Explore Single-Period Asset Arbitrage
This example explores a few basic concepts related to arbitrage in single-period, two-state asset portfolio. It uses these Symbolic Math Toolbox functions:
- equationsToMatrix to convert a linear system of equations to a matrix.
- linsolve to solve the system.
- Symbolic equivalent of standard MATLAB functions, such as diag.
The portfolio consists of a bond, a long stock, and a long call option on the stock. 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 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 = 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 + 1, r + 1] [ SD, SU] [ 0, CU]
CU is worth SU - K in state U. Substitute this value in payoff.
payoff = subs(payoff, CU, SU - K)
payoff = [ r + 1, r + 1] [ SD, SU] [ 0, SU - K]
Define the probabilities of reaching states U and D.
syms pU pD real
Under no-arbitrage, eqns must always hold true with positive pU and pD.
eqns = payoff*[pD; pU] - prices
eqns = 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 - 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 = [ 1, 1, 0] [ SD/(r + 1), SU/(r + 1), 0] [ 0, -(K - SU)/(r + 1), -1] b = 1 S0 0
Using linsolve, find the solution for the risk-neutral probablities and call price.
x = linsolve(A, b)
x = (S0 - SU + S0*r)/(SD - SU) -(S0 - SD + S0*r)/(SD - SU) ((K - SU)*(S0 - SD + S0*r))/((SD - SU)*(r + 1))
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 = r r r
To test for 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 a any call price other than xSol(3), there is arbitrage.
xSol = simplify(subs(x, [r,S0,K], [0.05,100,100]))
xSol = -(SU - 105)/(SD - SU) (SD - 105)/(SD - SU) (20*(SD - 105)*(SU - 100))/(21*(SD - SU))
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, S1 = 50, S2 = 150.
Advanced Derivatives, Pricing and Risk Management: Theory, Tools and Programming Applications by Albanese, C., Campolieti, G.