# Stochastic Processes and Advanced Mathematical Finance. Multiperiod Binomial Tree Models

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1 Steven R. Dunbar Department of Mathematics 203 Avery Hall University of Nebraska-Lincoln Lincoln, NE Voice: Fax: Stochastic Processes and Advanced Mathematical Finance Multiperiod Binomial Tree Models Rating Student: contains scenes of mild algebra or calculus that may require guidance. 1

2 Section Starter Question Suppose that you owned a 3-month option, and that you tracked the value of the underlying security at the end of each month. Suppose you were forced to sell the option at the end of two months. How would you find a fair price for the option at that time? What simple modeling assumptions would you make? Key Concepts 1. A multiperiod binomial derivative model can be valued by dynamic programming computing the replicating portfolio and corresponding portfolio values back one period at a time from the claim values to the starting time. Vocabulary 1. The multiperiod binomial model for pricing derivatives of a risky security is also called the Cox-Ross-Rubenstein model or CRR model for short, after those who introduced it in

3 Mathematical Ideas The Binomial Tree model The multiperiod binomial model has N time intervals created by N + 1 trading times t 0 = 0, t 1,..., t N = T. The spacing between time intervals is t i = t i t i 1, and typically the spacing is equal, although it is not necessary. The time intervals can be any convenient time length appropriate for the model, e.g. months, days, minutes, even seconds. Later, we will take them to be relatively short compared to T. We model a limited market where a trader can buy or short-sell a risky security (for instance a stock) and lend or borrow money at a riskless rate r. For simplicity we assume r is constant over [0, T ]. This assumption of constant r is not necessary, taking r to be r i on [t i, t i 1 ] only makes calculations messier. S n denotes the price of the risky security at time t n for n = 0, 1,... N. This price changes according to the rule S n+1 = S n H n+1, 0 n N 1 where H n+1 is a Bernoulli (two-valued) random variable such that { U, with probability p H n+1 = D, with probability q = 1 p. Again for simplicity we assume U and D are constant over [0, T ]. This assumption of constant r is not necessary, for example, taking U to be U i for i = 0, 1,..., N only makes calculations messier. A binomial tree is a way to visualize the multiperiod binomial model, as in Figure 1. A pair of integers (n, j), with n = 0,... N and j = 0,..., n identifies each node in the tree. We use the convention that node (n, j) leads to nodes (n + 1, j) and (n + 1, j + 1) at the next trading time, with the up change corresponding to (n + 1, j + 1) and the down change corresponding to (n + 1, j). The index j counts the number of up changes to that time, so n j is the number of down changes. Several paths lead to node (n, j), in fact ( n j) of them. The price of the risky underlying asset at trading time tn is then SU j D n j. The probability of going from price S to price SU j D n j is ( ) n p n,j = p j (1 p) n j. j 3

4 Value SU 4 SU 3 SU 2 SU 3 D SU SU 2 D S SUD SU 2 D 2 SD SUD 2 SD 2 SUD 3 SD 3 SD 4 t 1 t 2 t 3 t 4 t Figure 1: A binomial tree. 4

5 To value a derivative with payout f(s N ), the key idea is that of dynamic programming extending the replicating portfolio and corresponding portfolio values back one period at a time from the claim values to the starting time. An example will make this clear. Consider a binomial tree on the times t 0, t 1, t 2. Assume U = 1.05, D = 0.95, and exp(r t i ) = 1.02, so the effective interest rate on each time interval is 2%. We take S 0 = 100. We value a European call option with strike price K = 100. Using the formula derived in the previous section π = = 0.7 and 1 π = 0.3. Then concentrating on the single period binomial branch in the large square box, the value of the option at node (1, 1) is \$7.03 (rounded to cents). Likewise, the value of the option at node (1, 0) is \$0. Then we work back one step and value a derivative with potential payouts \$7.03 and \$0 on the single period binomial branch at (0, 0). This uses the same arithmetic to obtain the value \$4.83 (rounded to cents) at time 0. In the figure, the values of the security at each node are in the circles, the value of the option at each node is in the small box beside the circle. As another example, consider a European put on the same security. The strike price is again 100. All of the other parameters are the same. We work backward again through the tree to obtain the value at time 0 as \$ In the figure, the values of the security at each node are in the circles, the value of the option at each node is in the small box beside the circle. The multiperiod binomial model for pricing derivatives of a risky security is also called the Cox-Ross-Rubenstein model or CRR model for short, after those who introduced it in Advantages and Disadvantages of the model The disadvantages of the binomial model are: 1. Trading times are not really at discrete times, trading goes on continuously. 2. Securities do not change value according to a Bernoulli (two-valued) distribution on a single time step, or a binomial distribution on multiple 5

6 Value K = t 1 t 2 t Figure 2: Pricing a European call. 6

7 Value K = t 1 t 2 t Figure 3: Pricing a European put. 7

8 time periods, they change over a range of values with a continuous distribution. 3. The calculations are tedious. 4. Developing a continuous theory will take detailed limit-taking considerations. The advantages of the model are: 1. It clearly reveals the construction of the replicating portfolio. 2. It clearly reveals that the probability distribution is not centrally involved, since expectations of outcomes aren t used to value the derivatives. 3. It is simple to calculate, although it can get tedious. 4. It reveals that we need more probability theory to get a complete understanding of path dependent probabilities of security prices. It is possible, with considerable attention to detail, to make a limiting argument and pass from the binomial tree model of Cox, Ross and Rubenstein to the Black-Scholes pricing formula. However, this approach is not the most instructive. Instead, we will back up from derivative pricing models, and consider simpler models with only risk, that is, gambling, to get a more complete understanding of stochastic processes before returning to pricing derivatives. Some caution is also needed when reading from other sources about the Cox-Ross-Rubenstein or Binomial Option Pricing Model. Many other sources derive the Binomial Option Pricing Model by discretizing the Black-Scholes Option Pricing Model. The discretization is different from building the model from scratch because the parameters have special and more restricted interpretations than the simple model. More sophisticated discretization procedures from the numerical analysis of partial differential equations also lead to additional discrete option pricing models that are hard to justify by building them from scratch. The discrete models derived from the Black-Scholes model are used for simple and rapid numerical evaluation of option prices rather than for motivation. 8

9 Sources This section is adapted from: Chapter 2, Discrete Processes in Financial Calculus by M. Baxter, A. Rennie [2] and Quantitative Modeling of Derivative Securities by M. Avellaneda and P. Laurence [1]. Algorithms, Scripts, Simulations Algorithm Comment Post: Set up and solve for the value of the European call option in a two period binomial model. Comment Post: Output the derivative security value. 1 Set values of S, U, D, r, T, and K. 2 Define the derivative security payoff function. (European call option) 3 Define the risk neutral measure π 4 Solve for derivative values at (1, 1) and (1, 0) with the risk neutral measure formula 5 Solve for the derivative value with the risk neutral measure formula linear solver 6 Print the derivative value Scripts Scripts R R script for multiperiod S < 100 factorup < 1.05 factordown < 0.95 B < 1 effr < 1.02 d e l t a t i < 1 9

10 K < 100 f < function ( x, s t r i k e ) { # European c a l l option max( x s t r i k e, 0) } riskneutralmeas < function ( fup, fdown, exprdt ) { # r i s k n e u t r a l measure pi ( exprdt fdown )/( fup fdown ) } pirnm < riskneutralmeas ( factorup, factordown, effr ) v11 < (1/ effr ) (pirnm f (S factorup factorup, K) + (1 pirn factordown, K) ) v10 < (1/ effr ) (pirnm f (S factorup factordown, K) + (1 pi factordown factordown, K) ) value < (1/ effr ) (pirnm v11 + (1 pirnm) v10 ) cat ( value :, value, \n ) Octave Octave script for multiperiod S = 100; factorup = ; factordown = ; B = 1 ; effr = ; # effr = exp ( r d e l t a t i ) d e l t a t i = 1 ; K = 100; function r e t v a l = f ( x, s t r i k e ) # European c a l l option r e t v a l = max( x s t r i k e, 0 ) ; endfunction function r e t v a l = riskneutralmeas ( fup, fdown, exprdt ) 10

11 r e t v a l = ( exprdt fdown ) / ( fup fdown ) ; #r i s k n e u t r a l measure p i endfunction pirnm = riskneutralmeas ( factorup, factordown, effr ) ; v11 = (1 / effr ) (pirnm f ( S factorup factorup, K) + (1 pirnm) f ( S factorup factordown, K) ) ; v10 = (1 / effr ) (pirnm f ( S factorup factordown, K) + ( 1 pirnm) f ( S factordown factordown, K) ) ; value = (1 / effr ) (pirnm v11 + (1 pirnm) v10 ) ; disp ( d e r i v a t i v e value : ), value Perl Perl PDL script for multiperiod use PDL : : N i c e S l i c e ; \$S = 100; \$factorup = ; \$factordown = ; \$B = 1 ; \$effr = ; # effr = exp ( r d e l t a t i ) \$ d e l t a t i = 1 ; \$K = 100; sub f { # European c a l l option my ( \$x, \$ s t r i k e ) ; return max( pdl [ \$x \$ s t r i k e, 0 ] ) ; } sub riskneutralmeas { my ( \$fup, \$fdown, \$exprdt ) ; return ( \$exprdt \$fdown ) / ( \$fup \$fdown ) ; #r i s k n e u t r a l measure pi } \$pirnm = riskneutralmeas ( \$factorup, \$factordown, \$effr ) ; 11

12 \$v11 = ( 1 / \$effr ) ( \$pirnm f ( \$S \$factorup \$factorup, \$K ) + ( 1 \$pirnm ) f ( \$S \$factorup \$factordown, \$K ) ) ; \$v10 = ( 1 / \$effr ) ( \$pirnm f ( \$S \$factorup \$factordown, \$K ) + ( 1 \$pirnm ) f ( \$S \$factordown \$factordown, \$K ) \$value = ( 1 / \$effr ) ( \$pirnm \$v11 + ( 1 \$pirnm ) \$v10 ) ; print value :, \$value, \n ; SciPy Scientific Python script for multiperiod import scipy S = 100 factorup = 1.05 factordown = 0.95 B = 1 effr = 1.02 effr = exp(r*delta t i )deltati = 1K = 100 def f(x, strike): European call option return max(x - strike, 0) def riskneutralmeas(fup, fdown, exprdt): return (exprdt - fdown) / (fup - fdown) risk neutral measure pi pirnm = riskneutralmeas(factorup, factordown, effr) v11 = 1 / effr * (pirnm * f(s * factorup * factorup, K) + (1 - pirnm) * f(s * factorup * factordown, K)) v10 = 1 / effr * (pirnm * f(s * factorup * factordown, K) + (1 - pirnm) * f(s * factordown * factordown, K)) value = 1 / effr * (pirnm * v11 + (1 - pirnm) * v10) print value:, value, 12

13 Problems to Work for Understanding 1. Consider a two-time-stage example. Each time stage is a year. A stock starts at 50. In each year, the stock can go up by 10% or down by 3%. The continuously compounded interest rate on a \$1 bond is constant at 6% each year. Find the price of a call option with exercise price 50, with exercise date at the end of the second year. Also, find the replicating portfolio at each node. 2. Consider a three-time-stage example. The first time interval is a month, then the second time interval is two months, finally, the third time interval is a month again. A stock starts at 50. In the first interval, the stock can go up by 10% or down by 3%, in the second interval the stock can go up by 5% or down by 5%, finally in the third time interval, the stock can go up by 6% or down by 3%. The continuously compounded interest rate on a \$1 bond is 2% in the fist period, 3% in the second period, and 4% in the third period. Find the price of a call option with exercise price 50, with exercise date at the end of the 4 months. Also, find the replicating portfolio at each node. 3. A European cash-or-nothing binary option pays a fixed amount of money if it expires with the underlying stock value above the strike price. The binary option pays nothing if it expires with the underlying stock value equal to or less than the strike price. A stock currently has price \$100 and goes up or down by 20% in each time period. What is the value of such a cash-or-nothing binary option with payoff \$100 at expiration 2 time units in the future and strike price \$100? Assume a simple interest rate of 10% in each time period. 4. A long strangle option pays max(k 1 S, 0, S K 2 ) if it expires when the underlying stock value is S. The parameters K 1 and K 2 are the lower strike price and the upper strike price, and K 1 < K 2. A stock currently has price \$100 and goes up or down by 20% in each time period. What is the value of such a long strangle option with lower strike 90 and upper strike 110 at expiration 2 time units in the future? Assume a simple interest rate of 10% in each time period. 5. A long straddle option pays S K if it expires when the underlying stock value is S. The option is a portfolio composed of a call and a 13

14 put on the same security with K as the strike price for both. A stock currently has price \$100 and goes up or down by 10% in each time period. What is the value of such a long straddle option with strike price K = 110 at expiration 2 time units in the future? Assume a simple interest rate of 5% in each time period. Reading Suggestion: References [1] Marco Allavenada and Peter Laurence. Quantitative Modeling of Derivative Securities. Chapman and Hall, HG 6024 A3A [2] M. Baxter and A. Rennie. Financial Calculus: An introduction to derivative pricing. Cambridge University Press, HG 6024 A2W554. [3] S. Benninga and Z. Wiener. The binomial option pricing model. Mathematical in Education and Research, 6(3):27 33, [4] Freddy Delbaen and Walter Schachermayer. What is a... free lunch. Notices of the American Mathematical Society, 51(5), [5] Paul Wilmott, S. Howison, and J. Dewynne. The Mathematics of Financial Derivatives. Cambridge University Press,

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