# Using the call option to solve this linear program

Hello I have to do a project for a finance class and the Professor has given us the following problem. I'm not a finance student and am just now being introduced to the subject. I do not understand this problem at all. All Im asking is if someone can better help me understand what the professor wants and point me in the right direction in how to solve this. Or if you could share relevant links that my help me out. I appreciate it greatly.

Figure 1 describes the stock price process (St) for t = 0, 1, 2, 3 by a binomial tree. The aim is to price a European call option with strike K = 100 whit maturity T = 3. Assume further that the interest rate is constant to zero.

1.Compute the expected, (discounted) payoff of the call option at time t = 0 using the physical measure given in Figure 1. That is you use pricing rule B to price this call option.

2.We now want to use pricing rule C the call option. Solve the following linear program

$$min_{(w,x,y)} w$$

$$s.t.: x_0 + y_0S_0 ≤ w$$

$$x_{n-} + y_{n−}S_n ≥ x_n + y_nS_n$$ for all interior nodes n

$$x_{n−} + y_{n−}S_n ≥ C_n$$ for all terminal nodes n

To do so you will have to find a matrix A and vectors b and b such that you can reformulate the problem as

$$min_x c^T x$$ s.t. $$A · x ≥ b.$$

Interprete the solution of the LP, i.e. the optimal values of w, x and y, very carefully. How to you have to invest in the bond and the stock to replicate the payoff of the option? Start at time t = 0 with your description. You can use the R-package LpSolve for solving the linear program. Note however that in the package every decision variable is assumed to be non-negative. You will have to find a way to change this restriction.

$$n-$$ is the preceeding node and $$n+$$ is the succesor node

3.Now, turn to the dual problem

$$max_{λ≥0} b^T x$$ s.t. $$A^T · λ = c.$$

Interpret the dual solution und understand the risk neutral probabilities and the martingale property.

4.Change the values 105 to 108, 108 to 110, 109 to 115 in Figure 1. How does the optimal investment resp. the risk neutral distribution change?

• Hello, can you please write your equations in LateX? That would help to understand your problem. – siou0107 Jan 9 '20 at 15:22
• Do you know exactly what a "node" represents? Is it some combination of time period and stock price level? Does each node have only one "predecessor" ( a non-recombining tree) or can it have multiple predecessors (a recombining tree or lattice). Do you have a diagram of what the tree looks like? How many "terminal nodes" are there? The answer below (super-replication) is on the right track, but some details in the problem statement remain unclear to me. – noob2 Jan 11 '20 at 15:45
• @noob2 Ive edited my post and post the entire worshsheet so its more understandable. Greatly appreciate your comment. – Lion Heart Jan 11 '20 at 17:57

This is kinda 'super-replication'. You want to find the minimum amount $$w$$ that you need to invest such that a portfolio of $$x$$ amount of bank account and $$y$$ units of the stock, which costs less than $$w$$ initially and which you can dynamically adjust subject to no new injection of money ($$x_{n-} + y_{n−}S_n ≥ x_n + y_nS_n$$), returns at least the option payoff at maturity.