Explore Single-Period Asset Arbitrage

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.

Define Parameters of the Portfolio

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 SD<=K<=SU.

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 = 

(1S0C0)[sym(1); S0; C0]

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 SD<=K).

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

(r+1r+1SDSU0CU)[r + 1, r + 1; SD, SU; sym(0), CU]

CU is worth SU - K in state U. Substitute this value in payoff.

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

(r+1r+1SDSU0SU-K)[r + 1, r + 1; SD, SU; sym(0), SU - K]

Solve for Risk-Neutral Probabilities

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 = 

(pDr+1+pUr+1-1SDpD-S0+SUpU-C0-pUK-SU)[pD*(r + 1) + pU*(r + 1) - 1; SD*pD - S0 + SU*pU; - C0 - pU*(K - SU)]

Transform equations to use risk-neutral probabilities.

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

(pDrn+pUrn-1SDpDrnr+1-S0+SUpUrnr+1-C0-pUrnK-SUr+1)[pDrn + pUrn - 1; (SD*pDrn)/(r + 1) - S0 + (SU*pUrn)/(r + 1); - C0 - (pUrn*(K - SU))/(r + 1)]

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 = 

(110SDr+1SUr+100-K-SUr+1-1)[sym(1), sym(1), sym(0); SD/(r + 1), SU/(r + 1), sym(0); sym(0), -(K - SU)/(r + 1), -sym(1)]

b = 

(1S00)[sym(1); S0; sym(0)]

Using linsolve, find the solution for the risk-neutral probabilities and call price.

x = linsolve(A, b)
x = 

(S0-SU+S0rSD-SU-S0-SD+S0rSD-SUK-SUS0-SD+S0rSD-SUr+1)[(S0 - SU + S0*r)/(SD - SU); -(S0 - SD + S0*r)/(SD - SU); ((K - SU)*(S0 - SD + S0*r))/((SD - SU)*(r + 1))]

Verify the Solution

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 = 

(rrr)[r; r; r]

Test for No-Arbitrage Violations

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 = 

(-SU-105SD-SUSD-105SD-SU20SD-105SU-10021SD-SU)[-(SU - 105)/(SD - SU); (SD - 105)/(SD - SU); (20*(SD - 105)*(SU - 100))/(21*(SD - SU))]

Plot Call Price as a Surface

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.