Binomial model: numerical algorithm

Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4 5,4 max X S0 u d,0 C \ 3 4,3 C \ S 0 u d / C \ 1,0 S0 d / C 3,1 \ C \ 3 S d /,0 0 3,0 S u d 0 S u d 0 / C 3 5,3 max X S0 u d,0 C \ 3 S0 S u d 4, 0 3 u d / C 3 5, max X S0 u d,0 C \ 4 4,1 C \ 4 S 0 S0 u d d / C 4 5,1 max X S0 u d,0 C \ 5 4,0 S 0 d C X S d 5 5,0 max 0,0

Binomial model: numerical algorithm The calculations are performed as follows: Start at the end of the tree (at time T). The lowest node has the value: S 0. d N. Set the boundary condition in this node with respect to the option type, (see below). For each node at the same time, calculate the next price by multiplying with u/d and use the same boundary condition C. Next, go backward in the tree and calculate all possible stock prices as in the figure above, and thereafter the option value, C. We get:

Binomial model: numerical algorithm American: C max X S u, e P C P C 4 rt 4,4 0 u 5,5 d 5,4 C max X S u d, e P C P C 3 rt 4,3 0 u 5,4 d 5,3 C max X S u d, e P C P C rt 4, 0 u 5,3 d 5, C max X S d, e P C P C 4 rt 4,0 0 u 5,1 d 5,0 European: C e P C P C rt 4,4 u 5,5 d 5,4 C e P C P C rt 4,3 u 5,4 d 5,3 C e P C P C rt 4,0 u 5,1 d 5,0

Binomial model: The Greeks C 1,1 1,0 0 0,,1,1,0 0 0 0 0 S u S u d S u d S d 1 S0u S0d,1 0 t C S u S d C C C C C C C0( ) C0( ) C0( r) C0( r r) r

Binomial model: The Greeks P P S S P P T P r

Boundary conditions At maturity we use the following conditions, depending on the option type with strike price X BC = max(s X, 0) BC = max(x - S, 0) Call option. Put option. The lowest price is put r T Pmin X e S call Pmin S X e r T

Example American Put Option Compute the price of an American put option with strike price K = 100 and exercise time T = years, using a binomial tree with two trading dates t 1 = 0 and t = 1 (your portfolio at time t 3 = is the same as your portfolio at time t = 1) and parameters s 0 = 100, u = 1.4, d = 0.8, r = 10%, and p = 0.75

Example Replicating Portfolio In the binomial tree below the price of a binary asset-or-nothing option with expiry in two years and payoff S() if S() 10 X 0 otherwise has been computed using the parameters s 0 = 80, u = 1.5, d = 0.5, r = 0, and p = 0.50. In the definition of the contract function S() denotes the stock price at time t =. Find the replicating portfolio for this option and verify that the option is self-financing.

Replicating Portfolio We can use the values to calculate the replicating portfolio. At t = 0 the following must hold: x y10 90 x y 40 0 Since regardless if the stock price goes up or down the value of the portfolio should equal the value of the option. This yields: x = -45 and y = 9/8. We can also use: 1 u ( d) d ( u) 11.50 0.590 x 45 1 r u d 1 1.5 0.5 1 ( u) ( d) 1 90 0 9 y S0 u d 80 1.5 0.5 8 The same calculations can be made to find the replicated portfolio in all the nodes, e.g., where S = 10: 11.50 0.5180 x 90 1 1.5 0.5 1 180 0 3 y 10 1.5 0.5

Probabilities in the model S S u S e n n dt max 0 0 S S d S e n n dt min 0 0 u e d e dt dt n. P S P resp P S P max u min n d 5, 5 1 path 4, 4 3, 3 5, 4 5 paths, 4, 3 1, 1 3, 5, 3 10 paths 0, 0, 1 4, 1, 0 3, 1 5, 10 paths, 0 4, 1 3, 0 5, 1 5 paths 4, 0 5, 0 1 path t = 0 t = 1 t = t = 3 t = 4 t = 5

Finite difference methods Parabolic boundary value problem of the Black- Scholes type : 1 C S C ( r ) S C rc t S S If we let x = ln(s) we can rewrite the PDE by use of: C C x 1 C S x S S x C C C C C C 1 1 1 1 1 1 S S S x S x S x S S x S x S x 1 C 1 C S x S x

Finite difference methods C 1 C 1 C C r rc t x x x t x x C 1 C C rc where ν=r δ ½σ By doing this we have removed the explicit dependencies of S and thereby get the coefficients independent of the stock price!!!

The explicit finite difference method C Ci 1, j1 Ci 1, j1 x x C C C C x x i1, j1 i1, j i1, j1 Backward differences Ci 1, j Ci, j 1 C i 1, j 1 Ci 1, j Ci 1, j 1 Ci 1, j 1 Ci 1, j 1 rc t x x 1 C p C p C p C 1 rt i, j u i 1, j 1 m i 1, j d i 1, j 1 i1, j p p p u m d 1 t x 1 t x 1 t x x x x 3t

The implicit finite difference method C x C x C C i, j1 i, j1 x C C C i, j1 i, j i, j1 x Forward differences Ci 1, j Ci, j 1 C i, j 1 Ci, j Ci, j 1 Ci, j 1 Ci, j 1 rc t x x i1, j p C p C p C C u i, j1 m i, j d i, j1 i1, j boundary conditions p u pm 1 t r t x p d 1 t x x 1 t x x S C C U i, N i, N j S i, N i, N U j C j1 C j1 i, N i, N L j1 j call put 0 L 0 L Si, N S j i, N j1 U

The implicit finite difference method Ci, N j 1 1 0......... 0 C j 1 pu pm p 0...... 0 d 0 p 0... 0 in, j u pm pd C........................ 0... 0 pu pm pd 0 C i, Nj 0...... 0 pu pm pd C i, Nj1 0......... 0 1 1 C i, Nj U C Ci... Ci Ci L i, N i1, N 1, N 1, N 1, N j1 j j j1

Crank-Nicholson C C Ci1, j1 Ci 1, j Ci 1, j1 Ci, j1 Ci, j Ci, j1 1 x i1, j i, j t Ci 1, j1 Ci 1, j1 Ci, j1 C i, j1 Ci1, j Ci, j r 4x p C p C p C p C p C p C u i, j1 m i, j d i, j1 u i1, j1 m i1, j d i1, j1 p p p u m d 1 t 4 x x 1 t rt x 1 t 4 x x The accuracy in the methods above are: O(Δx + Δt), O(Δx + Δt) and O(Δx + (Δt/) ) respectively.

Schema - Finite Difference

The Hopscotch method

The Hopscotch method

Monte-Carlo Simulations The stock price is simulated by a stochastic process: dst rstdt Stdzt For simplicity, study the natural logarithm of the stock price: x t = ln(s t ) which gives: dx dt dz t r 1 x x t z z tt t tt t t xt x i t t t i1 S exp t x i ti z t t z t t

Monte-Carlo Simulations In the figure below, we show 100 such simulations of the stock price during a half of a year divided into 100 intervals. At the starting time, the stock price is 100, the volatility is 40% and the risk-free interest rate 6%.

Monte-Carlo Simulations

Monte-Carlo Simulations (10 000)

Monte-Carlo Simulations For each scenario, we then calculate the profit of the call options as: max(s T X, 0). To find the theoretical option value we calculate the mean value of the discounted pay-off: N 1 C exp( rt ) max S X,0 0 Ti, N i 1 where X is the strike price of the option. The standard deviation (SD) and the standard error (SE) of the simulations is given by: (Remember: the annualized volatility σ is the standard deviation of the instrument's yearly logarithmic returns.) N N 1 1 SD C C rt N N SD SE N T, i T, i exp 1 i1 N i1

Introduction to probability theory Study a Binomial three with the following properties: u = => d = 1/u = 0.5, S 0 = 4 and P u = P d = ½. where S uu u S S ud uds ( ) 0, ( ) 0,...

Introduction to probability theory If we are tossing a coin one, two and tree times, we get the following sample space: Ω 1 = {u, d}= {ω 1 }, Ω = {uu, ud, du, dd}= {ω }, Ω 3 = {uuu, uud, udu, duu, udd, dud, ddu, ddd}= {ω 3 } Introduce the interest rate r: 1 CU (cash unit) -> (1 + r). 1 CU =1. R CU. The factor R must be in the interval: d R u because if R > u nobody should be interested to buy the stock, if R < d then r < 0 which is unrealistic. Statement: We say that the model above is free of arbitrage if: d R u.

Introduction to probability theory Example: Let us study a European call option with strike K at t = 1. On maturity, the value is given by: V ( ) ( S ( ) K) max( S ( ) K,0) 1 1 1 We are looking for the arbitrage-free price. The two possible outcome, u and d are given by: ( us0 K) if 1 u V1 ( ) ( ds0 K) if 1 d To hedge a short position of the option we have to buy 0 stocks. At t = 0 we have then sold the option, giving us V 0 cash units. But we also buy 0 stocks at S 0. We then have (V 0 Δ 0 S 0 ) cash units to put in the bank (or that s what we had to borrow, depending of the sign) at a rate of r, where R = 1 + r. The value process gives us two possible values on maturity:

Introduction to probability theory V ( u) S ( u) R ( V S ) We get 1 0 1 0 0 0 V ( d) S ( d) R ( V S ) 1 0 1 0 0 0 V1( u) V1( d) V 0 S ( u) S ( d) S 1 1 By inserting 0 into the equation above, we find the price of the option at t = 0: 1 R d R u 1 1 V V ( u) V ( d) q V ( u) q V ( d) E V R u d u d R R Q 0 1 1 u 1 d 1 1 where we have defined p and q as the risk-neutral probabilities: q u R d q u d d R u u d

Introduction to probability theory We let the expression X E X represent the arbitrage free price on the option on the contingent claim X with respect to the risk-neutral probability measure Q, the martingale measure. Similar, we get and so on.. 1 Q R 1 V( uu) V( ud) V1 ( u) pv ( uu) qv ( ud) ; 1( u) R S ( uu) S ( ud) 1 V( du) V( dd) V1 ( d) pv ( du) qv ( dd) ; 1( d) R S ( du) S ( dd)

Finite Probability Spaces Let F be the set of all subsets to the sample space: Ω (Ø, {ddd}, {uuu, uud, udu, ddd}, Ω are examples of some) where Ø is the empty set. Then, we define a probability measure P by a function mapping F into the interval [0, 1] with P(Ω) = 1, where P UAk P A k 1 k 1 k and A 1, A,... is a sequence of disjoint sets in F. Probability measures has the following interpretation: Let A be a subset of F. Imagine that Ω is the set of all possible outcomes of some random experiment. There is a certain probability, between 0 and 1, that when that experiment is performed, the outcome will lie in the set A. We think of P(A) as this probability. From now we will use P u = 1/3 and P d = /3.

σ-algebra Definition: A σ-algebra is a collection F of sub sets in Ω with the following three properties: F C A F A F A1, A... is a sequence of subspaces to F U A k k F It is essential to understand that, in probabilistic terms, the σ- algebra can be interpreted as "containing all relevant information" about a random variable.

-algebra Example: Some important σ-algebras to Ω above is: F 0 = {Ø, Ω} F 1 = {Ø, Ω, {uuu, uud, udu, udd}, {duu, dud, ddu, ddd}} F = {Ø, Ω, {uuu, uud}, {udu, udd}, {duu, dud}, {ddu, ddd}} and all unions of these} F 3 = F = the set of all sub sets of Ω. We say that F 3 is finer than F, which is finer than F 1. If we introduce the terms A u = {uuu, uud, udu, udd} = {u**}, A d = {d**}, A uu ={uu*} etc, we can write: F 1 = {Ø, Ω, A u, A d } F = {Ø, Ω, A u, A d, A uu, A ud, A du, A dd, A uu UA du, A uu UA dd, A ud UA du, A ud UA dd, A uuc, A ud c, A du c, A dd c }

Filtrations F 1 = {Ø, Ω, A u, A d } F = {Ø, Ω, A u, A d, A uu, A ud, A du, A dd, A uu UA du, A uu UA dd, A ud UA du, A ud UA dd, A uuc, A ud c, A du c, A dd c }

Measures Definition: A pair (X, F), where X is a set and F an σ-algebra on X is called a measurable space. The sub-spaces that exist in F are called F-measurable sets. In particular, if a random variable Y is a function of X, Y = Ф(X), then Y is F X -measurable. Definition: A finite measure μ on a measurable space is a function such as: ( i) ( A) 0, ( ii) ( ) 0, ( iii) If A F k 1,,... and A A for i j, then k i j ( UAk) ( Ak) k1 k1

Some definitions Definition: A filtration F = F = {F t ; t 0} is a sequence of σ- algebras F 0, F 1,..., F n such that F t contains all sets in F t-1 : F F t t 0 s t Fs Ft If we consider a finite probability space (Ω, F t, P) with the filtration of σ-algebras sometimes called σ-fields. Definition: X is F-adapted if X t is F t -measurable for all t 0. Definition: A function f: X R is said to be F measurable if for each interval I the set f -1 (I) is F measurable, i.e.: x X f ( x) I F Definition: A stochastic variable X is a mapping of Ω on R: X : Ω -> R so that X is F-measurable

Stochastic Process Definition: A stochastic process can be considered as a discrete set of time indexed random variables or, as in time, a continuous set. In many situations we consider such a process containing a drift μ and diffusion σ: X(t + Δt) X(t) = μ[t, X(t)] Δt + σ[t, X(t)]Z(t) Sometimes this is interpreted as a random process (a random walk) upon a deterministic drift. In the continuous limit the random process becomes a Wiener process.

Wiener Process Definition: A stochastic process {W(t); t 0} is called a Wiener process if: 1. W(0) = 0. (W(u) W(t)) and (W(s) W(r)) are independent (i.e. W have independent increments) r s t u. 3. W(t) W(s) is normal distributed N é 0, t - sù. 4. W(t) have continuous trajectories. ë û " t s A very important property of a Wiener process (a Brownian motion) is (dw) = dt. In risk neutral valuation, we have a risk-free bond and a stock following the process: ds( t) ( t) S( t) dt ( t) S( t) dw( t)

Expectation value Definition: The expectation value (or mean value) of X given (Ω, F, P) is: ( ) { } E X X P in the discrete case and ( ) { } E X X dp in the continuous case.

Variance Definition: The Variance of X: Var X X ( ) E X dp{ } n Var X X ( ) E X ( ) P x E X ( ) ( x ) k1 k X k ( ) ( ) ( ) ( ) E X E X E X E X

Example Example: Calculate E[S 3 ] 3 ( ) ( ) ( ) ( ) S ( duu) Pduu S ( ddu) Pddu S ( dud ) Pdud S ( ddd ) Pddd E S S uuu P uuu S uud P uud S udu P udu S udd P udd 16 P( A ) 4 P( A U A ) P( A ) uu ud du dd 16 P S 16 4 P S 4 P S 1 1 4 4 36 16 S 16 4 4 1 16 4 4 S S 9 9 9 9

Indicator function Definition: An indicator function I defined by: I A 0 ( x) 1 x A x A where A is called a set indicated by I A. Definition: A function h is called simple if n h( x) ckik( x) k 1

Probability spaces A Probability spaces is defined by (Ω, F, P), where: Ω is a non empty set, sample space, which contains all possible outcomes of some random experiment. F is a σ-algebra of all subsets of Ω. P is a probability measure on (Ω, F ) which assigns to each set A Î F a number P(A) = [0, 1], which represent the probability that the outcome of the random experiment lies in A. Given (Ω, F, P) and a stochastic variable X. If X is a indicator function (e.g., X(Ω) = I A (Ω) = 1 if ω A and 0 otherwise) then: If X is simple n XdP P( A) XdP c I dp c P( A ) XdP X I AdP k Ak k k k1 k1 n Î A