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ECS550NFBIntroduction to Numerical Methods using Matlab

Day 5

Lukas [email protected]

Department of Mathematics, University of Matej Bel

June 12, 2015
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Today

Derivatives pricing

Binomial lattices Monte Carlo simulation Solving a partial differential equation (e.g. finite differencing)

This lecture is based on chapters 2, 4, 7, 8 and 9 inBrandimarte, Paolo. Numerical methods in finance and economics: a

MATLAB-based introduction. John Wiley & Sons, 2013.
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Notation

C - coupon F - face value P - price P V - present value

rt - interest rate

P V =Tt=0

Ct(1 + rt)t

+ FT

(1 + rT)T

cashFlows = [0 3 3 3 3 103];interestRate = 0.03;

pvvar(cashFlows,interestRate)
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Notation

- internal rate of return irr(cashFlows), solution toP =

Tt=0

Ct(1+)t +

FT(1+)T

orP =

Tt=0

Ct/m(1+/m)t +

FT(1+/m)T

if coupon is payed m times a year.

D - is duration, D= PV(t0)t0++PV(tn)tnPV , where P V(ti) is a

present value of cashflow in time ti.cfdur(cashFlows, interestRate),

DM- is modified duration, DM=D/(1 + /m). Why is itinteresting?dP

d =

D

MP, so P

D

MP

C= 1Pd2Pd2, Why is it interesting?

d2Pd2

=P C, so P DMP + PC2 ()2bndprice(Yield, CouponRate, Settle, Maturity)
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Example: Portofolio with given duration and convexity

We are asked to create a portofolio with duration D and convexityC from 3 bonds.

How will we set the weights?

3i=1 Diwi=D

3i=1 Ciwi=C

3i=1 wi= 1(Note that this is an approximation)
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Mean-Variance portofolio optimization

We have n stocks with mean returns{r1, r2, rn} and covariance

matrix of their returns How will we set our weights in the portofolio to get on average r

and minimizing risk (measured by the variance)?

minwwTw

s.t.ni=1 wiri= r

ni=1 wi= 1wi0

Makes sense if returns are normally distributied and the utilityfunction is quadratic.

[PRisk, PRoR, PWts]= frontcon(returns,covMatrix,100);
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Derivatives

How do we find a right price? How do we insure against potential losses?

What computational methods can we make use of Solving a partial differential equation (e.g. finite differencing) Monte Carlo simulation Binomial lattices
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Derivatives Pricing - simplest case

We construct a portofolio: 0= S0+ that replicate the option payoff

u= S0u + ert =fu

d= S0d + ert =fd

we get , by no-arbitrage principle f0= 0 it is not the discounted expected value of the

payoff if we use risk-neutral measure, it can be

interpreted as discounted expected value ofthe payoff

Source: Brandimarte, p.105
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Derivatives pricing - continuous time

Price of an Option, whose payoff depends only on the current price ofan underlying asset, or its price at maturity, follows Black-Scholesdifferential equation (deterministic!).

ft

+ rSfS

+12

2S2 2

fS2

rf= 0This PDE captures the dynamic, but we need to supply initial orterminal conditions (f(S, T) = max{S K, 0}).

In most cases, this equation must be solved numerically (but not forvanilla European put/call options blsprice).
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Derivatives pricing - binomial lattices



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How do we calibrate the parameters of our binomial lattice? d,u,p We can e.g. match the first two moments ifSt follows

dS=rSdt + SdW

Cox, Ross, Rubinstein (CRR): we have one degree of freedom, sowe can set u= 1/d and reduce the computational burden and get

u= et, d= e

t, p= e

rtdud

Jarrow-Rudd: we set p= 12 and calculate u= e

r2

2

t+

t

and

d= er

2

2tt
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Left: Stock, Right: European Call option. Source: Brandimarte, p.407
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Binomial lattice - code

function [price, lattice] = LatticeEurCall(S0,K,r,T,sigma,N)

deltaT = T/N;u=exp(sigma * sqrt(deltaT));

d=1/u;

p=(exp(r*deltaT) - d)/(u-d);

lattice = zeros(N+1,N+1);

for i=0:Nlattice(i+1,N+1)=max(0 , S0*(u^i)*(d^(N-i)) - K);

end

for j=N-1:-1:0

for i=0:j

lattice(i+1,j+1) = exp(-r*deltaT) * ...

(p * lattice(i+2,j+2) + (1-p) * lattice(i+1,j+2));

end

end

price = lattice(1,1);
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Binomial lattice - code improvementInefficiencies

we repeat discounted probabilities exp(-r*deltaT)*p and

exp(-r*deltaT)*(1-p) we use a full matrix lattice - half of it is empty

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Pricing American Options

We compareimmediate payoffwithcontinuation value fi,j = max{K Si,j, ert(pfi+1,j+1+ (1p)fi,j+1)}

[AssetPrice, OptionValue] = binprice(Price, Strike, Rate,

Time, Increment, Volatility, Flag)

price = AmPutLattice(S0,K,r,T,sigma,N)
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Derivatives Pricing - trinomial lattice

We will set pu, pm, pd to match the momentsofX, where Xt= log St Choice ofx? Rule of thumb x =

t



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Round-up: Should you use lattices?

Simple Fast when problem dimensionality is small

Difficult for more complex path-dependent options Discretization is important - e.g. poor choice ofx may lead to

negative probabilities Incorporating extensions usually requires careful implementation


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Derivatives pricing - Monte Carlo Methods

simple, flexible

computationally intensive when facing a complex problem, sometimes the only way to go

M C l M h d A P h
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Monte Carlo Methods - Asset Paths

There are two sources of errors

sampling error discretization error

Euler scheme

Brownian motion

dSt =a(St, t)dt + b(St, t)dWt

St =a(St, t)t + b(St, t))t

Geometric Brownian motion

dSt =Stdt + StdWt

St+t = (1 + t)St+ Stt

Di i i G i B i M i
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Discretizing Geometric Brownian Motion

But if we generate Geometric Brownian motion using

St+t = (1 + t)St+ SttSt will be normal, rather than log-normal.

Solution? Using Itos lemma on dSt=Stdt + StdWt we getd log St=

1

2

2 dt + dWtwhich is integrated to St =S0exp

12 2

t +

t0dW()

, so we

can generate sample paths using

St+t =Stexp 1

22 t + t .

To speed things up, should we vectorize? In most cases, yes. The bestadvice is to give it a try.

E l Si l M t C l f E C ll
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Example: Simple Monte Carlo for European Call

Create many sample paths for Stock price. Discount and take average.

function [Price,CI] = BlsMC2(S0,K,r,T,sigma,NRepl)

nuT = (r - 0.5*sigma^2)*T;siT = sigma * sqrt(T) ;

DiscPayoff = exp(-r*T)*max(0, S0*exp(nuT+siT*randn(NRepl,1))-K) ;

[Price,Var, CI] = normfit(DiscPayoff);

end

M t C l M th d H d i t t i
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Monte Carlo Methods - Hedging strategies

Stop loss strategy - hold an asset ifSt> K, sell it ifSt< K Delta hedging - replicate the option by holding stocks

Comparison of the costs of the strategy

HedgingScript.m

NSteps = 10

true price = 4.732837

cost of stop/loss (S) = 4.826756

cost of delta-hedging = 4.736975

NSteps = 1000

cost of stop/loss (S) = 4.860783

cost of delta-hedging = 4.732051

E h O ti
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Exchange OptionAt time T, we may exchange U for V and receive max{VT UT, 0}.SupposeUt and Vt follow bidimensional geometric Brownian motion

with drift r and the two Wiener processes have instantaneouscorrelation

dU(t) = rU(t)dt + UU(t)dWU(t)

dV(t) = rV(t)dt + VV(t)dWV(t)

How to generate (correlated) sample paths?

=

1 1

, =LLT, L=

1 0

1 2

,

1 = Z1

2 = Z1+

1 2Z2

[p,ci] = ExchangeMC(V0,U0,sigmaV,sigmaU,rho,T,r,NRepl)

Down and out Put option
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Down-and-out Put option

Payoff is zero whenever the stock price hits the barrier level Sb.

Crude Monte Carlo Conditional Monte Carlo Importance sampling

Crude Monte Carlo Down and out Put option
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Crude Monte Carlo - Down-and-out Put option

function [P,CI,NCrossed] = DOPutMC(S0,K,r,T,sigma,Sb,NSteps,NRepl)

Payoff = zeros(NRepl,1);

NCrossed = 0;

for i=1:NRepl

Path=AssetPaths1(S0,r,sigma,T,NSteps,1);

crossed = any(Path


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Conditional Monte Carlo - Down-and-out Put option

Pdo=P Pdi Discretization of the time interval of width t, T =M t, price

path is S={S1, S2, . . . , S M} Pdi=e

rTE[I(S)(K SM)+], where I(S) = 1 iff the price path hitthe barrier (Sj < Sb for some j)

If the barrier is crossed before maturity:

E[I(S)(K SM)+

|j, Sj] =er(T

t)

Bp(Sj , K , T t), where j,where t=tj denotes the crossing time and Bp is the price of aput option. payoffI(S)ert

Bp(Sj , K , T t), where j, where

t=t j.

The problem is that we hit the barrier too few times (249 out of200000), in all other cases, the option price is zero.

[Pdo,CI,NCrossed] =

DOPutMCCond(S0,K,r,T,sigma,Sb,NSteps,NRepl)

Importance Sampling Down and out Put option
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Importance Sampling - Down-and-out Put option

We can change the sampling design, simulate paths that hit the barriermore often and then correct for this change.

for the path sampling we generate independent normal draws Zj

with expected value =

r 22

t and variance 2t, then set

log Sjlog Sj1=Zj

instead, we will now generate from density with expected value b and thus will cross the barrier more often

Eg

f(Z)I(S)(KSM)+

g(Z) |j, Sj

= f(z1,...,zj)

g(z1,...,zj)Ef[I(S)(K SM)+|j, Sj]

The likelihood ratio f(z1,...,zj)

g(z1,...,zj)is a correction term.

[Pdo,CI,NCrossed] =

DOPutMCCondIS(S0,K,r,T,sigma,Sb,NSteps,NRepl,bp)

Arithmetic Average Asian Option
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Arithmetic Average Asian Option

The call option payoff is

max

1

N

N

i=1

S(ti) K, 0

Crude Monte Carlo Control Covariates

sum of asset prices as control covariate geometric average option as a control covariate

Crude Monte Carlo - Arithmetic Average Asian Option
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Crude Monte Carlo Arithmetic Average Asian Option

function [P,CI] = AsianMC(S0,K,r,T,sigma,NSamples,NRepl)

Payoff = zeros(NRepl,1);

for i=1:NRepl

Path=AssetPaths(S0,r,sigma,T,NSamples,1);Payoff(i) = max(0, mean(Path(2:(NSamples+1))) - K);

end

[P,aux,CI] = normfit( exp(-r*T) * Payoff);

Control Variable - Arithmetic Average Asian Option
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Control Variable Arithmetic Average Asian Option

Sum of asset prices

Y =Ni=0 S(ti) E[Y] =E

Ni=0 S(ti)

=S(0) 1e

r(N+1)t

1ert

[P,CI] = AsianMCCV(S0, K,r,T,sigma,NSamples,NRepl,NPilot)

Control Variable (2) - Arithmetic Average Asian Option
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Control Variable (2) Arithmetic Average Asian Option

Geometric average option

Payoff is max(S(ti))1/N

K, 0

Formula for the price of the geometric average option is available

[P,CI] = AsianMCGeoCV(S0,K,r,T,sigma,NSamples,NRepl,NPilot)

Estimating Greeks by Monte Carlo
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Estimating Greeks by Monte Carlo

Option sensitivities.

= df(S0)dS0 = limS00f(S0+S0)f(S0)

S0

we can use limS0

0E [C(S0+S0,)][C(S0,)]

S0

or improve it using limS00E [C(S0+S0,)]E[C(S0S0,)]

2S0instead

and using common numbers another approach is to calculate option price and in one

simulation - pathwise estimator

Pathwise estimator - Greeks
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at w se est ato G ee s

discounted option payoff is a random variable

C=erT max

{ST

K, 0}

, where ST

=S0

e(r2/2)T+

TZ

CS0

= dCdSTdSTdS0

dSTdS0

= STS0

dCdST

=erTI{ST > K}

so the estimator for is erTST

S0 I{ST > K}

function [Delta, CI] = BlsDeltaMCPath(S0,K,r,T,sigma,NRepl)

nuT = (r - 0.5*sigma^2)*T;

siT = sigma * sqrt(T);VLogn = exp(nuT+siT*randn(NRepl,1));

SampleDelta = exp(-r*T) .* VLogn .* (S0*VLogn > K);

[Delta, dummy, CI] = normfit(SampleDelta);

Finite Difference Methods - Black Scholes formula
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Assumptions

riskless rate r stock price follows geometric Brownian motion with constant drift

and volatility there are no dividends no borrowing or lending constraints no transaction costs

The the price of a derivative which payoff depends only on time andstock price follows the Black-Scholes formula

f

t + rS

f

S+

1

22S2

2f

S2 rf= 0

Finite Difference Methods
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We will work with a discrete grid

S= 0,S, 2S,...,MS t= 0,t, 2t,...,Nt fi,j =f(iS,jt)

There are different ways how to approximate partial derivatives

forward difference fS = fi+1,jfi,j

S

backward difference fS = fi,jfi1,j

S

central difference fS = fi+1,jfi1,j

2S

We also need boundary conditions Call option: f(S, T) = max{S K, 0}, S Put option: f(S, T) = max{K S, 0}, S

Boundary conditions
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y

European vanilla put option fi,N= max{K iS, 0} f0,j =K e

r(Nj)t

fM,j = 0

European vanilla call option

fi,N= max{iS K, 0} f0,j = 0, fM,j =M S Ker(Nj)t

Finite Difference Methods
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Math preliminaries:

Explicit scheme - use forward approximation for the derivativewith respect to the time - may not be stable Implicit scheme - use backward approximation for the derivative

with respect to the time - unconditionally stable Crank-Nicholson method - combination of explicit and implicit

scheme - more precise approximation - unconditionally stable

Source: Brandimarte, chap. 5

Explicit scheme
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fi,j fi,j1t

+ riSfi+1,j fi1,j

2S

+ 1

22i2(S)2

fi+1,j2fi,j+ fi1,j(S)2

=rfi,j

So the explicit scheme takes the following form

fi,j1=afi1,j+ bfi,j+ cfi+1,j

price = EuPutExpl(SO,K,r,T,sigma,Smax.dS,dt)

Stability?

Stability - Explicit scheme
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Explicit scheme is basically a trinomial lattice, hence we must choosethe discretization parameters Sand t, so that the probabilities would

be positive.

Source: Brandimarte, p. 480

Implicit scheme
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fi,j+1 fi,jt

+ riSfi+1,j fi1,j

2S

+ 1

22i2(S)2

fi+1,j2fi,j+ fi1,j(S)2

=rfi,j

So the implicit scheme takes the following form

afi1,j+ bfi,j+ cfi+1,j =fi,j+1

price = EuPutImpl(SO,K,r,T,sigma,Smax.dS,dt)

Stability?

Stability - Implicit scheme
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Stability is not an issue at all. We can see that rewriting the system inmatrix form and inspecting the eigenvalues of the matrix, they are all

smaller than one in absolute value (Brandimarte p. 209).

Source: Brandimarte, p. 480

Crank-Nicolson scheme
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We use derivatives in between the grid values

2

fS2 (Si, tj+1/2) = 12

2

fS2 (Si, tj+1) + 2

fS2 (Si, tj)

+ O(S2)

ft (Si, tj+1/2) =

f(Si,tj+1)f(Si,tj)t + O(t

2)

M1fj1= M2fj

Pricing barrier put option using Crank-Nicholson scheme:Boundary conditions:

f(Smax, t) = 0 f(Sb, t) = 0

price = DOPutCK(S0,K,r,T,sigma,Sb,Smax,dS,dt)

American Options
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to account for free boundary is trivial in the Explicit scheme, thisis however subject to possible instabilities

we can use implicit scheme with iterative Gauss-Seidel method for

solving Ax= b we can also use Crank-Nicholson scheme

price = AmPutCK(S0,K,r,T,sigma,Smax,dS,dt,omega,tol)

Literature


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Brandimarte, Paolo. Numerical methods in finance and economics: a MATLAB-based introduction.John Wiley & Sons, 2013.

chapter 2 - Financial Theory chapter 4.5 Variance reduction techniques

chapter 7 - Option pricing by binomial and trinomial lattices chapter 8 - Option pricing by Monte Carlo methods chapter 9 - Option pricing by finite difference methods Hull, John C. Options, futures, and other derivatives. Pearson Education India, 2006.

The End
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Thank you for your attention!
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