On the Convergence of European Lookback Options with Floating Strike in the Binomial Model
Fabien Heuwelyckx
Universit´ede Mons (Belgium) [email protected]
Warsaw, June 12th 2013 Introduction State of the art The approximation Work in progress
1 Introduction
2 State of the art
3 The approximation
4 Work in progress
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 2 / 21 Introduction State of the art The approximation Work in progress An intriguing function
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 3 / 21 Introduction State of the art The approximation Work in progress Asymptotic expansion
Study of the behavior of the option price as a function of the number of steps n in the Cox-Ross-Rubinstein model. In particular, to write this price as Π Π 1 Πfl = Πfl + √1 + 2 + O , n BS n n n3/2
where the coefficients Πi are bounded functions of n.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 4 / 21 Introduction State of the art The approximation Work in progress Approximation for vanilla options
Diener-Diener (2004) The evaluation of the European vanilla call in the Cox-Ross-Rubinstein model satisfies the relation
d2 − 1 2 2 S e 2 A − 12σ T (∆ − 1) 1 C V = C V + 0√ n + O , n BS 24σ 2πT n n3/2 with √ ln(S0/K)−nσ T /n ∆n = 1 − 2 √ , 2σ T /n 2 2 2 2 2 2 2 A = −σ T (6 + d1 + d2 ) + 4rT (d1 − d2 ) − 12r T .
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 5 / 21 Introduction State of the art The approximation Work in progress Other results
Results for some other options are also known: binary options by Chang and Palmer (2007); barrier options by Lin and Palmer (2013).
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 6 / 21 Introduction State of the art The approximation Work in progress Lookback option with floating strike
In the case of the European option:
Payoff for the call: f (T ) = ST − min St , 0≤t≤T
Payoff for the put: f (T ) = max St − ST , 0≤t≤T where St is the price of the underlying at time t and T the time to maturity.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 7 / 21 Introduction State of the art The approximation Work in progress Other common notations
r the risk free interest rate, σ the volatility of the underlying, √ σ T /n un = e the proportional upward jump, −1 dn = un the proportional downward jump.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 8 / 21 Introduction State of the art The approximation Work in progress Cheuk-Vorst lattice (1997)
Modified tree V . Backward induction. Value associated with a specific node depends only on the time and on the difference (in powers of un) between the present and the lowest value of the underlying from time t = 0 to the present time. For the call this difference is the value j such that
j Sm = min Si un. 0≤i≤m
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 9 / 21 Introduction State of the art The approximation Work in progress Option evaluation
Value inside the tree is not the option price. This value is the number by which we have to multiply the underlying price to obtain the corresponding option price:
fl Cn (m) = Sm V (j, m).
A previous node is the expectation of the following two nodes with respect to the probability
− rT qn = pnune n ,
with pn the traditional risk-neutral probability.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 10 / 21 Introduction State of the art The approximation Work in progress Example for n = 4
V (4, 4)
V (3, 3) V (3, 4)
V (2, 2) V (2, 3) V (2, 4)
V (1, 1) V (1, 2) V (1, 3) V (1, 4)
V (0, 0) V (0, 1) V (0, 2) V (0, 3) V (0, 4)
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 11 / 21 Introduction State of the art The approximation Work in progress Value function of the lookback option
Example with S0 = 80, σ = 0.2, r = 0.08 and T = 1.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 12 / 21 Introduction State of the art The approximation Work in progress Difference with the traditional tree
Each final level can be reached by several different numbers of upward jumps. Because of this, Vn(0, 0) can be written as a double sum
n l X −j X k n−k Vn(0, 0) = (1 − un ) Λj,k,n qn (1 − qn) j=0 k=j
with jn + j k n n l = , Λ = − , 2 j,k,n k − j k − j − 1
if k > j and Λj,k,n = 1 if k = j.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 13 / 21 Introduction State of the art The approximation Work in progress Price for a fixed number n
The call price (with n steps) can be deduced from this construction and is S0 Vn(0, 0), with
−rT Qn(1 − dn) 1 e Vn(0, 0) = φ1 − φ2 + φ3, (1 − Qn)(1 − Qndn) 1 − Qn 1 − Qndn where Q = qn , n 1−qn
φ1 = Bn,qn (bn/2c) − Qn Bn,qn (bn/2c − 1),
φ2 = Qn Bn,1−qn (bn/2c) − Bn,1−qn (bn/2c − 1),
φ3 = Qndn Bn,1−pn (bn/2c) − un Bn,1−pn (bn/2c − 1),
and Bn,p the cumulative distribution function of the binomial distribution with parameters n and p. F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 14 / 21 Introduction State of the art The approximation Work in progress The results (1/2)
Approximation of the complementary cumulative function for some binomial distributions (H. 2013) Suppose that p = 1 + √α + β + γ + δ + O 1 and n 2 n n n3/2 n2 n5/2 √ j = n + a n + 1 + b + √c + d + O 1 , where the sequence (b ) is bounded. Then n 2 2 n n n n3/2 n n
n X n pk (1 − p )n−k k n n k=jn 2 e−A /2 B C −C B2 D −D B −D B3 1 = Φ(A) + √ √n + 0 2 n + 0 1 n 3 n + O , 2π n n n3/2 n2
2 2 where A = 2(α − a), Bn = 2(β − bn), C0 = 2(α A + γ − c) + (2α/3 − A/12)(1 − A ), 2 C2 = A/2, D0 = 2(2αβA + δ − d) + 2(1 − A )β/3, 2 4 2 3 2 D1 = (1 − 4A + A )/12 − 2α(α − A − αA + A /3) + 2A(γ − c), D3 = (1 − A )/6, and Φ is the cumulative distribution function of the standard normal distribution.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 15 / 21 Introduction State of the art The approximation Work in progress The results (2/2)
Asymptotics of the lookback call (H. 2013) If r 6= 0 then the asymptotic formula for the price of the European lookback call option with floating strike is √ σ T 1 C fl = C fl + (C fl − S )√ n BS 2 BS 0 n √ 2 2 −a1/2 hσ T fl h −rt 3 i σ T e i 1 + C + 2S0 Φ(a1) − e Φ(a2) − + S0 √ 12 BS 2 2 2π n 1 + O , n3/2 √ √ with a1 = (r/σ + σ/2) T and a2 = (r/σ − σ/2) T .
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 16 / 21 Introduction State of the art The approximation Work in progress Work in progress
Do not restrict the evaluation to time t = 0. At time t, we have the information for the minimal value of the underlying between times 0 and t. We divide the remaining time line in n steps. The form of the previous tree changes if this minimal value is less than the underlying price at time t.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 17 / 21 j = a + n
j = a + n − 2bac j = n − bac − 1 j = a + n − 2(bac + 1)
j = a + n − 2b(a + n)/2c j = 0
Introduction State of the art The approximation Work in progress Generalization of the tree
a(n) = log(√St /Smin) σ T /n
a
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 18 / 21 a(n) = log(√St /Smin) σ T /n j = n − bac − 1 j = a + n − 2(bac + 1)
j = a + n − 2b(a + n)/2c j = 0
Introduction State of the art The approximation Work in progress Generalization of the tree
j = a + n
j = a + n − 2bac
a
The number of paths to arrive at a black state can be found as in the binomial tree.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 18 / 21 j = a + n
log(S /S ) j = aa+(nn) =− 2ba√ct min σ T /n j = n − bac − 1
j = 0
Introduction State of the art The approximation Work in progress Generalization of the tree
a j = a + n − 2(bac + 1)
j = a + n − 2b(a + n)/2c
The number of paths to arrive at a green state can be found as in the binomial tree without forgetting to subtract the paths that have been absorbed in the red tree.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 18 / 21 j = a + n
log(S /S ) j = aa+(nn) =− 2ba√ct min σ T /n
j = a + n − 2(bac + 1)
j = a + n − 2b(a + n)/2c
Introduction State of the art The approximation Work in progress Generalization of the tree
j = n − bac − 1 a
j = 0
The number of paths to arrive at a red state can be found more or less as in the basic case where the option is evaluated at emission.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 18 / 21 Introduction State of the art The approximation Work in progress
Thank you for your attention!
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 19 / 21 Introduction State of the art The approximation Work in progress Bibliography
1 Chang L.B., and Palmer K. (2007): Smooth convergence in the binomial model. Finance Stoch. 11, 91-105. 2 Cheuk T.H.F, and Vorst T.C.F (1997): Currency lookback options and observation frequency: a binomial approach. J. Int. Money Finance 16, 173-187. 3 Diener F., and Diener M. (2004): Asymptotics of the price oscillations of a European call option in a tree model. Math. Finance 14, 271-293. 4 Heuwelyckx F. (submitted): Convergence of european lookback options with floating strike in the binomial model. http://arxiv.org/abs/1302.2312. 5 Lin J., and Palmer K. (2013): Convergence of barrier option prices in the binomial model. Math. Finance.
F. Heuwelyckx (UMONS) Lookback Options Warsaw 2013 20 / 21