代写 Scheme statistic STAD70 Statistics & Finance II

STAD70 Statistics & Finance II
5. Monte Carlo Methods
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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
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Early exercise
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Path dependence
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Multi-asset dependence
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Brownian Motion
 Brownian Motion (BM) is building block of continuous stochastic models
 Standard BM {Wt} is such that W0& WW|W~N(0,ts)
0tss
 Arithmetic BM {Xt} with drift μ & volatility σ is
X 0 & (X X )|X ~N(ts),2(ts) 0 tss
 In form of Stochastic Differential Equation (SDE)
dXdtdWXX t(WW) ttt0t0
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Example
 Find conditional distribution
wheredX dtdW tt
Xt |Xs x,ts
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Example (Brownian Bridge)
 Find conditional distribution
wheredX dtdW tt
Xs |Xt x,s(0,t)
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Geometric Brownian Motion (GBM)
 Expressed in terms of SDE:   2  dY YdtYdW dlogY 
dtdW ttttt2t
 Resulting distribution   logYlogY logY 2 tW 
t0tt Y2
0  Y 2
log t Xt ~N t,2t Y2
0   
Y Y expX ~Y logN2 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 rSdtSdW  S ~S logN r ttttt02
 Called risk-neutral (R-N) measure   rt
 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 erT G   E erT f (S ) 0TT

Example
 Show that under R-N measure
E[St ]  S0ert
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Example
 Find price of forward F0,T (no dividends)
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Estimating Expectations
 If E erT 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):
ˆ1n 
G  erT f S(T) EerT f(S ), withprob.1
0
n
iT
i1 
 Moreover, by Central Limit Theorem (CLT) GG 1n
appr.  2 rT  ˆ ˆ
00
~ N0,1,wheresGn1 i1e f S(T)G
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2 i0
sG n

Example
 Show estimator of E erT 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 SdtSdW tttt
 Payoff function for strike K & maturity T f (ST ) 
 Generate random asset price variates as:  2 
S(T)S(0)expr T TZ i 2i
   
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 17 Barrier Options  Barrier options depend on their paths only through their max/min value before expiry LetM maxS &m minS T t 0tT T t 0tT  Can express barrier option prices as C erTES K     U&O CD&O   T  {MTB} C U&I CD&I  18 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)i0 whereti imit,i1, ,m t 0tT   For GBM dS  rS dt S dW (R-N measure) tttt  2   2 t  T / m ~iid N(0,1) S(ti )  S(ti1)expr  t  t Zi  where Z i    i1, ,m  19 Example  Sample GMB paths St t 20 Example  Price CU&O with K=80, B=90 St Which paths have non-zero payoff? B K t 21 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) 22 Example  For standard BM {Wt}, find the distribution of the maximum by time T: M  maxW  tt 0tT 23 24 Example  Find the probability that standard BM {Wt} hits barrier B=1 before time T=1 25 Example  MC estimates of P max{W }  1 using path  t 0t1 discretization w/ different n,m (n×m=100,000) Prob. (true prob.)  n100  n5000 mn ratio 26 m1000  m20    Example  Estimate prob. that standard BM hits 1 before time 1, with MC but without bias?  Hint: Use distribution of maximum 27 Example  MC estimates of P max{W }  1 using  t 0t1 direct simulation of max{Wt} w/ n=100,000 28 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 29 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)1exp2m(mb),m0b TTT  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 31 Extrema of Brownian Motion  Procedure for simulating maxima of arithmetic BM: 2 1.GenerateX ~NT,T T 2. Generate U ~ Uniform(0,1) 3. Calculate M | X  TT X  X222Tlog(U) TT 2  For maxima of geometric BM, exponentiate arithmetic BM result 32 Extrema of Brownian Motion  Proof of M|X TT X  X222Tlog(U) TT 2 33 34 Example  Up-and-out Call price (K