January 23, 2024
A derivative is a legal contract. The buyer chooses a seller agree to exchange future cash flows. The seller quotes a price to the buyer and both parties are obligated to make good on the agreed cash flows after the trade.
Traders buy and sell shares of instruments and eventually close out their position.
Market instruments can be bought or sold at a price and ownership entails cash flows. Shares of instruments can be traded based on available information and accumulate to positions. The mark-to-market value and amounts involved with trading correspond to price and cash flows respectively. The Unified Model demonstrates the connection between dynamic trading and how to value, hedge, and manage the risk of any portfolio.
Every arbitrage-free model of prices and cash flows is parameterized by a vector-valued martingale whose components are indexed by market instruments and a positive, adapted process called a deflator. If repurchase agreements are available the canonical deflator is the reciprocal of the money market account.
A derivative security is a contract between two parties: I will give you this on these dates if you will give me that on those dates. Derivatives must have existed since before recorded history. The Nobel prize winning breakthrough of Black, Scholes, and Merton was how to synthesize derivatives by dynamically trading market instruments based on the borrowing cost used to fund the hedge instead of trying to estimate the growth rate of the underlying securities.
This paper provides a unified model for valuing, hedging, and managing the risk of any derivative security. It shows how they can be synthesized by trading market instruments and turns the spotlight on what may be the next Nobel prize winning problem: how should you hedge if you can’t do it continuously?
The Unified Model can be used for any portfolio of stocks, bonds, currencies, commodities, and even other derivatives in the portfolio.
We assume the usual setup, \langle Ω, P, (\mathcal{A}_t)_{t\in T}\rangle, of a sample space Ω, a probability measure P, and an increasing filtration of algebras (\mathcal{A}_t) over the set of trading times T. If you are not familiar with this see the Notation section below.
Every instrument has a price X_t and a cash flow C_t at any trading time t\in T. Instruments are assumed to be perfectly liquid: they can be bought or sold at the given price in any amount. Cash flows are associated with owning an instrument and are almost always 0; stocks have dividends, bonds have coupons, currencies have no cash flows, commodities have storage costs. European options have exactly one cash flow at expiration.
A market model specifies prices X_{t}\colon\mathcal{A}_{t} \rightarrow \mathbf{R}^{I}, and cash flows C_{t}\colon\mathcal{A}_{t} \rightarrow \mathbf{R}^{I}, where I is the set of available market instruments.
We use the notation X\colon\mathcal{A}\to\mathbf{R} to indicate X\colon Ω\to\mathbf{R} is \mathcal{A}-measurable. If \mathcal{A} is finite then it is generated by its atoms and X is a function from the atoms of \mathcal{A} to \mathbf{R}.
A trading strategy is a finite collection of strictly increasing stopping times (τ_j), and trades (\Gamma_j) where \Gamma_j:\mathcal{A}_{τ_j}\rightarrow \mathbf{R}^{I} indicating the number of shares to trade in each instrument at time τ_j. Trades accumulate to a position \Delta_t = \sum_{τ_j < t}\Gamma_j = \sum_{s < t}\Gamma_s where \Gamma_s = \Gamma_j when s = τ_j. A trade at time t is not included in the position at time t; it takes some time for trades to settle.
The value (or mark-to-market) of a position at time t is V_t = \left( \Delta_t + \Gamma_t \right) \cdot X_t. This is what you would get from liquidating your existing position and trades just executed at the current market price.
The amount generated by the trading strategy at time t is A_t = \Delta_t \cdot C_t - \Gamma_t \cdot X_t. You receive cash flows proportional to your existing position and pay for the trades just executed.
Consider a two period model with T = \{t_0,t_1,t_2\} with two instruments, a bond and a stock paying no dividends. Assume the bond price R_t = 1 and stock price S_t = 100 are constant so X_t = (R_t, S_t) = (1, 100) for all t. We assume an initial position \Delta_{t_0} = (0,0). Consider the trading strategy with \tau_0 = t_0, \Gamma_0 = (-100,1) and \tau_1 = t_2, \Gamma_1 = (100, -1).
We have V_{t_0} = \Gamma_0\cdot X_{t_0} = -100\times 1 + 1\times 100 = 0 and A_0 = -\Gamma_0\cdot X_0 = -V_{t_0} = 0. The position at time t_1 is \Delta_{t_1} = \sum{\tau_j < 1}\Gamma_j = \Gamma_0 = (-100,1). Since \Gamma_t = \Gamma_j when t = \tau_j we have \Gamma_{t_1} = (0,0) so V_{t_1} = (\Delta_{t_1} + \Gamma_{t_1}\cdot X_{t_1} = 0 and A_1 = \Delta_{t_1}\cdot C_{t_1} - \Gamma_{t_1}\cdot X_{t_1} = 0. Note \Delta_{t_2} = -\Gamma_{t_2} so V_{t_2} = 0 and A_{t_2} = -\Gamma_{t_2}\cdot X_{t_2} = (100, -1)\cdot (1, 100) = 0.
Now suppose the stock pays a 2 dollar dividend at time t_1 so C_{t_1} = (0, 2) is the only nozero cash flow. As above, V_t = 0 and A_t = 0 for all t except A_1 = \Delta_{t_1}\cdot C_{t_1} - \Gamma_{t_1}\cdot X_{t_1} = (-100,1)\cdot (0,2) - 0 = 2. This model is not arbitrage free.
Arbitrage exists if there is a trading strategy with A_{τ_0} > 0, A_t \ge 0 for t > τ_0, and \sum_j \Gamma_j = 0; you make money on the first trade and never lose until the position is closed out.
The Fundamental Theorem of Asset Pricing (FTAP) states there is no arbitrage if and only if there exists a deflator, D_t:\mathcal{A}_t \rightarrow \left(0,\infty \right), with X_t D_t = E_t\bigl[X_v D_v + \sum_{t < u \le v} C_u D_u\bigr]. We can assume D_0 = 1 since if D_t is a deflator then so is D_t/D_0.
Note that if there are no cash flows, C_t = 0 for all t \in T, then X_t D_t is a martingale. For an infinite time horizon where the price times the deflator goes to 0, the current price is the expected value of discounted future cash flows, just as in classical Graham and Dodd valuation.
A consequence of the above using the definitions of value and amount is V_t D_t = E_t\bigl[V_v D_v + \sum_{t < u \le v}A_u D_u\bigr]. Note the similarity to the previous displated equation. Value corresponds to price and amount corresponds to cash flow. This is the skeleton key for valuing derivative securities. It shows how dynamic trading creates synthetic market instruments.
Proof. If u > t is sufficiently small then X_t D_t = E_t[(X_u + C_u) D_u] and \Delta_t + \Gamma_t = \Delta_u \begin{aligned} V_t D_t &= (\Delta_t + \Gamma_t)\cdot X_t D_t\\ &= \Delta_u\cdot E_t[(X_u + C_u) D_u]\\ &= E_t[(\Delta_u\cdot X_u + \Gamma_u\cdot X_u + A_u) D_u] \\ &= E_t[(V_u + A_u)D_u],\\ \end{aligned} where we use \Delta_u\cdot C_u = \Gamma_u\cdot X_u + A_u. The displayed formula above follows by induction.
For a trading strategy that closes out V_{τ_0} D_{τ_0} = E_{τ_0}[\sum_{t > τ_{0}}{A_{t}D_{t}] \ge 0}. Since V_{τ_0} = \Gamma_{τ_0} \cdot X_{τ_0}, A_{τ_0} = - \Gamma_{τ_0} \cdot X_{τ_0} and D_{τ_0} > 0 we have A_{τ_0} \le 0. This proves the “easy” direction of the FTAP.
There is no need to prove the “hard” direction since we have a large supply of arbitrage free models. All models of the form X_t D_t = M_t - \sum_{s \le t}{C_s D_s} where M_t:\mathcal{A}_t \rightarrow \mathbf{R}^{I} is a martingale and D_t:\mathcal{A}_t \rightarrow (0,\infty) are arbitrage free. This follows from substituting X_v D_v = M_v - \sum_{u\le v} C_u D_u to obtain the arbitrage-free condition: \begin{aligned} E_t\bigl[X_v D_v + \sum_{t < u \le v} C_u D_u\bigr] &= E_t\bigl[M_v - \sum_{u\le v} C_u D_u + \sum_{t < u \le v} C_u D_u\bigr] \\ &= E_t\bigl[M_v - \sum_{u\le t} C_u D_u \bigr] \\ &= M_t - \sum_{u\le t} C_u D_u \\ &= X_t D_t \\ \end{aligned}
If a derivative security pays amounts \bar{A}_j at times \bar{τ}_j and there is a hedge, (\bar{\Gamma}_t)_{t\in T}, that replicates these amounts then the value of the derivative is the cost of setting up the initial hedge: \bar{\Gamma}_0\cdot X_0. The hedge must satisfy A_t = 0 if t ≠ \bar{τ}_j for all j (self financing) and A_t = \bar{A}_j if t = \bar{τ}_j for some j. The formula V_0 = E[\sum_j \bar{A}_j D_{\bar{τ}_j}] gives the cost of the initial hedge since V_0 = \bar{\Gamma}_0\cdot X_0.
European options have a single payment, \bar{A}_T, at a fixed time T and have value V_0 = E[\bar{A}_T D_T]. Sometimes it is useful to compute this as E[\bar{A}_T D_T] = E^*[\bar{A}_T] E[D_T], where E^* is the expected value under the probability measure defined by dP^*/dP = D_T/E[D_T]. This is called the forward measure at time T.
We can compute V_0 using derivative cash flows and the deflator. Since \Gamma_0\cdot X_0 = V_0 the initial hedge is \Gamma_0 = dV_0/dX_0.
The hedge at time t is similarly determined. Since (\Delta_t + \Gamma_t)\cdot X_t = V_t we have \Delta_t + \Gamma_t = dV_t/dX_t, where the last term is the Fréchet derivative. Since the position \Delta_t is known at time t this determines the trades: \Gamma_t = dV_t/dX_t - \Delta_t.
In the continuous time case where stocks are modeled by geometric Brownian motion, this becomes classical Black-Scholes/Merton delta hedging where delta is \Delta and gamma is \Gamma. Under their mathematical assumptions the hedge perfectly replicates the derivative.
In the real world it is not possible to perfectly replicate the derivative security. It is the job of a quant to advise traders on when to hedge and how to manage the associated risk of imperfect replication.
The Black-Scholes/Merton model of a stock with no dividends (C_t = 0 for all t) is specified by the martingale M_t = (r, s\exp(\sigma B_t - \sigma^2t/2)) and deflator D_t = \exp(-\rho t). No need for Ito’s lemma, partial differential equations, or the Hahn-Banach theorem.
We can easily add fixed (C_j = d_j constant) or proportional (C_j = S_j p_j) dividends. Stock price “jumping” by dividend amount is a consequence of the model, not ad hoc no arbitrage arguments.
There is a canonical choice for a deflator if repurchase agreements are available.
Assume trades occur at discrete times, as they actually do, so T = \{t_j\} where t_i < t_j if i
< j.
A repurchase agreement at time t_j, R_j,
has price X^{R_j}_{t_j} = 1 and cash
flow C^{R_j}_{t_{j+1}} = R_j so for any
arbitrage free model D_{t_j} = E_{t_j}[R_j
D_{t_{j+1}}]. We assume D_{t_{j+1}} is \mathcal{A}_{t_j} measurable so D_{t_j} = R_j D_{t_{j+1}} and D_{t_j} = \Pi_{j<n} R_j^{-1} if D_{t_0} = 1.
Define the forward repo rate, f_j, by R_j = \exp(f_j\,\Delta t_j) where \Delta t_j = t_{j+1} - t_j, so D_{t_j} = \exp(-\sum_{j<n} f_j \Delta_{t_j}). The continuous time version of the canonical deflator is D_t = \exp(-\int_0^t f_s\,ds).
The price dynamics of all (non-risky) fixed income instruments are determined by the deflator.
Futures contracts are typically traded on exchanges. The exchange specifies times t_j for margin account adjustments. The price of a futures contract is always 0 for any exchange customer with a margin account in good standing. Each contract has an underlying S and an expiration T. The futures quote at time t, Φ_t = Φ^{S,T}_t, must satisfy Φ_T = S_T. Prior to that it is determined by the limit and market orders traded on the exchange. Futures have cash flows C_{t_j} = Φ_{t_j} - Φ_{t_{j-1}} for 0 < j \le n and C_{t_0} = 0 where t_0 is the time the contract is entered.
In an arbitrage-free model 0 = E_{t_j}[(Φ_{t_{j+1}} - Φ_{t_j}) D_{t_{j + 1}}] so we have Φ_{t_j} = E_{t_j}[Φ_{t_{j+1}}] since D_{t_{j + 1}} is t_j measurable and non-zero. This shows futures quotes are martingales, at least at the adjustment times. Assuming Φ_t^{S,T} = E_t[S_T] for all t \le T provides an arbitrage-free model of futures quotes.
A forward contract on underlying S with strike k expiring at time t pays C_t = C_t^{S,k,t} = S_t - k. It has initial value V_0 = E[(S_t - k)D_t] = S_0 - kE[D_t] in an arbitrage-free model. The par forward, f, is the strike that makes the initial value equal to zero: 0 = V_0 = E[(S_t - f)D_t] so S_0 = fE[D_t]. This formula is called the cost of carry.
The first thing every trader checks when using a new model is put-call parity. A (European) put option on underlying S with strike k expiring at time t pays C^p_t = \max\{k - S_t,0\} at t. A call option on underlying S with strike k expiring at time t pays C^c_t = \max\{S_t - k, 0\} at t. Since C^c_t - C^p_t = S_t - k we have V^c_0 - V^p_0 = c - p = S_0 - k E[D_t], where c and p are the value of the call and put at time 0. Put-call parity holds for any (arbitrage-free) model.
An American option with strike k and expiration t pays C_τ at a stopping time τ\le t at the discretion of the option holder. In the unified model this is represented by extending the sample space of the underlying, Ω, to Ω\times (0,t]. The outcome (ω,τ)\in Ω\times (0,t] represents exercising at time τ when ω occurs. Note that the model does not assume the option is exercised at the “optimal” time. In practice, not every market participant does this. Models should have knobs that reflect reality and choosing the optimal stopping time should not be implicit.
American options do not satisfy put-call parity in general. The exercise time of the put and the call can be different.
A zero coupon bond, D(u), pays one unit at maturity u so C^{D(u)}_u = 1 is the only cash flow. We write D_t(u) for the price X_t^{D(u)} of the zero coupon bond at time t. An arbitrage free model requires the price at time t to satisfy D_t(u)D_t = E_t[D_u] so D_t(u) = E_t[D_u]/D_t.
In the continuous time case the forward curve, f(u), is defined by D_0(u) = \exp(-\int_0^u f(s)\,ds). The forward curve at time t, f_t(u), is defined by D_t(u) = \exp(-\int_t^u f_t(s)\,ds).
A risky zero coupon bond with recovery R and default time T has a single cash flow C_u = 1 if default occurs after maturity or C_T = R if T \le u. It is customary to assume R is constant. As with American options, we must expand the sample space to Ω\times (0,\infty] where (ω,t)\in Ω\times (0,\infty] indicates default occurred at time t. The partition of (0,\infty] representing information available at time t for the default time is \{(t,\infty]\} \cup \{\{s\}:s \le t\}: if default has not occurred prior to t we only know T > t; if default occurred prior to time t we know exactly when it happened.
We write D_t^{R,T}(u) for the price X_t^{D^{R,T}(u)} of the risky zero coupon bond at time t. The dynamics of a risky zero are determined by D_t^{R,T}(u) D_t = E_t[R 1(T \le u) D_T + 1(T > u) D_u]. The credit spread s_t = s_t^{R,T}(u) defined by D_t^{R,T}(u) = D_t(u) e^{-u s_t} incorporates both recovery and default.
If rates are zero then D_t = 1 for all t and this simplifies to D_0^{R,T}(u) = R P(T \le u) + P(T > u) when t = 0. If T is exponentially distributed with hazard rate λ then P(T > t) = e^{-λ t} and D_0^{R,T}(u) = R + (1 - R)e^{-λu}. When λ = 0 the right hand side is 1. When R = 0 the credit spread equals the hazard rate. If λu is small then the approximation e^x \approx 1 + x for small x gives the rule of thumb s = λ(1 - R) where s = s_0 = s_0^{R,T}(u) is the credit spread at time 0.
For general t we have D_t^{R,T}(u) = R P(T \le u | T > t) 1(t < T \le u) + P(T > u | T > t) 1(T > u).
Unlike in the credit default swap market, mathematical finance literature likes to assume recovery is delayed until maturity. It is also popular to make the unrealistic assumption that default time is independent of the deflator. Under these assumptions we have D_t^{R,T}(u) = D_t(u)\bigl(R P(T \le u | T > t) 1(t < T \le u) + P(T > u | T > t) 1(T > u)\bigr). In principal R could be random and joint distributions involving the default time and deflators could be specified, but computations become more difficult.
See Fixed Income for details on how the deflator determines the dynamics of instruments.
The price of an instrument is not a number. Prior to a trade being executed, it depends on what instrument is being exhanged in the trade, the amount being purchased, and the legal entities involved.
The atoms of finance are holdings (a,i,e) indicating entity e owns amount a of instrument i. A trade involves the exchange of holdings at some time: (t;a,i,c;a',i',c') where t is the time of the exchange, a is the amount of instrument i the buyer, e, exchanges for the amount a' in instrument i' from the seller, e'. The price of the trade is X = a/a' after the fact.
Price can be modeled more realistically as a function X\colon T\times A\times I\times E\times I\times E\to \mathbf{R}, where T is the set of trading times, A is the set of amounts that can be traded, I is the set of market instruments, and E is the set of legal entities. At time t the trade (t;a'X_t,i,e;a',i',e'), where X_t = X_t(a',i,e,i',e'), is available to the buyer. The seller e' specifies the price X_t. The buyer e decides when to trade. After the trade the buyer holds (a', i', e) and the seller holds (a'X_t, i, e').
In the one-period model T = \{t_0, t_1\} and we let x = X_{t_0}, X = X_{t_1}. The cost of acquiring γ shares of each instrument at the beginning of the period is γ\cdot x. If that is negative then you make money. At the end of the period you close the position by selling all of your shares. This results in a cash flow of γ\cdot X to your account. If that is non-negative then you don’t lose money. A one period model has arbitrage if there exists γ\in \mathbf{R}^I with γ\cdot x < 0 and γ\cdot X(\omega)\ge0, \omega\in\Omega.
The one-period Fundamental Theorem of Asset Pricing states that there is no arbitrage if and only if there exists a positive measure \Pi on \Omega such that x = \int_\Omega X\,d\Pi. If such a measure exists and γ\cdot X\ge 0 then γ\cdot x \ge 0, so there is no arbitrage.
Lemma. If x\in\boldsymbol{R}^n and C is a closed cone in \boldsymbol{R}^n with x\not\in C then there exists ξ\in\boldsymbol{R}^n with ξ\cdot x < 0 and ξ\cdot y \ge0 for y\in C.
Recall that a cone is a subset of a vector space closed under addition and multiplication by a positive scalar, that is, C + C\subseteq C and tC\subseteq C for t > 0. For example, the set of arbitrage positions is a cone.
Proof. Since C is closed and convex there exists x^*\in C with 0 < ||x^* - x|| \le ||y - x|| for all y\in C. Let ξ = x^* - x. For any y\in C and t > 0 we have ty + x^*\in C so ||ξ|| \le ||ty + ξ||. Simplifying gives t^2||y||^2 + 2tξ\cdot y\ge 0. Dividing by t > 0 and letting t decrease to 0 shows ξ\cdot y\ge 0. Take y = x^* then tx^* + x^*\in C for t \ge -1. By similar reasoning, letting t increase to 0 shows ξ\cdot x^*\le 0 so ξ\cdot x^* = 0. Since 0 < ||ξ||^2 = ξ\cdot (x^* - x) = -ξ\cdot x we have ξ\cdot x < 0. \blacksquare
The set of non-negative finitely additive measures is a closed cone and X\mapsto \int_\Omega X\,d\Pi is positive, linear, and continuous so C = \{\int_\Omega X\,d\Pi \mid \Pi\ge 0\} is also a closed cone. The contrapositive of the FTAP follows from the lemma.
Note that the lemma is a purely geometric fact that does not involve the actual probability of anything. It is also called Farkas’ lemma and is a special case of the Hahn-Banach theorem in finite dimensional space.
The proof also shows how to find an arbitrage when one exists.
If a zero coupon bond exists, i.e., there is a ζ\in\mathbf{R}^I with ζ\cdot X(ω) = 1 for all ω\in Ω, then ζ\cdot x = \int_Ω ζ\cdot X\,d\Pi = ||\Pi|| is the price of the zero coupon bond and P = Π/ζ\cdot x is a probability measure. The discount is D = ζ\cdot x and we get the formula x = E[X]D for current price equaling the expected value of the discounted future price.
If \mathcal{A} is an algebra on the set Ω we write X\colon\mathcal{A}\to\mathbf{R} to indicate X\colonΩ\to\mathbf{R} is \mathcal{A}-measurable. If \mathcal{A} is finite then the atoms of \mathcal{A} form a partition of Ω and being measurable is equivalent to being constant on atoms. In this case X is indeed a function on the atoms.
If \mathcal{A} is an algebra of sets, the conditional expectation of X given \mathcal{A} is defined by Y = E\left\lbrack X | \mathcal{A}\right\rbrack if and only if Y is \mathcal{A} measurable and \int_A Y\,dP = \int_A X\,dP for all A\in\mathcal{A}. This is equivalent to Y(P|_{\mathcal{A}}) = (XP)|_{\mathcal{A}} where the vertical bar indicates restriction of a measure. In particular, E[X|\mathcal{A}](ω) = \sum \{X(α) P(α): α\subseteq ω, α\in\mathcal{A}\}
A filtration indexed by T\subseteq [0,\infty) is an increasing collection of algebras, (\mathcal{A}_t)_{t\in T}. A process M_{t}\colon\mathcal{A}_{t} \rightarrow \mathbf{R}, t\in T, is a martingale if M_t = E[M_u | \mathcal{A}_t] = E_t[M_u] for t\le u.
A stopping time is a function τ\colonΩ\to T such that \{ω\inΩ\mid τ(ω) \le t\} belongs to \mathcal{A}_t, t\in T.