The example question for these solutions can be found on my website (click here).

#### 5.1 Binomial Tree For Option Pricing

The two most popular models for using binomial trees to price options are

- Cox et al. (1979) (CRR for short) whose extra degree of freedom is to set
thus

- Rendleman and Bartter (1979) who choose:
and so

We wish to generate a stock price tree, so denote the value of the underlying asset after timestep i
and upstate j by S_{ij} and we have that:

First download and start with this template code:

#include<iostream>#include<cmath>#include<vector>usingnamespacestd;/* Template code for the binomial tree* Instructions to build the code are here*http://www.maths.manchester.ac.uk/~pjohnson/Tutorials/node1.html*/intmain() {// declare and initialise Black Scholes parameters// declare and initialise tree paramaters (steps in tree)// declare and initialise local variables (u,d,p)// create storage for the stock price tree and option price tree// setup and initialise the stock price tree// setup and initialise the final conditions on the option price tree// loop through time levels, setting the option price at each node in the tree// output the estimated option price}

First declare and initialise the Black Scholes parameters for your chosen problem. Here we are
going to value a Black Scholes vanilla European call option with, S_{0} = 100, X = 100, T = 1,
r = 0.06 and σ = 0.2, so declare variables for each of these. Next add in an integer to store the
number of steps in the tree and call it n. Finally add in some local variable to describe
the tree, so we have the timestep length dt, u, d and q. Your code should look like

// declare and initialise Black Scholes parametersdouble S0=100.,X=100.,T=1.,r=0.06,sigma=0.2;// declare and initialise tree paramaters (steps in tree)int n=3;// declare and initialise local variables (u,d,q)double dt,u,d,q;

Then calculate the values for dt, u, d and q, using the appropriate formula, you should check the value of these against those found in my lecture notes (click here, slide 19).

Next we need to create some storage for the values of the stock at each node in the tree. Declare a vector of vectors stockTree, into which we will place our stock price nodes.

// create storage for the stock price tree and option price treevector<vector<double>>stockTree(n+1,vector<double>(n+1));

where this has been intialised to an n + 1 × n + 1 2D array, and n is the number of nodes in the tree.

Now use a for loop and the function pow to input the value of the stock at each node in the
tree, where S_{i,j} →stockTree[i][j].

Now print out the vector stockTree[i][j] to the screen. Your code might look something like this

#include<iostream>#include<cmath>#include<vector>usingnamespacestd;/* Template code for the binomial tree* Instructions to build the code are here*http://www.maths.manchester.ac.uk/~pjohnson/Tutorials/node1.html*/intmain() {// declare and initialise Black Scholes parameters// declare and initialise Black Scholes parametersdouble S0=100.,X=100.,T=1.,r=0.06,sigma=0.2;// declare and initialise tree paramaters (steps in tree)int n=3;// declare and initialise local variables (u,d,q)double dt,u,d,q; dt = T/n; u =exp(sigma*sqrt(dt)); d =exp(-sigma*sqrt(dt)); q = (exp(r*dt)-d)/(u-d);// create storage for the stock price tree and option price treevector<vector<double>>stockTree(n+1,vector<double>(n+1));// setup and initialise the stock price treefor(int i=0;i<=n;i++) {for(int j=0;j<=i;j++) { stockTree[i][j]=S0*pow(u,j)*pow(d,i-j); cout << i << " " << j << " " << stockTree[i][j] << endl; } cout << endl; }/** OUTPUT0 0 1001 0 89.09471 1 112.242 0 79.37872 1 1002 2 125.9783 0 70.72223 1 89.09473 2 112.243 3 141.398*/}

Compare the values with those from the example in the lectures.

#### 5.2 The Option Value Tree

Let us generate a simple example so that we can compare results at every stage to something that we can work out on paper. This is an important idea in debugging, to solve the problem and do all of your bug checking on a small scale before attempting the full problem.

First declare a vector of vectors which shall hold the values of the option

// create storage for the stock price tree and option price treevector<vector<double>>valueTree(n+1,vector<double>(n+1));

and set it to the same size as stockTree. Here we use the same relation

V _{i,j} →valueTree[i][j].

Now fill in the final values of the tree, given that we have already first generated the stock tree:

for(int j=0;j<=n;j++) { valueTree[n][j]=max(stockTree[n][j]-X,0.); cout << n << " " << j << " " << valueTree[n][j] << endl; }

Now we need to loop backwards through the tree to generate the value at each node using the equation:

- Write for loops to move backwards through the vector array caculating the value of the option at each node.
- Print out the tree to screen (with n=3) and compare to the simple example (click here, slide 19) to check your code is working.
- If your values don’t match - try to work out why!!
- Now print out the value at (0, 0) increasing the number of steps in the tree. Do the results look feasible? Compare them against the exact values from the formula.

Finally your code should look like this

#include<iostream>#include<cmath>#include<vector>usingnamespacestd;/* Template code for the binomial tree* Instructions to build the code are here*http://www.maths.manchester.ac.uk/~pjohnson/Tutorials/node1.html*/intmain() {// declare and initialise Black Scholes parametersdouble S0=100.,X=100.,T=1.,r=0.06,sigma=0.2;// declare and initialise tree paramaters (steps in tree)int n=3;// declare and initialise local variables (u,d,q)double dt,u,d,q; dt = T/n; u =exp(sigma*sqrt(dt)); d =exp(-sigma*sqrt(dt)); q = (exp(r*dt)-d)/(u-d);// create storage for the stock price tree and option price treevector<vector<double>>stockTree(n+1,vector<double>(n+1));// setup and initialise the stock price treefor(int i=0;i<=n;i++) {for(int j=0;j<=i;j++) { stockTree[i][j]=S0*pow(u,j)*pow(d,i-j); } } vector<vector<double>>valueTree(n+1,vector<double>(n+1));for(int j=0;j<=n;j++) { valueTree[n][j]=max(stockTree[n][j]-X,0.); }for(int i=n-1;i>=0;i--) {for(int j=0;j<=i;j++) { valueTree[i][j] =exp(-r*dt)*( q*valueTree[i+1][j+1] + (1-q)*valueTree[i+1][j]); } } cout << " V(S="<<S0<<",t=0) = " << valueTree[0][0] << endl;/** OUTPUTV(S=100,t=0) = 11.552*/}

Finally, you could try to improve your code with the following

- Create a function returning the value of the binomial tree for a set of given parameters.
- Write a code storing two time-levels, and compare (at every stage if needed) with the previous code.
- Is it possible to store just one time-level? Try to write a code for this.
- Do you notice any difference (time taken for computation) between the codes with different storage requirements?

#### 5.3 Evaluating Convergence Properties

Taking the code from last time, we can finish this off by taking the main algorithm out into a function and allowing the payoff function to be set independently. A representative code might look like the following:

#include<iostream>#include<fstream>#include<cmath>#include<vector>#include<algorithm>usingnamespacestd;/* Template code for the binomial tree* Instructions to build the code are here*http://www.maths.manchester.ac.uk/~pjohnson/Tutorials/node1.html*/// return the payoff of the function you want to evaluatedoublepayoff(double S,double X) {returnmax(S-X,0.); }// return the value of the binomial treedoublebinomialTree(double S0,double X,double T,double r,double sigma,int n) {// declare and initialise local variables (u,d,q)double dt,u,d,q; dt = T/n; u =exp(sigma*sqrt(dt)); d =exp(-sigma*sqrt(dt)); q = (exp(r*dt)-d)/(u-d);// create storage for the stock price tree and option price treevector<vector<double>>stockTree(n+1,vector<double>(n+1));// setup and initialise the stock price treefor(int i=0;i<=n;i++) {for(int j=0;j<=i;j++) { stockTree[i][j]=S0*pow(u,j)*pow(d,i-j); } } vector<vector<double>>valueTree(n+1,vector<double>(n+1));for(int j=0;j<=n;j++) { valueTree[n][j]=payoff(stockTree[n][j],X); }for(int i=n-1;i>=0;i--) {for(int j=0;j<=i;j++) { valueTree[i][j] =exp(-r*dt)*( q*valueTree[i+1][j+1] + (1-q)*valueTree[i+1][j]); } }returnvalueTree[0][0]; } intmain() {// declare and initialise Black Scholes parameters// declare and initialise Black Scholes parametersdouble S0=100.,X=100.,T=1.,r=0.06,sigma=0.2;// declare and initialise tree paramaters (steps in tree)int n=3; cout << " V(S="<<S0<<",t=0) = " <<binomialTree(S0,X,T,r,sigma,n) << endl;/** OUTPUTV(S=100,t=0) = 11.552*/}

Now we wish to investigate what happens when N is increasing. Write the following loop in your code to output some results to file, you will need to adjust the output filename depending on you preference for where the file should be saved

// open output streamofstreamoutput("binomial-convergence.csv");// check it is openif(output.is_open()) { cout << "File opened successfully" << endl;// output various n and V_n to filefor(int n=3;n<=1000;n++) { output << n<<" , " <<binomialTree(S0,X,T,r,sigma,n) << endl; } cout << "File write complete" << endl; }else{ cout << "File could not be opened for some reason.\n"; }/** OUTPUTFile opened successfullyFile write complete*/

Run the code for S_{0} = 100 and X = 100 and then import the data into a ploting package and
visualise the results. You should see the following pattern

The pattern can be explained by realising that the nodes at terminal time will either be placed
exactly on the strike price, or above/below depending on whether the number of steps is odd or
even. We will see quite a different pattern if you choose S_{0}≠X, so for example set S_{0} = 97.3467 in
your code and rerun the results. After putting this in a plotting package you should see
something like this

To produce this graph I have set n = 25 as the minimum grid size to show up the pattern better. This time we see humps as well as an odd-even effect. The humps are again caused by the positioning of the grid nodes relative to the strike price. Although the convergence of the tree is O(1∕n), the humps make it very difficult to achieve high accuracy. There are several papers that try to address this, including Leisen and Reimer (1996) and Heston and Zhou (2000), whilst Joshi (2007) gives a good comparison of different methods when American options are being priced.

### References

Cox, John C, Stephen A Ross, and Mark Rubinstein. Option pricing: A simplified approach. Journal of financial Economics, 7(3):229–263, 1979.

Heston, Steve and Guofu Zhou. On the rate of convergence of discrete-time contingent claims. Mathematical Finance, 10(1):53–75, 2000.

Joshi, Mark S. The convergence of binomial trees for pricing the american put. Available at SSRN 1030143, 2007. URL http://fbe.unimelb.edu.au/__data/assets/pdf_file/0006/806280/170.pdf.

Leisen, Dietmar PJ and Matthias Reimer. Binomial models for option valuation-examining and improving convergence. Applied Mathematical Finance, 3(4): 319–346, 1996.

Rendleman, Richard J and Brit J Bartter. Two-state option pricing. The Journal of Finance, 34(5):1093–1110, 1979.

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