April 25, 2024
Every arbitrage-free model is parameterized by a positive measure process. There is no need to restrict models to Ito processes and use partial differential equations. There is also no need for utility functions or market equilibrium assumptions. In fact, there is no need for probability measures. The Unified Model involves only geometry, as was first pointed out by Stephen Ross. We do need the notion of sample spaces and algebras of sets in order to mathematically model partial information.
It is time to banish the mathematical fiction of continuous time trading and focus on problems practitioners find useful: when to re-hedge and how well the hedge performs.
There is a clear trajectory in mathematical finance starting from Black, Scholes, and Merton valuing an option to expanding the universe of instruments, incorporate credit and liquidity considerations, and make model assumptions explicit. Realistic models must allow for the empirical fact that not all market participants act optimally.
Scholes and Merton won a Nobel Prize for a new method of valuing derivative securities that did not require estimating the return on the underlying. Under the assumption that stock prices can be modeled by geometric Brownian motion and it was possible to trade in continuous time, all one needed to know was the volatility and the funding rate.
The prices of n instruments at the beginning of the period can be represented by a vector x\in 𝑹^n. The prices at the end of the period are X(\omega)\in 𝑹^n if \omega\in\Omega occured where X\colon\Omega\to 𝑹^n
Arbitrage exists in a one-period model if it is possible to buy a portfolio of instruments at negative cost and sell them at the end of the period without losing money. The cost of purchasing \gamma\in 𝑹^n in each instrument at the beginning of the period is the dot product \gamma\cdot x. Assuming all positions are completely unwound at the end of the period, it will pay \gamma\cdot X(\omega) if \omega\in\Omega occurs. Arbitrage exists if there is a \gamma\in 𝑹^n with \gamma\cdot x < 0 and \gamma\cdot X(\omega)\ge0 for all \omega\in\Omega.
Stephan Ross was the first to show a one-period model is arbitrage-free if and only if x belongs to the smallest closed cone containing the range of X: X(\Omega) = \{X(\omega)\mid\omega\in\Omega\}.
Recall a cone is a subset of a vector space that is closed under addition and multiplication by positive scalars.
Exercise. The set of arbitrage portfolios for a one-period model is a cone.
Exercise. The set \{\sum_{j=1}^n p_j v_j\mid p_j \ge 0, p_j\in 𝑹\} is the smallest closed cone containg the vectors \{v_j\}_{j=1}^n.
Consider a one-period market having a bond that doubles in value over the period, a stock with initial price 1 that either stays the same or triples in value, and a call option on the stock with strike 2 and price c. This is modeled by x = (1,1,c), X(\omega) = (2,\omega,\max\{\omega - 2, 0\}) where \Omega = \{1, 3\}. For the model to be arbitrage-free there must exist p,q\ge0 with x = X(1)p + X(3)q and p,q\ge0. The constraint on the bond and stock give 1 = 2p + 2q and 1 = p + 3q respectively.
These equations have the unique solution p = q = 1/4 so c = 0\times 1/4 + 1\times 1/4 = 1/4. Note p + q = 1/2 is the discount over the period.
Consider a one-period market having a bond with zero interest over the period, a stock with initial price 100 that can go to 90, 100, or 110 at the end of the period, and a call option on the stock with strike 100 and price c. This is modeled by x = (1,100,c), X(\omega) = (1,\omega,\max\{\omega - 100, 0\}) where \Omega = \{90, 100, 110\}. For the model to be arbitrage-free there must exist p,q,r\ge0 with x = X(90)p + X(100)q + X(110)r.
The equation for the bond, stock, and option are 1 = p + q + r, 100 = 90p + 100q + 110r, and c = 10r. The first two equations can be written in matrix notation as \begin{bmatrix} 1 \\ 100 \\ \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 90 & 100 \\ \end{bmatrix} \begin{bmatrix} p \\ q \\ \end{bmatrix} + \begin{bmatrix} r \\ 110r \\ \end{bmatrix}. Solving for p and q gives \begin{bmatrix} p \\ q \\ \end{bmatrix} = \begin{bmatrix} r \\ 1 - 2r \\ \end{bmatrix} The conditions p,q\ge0 imply r\ge0 and 1-2r\ge0 so 0\le r\le 1/2. Since c = 10r we have 0\le c\le 5 if the model is arbitrage-free.
Note the option price is not unique in this example. This is the case for any realisitic model of what can occur in the market.
Exercise. If x = (100, c) and X(\omega) = (\omega, \max\{\omega - 100\}) for \omega\in\{90,100,110\} show the model is arbitrage-free if and only if 0\le c\le 100/11.
Even if the bond is not available for trading the option price must be strictly less than 10 in an arbitrage-free model.
The binomial model is where a bond has realized return R and the stock has initial price s and ends at either a low price L or high price H, x = (1, s), X(\omega) = (R, \omega) where \omega\in\Omega = \{L,H\}. The no arbitrage condition is x = X(L)p + X(H)q for some p,q\ge0. This has a unique solution.
Exercise. Show p = (H - Rs)/R(H - L) and q = (Rs - L)/R(H - L).
Exercise. Show the conditions p,q\ge0 imply L \le Rs \le H.
If an option pays \phi(\omega) then it pays either \phi(L) or \phi(H) at expiration. Every European option in a binomial model is a linear function and its arbitrage-free price is determined by the no arbitrage condition the bond and stock satisfy.
Exercise. If x = (1, s, c) and X(\omega) = (R, \omega, \max\{\omega - K, 0\}) where \omega\in [L, H] what are the no-arbitrage constraints on c?
Assume \Omega is the sample space of possible outcomes and \mathcal{A}_t is a partition of \Omega representing the information available at each trading time t\in T.
A market model for a set I of instruments is specified by functions for prices X_t\colon\mathcal{A}_t\to 𝑹^I and cash flows C_t\colon\mathcal{A}_t\to 𝑹^I, t\in T. 1
We assume any amount of each instrument can be bought or sold at price X_t at time t\in T. Cash flows are almost always 0. Stocks pay dividends on dividend dates, bonds pay periodic coupons, European options have a single cash flow at expiration, futures have daily margin adjustments as cash flows. The price of a futures is always 0.
A trading strategy is a finite set of increasing stopping times (\tau_j) and how much of each instrument to trade \Gamma_j\colon\mathcal{A}_{\tau_j}\to 𝑹^I at each time. Trades accumulate to a position \Delta_t = \sum_{\tau_j < t}\Gamma_j = \sum_{s < t} \Gamma_s where \Gamma_s = \Gamma_j if s = \tau_j. Note the strict inequality. This reflects the reality that it takes some time after a trade is executed for it to settle.
We do not assume there is a money market account available to finance trading or require a strategy to be self-financing.
The value, or mark-to-market of a trading strategy is V_t = (\Delta_t + \Gamma_t)\cdot X_t. It the the amount generated by unwinding the existing position and trades just executed at the prevailing market price, assuming that is possible.
Trading strategies involve cash flows to the trading account A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t. At time t cash flows are received in proportion to the existing position and the trades just executed are debited from the trading account.
Arbitrage exists for a model if there is a trading strategy (\tau_j, \Gamma_j) with \sum_j \Gamma_j = 0, A_{\tau_0} > 0, A_t\ge0 for t > \tau_0. The strategy must close out, make money on the first trade, and never lose money thereafter.
Arbitrage-free models are parameterized by a function D_t\colon\mathcal{A}_t\to (0,\infty). We call D_t a deflator. If a money-market account is available to fund trading then D_t = 1/R_t where R_t is the realized return on the money-market account.
Let M_t = X_t D_t + sum_{s\le t}C_s D_s.
Lemma. A model is arbitrage-free if M_u|_{\mathcal{A}_t} = M_t.
Theorem. A model is arbitrage-free if M_u|_{\mathcal{A}_t} = M_t.
Lemma. Using value V_t = (\Delta_t + \Gamma_t)\cdot X_t and account A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t, V_t D_t = E[V_u D_u + \sum_{t < s \le u} A_s D_s\mid\mathcal{A}_t].\tag{3}
Note the similarity between equation (2) and (3). Price and cash flow in (2) correspond to value and account in (3). Every trading strategy produces a synthetic derivative instrument.
Theorem. (Fundamental Theorem of Asset Pricing) Every model parameterized by a vector-valued martingale (M_t)_{t\in T} and a positive, adapted function (D_t)_{t\in T} where X_t D_t = M_t - \sum_{s\le t}C_s D_s is arbitrage-free.
Proof. If (\tau_j, \Gamma_j) is a closed out trading strategy with A_t\ge0 for t > \tau_0 then V_{\tau_0} = E[ \sum_{t < s \le u} A_s D_s\mid\mathcal{A}_{\tau_0}] \ge 0. Since V_{\tau_0} = \Gamma_0\cdot X_{\tau_0} = -A_{\tau_0} and D_t is positive we have A_{\tau_0} \le 0 so there is no arbitrage.
If a money market account is available with return R_t, then D_t = 1/R_t. In this case we call D_t the stochastic discount instead of deflator. A money market account can be used to fund trading strategies to make them self-financing.
Let M_t = se^{\sigma B_t - \sigma^2t/2} and D_t = e^{-\rho t}. Since S_tD_t = se^{\sigma B_t - \sigma^2t/2} we have S_t = se^{\rho t + \sigma B_t - \sigma^2t/2}.
There is no need to restrict models to Ito processes, use partial differential equations, or consider self-financing portfolios.
We can allow \sigma to be a function of time and use the martingale e^{\int_0^t \sigma(s)\,dB - (1/2)\int_0^t \sigma^2(s)\,ds}.
If the stock pays dividends d_j at times t_j then these are cash flows and equation (1) gives S_t = se^{\rho t + \sigma B_t - \sigma^2t/2} - \sum_{t_j\le t} d_j e^{\rho(t - t_j)}. It is valid for any dividends d_j that depend only on the information available at time t_j. For discrete dividends d_j is constant. For proportional dividend d_j = p_j S_{t_j}.
A zero coupon bond with maturity u has a single cash flow of 1 at time u. By equation (2) we have D_t(u)D_t = E_t[D_u] where D_t(u) is the price at time t of the zero coupon bond.
If a zero coupon bond defaults at the random time T and pays random recovery R when it defaults then by equation (2) D^{T,R}_t(u)D_t = E_t[R1(t < T \le u)D_T + 1(T > u)D_u] where D^{T,R}_t(u) is the price at time t of the risky zero coupon bond.
Exercise. If P(T > t) = e^{-\lambda t} is independent of D_t, t\ge 0,….
If \mathcal{A} is an algebra of sets on the set \Omega we write X\colon\mathcal{A}\to 𝑹 to indicate X\colon\Omega\to 𝑹 is \mathcal{A}-measurable. If \mathcal{A} is finite then the atoms of \mathcal{A} form a partition of \Omega and X is \mathcal{A}-measurable if and only if it is constant on atoms of \mathcal{A}. In this case X is indeed a function on the atoms.↩︎