CS476/676 Numeric Computation for Financial Modelling Lecture 5
• European option pricing under a binomial lattice and dividend
• Pricing American option under a binomial lattice
• Convergence (and convergence rate ) of option values under a binomial lattice
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• Black-Scholes formula for European calls and puts
immediately before t immediately after t.
Assume that, at t = 0, it is known that the underlying pays dividendD,e.g.,D=ρS(t−d),att=td, 0≤td ≤T
premium Today
payoff(ST ) T
No arbitrage implies
S ( t +d ) = S ( t −d ) − D
However, this does not affect option holder/writer, i.e., option is
dividend protected.
No arbitrage implies that V (S(t), t) (≡ V ̃ (t)) is a continuous
function of t, i.e.,
− − + + − + V S(td ),td =V S(td ),td =V S(td )−D,td
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Pricing under dividend payment: assuming td is dividend date for n=N−1,N−2,…,0do
for j = 0,1,…,n do
Vn =e−∆tq∗Vn+1 +(1−q∗)Vn+1
j j+1 j end
if tn is the closest time point to td
compute Vjn at t−n by (1) and interpolation
Note: interpolation is required since S(t−d ) − D may not equal to any node price Sjn
Pricing American Options
Consider an American option which can be exercised at the discrete time tn, n = 0,··· ,N.
Given a no-arbitrage lattice, the holder needs to optimally decide, at any tn, n = 0,1,··· ,N, whether to exercise or to continue to hold.
At the expiry T = tN , the holder exercises if and only of payoff(SjN ) > 0 and VjN = payoff(SjN ).
Dynamic Programming =⇒ optimal strategy needs to be optimal at any node, assuming optimal decision starting from that node forward.
Hence a solution can be made one step a time, in backwards.
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Backward iteration:
• At the expiry T = tN, VjN = payoff(SjN)
• At (tn,Sjn), the option has (continue-to-hold ) continuation value:
(V n)∗ = e−r∆tEQ(V n+1|Sn) = e−r∆t(q∗V n+1 + (1 − q∗)V n+1), j j j+1 j
and payoff(Sjn), if exercised. Hence the fair option value is Vjn = max((Vjn)∗, payoff(Sjn)), j = 0, 1, · · · , n
i.e., the optimal exercise strategy is to exercise if the continuation value is less than the payoff.
Pricing American option:
for n=N−1,N−2,…,0do for j = 0,1,…,n do
(V n)∗ = e−∆t q∗V n+1 + (1 − q∗)V n+1 j j+1 j
Vjn = max (Vjn)∗, payoff(Sjn) end
%continuation value
Computational Costs
flops (floating point operations): O(N2).
time efficiency: vectorization Matlab code for efficiency, see §2.5
course notes
space cost: if only V0 is needed, we can compute using O(N) space
BS Formula
Central limit theorem implies that log St − log S0 has a normal distribution with mean αt and variance σ2t.
It can be shown that the binomial lattice standard option value functions converges to the BS formulae.
Let 0 ≤ t ≤ T. As N → +∞, the call option value C(S,t) is C(S, t) = SNcdf (d1) − Ke−r(T −t)Ncdf (d2)
T − t: time to expiry
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Ncdf (d) : cumulative distribution function for a standard normal 1d12
log(S/K) + (r + 0.5σ2)(T − t) σ√T − t
log(S/K) + (r − 0.5σ2)(T − t) σ√T − t
Ncdf(d)=√2π
• Option value only depends on σ, not the average rate of return. • There is no analytic formula for American option pricing.
Convergence Rate
LetVexact betheBSvaluewhent=0andS=S0,i.e., 0
Vexact =V(S0,0). 0
Let V tree(∆t) be the binomial lattice value at t = 0 and S = S0, 0
when ∆t = T/N.
PDE theory shows (later) that, if strike price is at a binomial
Otherwise, smoothing of payoff is required, see §5.4.
V tree(∆t) = V exact + const · ∆t + small o(∆t) 00
If convergence rate is linear, then the ratio
V tree((∆t)/2) − V tree(∆t)
∆t→0 V tree((∆t)/4) − V tree((∆t)/2)
would approach 2.
If convergence rate is quadratic, then
V tree(∆t) = V exact + α(∆t)2 + o((∆t)2) 00
where α is some constant independent of ∆t.
What would the ratio (2) approaches to if convergence rate is
quadratic?
This allows us to computationally verify convergence rate of a method.