Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

This example explores basic arbitrage concepts in a single-period, two-state asset portfolio. The portfolio consists of a bond, a long stock, and a long call option on the stock.

It uses these Symbolic Math Toolbox™ functions:

`equationsToMatrix`

to convert a linear system of equations to a matrix.`linsolve`

to solve the system.Symbolic equivalents of standard MATLAB® functions, such as

`diag`

.

This example symbolically derives the risk-neutral probabilities and call price for a single-period, two-state scenario.

Create the symbolic variable `r`

representing the risk-free rate over the period. Set the assumption that `r`

is a positive value.

syms r positive

Define the parameters for the beginning of a single period, `time = 0`

. Here `S0`

is the stock price, and `C0`

is the call option price with strike, `K`

.

syms S0 C0 K positive

Now, define the parameters for the end of a period, `time = 1`

. Label the two possible states at the end of the period as U (the stock price over this period goes up) and D (the stock price over this period goes down). Thus, `SU`

and `SD`

are the stock prices at states U and D, and `CU`

is the value of the call at state U. Note that .

syms SU SD CU positive

The bond price at `time = 0`

is 1. Note that this example ignores friction costs.

Collect the prices at `time = 0`

into a column vector.

prices = [1 S0 C0]'

prices =

Collect the payoffs of the portfolio at `time = 1`

into the `payoff`

matrix. The columns of `payoff`

correspond to payoffs for states D and U. The rows correspond to payoffs for bond, stock, and call. The payoff for the bond is `1 + r`

. The payoff for the call in state D is zero since it is not exercised (because ).

payoff = [(1 + r), (1 + r); SD, SU; 0, CU]

payoff =

`CU`

is worth `SU - K`

in state U. Substitute this value in `payoff`

.

payoff = subs(payoff, CU, SU - K)

payoff =

Define the probabilities of reaching states U and D.

syms pU pD real

Under no-arbitrage, `eqns == 0`

must always hold true with positive `pU`

and `pD`

.

eqns = payoff*[pD; pU] - prices

eqns =

Transform equations to use risk-neutral probabilities.

syms pDrn pUrn real; eqns = subs(eqns, [pD; pU], [pDrn; pUrn]/(1 + r))

eqns =

The unknown variables are `pDrn`

, `pUrn`

, and `C0`

. Transform the linear system to a matrix form using these unknown variables.

[A, b] = equationsToMatrix(eqns, [pDrn, pUrn, C0]')

A =

b =

Using `linsolve`

, find the solution for the risk-neutral probablities and call price.

x = linsolve(A, b)

x =

Verify that under risk-neutral probabilities, `x(1:2)`

, the expected rate of return for the portfolio, `E_return`

equals the risk-free rate, `r`

.

E_return = diag(prices)\(payoff - [prices,prices])*x(1:2); E_return = simplify(subs(E_return, C0, x(3)))

E_return =

As an example of testing no-arbitrage violations, use the following values: `r = 5%`

, `S0 = 100`

, and `K = 100`

. For `SU < 105`

, the no-arbitrage condition is violated because `pDrn = xSol(1)`

is negative (`SU >= SD`

). Further, for any call price other than `xSol(3)`

, there is arbitrage.

xSol = simplify(subs(x, [r,S0,K], [0.05,100,100]))

xSol =

Plot the call price, `C0 = xSol(3)`

, for `50 <= SD <= 100`

and `105 <= SU <= 150`

. Note that the call is worth more when the "variance" of the underlying stock price is higher for example, `SD = 50, SU = 150`

.

fsurf(xSol(3), [50,100,105,150]) xlabel SD ylabel SU title 'Call Price'

`Advanced Derivatives, Pricing and Risk Management: Theory, Tools and Programming Applications`

by Albanese, C., Campolieti, G.

Was this topic helpful?