Tree Pricing_Draft

Binomial Tree Option Pricing

Author:Brian Bannon

Contents

1 Introduction 1

2 Theory of Binomial Tree Pricing Models 12.1 Black Scholes Merton Model . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Binomial Tree Approximation Model . . . . . . . . . . . . . . . . . . . . . . 32.3 Leisen Reimer Binomial Tree Model . . . . . . . . . . . . . . . . . . . . . . . 52.4 Richardson Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Methodology of Testing 73.1 Selection of Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Conclusions 84.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 References 9

6 Appendix 106.1 MatLab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6.1.1 Assignment 3.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.1.2 binoTree.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.1.3 ppInv.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.1.4 LR BinoTree.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

i

1 Introduction

In this report we are going to look at Binomial Tree option pricing and extensions onto thetree pricing option for plain vanilla options.We will be implementing the Leisen Reimer (LR) Binomial Tree and comparing the pricingresults with normal binomial tree methods along with the standard Black Scholes models tocompare our models. Lastly we will be performing variance reduction techniques to ensurethat the error within these models are reduced to can obtain a value close to the stock’sintrinsic

2 Theory of Binomial Tree Pricing Models

2.1 Black Scholes Merton Model

The Black-Scholes-Merton model is mathematical model used to give a theoretical estimateof the price of a European-style option.Testing over the years have shown that the Black-Scholes Merton model gives good approximations of the observed price and is widely usedwithin industry as an approximation into the price of a given option.This model assumes certain constraints in the market to price options effectively.It assumesthat the market has at least one risky asset called the stock and one risk-less asset calledthe money market, cash or bond.The asset is assumed to have a risk-less rate(risk-free in-terest rate), moves in a geometric Brownian motion(random walk) with drift and assume itsdrift and volatility are constant.Lastly the stock does not pay dividend( For our model inthis report the stock does pay dividends, which is just a trivial extension onto the originalformula).Assumptions on the market have to be made as well. There is to be no arbitrage opportu-nities, it is possible to borrow and lend any amount t the risk free rate,it is possible to buyand sell any amount of stock, and transactions fees and costs do not apply.With these assumptions holding.Then there is a option trading in the market.The option hasa current price now (spot price) and will reach a certain price(strike price) with a certainpay-off at a specified date(Maturity) in the future.The future price of this option is com-pletely determined by current price of the option even though we know nothing about thepath and future stock price of the option.From creating a dynamic hedging position of a longstock position and a short option position a differential equation called the Black-Scholesformula was developed, which governed he price of a option.

∂V

∂t+

1

2σ2S2∂

2V

∂S2+ rS

∂V

∂S− rV = 0 (1)

Where S is the price of the stock,V (S, t) is the price of the derivative as a function of timeand stock price,K is the strike price,r is the risk free rate( expressed in terms of continuouscompounding),σ is the volatility of the returns of the assets.The value for a non-dividend paying call option is:

C(S, t) = N(d1)S −N(d2)Ke−r(T−t) (2)

1

And the price for a put option is

P (S, t) = Ke−r(T−t) − S + C(S, t) = N(−d2)Ke−r(T−t) −N(−d1)S (3)

where

d1 =1

σ√T − t

[ln(S

K) + (r +

σ2

2)(T − t)] (4)

d2 =1

σ√T − t

[ln(S

K) + (r − σ2

2)(T − t)] (5)

As above where T − t is the time to maturity t = now and T = maturity,and N(x) isthe cumulative distribution function of the standard normal distribution.For a European calloption to be in the money, this mean that the spot price of the option at maturity must begreater that the strike price at the current time of writing the option.Thus when it comes tothe maturity date the issuer will exercise the right to buy the option for the strike price andmake a profit from buying the options at a cheaper price than market price.If the spot priceis lower than the strike price the holder will choose the right not to exercise the option.Thepayout for this is written as follows:

callpayoff = max(S −K, 0) =

{S −K, if S > K.

0, if S < K.(6)

For a put option the payout for this is

putpayoff = max(K − S, 0) =

{K − S, if S < K.

0, if S > K.(7)

where the holder will exercise the option if the spot price is lower than the strike price.The Black-Scholes Model has some problems when it comes to look into pricing AmericanOptions. These are options that can be exercised at any point during the time range fromissue to maturity. For this instance the Black-Scholes equation changes into an inequalityand not given an exact approximation.For this report we will be interested in pricing ”plain vanilla” options, which are standardEuropean Call/Put options.These options are constricted to only being exercised on thematurity date given.Even though this report is about Binomial Tree approaches to pricingoptions.It is good to give a strong foundation on which these Tree Approach models are builtoff so confusion does not incur latter on in the report.

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2.2 Binomial Tree Approximation Model

The binomial tree model is a numerical method for pricing options.The binomial model wasfirst proposed by Cox, Ross and Rubinstein in 1979.Essentially, the model uses a discrete-time (lattice based) model of the varying price over time of the underlying financial instru-ment.This model traces the evolution of the option in discrete-time. This by means of thelattice approach. Thus for a number of times steps between valuation of the option andexpiration date, each of the nodes representing a discrete time step gives a possible price ofthe underlying options at a given point in time.The model is conducted through an iterative process, starting at each of the final expirationdate nodes and working backwards discounting through the tree towards the initial startingnode.The value of the option at each point, or node on the tree is evaluated at each discretepoint.In summary we must generate the tree, calculate the option at each of the final nodesand work backward to find the option value at the preceding point.

To generate the tree, at each point it must be assumed that the option will move up (u)or down(d) by a specific factor per step of the tree.if S is the current price, then the nextfollowing price will be:

Sup = S ∗ u (8)

Sdown = S ∗ d (9)

These up and down factors are calculated using the volatility σ and the time t, measured inyears.

u = eσ√t (10)

d = e−σ√t (11)

u =1

d(12)

S.u.d = S.d.u. = S (13)

The CRR method ensures that the tree is recombinant, i.e. if the underlying asset moves upand then down (u,d), the price will be the same as if it had moved down and then up (d,u),here the two paths merge or recombine. This property reduces the number of tree nodes,and thus accelerates the computation of the option price.

Again as in the Section(2.1), the option values at each point is the options exercisevalue.Max[(Sn −K), 0] for a call option, Max[(K − Sn), 0] for a put option.

3

S0u2

S0u

S0 S0

S0d

S0d2

Fig(1): Graphic representation of the Binomial Tree approach.

Now you obtain the penultimate time step and work backwards to the valuation pointwhere the value of the option can be obtained.To find this value this model used a riskneutrality assumption, that today’s fair price of the a derivative is equal to the expectedvalue of its future payoff discounted by the risk free rate.Thus the option value at a nodeis calculated from the two later nodes one time step on and weighted by their respectiveprobabilities.p is the probability of an up movement and (1− p) is the probability of a downmovement.Thus the binomial tree value is the (probability of moving up)(Option price ofup one step) + (probability of moving down)(Option Price of down one time step)*DiscountFactor.This is shown in Eq(14).

C = e−rtn (pCu + (1− p)Cd) (14)

where the probability of moving up is

p =e(r−q) t

n − du− d

(15)

This is chosen such that the related binomial distribution simulates the geometric BrownianMotion of the underlying stock with parameters r and σ.The binomial model assumes thatmovements in the price follow a binomial distribution; for many trials, this binomial distri-bution approaches the normal distribution assumed by BlackScholes.q is the dividend yieldof the underlying option.This binomial tree approach provides a discrete time approximation to the continuous pro-cess underlying the Black-Scholes model.In European options without dividends the binomialtree method converges to the Black-Scholes Price. We will be looking more into that in thenext section of the report

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2.3 Leisen Reimer Binomial Tree Model

Leisen and Reimer developed a model with the purpose of improving the rate of convergenceof their binomial tree. As mentioned above most tree models converge to the Black Scholessolution in the limit as the size of the time step ∆t is reduced t zero. But however in binomialtree models the convergence is not smoothThis model converges quadratically in the number of time steps, at least for EuropeanCall and American Call Options. While the binomial tree approach has linear convergence.Therefore the accuracy is much better. Since there are no small oscillations in this model,due to the selection of of odd times steps, compared to no constraints on time step selectionin the normal binomial tree, Richardson extrapolation is easily implemented with this modelto increase the accuracy even more.

d1 =ln( s

k) + (r − q + σ2/2)T

σ√T

(16)

d2 =ln( s

k) + (r − q − σ2/2)T

σ√T

(17)

p = f(d2, N) (18)

p′ = f(d1, N) (19)

u =p′

pe(r−q)∆T (20)

d =1− p′

1− pe(r−q)∆T (21)

Leisen-Reimer developed a method in which the parameters u, d,and p of the binomialtree can be altered in order to get better convergence behaviour.Instead of choosing theparameters p, u and d to get convergence to the normal distribution Leisen-Reimer suggestto use the inversion formulae called the Peizer-Pratt Inversion, see Eq(22).

f(x, n) =1

2+

1

2sign(x)

√1− exp[−(

x

n+ 13

+ 110(n+1)

)2(n+1

6)] (22)

This inversion formulae reverts the standard method and they use the normal approxima-tions to determine the Binomial distribution, and suggest Eq(22) to replace p(the probabilityof going up).

5

2.4 Richardson Extrapolation

In numerical analysis Richardson Extrapolation is a sequence acceleration method, usedto improve the rate of convergence of a sequence.For us in this report as mentioned in theprevious sections Tree approach models do converge to the Black Scholes Price as the numberof time steps tend to infinity.But this convergence can be slow and non-monotonic.Thesetree approaches converge to the normal distribution defined by the Black Scholes model.Ifthe convergence is linear you need to double the steps to get double the accuracy(half theexpected error) but the nodes grow at a rate of N2. The Richardson Extrapolation is a shortcut to this convergence cutting down the execution time.This method improves estimates estimates when the error is known to have a predictableform;

v = f(h) + a1h+ a2h2 + a3h

3 + ..... (23)

here v is assumed to be the true value of the option and f(h) the value returned from the treewith a step size h. Under the assumptions of the technique the accuracy of the estimate f(h)will improve as h tends to zero.as the error term will clearly decrease. Suppose we doublethe steps to half the size.

v = f(h

2) + a1(

h

2)2 + a3(

h

2)3 + ..... (24)

Subtract Eq(23) from twice Eq(24) and get

v = 2f(h

2)− f(h) + a2h

2 + .... (25)

And in doing so cancels the errors in h.Repeated applications of Richardson’s Extrapolationmay be used to combine more estimates in such a way to eliminate higher order terms in theerror expansion.It’s disadvantage is that is can easily be misused and may make estimatesworst.But this is an iterative process and so can be used to eliminator the h2 and improveestimates even more.

6

3 Methodology of Testing

3.1 Selection of Test Cases

For this report we are going to model the Leisen-Reimer Binomial Tree and compare it tothe normal Binomial Tree Approach and compare the convergence of these two approachesover a number of time steps and comment on their rate of convergence to the Black ScholesPrice. This will be all conducted through the programming application of MatLab.We willcreate function files to model the dynamics of both tree approaches and use the in-builtfunction of blsprice for the Black Scholes option price to test the convergence.

To test both these models we will graphically present the results from the models to bestconvey the accuracy of Normal Binomial Tree and the Leisen-Reimer Binomial Tree.To testthe robustness of these computational models we must note their performance and accuracyat critical points in the time line of an option.Thus for example

• An option deep in the money

• An option deep out of the money

• An option where the strike price is equal to the spot price

• An option where it is just slightly above the strike

• An option just slightly below the strike

• To examine the same critical points for a put option

• Examine the models as the number of time steps increase

• Change in fundamental parameters i.e. volatility, expiration time, dividend yield etc.

Note: In our analysis of these tree approaches we only conducted test conditions onEuropean Options to keep things simple for the models. American options tree approachescould be conducted, but was not part of the idea behind this report to introduce the reader tothe use of the Liesen-Reimer binomial tree approach.

In the following section we will present the results from what we have found from ourresearch into binomial tree approaches

7

Results:

Empirical Results: Firstly we are going to look at a European Call option at different stages in the option pricing time line and at different number of time steps to increase accuracy. We use a model template for the following results in Table (1) and Table (2), such that we keep all the variables constant and just change the position of the market price with respect to the strike price. In this model r = 0.02, T = 0.5, vol = 0.3, and q = 0.05 with a times steps of N=199.We limit the time steps to odd time steps to smooth the oscillations and it’s a part of the constraints of the model. If we used even number of time steps the normal Binomial model would produce a better result closer to the Black Scholes Price

Deep in The money

Deep out of the money

At the money

$1 above Strike

$1 below Strike

Black Scholes Price 5.727109 0.000921 0.682593 1.279708 0.285079 Binomial Tree

Price 5.727024 0.000882 0.681658 1.278727 0.285695

Leisen-‐Reimer Tree Price

5.727109 0.000921 0.680748 1.279707 0.285079

Table (1): Table of the different models at different positions in the life cycle of a call option

We can see from this table that the Leisen Reimer Tree model is closer to the Black Scholes price in all of the cases expect for at the money.

Next we are going to do the same but for a put option.

Deep out The money

Deep in of the money

At the money

$1 above Strike

$1 below Strike

Black Scholes Price 0.007908 4.034819 0.815252 0.437057 1.393049 Binomial Tree

Price 0.007824 4.034781 0.816178 0.436077 1.393664

Leisen-‐Reimer Tree Price

0.007909 4.034820 0.815252 0.437056 1.393048

Table (2): Table of the different models at different positions in the life cycle of a put option

For the put option Table (2) we see that all the values for the Leisen Reimer Tree model are close to the Black Scholes price in all situations. I think for Table (1) for at the money it could have been a systematic error that may have made the values not in the same range.

Next we will test the convergence of the binomial model and Leisen Reimer model by increasing the number of time steps of a European Call option to see which converges to the Black Scholes value quicker and more accurately.

Number of Time Steps Binomial Model LR Binomial Model Black Scholes Price

11 4.757207 4.763223 4.763037 21 4.761255 4.763091 4.763037 51 4.76183 4.763046 4.763037 75 4.762808 4.763041 4.763037

101 4.762351 4.763039 4.763037 201 4.762874 4.763038 4.763037 401 4.762979 4.763037 4.763037 499 4.762972 4.763037 4.763037

Table (3): Table of the different models at different time steps in the life cycle of a call option

We see from Table (3) as you increase the number of time steps both tree models get closer to the Black Scholes price which is true from theoretical assumptions. We can see that the Leisen Reimer Model reached the Black Scholes Price at around 400 time steps while the CRR binomial tree after 500 time steps has not reached the Black Scholes Value, proving that the Leisen Reimer Model reduces execution time and is more accurate.

Next we are going to look at what affect the time to expiry has on these tree models, by taking a European call option and changing the maturity time

Time Range (Years) Binomial Model LR Binomial Model Black Scholes Price 0.25 4.869392 4.86944 4.869439 0.5 4.762822 4.763038 4.763037 1 4.62036 4.620992 4.620991

1.25 4.571736 4.572024 4.572024 1.5 4.529081 4.530651 4.53065 5 4.103011 4.103961 4.103963 10 3.483733 3.483696 3.483699 20 0.620608 0.622438 0.622438

Table (4): Table of the different models at different expiry dates in the life cycle of a call option

If we look at Table (4) we can see that all models price drop due the extension of the maturity date. Also we see that the Leisen Reimer Model still is the most accurate to the Black Scholes price through this these time ranges. This makes sense as the longer you hold the option the more the price will go down.

Volatility Binomial Model LR Binomial Model Black Scholes Price

0.1 4.74389 4.74389 4.74389 0.2 4.744428 4.744448 4.744448 0.3 4.762822 4.763038 4.763037 0.4 4.832861 4.832801 4.8328 0.5 4.957288 4.957177 4.957177

Table (5): Table of the different models at different volatilities in the life cycle of a call option

Lastly the volatility, across all models as the volatility increases the price of all the models increases. This makes sense as the volatility increase the holder has a greater chance of making or losing money so the price of the option must increase.

If we look at other variables in the models such as dividend yield and the risk free rate of return we get, increase the dividend yield the price goes down and as you decrease the risk-‐free rate of return the price of the models goes up.

Graphical Results: In this section we are going to look at some graphs of the performance of the models in different positions and comment on the output of these graphs.

Fig (2): Graph of a call option deep in the money we can see the LR Model and CRR model converge to Black Scholes Price

As we can see here from Fig(2) that the deep in the money call option position where the spot price is greater than the strike shows that the LR model converges quickly and accurately to the desired Black Scholes Price. It does not truly converge till about 400-‐500 time steps in. As we can see with the CRR model the price oscillates around the Black Scholes price but does not converge completely.

Fig (3): Graph of a call option deep out the money we can see the LR Model and CRR model converge to Black Scholes Price

Fig(3) show the graph call option deep out of the money meaning that the current market spot price is under the strike price, and just the holder will not exercise the option. Again we see that the LR model converges quicker than the CRR model, and again the CRR oscillates close to but does not converge with the Black Scholes Price. Which in this case is close to zero because the option is out of the money the holder would pay zero for the option as he can buy it cheaper in the market and thus will not exercise the option to buy.

In Fig (4) we see a call option at the money, by far the most interesting graphical result produced through this report .We see that both model now converge to the Black Scholes Price. Yes the LR model does take less execution time and converges quicker, but still both converge. Accuracy was a strong result of the LR model, something, which overpowered the CRR model. The only theory I have for this to happen is that, because the spot price is the

same as the strike price, the holder is just going to buy the option because it is the market value of the option. Thus this is the lowest critical point at which the holder will execute because the option is worth market value.

Fig (4): Graph of a call option at the money we can see the LR Model and CRR model converge to Black Scholes Price

Fig (5): Graph of a call option at the money we can see the LR Model and CRR model converge to Black Scholes Price. With the LR Model converging a lot quicker than the CRR

In Fig (5) we take a close up of the convergence happening to see if we can get a better picture from the result.

Fig (6): Graph of a call option very deep the money we can see the LR Model and CRR model converge to Black Scholes Price. This model was model was set with S = 30, K =20, r = 0.05, T = 1, Vol = 0.3, q = 0.05, and N=99.

Fig (6) shows a European call option very deep into the money, more so than Fig (2). We see if we increase some of the option fundamental parameters such as risk free rate and time but mostly how deep in the position the holder is the CRR model shifts closer to the Black Scholes Price. Bisecting the Black Scholes Price line on several occasions. Showing during certain time steps the CRR model does give the Black Scholes price. We also see that the period between crests and troughs (. i.e. Frequency) is getting longer and CRR line is smoothing out. It may look like the CRR model will converge to the Black Scholes Price, but if you increase the time steps you see it just oscillates above and below this price.

1 Conclusions

1.1 Summary of Results

In conclusion, this report has presented the reader with hopefully a clear understandingof the topic of Binomial Tree approximations. We have discussed the theory behind thisapproach. Discussed some key mathematical and financial assumptions about the models.

We have presented a way in which to test and model these binomial tree approachesusing the programming application of MatLab.Finally we have presented the results fromour testing to the reader.

From the results, mostly all of the graphical and empirical results had proven the theoret-ical predictions described in the previous sections of this report. The Liesen-Reimer binomialtree approach had out performed the CRR binomial tree approach on several different posi-tions within the option time line .The Liesen-Reimer binomial tree approach had producedmore accurate results, closer to the Black Scholes price and we had presented graphical ev-idence that the Liesen-Reimer binomial tree approach did converge quicker to the desiredreference price. In the graphical results section further investigate would be needed into ”atthe money” option critical points to describe the nature of the CRR behaviour at that point,and also at points which are highly deep in the money. The author had given his suggestionon what this may be but further analysis and investigation in Liesen-Reimer binomial treeapproach methodology would present a clearly picture of why this is happening.

There was some limitations to my analysis. I was not able to present the case of Europeanput options to the reader as through some programming error I believe the results where notoutputting intuitive results for option positions. The code has been provided in this reportso further analysis will be done to correct this error.

Lastly I presented some results in changing the fundamental parameters and what effectthat had on the models results. I presented some cases where I changed what I believed to beimportant variables associated with the approximation price, but a wide number of differentcombinations of parameters could be tried. Unfortunately I did not have time to present aconcise way to present all this information, so I just showed the changes in variables thatthe reader would associate with option pricing.

1

2 References

• Hull, J. C. (2006). Options, futures, and other derivatives. Pearson Education India.

• Hanna A.(2015). Computational Methods in Finance: Companion Notes.Queens Uni-versity Belfast Management School

• Hanna A. (2015). Computational Methods in Finance: Lecture Notes: Tree Ap-proaches, Queens University Belfast Management School.

• Wallner, C.and Uwe, W.Efficient computation of option price sensitivities for optionsof American Style. No.1, CPQF, Working Paper Series,2004

• Wilmott, P. (2013). Paul Wilmott on quantitative finance. John Wiley & Sons.

2

3 Appendix

3.1 MatLab Code

3.1.1 Assignment 3.m

1 %%2 %Assignment 3 Le i sen Reimer (LR) Binomial Tree3

4 S = 14 ;5 K = 9 ;6 r = 0 . 0 2 ;7 T = 0 . 5 ;8 vo l = 0 . 3 ;9 q = 0 . 0 5 ;

10 c a l l p u t = ’ c a l l ’ ; % Enter ”Put” in f o r put money t e s t s11 Option = ’E ’ ;12 N = 199 ;13

14

15

16 LR Price = LR BinoTree ( S , K , r , T, vol , q , ca l l put , N) ;17

18

19

20 BinoTree = binoTree (S , K, r , T, vol , q , ca l l pu t , Option , N) ;21

22 [ Cal l , Put ] = b l s p r i c e (S , K, r , T, vol , q ) ;23

24 [ LR Price , BinoTree , Ca l l ] ’ ;25 s p r i n t f ( ’ Pr iceBlk : \ t%f \nPriceBino : \ t%f \nPriceLR :\ t%f \n ’ , Cal l ,

BinoTree , LR Price )26

27

28 %%29 % Plo t t i ng the opt ion p r i c e s30 a = 1 ; b = 499 ;31 N = [ a : 2 : b ] ;32 P = N∗0 ;33 B = P + b l s p r i c e (S , K, r , T, vol , q ) ;34 LR = N∗0 ;35

36 f o r i =1: l ength (N)37

38 P( i ) = binoTree (S ,K, r ,T, vol , q , ca l l put , Option ,N( i ) ) ;

3

39 LR( i ) = LR BinoTree ( S , K , r , T, vol , q , ca l l put , N( i ) ) ;40

41

42 end43

44 p lo t (N,P, ’−r ’ ,N,B, ’b ’ ,N,LR, ’ g ’ ) ;45 t i t l e ( ’ Black Scholes , Binomial Tree and Le i sen Reimer Binomial Tree

Option Pr i ce ’ )46 x l a b e l ( ’Time Steps ’ )47 y l a b e l ( ’ Option Pr i ce ’ )48 l egend ( ’ Binomial Tree Approx . Pr i ce ’ , ’ Black Scho l e s Pr i ce ’ , ’ Le i sen

Reimer Binomial Tree Approx . Pr i ce ’ , ’ Locat ion ’ , ’ nor theas t ’ )49

50 %%51 %Star t o f t e s t ca s e s ; In the f o l l o w i n g s s e c t i o n s we are going to

look at52 %European c a l l opt ion in or around the money53

54 %Test 155 %Deep in the money56 S = 30 ;57 K = 20 ;58 r = 0 . 0 5 ;59 T = 1 ;60 vo l = 0 . 3 ;61 q = 0 . 0 5 ;62 c a l l p u t = ’ c a l l ’ ;% Enter ”Put” in f o r put money t e s t s63 Option = ’E ’ ;64 N = 999 ;65

66

67




74 [ Cal l , BinoTree , LR Price ] ’ ;75 s p r i n t f ( ’ Pr iceBlk : \ t%f \nPriceBino : \ t%f \nPriceLR :\ t%f \n ’ , Cal l ,


77

78

79 a = 1 ; b = 99 ;

4

80 N = [ a : 2 : b ] ;81 P = N∗0 ;82 B = P + b l s p r i c e (S , K, r , T, vol , q ) ;83 LR = N∗0 ;84

85 f o r i =1: l ength (N)86

87 P( i ) = binoTree (S ,K, r ,T, vol , q , ca l l put , Option ,N( i ) ) ;88 LR( i ) = LR BinoTree ( S , K , r , T, vol , q , ca l l put , N( i ) ) ;89

90

91 end92



Reimer Binomial Tree Approx . Pr i ce ’ , ’ Locat ion ’ , ’ s outheas t ’ )98

99 %%100 %Test 2101 %Deep out the money102 S = 5 ;103 K = 9 ;104 r = 0 . 0 2 ;105 T = 0 . 5 ;106 vo l = 0 . 3 ;107 q = 0 . 0 5 ;108 c a l l p u t = ’ c a l l ’ ;% Enter ”Put” in f o r put money t e s t s109 Option = ’E ’ ;110 N = 199 ;111

112

113





BinoTree , LR Price )

5

122

123

124

125 a = 1 ; b = 199 ;126 N = [ a : 2 : b ] ;127 P = N∗0 ;128 B = P + b l s p r i c e (S , K, r , T, vol , q ) ;129 LR = N∗0 ;130

131 f o r i =1: l ength (N)132


136

137 end138




145 %%146 %%%Test 3147 %At the money148 S = 9 ;149 K = 9 ;150 r = 0 . 0 2 ;151 T = 0 . 5 ;152 vo l = 0 . 3 ;153 q = 0 . 0 5 ;154 c a l l p u t = ’ c a l l ’ ; % Enter ”Put” in f o r put money t e s t s155 Option = ’E ’ ;156 N = 200 ;157

158

159



164 [ Cal l , Put ] = b l s p r i c e (S , K, r , T, vol , q ) ;

6

165

166 [ LR Price , BinoTree , Ca l l ] ’ ;167

168 s p r i n t f ( ’ Pr iceBlk : \ t%f \nPriceBino : \ t%f \nPriceLR :\ t%f \n ’ , Cal l ,BinoTree , LR Price )

169

170


177 f o r i =1: l ength (N)178


182

183 end184




191 %%192 %Test 4193 %Just $1 above the money194 S = 10 ;195 K = 9 ;196 r = 0 . 0 2 ;197 T = 0 . 5 ;198 vo l = 0 . 3 ;199 q = 0 . 0 5 ;200 c a l l p u t = ’ c a l l ’ ;% Enter ”Put” in f o r put money t e s t s201 Option = ’E ’ ;202 N = 199 ;203

204

205

206 LR Price = LR BinoTree ( S , K , r , T, vol , q , ca l l put , N) ;

7

207



212 [ LR Price , BinoTree , Ca l l ] ’ ;213

214 s p r i n t f ( ’ Pr iceBlk : \ t%f \nPriceBino : \ t%f \nPriceLR :\ t%f \n ’ , Cal l ,BinoTree , LR Price )

215

216


223 f o r i =1: l ength (N)224


228

229 end230




237 %%238 %Test 5239 %Just $1 below the money240 S = 8 ;241 K = 9 ;242 r = 0 . 0 2 ;243 T = 0 . 5 ;244 vo l = 0 . 3 ;245 q = 0 . 0 5 ;246 c a l l p u t = ’ c a l l ’ ; % Enter ”Put” in f o r put money t e s t s247 Option = ’E ’ ;248 N = 199 ;

8

249

250

251






261

262


269 f o r i =1: l ength (N)270


274

275 end276




283 %%284 %Test 6285 %Deep out o f the money286 S = 15 ;287 K = 9 ;288 r = 0 . 0 2 ;289 T = 0 . 5 ;290 vo l = 0 . 3 ;

9

291 q = 0 . 0 5 ;292 c a l l p u t = ’ put ’ ;293 Option = ’E ’ ;294 N = 199 ;295

296

297




304 [ Cal l , BinoTree , LR Price ] ’ ;305 s p r i n t f ( ’ Pr iceBlk : \ t%f \nPriceBino : \ t%f \nPriceLR :\ t%f \n ’ , Put ,


307

308


315 f o r i =1: l ength (N)316


320

321 end322




329 %%330 %Test 7331 %Deep in the money332 S = 5 ;

10

333 K = 9 ;334 r = 0 . 0 2 ;335 T = 0 . 5 ;336 vo l = 0 . 3 ;337 q = 0 . 0 5 ;338 c a l l p u t = ’ put ’ ;% Enter ”Put” in f o r put money t e s t s339 Option = ’E ’ ;340 N = 199 ;341

342

343




350 [ LR Price , BinoTree , Ca l l ] ’ ;351 s p r i n t f ( ’ Pr iceBlk : \ t%f \nPriceBino : \ t%f \nPriceLR :\ t%f \n ’ , Put ,


353

354


361 f o r i =1: l ength (N)362


366

367 end368




11

375 %%376 %Test 8377 %At the money378 S = 9 ;379 K = 9 ;380 r = 0 . 0 2 ;381 T = 0 . 5 ;382 vo l = 0 . 3 ;383 q = 0 . 0 5 ;384 c a l l p u t = ’ put ’ ;% Enter ”Put” in f o r put money t e s t s385 Option = ’E ’ ;386 N = 199 ;387

388

389






399

400


407 f o r i =1: l ength (N)408


412

413 end414


Option Pr i ce ’ )417 x l a b e l ( ’Time Steps ’ )

12

418 y l a b e l ( ’ Option Pr i ce ’ )419 l egend ( ’ Binomial Tree Approx . Pr i ce ’ , ’ Black Scho l e s Pr i ce ’ , ’ Le i sen


421 %%422 %Test 9423 % $1 Above the money424 S = 10 ;425 K = 9 ;426 r = 0 . 0 2 ;427 T = 0 . 5 ;428 vo l = 0 . 3 ;429 q = 0 . 0 5 ;430 c a l l p u t = ’ put ’ ;% Enter ”Put” in f o r put money t e s t s431 Option = ’E ’ ;432 N = 199 ;433

434

435






445

446


453 f o r i =1: l ength (N)454


458

459 end460

13




467 %%468 %Test 10469 % $1 below the money470 S = 8 ;471 K = 9 ;472 r = 0 . 0 2 ;473 T = 0 . 5 ;474 vo l = 0 . 3 ;475 q = 0 . 0 5 ;476 c a l l p u t = ’ put ’ ;% Enter ”Put” in f o r put money t e s t s477 Option = ’E ’ ;478 N = 199 ;479

480

481






491

492


499 f o r i =1: l ength (N)500

501 P( i ) = binoTree (S ,K, r ,T, vol , q , ca l l put , Option ,N( i ) ) ;502 LR( i ) = LR BinoTree ( S , K , r , T, vol , q , ca l l put , N( i ) ) ;

14

503

504

505 end506



Reimer Binomial Tree Approx . Pr i ce ’ , ’ Locat ion ’ , ’ nor theas t ’ )

15

3.1.2 binoTree.m

1 f unc t i on [ pr i c e , spotTree , p r i c eTree ] = binoTree (S , K, r , T, vol ,q , ca l l pu t , AE, N)

2 %binoTree Option p r i c e us ing binomial t r e e3 % Arguments are spot , s t r i k e , rate , time , vol , y i e ld , c a l l /put

s t r i ng , AE s t r i n g4 % time s t ep s5

6

7 %I n i t i a l i s e v a r i a b l e s8 dt = T / N;9 u = exp ( vo l ∗ s q r t ( dt ) ) ;

10 d = 1 / u ;11 p = ( exp ( ( r−q ) ∗ dt ) − d) / (u − d) ;12 df = exp(−r ∗ dt ) ;13 %c a l l /put switch14

15 i f upper ( c a l l p u t (1 ) )==’C ’16 alpha = 1 ;17 e l s e18 alpha = −1;19 end20

21 %european / american switch22 i f upper (AE(1) )==’A ’23 beta = 1 ;24 e l s e25 beta = 0 ;26 end27 %I n i t i a l i s e and s i z e working v a r i a b l e s28

29 p r i c e = 0 ;30 FF = ze ro s (N+1,N+1) ;31 SS = ze ro s (N+1,N+1) ;32 SS (1 , 1 )=S ;33

34 %Evolve spot p r i c e s forward35 f o r i =2:N+136 SS( i , 1) = SS( i − 1 , 1) ∗ d ;37 f o r j =2: i38 SS( i , j ) = SS( i − 1 , j − 1) ∗ u ;39 end40 end41

16

42 %Set boundary c o n d i t i o n s f o r payo f f43 f o r j =1:1 :N+144 FF(N+1, j )=max( 0 , ( SS(N+1, j ) − K) ∗ alpha ) ;45 end46 f o r i=N:−1:147 f o r j =1: i48 FF( i , j ) = df ∗ (FF( i + 1 , j ) ∗ (1 − p) + FF( i + 1 , j + 1)

∗ p) ;49 i f beta %American50 i f ( SS ( i , j ) − K) ∗ alpha > FF( i , j )51 FF( i , j ) = (SS( i , j ) − K) ∗ alpha ;52 end53 end54 end55 end56

57 p r i c e = FF(1 , 1 ) ;58 spotTree = SS ;59 pr i c eTree = FF;60 end

17

3.1.3 ppInv.m

1 f unc t i on [ y ] = ppInv ( x , n )2 %Returns the Peizer−Pratt In v e r s i on3 %Defined f o r odd va lues o f n only4

5 t = exp (−((x / (n + 1 / 3 + 0 .1 / (n + 1) ) ) ˆ 2) ∗ (n + 1 / 6) ) ;6 y = 0 .5 ∗(1 + s i gn ( x ) ∗ s q r t (1 − t ) ) ;7

8 end

18

3.1.4 LR BinoTree.m

1 f unc t i on [ Pr i ce ] = LR BinoTree ( S , K , r , T, vol , q , ca l l put , N)2 % Binomial t r e e implementation3

4 % I n i t i a l i s e v a r i a b l e s5 dt = T / N;6 d1 = ( log (S/K) + ( r − q + 0.5∗ vo l ˆ2)∗T) /( vo l ∗ s q r t (T) ) ;7 d2 = ( log (S/K) + ( r − q − 0 .5∗ vo l ˆ2)∗T) /( vo l ∗ s q r t (T) ) ;8 p = ppInv ( d2 ,N) ;9 p1 = ppInv ( d1 ,N) ;

10 u = ( p1/p)∗exp ( ( r−q )∗dt ) ;11 d = ((1−p1 ) /(1−p) )∗exp ( ( r−q )∗dt ) ;12 df = exp(−r ∗ dt ) ;13

14 FF = ze ro s (N+1,N+1) ;15 SS = ze ro s (N+1,N+1) ;16

17 %c a l l /put switch18

19 i f upper ( c a l l p u t (1 ) ) ==’C ’20 alpha = 1 ;21 e l s e22 alpha = −1;23 end24

25

26

27 % Evolve Spot p r i c e s foward28 SS (1 , 1 ) = S ;29 % quick note ; ” i ” i s loop counter . From 2 to N+1.30 f o r i =2:N+131 f o r j =2: i32 SS( i , j ) = SS( i −1, j−1)∗u ;33 end34 SS( i , 1 ) = SS( i −1 ,1)∗d ;35 end36

37 % Set boundary c o n d i t i o n s f o r payo f f . j r ep r e s en s increments inv e r t i c a l

38 % space . i i s increments in time .39 f o r j = 1 :N+140 FF(N+1, j ) = max( ( SS( i , j ) − K)∗alpha , 0 ) ;41 end42

19

43 % Discount the opt ion p r i c e back through the t r e e44

45 f o r i = N:−1:146 f o r j =1: i47 FF( i , j ) = df ∗(p∗FF( i +1, j +1) + (1−p)∗FF( i +1, j ) ) ;48 end49 end50

51 Pr i ce = FF(1 , 1 ) ;52

53 end

20

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