April 25, 2024
There has been considerable progress since the latter two of Fischer Black, Myron Scholes, and Robert Merton III won Nobel Prizes for their theory of option pricing. However, there are fundamental problems their mathematical theory did not address. What volatiltiy should be used for valuing an option? After putting on a hedge, when should that be adjusted? How well will a hedging strategy perform?
This note does not solve these problems. It only proposes a rigorous mathematical framework to allow for quantitative discussions. We use stochastic processes to model market instrument prices and cash flows but have no need for Brownian motion, much less Ito processes. Only familiarity with vector spaces and linear transformations is required. The Fundamental Theorem of Asset Pricing shows arbitrage-free models admit a positive measure that can be used to value derivatives, as first pointed out by Stephen Ross. The hedge is the derivative of the value with respect to prices. Any positive measure divided by its total mass, if finite, is a “probability” measure, however the FTAP is a purely geometric result. The “risk-neutral probability measure” is only marginally useful for the difficult problem of calculating real world probabilities needed for managing risk.
When a mathematical model in physics does not agree with observation it is time to come up with a new model. This does not seem to be the case in mathematical finance. Toward the end of the 19th century physicists seemed to think Newton’s program had been successfully carried out and it was just a matter of adding decimal places to physical constants. The ultraviolet catastrophe could not be explained using that theory: heating up a black body should result in unbounded energy being emitted. This led Max Plank in 1900 to postulate electromagnetic radiation might only be emitted in integer multiples of the Planck constant to fit observed data. The price of an instrument is an integer multiple of the smallest trading unit determined by the issuer. There is no need to find a parameter that fits that to data, the offering document specifies it.
Rutherford shot alpha particles through a thin gold sheet in 1911 to upset Thomson’s 1904 “plum pudding” model of atoms. Most of them went through but some bounced back. Atoms must have a tiny nucleus of positive particles and a cloud of negatively charged electrons somehow floating around without pudding. If an electron is orbiting a proton in a hydrogen atom then the classical model would require it to be continuously emit radiation (bremsstrahlung). Niels Bohr leveraged Plank’s observation to come up with a model for explaining the discrete spectral lines emitted by hydrogen atoms Balmer observed in 1885. Electrons are “waves” that can only orbit in integral multiples of their wavelength.
Werner Heisenberg added the time dimension for electrons in Bohr orbits. An electron can jump from level i to j, represented by e_{ij}. Consecutive jumps can only occur at compatible levels, e_{ij}e_{kl} = 1 when j = k and is 0 when j\not=k. His colaborators Max Born and Pascual Jordan pointed out this was equivalent to matrix muliplication. A consequence of Heisenberg’s theory is that it is not possible to simultaneously measure both position and momemtum to arbitray precision. There is an analog of the Heisenberg uncertainty principle in finance. A market order is executed immediately, but there is slippage involed with its price. A limit order guarantees the price, but it is uncertain when, or if, it will get executed.
The classical theory of mathematical finance does not account for many trading realities. Every transaction has a buyer and a seller. The seller offers possible exchanges and the buyer decides which to take. Issuers make securities available for trading. Exchanges and brokers facilitate trading.
We consider the fundamental problem of how to devise a trading strategy that replicates derivative payoffs. Continuous time trading is a mathematical fiction. Traders need to execute a finite number of trades and manage the associated risks.
Market instruments have prices and cash flows. Trading involves valuing positions and accounting for transaction costs. Quants provide arbitrage-free models of market dynamics.
Most models assume there is a money market account available to finance trading strategies. The Unified Model does not. All instruments used in trading must be explicitly specified. Realistic models must allow for non-optimal trading strategies.
Traders decide when and how much to trade at discrete times based on available information. Managing risk involves trading strategies used for hedging. The Unified Model forces you to consider the decisions traders must make every day.
See Mathematical Prerequisites for the elementary mathematics required.
Instruments have prices and cash flows. We make the usual unrealistic assumption that any instrument can be bought or sold at any time in any amount at the market price. Owning an instrument entails cash flows from the issuer or exchange. Unlike prices, cash flows are almost always 0. Stocks pay dividends, bonds pay coupons, futures pay the daily difference of their market quotes and always have price 0. Futures quotes are naturally occurring martingales.
A model of the market assumes a set of outcomes \Omega of all possible things that can occur. Information available at trading time t\in T is modeled by a partition {\mathcal{A}}_t of \Omega.
Let I be the set of all instruments, X_t\colon{\mathcal{A}}_t\to \boldsymbol{R}^I be the prices, and C_t\colon{\mathcal{A}}_t\to \boldsymbol{R}^I be the cash flows of each instrument given information {\mathcal{A}}_t at trading time t. In particular, X_t(i) = X_t^i is the price at time t of instrument i\in I.
Trades are based on information available at the time of the trade. The buyer decides how much of each instrument to buy or sell at offered market prices. Trades accumulate to positions and cash flows proportional to existing positions are credited.
A trading strategy is a finite number of increasing stopping times1 (\tau_j) and corresponding trades {\Gamma_j\colon{\mathcal{A}}_{\tau_j}\to \boldsymbol{R}^I} in each instrument with {\sum_j \Gamma_j = 0}. Trading strategies must eventually close out.
Trades accumulate to positions {\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. Trades done at time t must settle before belonging to a position.
The value, or mark-to-market, of a trading strategy at time t is V_t = (\Delta_t + \Gamma_t)\cdot X_t. It is the value of liquidating the existing positions, and the trades just done, at current prices.
The (brokerage) account, or (trade) blotter, is A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t. Cash flows are credited in proportion to existing positions and trades just executed are debited.
The net profit and loss over the period from time t to time u is {N(t,u) = V_u - V_t + \sum_{t < s \le u} A_s}. A strategy is self-financing at t if A_t = 0. The Unified Model does not assume account payments are reinvested in a money-market account.
A model admits arbitrage if there is a trading strategy with A_{\tau_0} > 0 and A_t\ge0 for t > \tau_0: you make money on the first trade and never lose money. Note that this definition does not require a “probability” measure. Traders and risk managers also compare {A_{\tau_0} = -\Gamma_{\tau_0}\cdot X_{\tau_0}} to |\Gamma_{\tau_0}|\cdot |X_{\tau_0}| as a gauge of return on investment. If they are savvy, they will also consider A_t over the life of the trading strategy.
Every arbitrage-free model of prices and cash flows is parameterized by positive discount measures {D_t\in ba({\mathcal{A}}_t)} and functions {M_t\colon{\mathcal{A}}_t\to\boldsymbol{R}^I} satisfying {M_tD_t = (M_u D_u)|_{{\mathcal{A}}_t}}, t \le u. If prices and cash flows satisfy \tag{0} X_t D_t = M_t D_0 - \sum_{s\le t} C_s D_s then the model is arbitrage-free. We can multiply the discount measure by a positive constant without affecting this equation so we assume D_0 has mass 1.
Lemma. Under this parameterization \tag{1} X_t D_t = (X_u D_u + \sum_{t < s \le u}C_s D_s)|_{{\mathcal{A}}_t}, u \ge t,
This follows immediately from replacing X_u D_u by {M_u - \sum_{s\le u}C_s D_s}.
Theorem. Under this parameterization the model of prices X_t and cash flows C_t does not admit arbitrage.
Proof. If (\tau_j, \Gamma_j) is a trading strategy with A_t\ge 0 for t > \tau_0 then V_{\tau_0} D_{\tau_0} = E_{\tau_0}[\sum_{t > \tau_0} A_t D_t] \ge 0 since D_t > 0. Using V_{\tau_0} = \Gamma_{\tau_0}\cdot X_{\tau_0} = -A_{\tau_0} and D_{\tau_0} > 0 we have A_{\tau_0} \le 0.
The Fundamental Theorem of Asset Pricing states the converse is true, but who cares? The job of a quant is to provide arbitrage-free models to their employers. The trick is to find martingales that can fit market prices.
Lemma. Using the above definitions \tag{2} V_t D_t = (V_u D_u + \sum_{t < s \le u}A_s D_s)|_{{\mathcal{A}}_t}. Note how trading strategies create synthetic instruments where value V_t corresponds to prices X_t and account A_t corresponds to cash flows C_t.
Proof. Assuming discrete time (t_j) we only need V_j D_j = (V_{j+1} + A_{j+1})D_{j+1}|_{{\mathcal{A}}_{j}} and induction. Using \Delta_{j+1}\cdot C_{j+1} = \Gamma_{j+1}\cdot X_{j+1} + A_{j+1} we have \begin{aligned} V_j D_j &= (\Delta_j + \Gamma_j)\cdot X_j D_j \\ &= \Delta_{j+1}\cdot X_j D_j \\ &= \Delta_{j+1}\cdot (X_{j+1} + C_{j+1})D_{j+1}|_{{\mathcal{A}}_{j}} \\ &= (\Delta_{j+1}\cdot X_{j+1} + \Gamma_{j+1}\cdot X_{j+1} + A_{j+1}) D_{j+1}|_{{\mathcal{A}}_{j}} \\ &= (V_{j+1} + A_{j+1})D_{j+1}|_{{\mathcal{A}}_{j}} \\ \end{aligned}
In continuous time the same argument holds path-wise if we take u > t sufficiently small.
Formulas (1) and (2) are the foundation of derivative pricing. A derivative is a contract specifying payments A_j at times \tau_j. If there is a trading strategy replicating these payments, and A_t = 0 when t\not=\tau_j, then its value is the cost of setting up the initial hedge.
Given a derivative instrument, how does one find a trading strategy that replicates its payments? Let’s assume trades occur at discrete times (t_j). Since {V_{t_0} = (\sum_{t_j > t_0} A_{t_j} D_{t_j})|_{{\mathcal{A}}_t}} and {V_{t_0} = \Gamma_{t_0}\cdot X_{t_0}} we have \Gamma_0 = \frac{d}{dX_{0}}(\sum_{j > 0} A_j D_j)|_{{\mathcal{A}}_0}, where d/dX_0 is the Fréchet derivative.
At subsequent times we have {V_j D_j = E_j[\sum_{k > j} A_k D_k]} and {V_j = (\Delta_j + \Gamma_j)\cdot X_j} so (\Delta_j + \Gamma_j) D_j = \frac{d}{dX_{j}}\sum_{k > j} A_k D_k. Since \Delta_j = \sum_{i < j}\Gamma_i this determines \Gamma_j by induction. Note \Delta is delta and \Gamma is gamma.
Of course, this does not guarantee the trading strategy will replicate the payments. More work remains to be done in assessing the risk involved with realistic trading strategies.
Repurchase agreements determine a canonical discount measure.
If a repurchase agreement has price X_t = 1 and cash flow {C_{t + \Delta t} = \exp(f_t\Delta t)} then
by equation (1) {1D_t = \exp(f_t\Delta t)D_{t
+ \Delta t}|_{{\mathcal{A}}_t}}. If D_{t + \Delta t} is {\mathcal{A}}_t measurable then D_{t + \Delta t}/D_t = \exp(-f_t\Delta t). In
discrete time D_j = \exp(-\sum_{i < j}
f_i\Delta t_i)D_0 so the canonical discount measure. is
determined by repo rates.
The continuous time version of this is D_t =
\exp(-\int_0^t f_s\,ds) where f_t is the (continuously compounded)
stochastic forward rate.
Traders assume there is a money market account available for funding, but other traders on the repo desk get paid a couple of basis points for providing that abstraction. If you are doing day trading funded by your credit card then you need to use a discount based on the APR you are being charged.
Applications of formula (1). We use the customary notation E_t[X] = E[X\mid{\mathcal{A}}_t] for (XP)|_{{\mathcal{A}}_t} when P is a positive measure having mass 1. Unlike classical models E_t[X] is a measure, not a random variable.
The Black, Scholes, and Merton model is D_t = e^{-\rho t}P where P is Wiener measure on the sample space of continuous functions C[0,\infty) and {M_t = (r, se^{\sigma B_t - \sigma^2t/2})}. There is no need for Ito’s formula or partial differential equations when using the Unified Model.
Purchasing a shares of instrument X having price X_t at time t is a cash flow {C_t = (-aX_t, a)} involving a currency and X. This is a mathematical idealization. Transactions in the real world occur on exchanges and involve order books, limit orders, and market orders.
A limit order L^{x,a,t} is specified by a level x of an instrument X, an amount a in shares of X, and the time t it is made available. After the limit order is submitted the buyer must wait until the underlying instrument reaches level x. When it does the buyer receives a shares of X in exchange for ax dollers. Limit orders always have price 0 and a single cash flow C^L_\tau = (-ax, a) at the first time the underlying reaches level x. If X_t < x then {\tau = \inf\{u > t\mid X_u \ge x\}}. If X_t > x then {\tau = \inf\{u > t\mid X_u \le x\}}.
Limit orders guarantee the price but not the time of transaction.
A market order M^{X,a,t} is specified by an instrument X, an amount a in shares of X, and the time it is executed. After the market order is submited the buyer receives a share of X at time t in exchange for a(X_t + \Delta X_t) dollars where \Delta X_t is the slippage that depends on the mechanics of how order books operate. Market orders always have price 0 and a single cash flow {C^M_t = (-a(X_t + \Delta X_t), a)}.
Market orders guarantee the time but not the price of the transaction.
Let S_t be the stock price at time t. A forward contract with strike k expiring at time t has a single non-zero cash flow {C_t = S_t - k}. From formula (1) its value at time t = 0 is E[(S_t - k)D_t]. The (at-the-money) forward f(t) is the strike making this value 0, hence f(t) D(t) = E[S_t D_t] = S_0 if there are no dividends. This is called the cost of carry.
It becomes more complicated if the stock has dividends d_j at times t_j. By formula (0) we have {S_tD_t = M_t - \sum_{t_j\le t} d_j D_{t_j}} where M_t is a martingale with E[M_t] = S_0 hence {f(t) D(t) = S_0 - \sum_{t_j\le t}E[d_jD_{t_j}]}.
If the dividends are constant then {f(t) D(t) = S_0 - \sum_{t_j\le t} d_j D(t_j)}
If the dividends are proportional with d_j = p_j S_{t_j} then {f(t)D(t) = S_0 - \sum_{t_j\le t} p_j E[S_{t_j} D_{t_j}]}. Let {f_j = E[S_{t_j} D_{t_j}] = S_0 - \sum_{i\le j} p_i E[S_{t_i} D_{t_i}]}. We have f_0 = S_0 and {f_{j} - f_{j-1} = -p_j E[S_{t_j} D_{t_j}] = p_j f_j}, j > 0 so f_j = f_{j-1}/(1 + p_j). This shows \begin{aligned} f(t)D(t) &= S_0 - \sum_{t_j\le t} p_j E[S_{t_j} D_{t_j}] \\ &= S_0 - \sum_{t_j\le t} p_jS_0/\prod_{i\le j} (1 + p_i) \\ \end{aligned}
These formulas are used for dividend adjusted stock prices.
A futures contract \Phi^{X,T} is specified by an instrument X, an expiration T, and a set of calculation times t_0 < \cdots < t_n where t_n = T. The futures quote \Phi_{t_n} is equal to the instrument price X_{t_n} at expiration. Prior to that the quote is determined by the market. The price of a futures is always 0 and has cash flows C_j = \Phi_{t_j} - \Phi_{t_{j-1}} at time t_j, 0 < j \le n. In an arbitrage-free model {0D_{t_j} = E_{t_j}[(\Phi_{t_{j+1}} - \Phi_{t_j}) D_{t_{j+1}}]}. If D_t is the canonical discount then D_{t_{j+1}} is known at time t_j so \Phi_{t_j} = E_{t_j}[\Phi_{t_{j+1}}]. Futures are naturally occuring martingales. ### Zero Coupon Bonds
Let D(u) be (the symbol for) a zero coupon bond with a single non-zero cash flow {C^{D(u)}_u = 1} at maturity u. If D_t is a discount measure then the price at time t of D(u) satisfies X^{D(u)}_t D_t = (C^{D(u)}_u D_u)|_{{\mathcal{A}}_t} = (D_u)|_{{\mathcal{A}}_t} for t \le u so D_t(u) = X^{D(u)}_t = E_t[D_u]/D_t. We use the helpfully confusing notation D(u) for D_0(u) = E[D_u].
The discount measure/stochastic discount determines the price dynamics of zero coupon bonds.
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_u = R if T \le u. It is customary to assume R is constant. 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_u + 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 default is independent of the discount measure we have at time t = 0 {D_0^{R,T}(u) = (RP(T\le u) + P(T > u))D(u)}. The credit spread is 0 when R = 1 or T = \infty. If T is exponentially distributed with hazard rate \lambda then P(T > t) = e^{-\lambda t}.
Exercise. Show s_0^{T,R}\approx \lambda(1 - R) for small \lambda.
We can also consider risky forward rate agreements.
A forward rate agreement F^\delta(u,v) specifies a rate f, an effective date u, a termination date v and a day count basis \delta. It has cash flows C^{F^\delta(u,v)}_u = -1 and C^{F^\delta(u,v)}_v = 1 + f\delta(u,v) where \delta(u,v) is the day count fraction.2 The par coupon is the value of f for which {X^{F^\delta(u,v)}_t = E_t[-D_u + (1 + f\delta(u,v)D_v] = 0} F_t^\delta(u,v) = \frac{1}{\delta(u,v)}\left(\frac{D_t(u)}{D_t(v)} - 1\right).
Exercise. Show 0 = E_t[(F_t^\delta(u,v) - F_u^\delta(u,v))\delta(u,v)D_v] for t\le u < v.
Hint: Note F_t^\delta(u,v)\delta(u,v)E_t[D_v] = E_t[-D_u + D_v)] so F_u\delta E_u[D_v] = E_u[-D_u + D_v]. Use E_t[E_u[X]] = E_t[X] if t \le u.
This show a forward rate agreement with a single cash flow {C_v = (F_t^\delta(u,v) - F_u^\delta(u,v))\delta(u,v)} has the same price as an FRA with an exchange of principal but they have different risk profiles if default occures between time u and v.
Convexity is the difference between the quote on interest rates futures and the corresponding par forward rate. If F = F^{\delta(u,v)}_u is the forward rate over [u,v] at time u then the corresponding futures quote is \phi = E[F]. Let f = F^{\delta(u,v)}_0 be the par forward.
Exercise. Show {\phi - f = -\operatorname{Cov}(F,D_v)/D(v)}.
Hint: 0 = E[(F - f)D_v]
Since foward rates and discounts are negatively correlated the convexity is positive.
A (interest rate) swap F^\delta(\{t_j\}) specifies a coupon c, calculation dates \{t_j\}_{j=0}^n, and a day count basis \delta. It has cash flows C^{F^\delta(\{t_j\})}_{t_0} = -1 and C^{F^\delta(\{t_j\})}_{t_j} = c\delta(t_{j-1},t_j) for 0 < j < n, and C^{F^\delta(\{t_j\})}_{t_n} = 1 + c\delta(t_{n-1},t_n). The par coupon F^\delta(\{t_j\})_t is the value of c for which the price of the swap is 0 at t.
Exercise. Show F^\delta(\{t_j\})_0 = (1 - D(t_n))/(\sum_{0<j\le n} \delta(t_{j-1},t_j) D(t_j)).
Post 2008, swaps having different payment frequencies started being quoted at different par coupons. A simple default model having constant recovery and hazard rate can fit market prices remakably well.
If A and B are sets then B^A = \{f\colon A\to B\} is the set of all functions from A to B. We can identify \{0,1\}^A = 2^A with the the set of all subsets of A. If B\subseteq A let 1_B(x) = 1 if x\in B and 1_B(x) = 0 if x\notin B be the indicator function of B.
Instead of countably additive “probability” measures and martingales that involve conditional expectation we use the more elementary notion of finitely additive measures and restriction of measures to subalgebras of sets.
Dunford and Schwartz use ba(\Omega) to denote the Banach space of bounded finitely additive measures on \Omega. It is the dual of B(\Omega), the Banach algebra of bounded functions on \Omega having norm {\|f\| = \sup_{\omega\in\Omega} |f(\omega)|}.
If L\colon B(\Omega)\to\boldsymbol{R} is a bounded linear functional then we can define a measure \lambda\in ba(\Omega) by \lambda(A) = L1_A.
For A,B\subseteq\Omega we have 1_{A\cup B} = 1_A + 1_B - 1_{A\cap B} so \lambda(A\cup B) = \lambda(A) + \lambda(B) - \lambda(A\cap B). Since \lambda(\emptyset) = L1_\emptyset = L0 = 0 this shows \lambda is a finitely additive measure.
If \lambda\in ba(\Omega) we can define a linear functional on elementary functions, finite linear combinations of indicator functions, by {L(\sum_j a_j 1_{A_j}) = \sum_j a_j\lambda(A_j)}, a_j\in\boldsymbol{R}, A_j\subseteq\Omega. If (A_j) are pairwise disjoint this is well defined.
Exercise. Show every elementary function \sum_j a_j 1_{A_j} = \sum_k b_k 1_{B_k} where (B_k) are pairwise disjoint.
Hint: a1_A + b1_B = a1_{A\setminus B} + (a + b)1_{A\cap B} + b1_{B\setminus A}.
If f = \sum_j a_j 1_{A_j} where (A_j) are pairwise disjoint then {|Lf| \le \max_j\{|a_j|\} \sum_j |\lambda(A_j)| \le \|f\|\|\lambda\|} so L is continuous. Elementary functions are norm dense in B(\Omega) so we can extend L to the entire space.
Exercise. Show \|L\| = \|\lambda\|.
Hint: \|\lambda\| = \sup \sum_j |\lambda(A_j)| where (A_j) are disjoint subsets of \Omega.
This shows B(\Omega)^* is isometrically isomorphic to ba(\Omega).
For any g\in B(\Omega) the multiplication operator M_g\colon B(\Omega)\to B(\Omega) defined by M_gf = fg is bounded with norm \|g\|. Its adjoint M_g^*\colon ba(\Omega)\to ba(\Omega) defines multiplication of a measure by a bounded function. We write g\lambda for M_g^*\lambda, g\in B(\Omega), \lambda\in ba(\Omega).
Information at time t is modeled by an algebra {\mathcal{A}} of sets on the sample space \Omega of all possible outcomes. If {\mathcal{A}} is finite then its atoms form a partition of \Omega and generate the algebra. Complete information is knowing \omega\in\Omega. Partial information is knowing only which atom \omega belongs to. No information is knowing only \omega\in\Omega.
A partition {\mathcal{A}} of \Omega is also a set and B({\mathcal{A}}) is the Banach algebra of {\mathcal{A}}-measurable sets. If \lambda is a measure and X\in B(\Omega) define Y\in B({\mathcal{A}}) by Y(\lambda|_{\mathcal{A}}) = (X\lambda)|_{\mathcal{A}}. If \lambda is a positive measure with mass 1 this can be written Y = E[X\mid{\mathcal{A}}], the conditional expectation of X with respect to the algebra {\mathcal{A}}, where Y is the Radon-Nykodym derivative.
The main impetus for this work was Peter Carr who invited me back into the academic world in 2008 to teach Derivative Securities at NYU Courant. I was chagrined to find there was no proof of the Fundamental Theorem of Asset Pricing accessible at the masters level, so I came up with one.
I thank Kevin Atteson, Bill Goff, Stewart Inglis, Andrew Kalotay, Alex Mayus, and Robert Merton III for helpful insights and feedback.
A function \tau\colon\Omega\to\boldsymbol{R} is a stopping time if \{\tau\le t\} is {\mathcal{A}}_t measurable for all t. If ({\mathcal{A}}_t)_{t\in T} are algebras indexed by time then {\mathcal{A}}_\tau = \{A\cap\{\tau\le t\}\mid A\in{\mathcal{A}}_t, t\in T\}.↩︎
The day count fraction \delta(u, v) is approximately equal to time from u to v in years. For example, the Actual/360 day count is the number of days from u to v divided by 360.↩︎