X(0) = 0, unless stated otherwise

2. for any 0 ≤ t₀ < t₁ < · · · < 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 σ²(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 σ².

• 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 − σ²/2) 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)

Local-Volatility 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 exponential-sized 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 local-volatility function σ(S, V ) is separable^b if σ(S, t) = σ₁(S) σ₂(t).

• A general solution is close.^c

aLok (D99922028) & Lyuu (2015, 2016).

bRebonato (2004); Brace, G¸atarek, & Musiela (1997).

cLok (D99922028) & Lyuu (2016).

The Hull-White Model

• Hull and White (1987) postulate the following model, dS

S = r dt + √

V dW₁, dV = μ_vV dt + bV dW₂.

• 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^θV dW₁, dV = bV dW₂,

for 0 ≤ θ ≤ 1.

The Hilliard-Schwartz Model

• Hilliard and Schwartz (1996) postulate the following general model,

dS

S = r dt + f (S)V ^a dW₁, dV = μ(V ) dt + bV dW₂,

for some well-behaved function f (S) and constant a.

Heston’s Stochastic-Volatility Model

• Heston (1993) assumes the stock price follows dS

S = (μ − q) dt + √

V dW₁, (16)

dV = κ(θ − V ) dt + σ√

V dW₂. (17) – V is the instantaneous variance, which follows a

square-root process.

– dW1 and dW2 have correlation ρ.

– The riskless rate r is constant.

Heston’s Stochastic-Volatility Model (concluded)

• It may be the most popular continuous-time stochastic-volatility model.^a

• For American options, we will need a tree for Heston’s model.^b

• They are all O(n³)-sized.

aChristoffersen, Heston, & Jacobs (2009).

bLeisen (2010); Beliaeva & Nawalka (2010); Chou (R02723073) (2015).

Why Are Trees for Stochastic-Volatility Models Difficult?

• The CRR tree is 2-dimensional.^a

• The constant volatility makes the span from any node fixed.

• But a tree for a stochastic-volatility model must be 3-dimensional.

– Every node is associated with a pair of stock price and a volatility.

Why Are Trees for Stochastic-Volatility Models

Difficult: Binomial Case?

Why Are Trees for Stochastic-Volatility Models

Difficult: Trinomial Case?

Why Are Trees for Stochastic-Volatility Models Difficult? (concluded)

• Locally, the tree looks fine for one time step.

• But the volatility regulates the spans of the nodes on the stock-price 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 Stochastic-Volatility Models

• A few stochastic-volatility models suffer from subexponential (c^√n) tree size.

• Examples include the Hull-White (1987),

Hilliard-Schwartz (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 S²V , 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 ≡ M²(e^σ2Δt − 1).

• Impose the matching of mean and that of variance:

1 = pu + pm + pd,

SM = (puu + pm + (pd/u)) S,

S²V = 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 + M² − M

− (M − 1) (u − 1) (u² − 1) , p_d = u² 

V + M² − M

− u³(M − 1) (u − 1) (u² − 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

2λ² +

r + σ² √ Δt

2λσ ,

p_d → 1

2λ² −

r − 2σ² √ 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

Down-and-in 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_iS_i(τ ) − X, 0

 .

Multivariate Contingent Claims (continued)

^a

Name Payoff

Exchange option max(S1(τ) − S2(τ), 0) Better-off option max(S1(τ), . . . , S_k(τ), 0) Worst-off option min(S₁(τ), . . . , S_k(τ), 0)

Binary maximum option I{ max(S₁(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) Multi-strike option max(S1(τ) − X1, . . . , S_k(τ) − X_k, 0)

Pyramid rainbow option max(| S₁(τ) − X₁ | + · · · + | 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 early-exercise 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 finite-difference 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 finite-difference formula for gamma ∂²f /∂S² 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¨ıt-Sahalia and Lo (1998); Bondarenko (2003).

Gamma (continued)

• In general, suppose

∂ⁱ

∂θⁱe^−rτE[ P (S) ] = e^−rτE

 ∂ⁱP (S)

∂θⁱ

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 long-term mean level μ at rate β.

• Superimposed on this “pull” is a normally distributed stochastic term σ dW .

aVasicek (1977).

The Cox-Ingersoll-Ross Model

^a

• It is the following square-root 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 < σ².

aCox, Ingersoll, and Ross (1985).

The Ho-Lee Model

^a

• The continuous-time limit of the Ho-Lee model is dr = θ(t) dt + σ dW.

• This is Vasicek’s model with the mean-reverting drift replaced by a deterministic, time-dependent 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 Black-Derman-Toy Model

^a

• The continuous-time 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 Black-Karasinski 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 mean-reverting 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 Ho-Lee model, this is a normal model.

• Many European-style securities can be evaluated analytically.

• Efficient numerical procedures can be developed for American-style securities.

aHull and White (1990).

The Hull-White Model

• The Hull-White 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 rate-sensitive securities.

在文檔中 Toward the Black-Scholes Formula (頁 34-86)