92922
0.85993
0.88100
0.89973
0.91621
0.93056
0.86214
0.88298
0.90147
0.91774
0.93189
1.5
1.6
1.7
1.8
1.9
0.93319
0.94520
0.95543
0.96407
0.97128
0.93448
0.94630
0.95637
0.96485
0.97193
0.93574
0.94738
0.95728
0.96562
0.97257
0.93699
0.94845
0.95818
0.96638
0.97320
0.93822
0.94950
0.95907
0.96712
0.97381
0.93943
0.95053
0.95994
0.96784
0.97441
0.94062
0.95154
0.96080
0.96856
0.97500
0.94179
0.95254
0.96164
0.96926
0.97558
0.94295
0.95352
0.96246
0.96995
0.97615
0.94408
0.95449
0.96327
0.97062
0.97670
2.0
2.1
2.2
2.3
2.4
0.97725
0.98214
0.98610
0.98928
0.99180
0.97778
0.98257
0.98645
0.98956
0.99202
0.97831
0.98300
0.98679
0.98983
0.99224
0.97882
0.98341
0.98713
0.99010
0.99245
0.97932
0.98382
0.98745
0.99036
0.99266
0.97982
0.98422
0.98778
0.99061
0.99286
0.98030
0.98461
0.98809
0.99086
0.99305
0.98077
0.98500
0.98840
0.99111
0.99324
0.98124
0.98537
0.98870
0.99134
0.99343
0.98169
0.98574
0.98899
0.99158
0.99361
2.5
2.6
2.7
2.8
2.9
0.99379
0.99534
0.99653
0.99744
0.99813
0.99396
0.99547
0.99664
0.99752
0.99819
0.99413
0.99560
0.99674
0.99760
0.99825
0.99430
0.99573
0.99683
0.99767
0.99831
0.99446
0.99585
0.99693
0.99774
0.99836
0.99461
0.99598
0.99702
0.99781
0.99841
0.99477
0.99609
0.99711
0.99788
0.99846
0.99492
0.99621
0.99720
0.99795
0.99851
0.99506
0.99632
0.99728
0.99801
0.99856
0.99520
0.99643
0.99736
0.99807
0.99861
3.0
3.1
3.2
3.3
3.4
0.99865
0.99903
0.99931
0.99952
0.99966
0.99869
0.99906
0.99934
0.99953
0.99968
0.99874
0.99910
0.99936
0.99955
0.99969
0.99878
0.99913
0.99938
0.99957
0.99970
0.99882
0.99916
0.99940
0.99958
0.99971
0.99886
0.99918
0.99942
0.99960
0.99972
0.99889
0.99921
0.99944
0.99961
0.99973
0.99893
0.99924
0.99946
0.99962
0.99974
0.99896
0.99926
0.99948
0.99964
0.99975
0.99900
0.99929
0.99950
0.99965
0.99976
3.5
3.6
3.7
3.8
3.9
0.99977
0.99984
0.99989
0.99993
0.99995
0.99978
0.99985
0.99990
0.99993
0.99995
0.99978
0.99985
0.99990
0.99993
0.99996
0.99979
0.99986
0.99990
0.99994
0.99996
0.99980
0.99986
0.99991
0.99994
0.99996
0.99981
0.99987
0.99991
0.99994
0.99996
0.99981
0.99987
0.99992
0.99994
0.99996
0.99982
0.99988
0.99992
0.99995
0.99996
0.99983
0.99988
0.99992
0.99995
0.99997
0.99983
0.99989
0.99992
0.99995
0.99997
P(Z<z)
z
0.800
0.84162
0.850
1.03643
0.900
1.28155
0.950
1.64485
0.975
1.95996
0.990
2.32635
0.995
2.57583
0.999
3.09023
Inverse lookup: