May 3, 2024
This note assumes you are familiar with measure theory and stochastic processes, but are not necessarily an expert. We provide a mathematically rigorous model that extends (Ross 1978) without involving the Hahn-Banach theorem. It also does not involve probability measures, sigma algebras, Brownian motion, the Ito formula, or partial differential equations. As Ross observed, the Fundamental Theorem of Asset Pricing involves only geometry, not probability. The Unified Model can be used for any set of instruments to value, hedge, and understand how poorly risk-neutral pricing can be used for managing risk. It does not provide a solution, only an initial framework for further research. For proofs and more details see Unified Model.
Let T be the set of trading times, I the set of all market instruments, \Omega the sample space of possible outcomes, and ({\mathcal{A}}_t)_{t\in T} be algebras of sets on \Omega indicating the information available at each trading time.
If {\mathcal{A}} is a finite algebra of sets on \Omega then [\omega] = \cap\{A\in{\mathcal{A}}\mid\omega\in{\mathcal{A}}\} is the atom of {\mathcal{A}} containing \omega\in\Omega.
Exercise. If B\subseteq[\omega] and B\in{\mathcal{A}} then B = \emptyset or B = [\omega].
Exercise. The atoms of {\mathcal{A}}, [{\mathcal{A}}], form a partition of \Omega.
A partition represents partial information. Complete information is knowing \omega\in\Omega. Partial information is knowing only which atom of the partition \omega belongs to.
A function X\colon\Omega\to\boldsymbol{R} is {\mathcal{A}} measurable if and only if it is constant on atoms of {\mathcal{A}} so X\colon[{\mathcal{A}}]\to\boldsymbol{R} is a function.
If P is a positive measure with mass 1, aka a probability measure, on \Omega and X\colon\Omega\to\boldsymbol{R} is a random variable then conditional expectation Y = E[X|{\mathcal{A}}] is equivalent to restriction of measure Y(P|{\mathcal{A}}) = (XP)|{\mathcal{A}}.
Price – X_t\colon[{\mathcal{A}}_t]\to\boldsymbol{R}^I market prices assuming perfect liquidity.
Cash flow – C_t\colon[{\mathcal{A}}_t]\to\boldsymbol{R}^I dividends, coupons, margin adjustments for futures.
Trading Strategy – \tau_0 < \cdots < \tau_n stopping times and trades \Gamma_j\colon[{\mathcal{A}}_{\tau_j}]\to\boldsymbol{R}^I
Position – \Delta_t = \sum_{\tau_j < t}\Gamma_j = \sum_{s < t} \Gamma_s accumulate trades not including last trades.
Value – V_t = (\Delta_t + \Gamma_t)\cdot X_t mark-to-market existing positions and current trades at current prices.
Account – A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t receive cash flows proportional to existing positions and pay for trades just executed.
Arbitrage exists if there is a trading strategy with A_{\tau_0} > 0, A_t \ge 0, t > \tau_0, and \sum_{j} \Gamma_j = 0. The first trade makes money and subsequent trades never lose money.
The Fundamental Theorem of Asset Pricing states there is no arbitrage if and only if there exist deflators, positive measures D_t on {\mathcal{A}}_t, {t\in T}, with \tag{1} X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|{{\mathcal{A}}_t} A martingale measure satisfies M_t = M_u|{\mathcal{A}}_t for t \le u. Note if cash flows are zero then deflated prices are a martingale measure. If X_u D_u goes to zero as u goes to infinity then prices are determined by deflated future cash flows.
Lemma. If X_t D_t = M_t - \sum_{s\le t} C_s D_s where M_t is a martingale measue then there is no arbitrage.
Lemma. For any arbitrage free model and any trading strategy \tag{2} V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|{{\mathcal{A}}_t}
Note how the value V_t corresponds to price X_t and account A_t corresponds to C_t in equations (2) and (1) respectively. Trading strategies create synthetic market instruments. This is the skeleton key to pricing derivative securities.
Suppose a derivative security specifies amounts \overline{A}_j be paid at times \overline{\tau}_j. If there is a trading strategy (\tau_j, \Gamma_j) with A_{\overline{\tau}_j} = \overline{A}_j for all j and A_t = 0 otherwise (aka self-financing) then a “perfect hedge” exists1.
Note V_t D_t= (\sum_{\tau_j > t} \overline{A}_j D_{\overline{\tau}_j})|{\mathcal{A}}_t can be computed from the derivative contract specification and the deflators D_t. Since we also have V_t = (\Delta_t + \Gamma_t)\cdot X_t the Frechet derivative D_{X_t}V_t of option value with respect to X_t is \Delta_t + \Gamma_t.
If time T = \{t_j\} is discrete we can compute a possible hedge at each time, \Gamma_j = D_{X_j}V_j - \Delta_j, since \Delta_j is known at t_{j-1}. In general this hedge will not exactly replicate the derivative contract obligation.
Note the Unified Model does not require Ito’s formula, much less a proof involving partial differential equations and change of measure. One simply writes down a martingale and deflator then uses equation (2) to value, hedge, and manage the risk of realistic trading strategies. The notion of “continuous time” hedging is a mathematical myth.
The Black-Scholes/Merton model uses M_t = (r, s\exp(\sigma B_t - \sigma^2t/2)P, where B_t is Brownian motion, P is Wiener measure, and the deflator is D_t = \exp(-\rho t)P.
If repurchase agreements are available then a canonical deflator exists. A repurchase agreement over the interval [t_j, t_{j+1}] is specified by a rate f_j known at time t_j. The price at t_j is 1 and it has a cash flow of {\exp(f_j(t_{j+1} - t_j))} at time t_{j+1}. By equation (1) we have {D_j = \exp(f_j\Delta t_j)D_{j+1}|{\mathcal{A}}_j}. If D_{j+1} is known at time t_j then {D_{j+1}/D_j = \exp(-f_j\Delta t_j)} and {D_j = \exp(-\sum_{i < j}f_i\Delta t_i)} is the canonical deflator.
The continuous time analog is D_t = \exp(-\int_0^t f(s)\,ds) where f is the continuously compounded instantaneous forward rate.
A zero coupon bond maturing at time u has a unit cash flow at u. In an arbitrage free model its price at time t\le u, D_t(u) satisfies D_t(u) D_t = D_u|{\mathcal{A}}_t so D_t(u) is the Radon-Nikodym derivative of D_u|{\mathcal{A}}_t with respect to D_t.
A fixed income instrument is a portfolio of zero coupon bonds. If a bond pays c_j at time t_j its present value at time t is P_t = \sum_{t_j > t} c_j D_t(u_j).
Zero coupon bond prices are determined by the deflators.
A forward rate agreement with coupon f over the interval [u,v] having day count convention \delta2 has two cash flows: -1 at time t and 1 + f\delta(u,v) at time u. The par forward coupon at time t, F_t^\delta(u,v) is the coupon for which the price is 0 at time t. By equation (1) we have 0 = (-D_u + (1 + F_t^\delta(u,v))D_v|{\mathcal{A}}_t so F_t^\delta(u,v) = \frac{1}{\delta(u,v)}\bigl(\frac{D_t(u)}{D_t(v)} - 1\bigr).
A swap is a collection of back-to-back forward rate agreements involving times (t_j). The swap par coupon making the price 0 at time t is F_t^\delta(t_0,\dots,t_n) = \frac{D_t(t_0) - D_t(t_n)}{\sum_{j=1}^n\delta(t_{j-1},t_j)D_t(t_j)}.
Exercise. Show if n = 2 the swap par coupon is the same as the par forward coupon.
Companies can default and may pay only a fraction of the notional owed on bonds they issued. A simple model3 for this is to assume the time of default T and recovery R are random variables. The sample space for the default time is [0,\infty) indicating the time of default. The information available at time t is the partition consisting of singletons \{s\}, s < t and the set [t, \infty). If default occurs prior to t then we know exactly when it happened. If default has not occurred by time t then we only know it can occur any time after that.
The cash flows for a risky bond D^{T,R}(u) are 1 at time u if T > t and RD_T at time T if T \le t. For the model to be arbitrage-free we must have D_t^{R,T}(u) D_t = \bigl(1(T > u)D_u + 1(T\le u)R D_T\bigr)|{\mathcal{A}}_t. The credit spread \lambda_t^{T,R}(u) is defined by {D_t^{T,R}(u) = D_t(u)\exp(-\lambda_t^{T,R}(u) (u - t))}. Note if T = \infty or R = 1 then the credit spread is zero.
If the deflators are independent of the stopping time and recovery is RD_T(u) at T\le u for some constant R then using D_T(u)D_T = D_u|{\mathcal{A}}_t for T \le u we have D_t^{T,R}(u) D_t P(T > t) = \bigl(P(T > u) + R P(t < T \le u)\bigr)D_u|{\mathcal{A}}_t and the credit spread is \lambda_t^{T,R}(u) = -\frac{\log(P(T > u) + R P(t < T \le u))}{u - t}.