STAD70 Statistics & Finance II
5. Monte Carlo Methods
1
Numerical Option Pricing
1. 2.
3.
Three basic numerical option pricing methods
Binomial Trees (BT)
Finite Difference (FD)
Based on Black-Scholes PDE
Monte-Carlo (MC) simulation
Based on SDE for asset prices & risk-neutral valuation
BT
FD
MC
European options
Early exercise
Path dependence
Multi-asset dependence
2
Brownian Motion
Brownian Motion (BM) is building block of continuous stochastic models
Standard BM {Wt} is such that W0& WW|W~N(0,ts)
0tss
Arithmetic BM {Xt} with drift μ & volatility σ is
X 0 & (X X )|X ~N(ts),2(ts) 0 tss
In form of Stochastic Differential Equation (SDE)
dXdtdWXX t(WW) ttt0t0
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Example
Find conditional distribution
wheredX dtdW tt
Xt |Xs x,ts
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Example (Brownian Bridge)
Find conditional distribution
wheredX dtdW tt
Xs |Xt x,s(0,t)
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Geometric Brownian Motion (GBM)
Expressed in terms of SDE: 2 dY YdtYdW dlogY
dtdW ttttt2t
Resulting distribution logYlogY logY 2 tW
t0tt Y2
0 Y 2
log t Xt ~N t,2t Y2
0
Y Y expX ~Y logN2 t,2t
t0 t0 2
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Risk-Neutral Pricing
Assuming Geometric BM for {St} there exists specialdistribution,sothat 2
t,2t
dS rSdtSdW S ~S logN r ttttt02
Called risk-neutral (R-N) measure rt
Under R-N measure: E[St ] S0e , where r is risk- free interest rate
Price of any European derivative with payoff
GT=f (ST) given by discounted expectation
w.r.t. R-N measure: G E erT G E erT f (S ) 0TT
Example
Show that under R-N measure
E[St ] S0ert
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Example
Find price of forward F0,T (no dividends)
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Estimating Expectations
If E erT f (S ) cannot be explicitly computed, it T
can be estimated/approximated by simulation: Generate #N independent random variates Si(T),
i=1,…,n based on R-N measure
By Law of Large Numbers (SLLN):
ˆ1n
G erT f S(T) EerT f(S ), withprob.1
0
n
iT
i1
Moreover, by Central Limit Theorem (CLT) GG 1n
appr. 2 rT ˆ ˆ
00
~ N0,1,wheresGn1 i1e f S(T)G
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2 i0
sG n
Example
Show estimator of E erT f (S ) is consistent, T
and build 95% confidence interval for G0
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European Call
Estimating European call price w/ simulation Asset price dynamics:
dS SdtSdW tttt
Payoff function for strike K & maturity T f (ST )
Generate random asset price variates as: 2
S(T)S(0)expr T TZ i 2i
where Zi is standard Normal variate
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European Call
Simulated asset price variates at T
Histogram
● Q-Q plot vs log-Norm
– – log-Normal
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European Call
Convergence of MC price to true price
n (# random deviates used)
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Call price
Path Dependent Options
Path dependent options: payoff depends on (aspects of) entire asset path
In contrast to European options, whose payoff depend only on asset price at expiry ST
E.g. Barrier option: options payoff depends on whether asset crosses pre-set barrier level before expiration
Typically, payoff at expiration is equal to call/put, but option is activated / knocked-out when price hits barrier
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Barrier Options
4 types of Barrier options:
1. Up-and-out (U&O): price starts below barrier &
has to move up for option to be knocked out
2. Down-and-out (D&O): price starts above barrier & has to move down option to be knocked out
3. Up-and-in (U&I): price starts below barrier & has to move up for option to become activated
4. Down-and-in (D&I): price starts above barrier & has to move down for option to become activated
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Barrier Options
Let C/P be price of plain Euro call/put option, CD&O be that of Euro down-&-out call, etc
Find CU&O when barrier B < K strike
Find PU&I+PU&O, where options have same B, K, T, etc
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Barrier Options
Barrier options depend on their paths only
through their max/min value before expiry
LetM maxS &m minS
T t 0tT T t 0tT
Can express barrier option prices as C erTES K
U&O CD&O
T
{MTB}
C U&I
CD&I
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Simulating GBM Paths
To price path-dependent (e.g. barrier) options need to simulate asset price paths {S }
In practice, need to discretize time Simulate asset price at #m points
mT
S(ti)i0 whereti imit,i1, ,m
t 0tT
For GBM dS rS dt S dW (R-N measure)
tttt
2
2
t T / m
~iid N(0,1)
S(ti ) S(ti1)expr t t Zi where Z
i
i1, ,m
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Example
Sample GMB paths St
t
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Example
Price CU&O with K=80, B=90 St
Which paths have non-zero payoff?
B K
t
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Monte Carlo for Barrier Options
MC for barrier options based on simple discretization leads to biased prices!!
E.g. for CU&O will MC over/under-estimate price? More generally, for path-dependent payoffs
MC is not necessarily unbiased
Fortunately, bias can be reduced by increasing
number of steps (m) in time discretization
Trade-off between # paths (n) & # steps (m)
n↑ Var↓ & m↑ Bias↓ (Bias-Variance trade-off)
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Example
For standard BM {Wt}, find the distribution of
the maximum by time T: M maxW
tt
0tT
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Example
Find the probability that standard BM {Wt} hits barrier B=1 before time T=1
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Example
MC estimates of P max{W } 1 using path
t 0t1
discretization w/ different n,m (n×m=100,000)
Prob.
(true prob.)
n100 n5000
mn ratio
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m1000 m20
Example
Estimate prob. that standard BM hits 1 before time 1, with MC but without bias?
Hint: Use distribution of maximum
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Example
MC estimates of P max{W } 1 using
t 0t1
direct simulation of max{Wt} w/ n=100,000
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Extrema of Brownian Motion
For standard BM {Wt}, the maximum MT over (0,T) is distributed as |WT|
For arithmetic BM {Xt}, the distribution of the maximum is difficult to work with
Reflection principle does not work b/c of drift
However, one can easily simulate random
deviates of maximum using Brownian bridge
Construction allows for general treatment of extrema of various processes
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Extrema of Brownian Motion
Consider arithmetic BM: dX dt dW tt
Conditional on XT=b, the maximum (MT|XT)=maxt{Xt|XT} of the Brownian bridge process has a Rayleigh distribution:
P(M m|X b)1exp2m(mb),m0b TTT
Note that distribution of conditional maximum is independent of the drift, given XT=b
2
30
Extrema of Brownian Motion
Xt
Conditional distr. of M1 | X1 b
b t
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Extrema of Brownian Motion
Procedure for simulating maxima of arithmetic BM:
2 1.GenerateX ~NT,T
T
2. Generate U ~ Uniform(0,1)
3. Calculate M | X TT
X X222Tlog(U) TT
2
For maxima of geometric BM, exponentiate arithmetic BM result
32
Extrema of Brownian Motion
Proof of
M|X TT
X X222Tlog(U) TT
2
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Example
Up-and-out Call price (K