Toward the BlackScholes Formula
• As n increases, the stock price ranges over ever larger numbers of possible values, and trading takes place nearly continuously.
• Need to calibrate the BOPM’s parameters u, d, and R to make it converge to the continuoustime model.
• We now skim through the proof.
Toward the BlackScholes Formula (continued)
• Let τ denote the time to expiration of the option measured in years.
• Let r be the continuously compounded annual rate.
• With n periods during the option’s life, each period represents a time interval of τ /n.
• Need to adjust the periodbased u, d, and interest rate r to match the empirical results as n → ∞.ˆ
Toward the BlackScholes Formula (continued)
• First, ˆr = rτ/n.
– Each period is τ/n years long.
– The period gross return R = e^{ˆr}.
• Let
ln S_{τ} S
denote the continuously compounded rate of return of the stock.
Toward the BlackScholes Formula (continued)
• Assume the stock’s true continuously compounded rate of return has mean μτ and variance σ^{2}τ .
• Call σ the stock’s (annualized) volatility.
• We need one more condition to have a solution for u, d, q.
• Impose
ud = 1.
– It makes nodes at the same horizontal level of the tree have identical price.^{a}
aOther choices are possible (see text).
Toward the BlackScholes Formula (continued)
• Pick
u = e^{σ}
√τ /n, d = e^{−σ}
√τ /n, q = 1
2 + 1 2
μ σ
τ
n . (12)
• With Eqs. (12), it can be checked that the mean μτ is matched by the BOPM.
• Furthermore, the variance σ^{2}τ is asymptotically matched as well.
Toward the BlackScholes Formula (continued)
• The choices (12) result in the CRR binomial model.^{a}
• The noarbitrage inequalities d < R < u may not hold under Eqs. (12) on p. 80 or Eq. (8) on p. 56.
– If this happens, the probabilities lie outside [ 0, 1 ].
• The problem disappears if n is large enough.
aCox, Ross, and Rubinstein (1979).
Toward the BlackScholes Formula (continued)
• What is the limiting probabilistic distribution of the continuously compounded rate of return ln(S_{τ}/S)?
• It approaches N(μτ + ln S, σ^{2}τ ).
• Conclusion: S_{τ} has a lognormal distribution in the limit.
Toward the BlackScholes Formula (continued)
• In the riskneutral economy, pick q = R − d
u − d . by Eq. (8) on p. 56.
Lemma 1 The continuously compounded rate of return ln(S_{τ}/S) approaches the normal distribution with mean (r − σ^{2}/2) τ and variance σ^{2}τ in a riskneutral economy.^{a}
Toward the BlackScholes Formula (concluded)
Theorem 2 (The BlackScholes Formula) C = SN (x) − Xe^{−rτ}N (x − σ√
τ ), P = Xe^{−rτ}N (−x + σ√
τ ) − SN (−x), where
x ≡ ln(S/X) +
r + σ^{2}/2 τ σ√
τ .
BOPM and BlackScholes Model
• The BlackScholes formula needs 5 parameters: S, X, σ, τ , and r.
• Binomial tree algorithms take 6 inputs: S, X, u, d, ˆr, and n.
• The connections are
u = e^{σ}
√τ /n,
d = e^{−σ}
√τ /n,
5 10 15 20 25 30 35 n
11.5 12 12.5 13
Call value
0 10 20 30 40 50 60 n
15.1 15.2 15.3 15.4 15.5
Call value
• S = 100, X = 100 (left), and X = 95 (right).
BOPM and BlackScholes Model (concluded)
• The binomial tree algorithms converge reasonably fast.
• The error is O(1/n).^{a}
• Oscillations can be dealt with by the judicious choices of u and d.^{b}
aChang and Palmer (2007).
bSee Exercise 9.3.8 of the textbook.
Implied Volatility
• Volatility is the sole parameter not directly observable.
• The BlackScholes formula can be used to compute the market’s opinion of the volatility.
– Solve for σ given the option price, S, X, τ, and r with numerical methods.
• Implied volatility is
the wrong number to put in the wrong formula to get the right price of plainvanilla options.^{a}
• It is often preferred to historical volatility in practice.
aRebonato (2004).
Problems; the Smile
• Options written on the same underlying asset usually do not produce the same implied volatility.
• A typical pattern is a “smile” in relation to the strike price.
– The implied volatility is lowest for atthemoney options.
– It becomes higher the further the option is in or outofthemoney.
Binomial Tree Algorithms for American Puts
• Early exercise has to be considered.
• The binomial tree algorithm starts with the terminal payoﬀs
max(0, X − Su^{j}d^{n−j}) and applies backward induction.
• At each intermediate node, it compares the payoﬀ if exercised and the continuation value.
• It keeps the larger one.
Extensions of Options Theory
And the worst thing you can have is models and spreadsheets.
— Warren Buﬀet, May 3, 2008
Barrier Options
• Their payoﬀ depends on whether the underlying asset’s price reaches a certain price level H.
• A knockout option is like an ordinary European option.
• But it ceases to exist if the barrier H is reached by the price of its underlying asset.
H
Time Price
S Barrier hit
Barrier Options (concluded)
• A knockin option comes into existence if a certain barrier is reached.
• A downandin option is a call knockin option that comes into existence only when the barrier is reached and H < S.
A Formula for DownandIn Calls
^{a}• Assume X ≥ H.
• The value of a European downandin call on a stock paying a dividend yield of q is
Se^{−qτ}
H S
_{2λ}
N(x) − Xe^{−rτ}
H S
_{2λ−2}
N(x − σ√ τ),
(13)
– x ≡ ^{ln(H}^{2}/(SX))+(r−q+σ^{2}/2) τ σ√
τ .
– λ ≡ (r − q + σ^{2}/2)/σ^{2}.
aMerton (1973).
Binomial Tree Algorithms
• Barrier options can be priced by binomial tree algorithms.
• Below is for the downandout option.
0 H
H 8
16
4
32
8
2
64
16
4
1
4.992
12.48
1.6
27.2
4.0
0
58
10
0
0 X
0.0
S = 8, X = 6, H = 4, R = 1.25, u = 2, and d = 0.5.
Backwardinduction: C = (0.5 × C_{u} + 0.5 × C_{d})/1.25.
Binomial Tree Algorithms (continued)
• But convergence is erratic because H is not at a price level on the tree.^{a}
– The barrier H is moved to a node price.
– This “eﬀective barrier” changes as n increases.
• In fact, the binomial tree is O(1/√
n) convergent.^{b}
aBoyle and Lau (1994).
bLin (R95221010) (2008).
Binomial Tree Algorithms (concluded)
^{a}100 150 200 250 300 350 400
#Periods 3
3.5 4 4.5 5 5.5
Downandin call value
aLyuu (1998).
PathDependent Derivatives
• Let S0, S1, . . . , S_{n} denote the prices of the underlying asset over the life of the option.
• S0 is the known price at time zero.
• S_{n} is the price at expiration.
• The standard European call has a terminal value depending only on the last price, max(S_{n} − X, 0).
• Its value thus depends only on the underlying asset’s
PathDependent Derivatives (continued)
• Some derivatives are pathdependent in that their terminal payoﬀ depends critically on the path.
• The (arithmetic) averagerate call has this terminal value:
max
1 n + 1
n i=0
S_{i} − X, 0
.
• The averagerate put’s terminal value is given by max
X − 1
n + 1
n i=0
S_{i}, 0
.
PathDependent Derivatives (concluded)
• Averagerate options are also called Asian options.^{a}
• They are useful hedging tools for ﬁrms that will make a stream of purchases over a time period because the costs are likely to be linked to the average price.
• The averaging clause is also common in convertible bonds and structured notes.
aAs of the late 1990s, the outstanding volume was in the range of 5–10 billion U.S. dollars (Nielsen & Sandmann, 2003).
AverageRate Options
• Averagerate options are notoriously hard to price.
• The binomial tree for the averages does not combine (see next page).
• A naive algorithm enumerates the 2^{n} paths for an nperiod binomial tree and then averages the payoﬀs.^{a}
• But the complexity is exponential.
• The Monte Carlo method^{b} and approximation algorithms are some of the alternatives left.
aDai (B82506025, R86526008, D8852600) and Lyuu (2007) reduce it to 2^{O(}^{√}^{n )}. Hsu (R7526001, D89922012) and Lyuu (2004) reduce it to O(n^{2}) given some regularity assumptions.
bSee pp. 142ﬀ.
S
Su
Sd
Suu
Sud
Sdu
p
1−^{ p}
⎟⎠
⎜ ⎞
⎝
⎛ + + −
=
PD[ 6 6X 6XX ;
&_{::}
⎟⎠
⎜ ⎞
⎝
⎛ + + −
=
PD[ 6 6X 6XG ;
&_{:/}
⎟⎠
⎜ ⎞
⎝
⎛ + + −
=
PD[ 6 6G 6GX ;
&_{/:}
⎞
⎛6 +6G +6GG
( )
7 :/
: ::
H
&
S S&
& = + −
( )
7 //
/ /:
H
&
S S&
& = + −
( )
7 /
:
H
&
S S&
&= + − p
1−^{ p} p
1−^{ p} p
1−^{ p}
p
1−^{ p} p
1−^{ p}
ContinuousTime Financial Mathematics
A proof is that which convinces a reasonable man;
a rigorous proof is that which convinces an unreasonable man.
— Mark Kac (1914–1984)
Brownian Motion
^{a}• Brownian motion is a stochastic process { X(t), t ≥ 0 } with the following properties.
1. X(0) = 0, unless stated otherwise.
2. for any 0 ≤ t_{0} < t_{1} < · · · < t_{n}, the random variables X(t_{k}) − X(t_{k−1})
for 1 ≤ k ≤ n are independent.^{b}
3. for 0 ≤ s < t, X(t) − X(s) is normally distributed with mean μ(t − s) and variance σ^{2}(t − s), where μ and σ = 0 are real numbers.
aRobert Brown (1773–1858).
bSo X(t) − X(s) is independent of X(r) for r ≤ s < t.
Brownian Motion (concluded)
• The existence and uniqueness of such a process is guaranteed by Wiener’s theorem.^{a}
• This process will be called a (μ, σ) Brownian motion with drift μ and variance σ^{2}.
• The (0, 1) Brownian motion is called the Wiener process.
aNorbert Wiener (1894–1964). He received his Ph.D. from Harvard in 1912.
Ito Process
^{a}• A shorthand^{b} is the following stochastic diﬀerential equation for the Ito diﬀerential dX_{t},
dX_{t} = a(X_{t}, t) dt + b(X_{t}, t) dW_{t}. (14) – Or simply
dX_{t} = a_{t} dt + b_{t} dW_{t}.
aIto (1944).
bPaul Langevin (1872–1946) in 1904.
Ito Process (concluded)
• dW is normally distributed with mean zero and variance dt.
• An equivalent form of Eq. (14) is dX_{t} = a_{t} dt + b_{t}√
dt ξ, (15)
where ξ ∼ N (0, 1).
Modeling Stock Prices
• The most popular stochastic model for stock prices has been the geometric Brownian motion,
dS
S = μ dt + σ dW.
• The continuously compounded rate of return X ≡ ln S follows
dX = (μ − σ^{2}/2) dt + σ dW by Ito’s lemma.^{a}
aConsistent with Lemma 1 (p. 84).
0.2 0.4 0.6 0.8 1 Time (t) 1
1 2 3 4 5 6 Y(t)
LocalVolatility Models
• The more general deterministic volatility model posits dS
S = (r_{t} − q_{t}) dt + σ(S, t) dW,
where instantaneous volatility σ(S, t) is called the local volatility function.^{a}
• One needs to recover σ(S, t) from the implied volatilities.
aDerman and Kani (1994); Dupire (1994).
0 0.5
1 1.5
2 2.5
3
0 0.2 0.4 0.6 0.8 1 20 30 40 50 60 70 80 90 100 110
Strike ($)
Implied Vol Surface
Time to Maturity (yr)
Implied Vol (%)
0 0.5
1 1.5
2 2.5
3
0 0.2 0.4 0.6 0.8 1 20 30 40 50 60 70 80 90 100 110
Stock ($)
Local Vol Surface
Time (yr)
Local Vol (%)
Implied Trees
• The trees for the local volatility model are called implied trees.^{a}
• Their construction requires an implied volatility surface.
• An exponentialsized implied tree exists.^{b}
• How to construct a valid implied tree with eﬃciency has been open for a long time.^{c}
aDerman & Kani (1994); Dupire (1994); Rubinstein (1994).
bCharalambousa, Christoﬁdesb, & Martzoukosa (2007).
cRubinstein (1994); Derman & Kani (1994); Derman, Kani, & Chriss (1996); Jackwerth & Rubinstein (1996); Jackwerth (1997); Coleman, Kim, Li, & Verma (2000); Li (2000/2001); Moriggia, Muzzioli, & Torri celli (2009).
Implied Trees (concluded)
• It is solved for separable local volatilities σ.^{a}
– The localvolatility function σ(S, V ) is separable^{b} if σ(S, t) = σ_{1}(S) σ_{2}(t).
• A general solution is close.^{c}
aLok (D99922028) & Lyuu (2015, 2016).
bRebonato (2004); Brace, G¸atarek, & Musiela (1997).
cLok (D99922028) & Lyuu (2016).
The HullWhite Model
• Hull and White (1987) postulate the following model, dS
S = r dt + √
V dW_{1}, dV = μ_{v}V dt + bV dW_{2}.
• Above, V is the instantaneous variance.
• They assume μv depends on V and t (but not S).
The SABR Model
• Hagan, Kumar, Lesniewski, and Woodward (2002) postulate the following model,
dS
S = r dt + S^{θ}V dW_{1}, dV = bV dW_{2},
for 0 ≤ θ ≤ 1.
The HilliardSchwartz Model
• Hilliard and Schwartz (1996) postulate the following general model,
dS
S = r dt + f (S)V ^{a} dW_{1}, dV = μ(V ) dt + bV dW_{2},
for some wellbehaved function f (S) and constant a.
Heston’s StochasticVolatility Model
• Heston (1993) assumes the stock price follows dS
S = (μ − q) dt + √
V dW_{1}, (16)
dV = κ(θ − V ) dt + σ√
V dW_{2}. (17) – V is the instantaneous variance, which follows a
squareroot process.
– dW1 and dW2 have correlation ρ.
– The riskless rate r is constant.
Heston’s StochasticVolatility Model (concluded)
• It may be the most popular continuoustime stochasticvolatility model.^{a}
• For American options, we will need a tree for Heston’s model.^{b}
• They are all O(n^{3})sized.
aChristoﬀersen, Heston, & Jacobs (2009).
bLeisen (2010); Beliaeva & Nawalka (2010); Chou (R02723073) (2015).
Why Are Trees for StochasticVolatility Models Diﬃcult?
• The CRR tree is 2dimensional.^{a}
• The constant volatility makes the span from any node ﬁxed.
• But a tree for a stochasticvolatility model must be 3dimensional.
– Every node is associated with a pair of stock price and a volatility.
Why Are Trees for StochasticVolatility Models
Diﬃcult: Binomial Case?
Why Are Trees for StochasticVolatility Models
Diﬃcult: Trinomial Case?
Why Are Trees for StochasticVolatility Models Diﬃcult? (concluded)
• Locally, the tree looks ﬁne for one time step.
• But the volatility regulates the spans of the nodes on the stockprice plane.
• Unfortunately, those spans diﬀer from node to node because the volatility varies.
• So two time steps from now, the branches will not combine!
Complexities of StochasticVolatility Models
• A few stochasticvolatility models suﬀer from subexponential (c^{√}^{n}) tree size.
• Examples include the HullWhite (1987),
HilliardSchwartz (1996), and SABR (2002) models.^{a}
aChiu (R98723059) (2012).
Trees
I love a tree more than a man.
— Ludwig van Beethoven (1770–1827)
Trinomial Tree
• Set up a trinomial approximation to the geometric Brownian motion^{a}
dS
S = r dt + σ dW.
• The three stock prices at time Δt are S, Su, and Sd, where ud = 1.
• Let the mean and variance of the stock price be SM and S^{2}V , respectively.
aBoyle (1988).
*  j
pu
pm
pd
Su S Sd S

Δt
*  j
*  j
*  j
*  j
Trinomial Tree (continued)
• By Eqs. (5) on p. 24,
M ≡ e^{rΔt},
V ≡ M^{2}(e^{σ}^{2}^{Δt} − 1).
• Impose the matching of mean and that of variance:
1 = pu + pm + pd,
SM = (puu + pm + (pd/u)) S,
S^{2}V = pu(Su − SM)^{2} + pm(S − SM)^{2} + pd(Sd − SM)^{2}.
Trinomial Tree (concluded)
• Use linear algebra to verify that p_{u} = u
V + M^{2} − M
− (M − 1) (u − 1) (u^{2} − 1) , p_{d} = u^{2}
V + M^{2} − M
− u^{3}(M − 1) (u − 1) (u^{2} − 1) .
– We must also make sure the probabilities lie between 0 and 1.
A Trinomial Tree
• Use u = e^{λσ}^{√}^{Δt}, where λ ≥ 1 is a tunable parameter.
• Then
p_{u} → 1
2λ^{2} +
r + σ^{2} √ Δt
2λσ ,
p_{d} → 1
2λ^{2} −
r − 2σ^{2} √ Δt
2λσ .
Barrier Options Priced by Trinomial Trees
0 50 100 150 200
5.61 5.62 5.63 5.64 5.65 5.66
Downandin call value
Multivariate Contingent Claims
• They depend on two or more underlying assets.
• The basket call on m assets has the terminal payoﬀ max
_{m}
i=1
α_{i}S_{i}(τ ) − X, 0
.
Multivariate Contingent Claims (continued)
^{a}Name Payoﬀ
Exchange option max(S1(τ) − S2(τ), 0) Betteroﬀ option max(S1(τ), . . . , S_{k}(τ), 0) Worstoﬀ option min(S_{1}(τ), . . . , S_{k}(τ), 0)
Binary maximum option I{ max(S_{1}(τ), . . . , S_{k}(τ)) > X } Maximum option max(max(S1(τ), . . . , S_{k}(τ)) − X, 0) Minimum option max(min(S1(τ), . . . , S_{k}(τ)) − X, 0) Spread option max(S1(τ) − S2(τ) − X, 0)
Basket average option max((S1(τ) + · · · + S_{k}(τ))/k − X, 0) Multistrike option max(S1(τ) − X1, . . . , S_{k}(τ) − X_{k}, 0)
Pyramid rainbow option max( S_{1}(τ) − X_{1}  + · · · +  S_{k}(τ) − X_{k}  − X, 0)
Multivariate Contingent Claims (concluded)
• Trees for multivariate contingent claims typically has size exponential in the number of assets.
• This is called the curse of dimensionality.
Numerical Methods
All science is dominated by the idea of approximation.
— Bertrand Russell
Monte Carlo Simulation
^{a}• Monte Carlo simulation is a sampling scheme.
• In many important applications within ﬁnance and without, Monte Carlo is one of the few feasible tools.
aA top 10 algorithm according to Dongarra and Sullivan (2000).
Monte Carlo Option Pricing
• For the pricing of European options, we sample the stock prices.
• Then we average the payoﬀs.
• The variance of the estimator is now 1/N of that of the original random variable.
How about American Options?
• Standard Monte Carlo simulation is inappropriate for American options because of early exercise.
• It is diﬃcult to determine the earlyexercise point based on one single path.
• But Monte Carlo simulation can be modiﬁed to price American options with small biases.^{a}
• The LSM can be easily parallelized.^{b}
aLongstaﬀ and Schwartz (2001).
bHuang (B96902079, R00922018) (2013); Chen (B97902046,
Delta and Common Random Numbers
• In estimating delta ∂f/∂S, it is natural to start with the ﬁnitediﬀerence estimate
e^{−rτ} E[ P (S + ) ] − E[ P (S − ) ]
2 .
– P (x) is the terminal payoﬀ of the derivative security when the underlying asset’s initial price equals x.
• Use simulation to estimate E[ P (S + ) ] ﬁrst.
• Use another simulation to estimate E[ P (S − ) ].
• Finally, apply the formula to approximate the delta.
Delta and Common Random Numbers (concluded)
• This method is not recommended because of its high variance.
• A much better approach is to use common random numbers to lower the variance:
e^{−rτ} E
P (S + ) − P (S − ) 2
.
• Here, the same random numbers are used for P (S + ) and P (S − ).
Gamma
• The ﬁnitediﬀerence formula for gamma ∂^{2}f /∂S^{2} is e^{−rτ} E
P (S + ) − 2 × P (S) + P (S − ) ^{2}
.
• Choosing an of the right magnitude can be challenging.
– If is too large, inaccurate Greeks result.
– If is too small, unstable Greeks result.
• This phenomenon is sometimes called the curse of diﬀerentiation.^{a}
aA¨ıtSahalia and Lo (1998); Bondarenko (2003).
Gamma (continued)
• In general, suppose
∂^{i}
∂θ^{i}e^{−rτ}E[ P (S) ] = e^{−rτ}E
∂^{i}P (S)
∂θ^{i}
holds for all i > 0, where θ is a parameter of interest.
– A common requirement is Lipschitz continuity.^{a}
• Then formulas for the Greeks become integrals.
• As a result, we avoid , ﬁnite diﬀerences, and resimulation.
Gamma (concluded)
• This is indeed possible for a broad class of payoﬀ functions.^{a}
• In queueing networks, this is called inﬁnitesimal perturbation analysis (IPA).^{b}
aTeng (R91723054) (2004) and Lyuu and Teng (R91723054) (2011).
bCao (1985); Ho and Cao (1985).
Interest Rate Models
[Meriwether] scoring especially high marks in mathematics — an indispensable subject for a bond trader.
— Roger Lowenstein, When Genius Failed (2000) Bond market terminology was designed less to convey meaning than to bewilder outsiders.
— Michael Lewis, The Big Short (2011)
The Vasicek Model
^{a}• The short rate follows
dr = β(μ − r) dt + σ dW.
• The short rate is pulled to the longterm mean level μ at rate β.
• Superimposed on this “pull” is a normally distributed stochastic term σ dW .
aVasicek (1977).
The CoxIngersollRoss Model
^{a}• It is the following squareroot short rate model:
dr = β(μ − r) dt + σ√
r dW. (18)
• The diﬀusion diﬀers from the Vasicek model by a multiplicative factor √
r .
• The parameter β determines the speed of adjustment.
• The short rate can reach zero only if 2βμ < σ^{2}.
aCox, Ingersoll, and Ross (1985).
The HoLee Model
^{a}• The continuoustime limit of the HoLee model is dr = θ(t) dt + σ dW.
• This is Vasicek’s model with the meanreverting drift replaced by a deterministic, timedependent drift.
• A nonﬂat term structure of volatilities can be achieved if the short rate volatility is also made time varying,
dr = θ(t) dt + σ(t) dW.
The BlackDermanToy Model
^{a}• The continuoustime limit of the BDT model is d ln r =
θ(t) + σ^{}(t)
σ(t) ln r
dt + σ(t) dW.
• This model is extensively used by practitioners.
• The BDT short rate process is the lognormal binomial interest rate process.
• Lognormal models preclude negative short rates.
aBlack, Derman, and Toy (BDT) (1990), but essentially ﬁnished in 1986 according to Mehrling (2005).
The BlackKarasinski Model
^{a}• The BK model stipulates that the short rate follows d ln r = κ(t)(θ(t) − ln r) dt + σ(t) dW.
• This explicitly meanreverting model depends on time through κ( · ), θ( · ), and σ( · ).
• The BK model hence has one more degree of freedom than the BDT model.
• The speed of mean reversion κ(t) and the short rate volatility σ(t) are independent.
The Extended Vasicek Model
^{a}• The extended Vasicek model adds time dependence to the original Vasicek model,
dr = (θ(t) − a(t) r) dt + σ(t) dW.
• Like the HoLee model, this is a normal model.
• Many Europeanstyle securities can be evaluated analytically.
• Eﬃcient numerical procedures can be developed for Americanstyle securities.
aHull and White (1990).
The HullWhite Model
• The HullWhite model is the following special case, dr = (θ(t) − ar) dt + σ dW.
*  j
(0, 0)
*  j
(1, 1)
*  j
(1, 0)
*
(1, −1) 
*  j
*  j
*  j
*  j
 j R
*  j
*  j
*  j
*  j
* 
Δt
6
?^{Δr}
The Extended CIR Model
• In the extended CIR model the short rate follows dr = (θ(t) − a(t) r) dt + σ(t)√
r dW.
• The functions θ(t), a(t), and σ(t) are implied from market observables.
• With constant parameters, there exist analytical solutions to a small set of interest ratesensitive securities.