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:
equationsToMatrixto convert a linear system of equations to a matrix.
Symbolic equivalents of standard MATLAB functions, such asdiag.
This example symbolically derives the risk-neutral probabilities and call price for a single-period, two-state scenario.
Create the symbolic variablerrepresenting the risk-free rate over the period. Set the assumption thatris a positive value.
Define the parameters for the beginning of a single period,time = 0. HereS0is the stock price, andC0is the call option price with strike,K.
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,SUandSDare the stock prices at states U and D, andCUis the value of the call at state U. Note thatSD=K=SU.
The bond price attime = 0is 1. Note that this example ignores friction costs.
Collect the prices attime = 0into a column vector.
Collect the payoffs of the portfolio attime = 1into thepayoffmatrix. The columns ofpayoffcorrespond to payoffs for states D and U. The rows correspond to payoffs for bond, stock, and call. The payoff for the bond is1 + r. The payoff for the call in state D is zero since it is not exercised (becauseSD=K).
CUis worthSU – Kin state U. Substitute this value inpayoff.
Define the probabilities of reaching states U and D.
Under no-arbitrage,eqns == 0must always hold true with positivepUandpD.
Transform equations to userisk-neutralprobabilities.
The unknown variables arepDrn,pUrn, andC0. Transform the linear system to a matrix form using these unknown variables.
Usinglinsolve, find the solution for the risk-neutral probabilities and call price.
Verify that under risk-neutral probabilities,x(1:2), the expected rate of return for the portfolio,E_returnequals the risk-free rate,r.
As an example of testing no-arbitrage violations, use the following values:r = 5%,S0 = 100, andK = 100. ForSU 105, the no-arbitrage condition is violated becausepDrn = xSol(1)is negative (SU = SD). Further, for any call price other thanxSol(3), there is arbitrage.
Plot the call price,C0 = xSol(3), for50 = SD = 100and105 = 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.
Advanced Derivatives, Pricing and Risk Management: Theory, Tools and Programming Applicationsby Albanese, C., Campolieti, G.
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