February 16, 2025
This note suggests improvements to Stephen Ross’s paper “A Simple Approach to the Valuation of Risky Streams” (Ross 1978). Ross expanded on the groundbreaking theory developed by Fischer Black, Myron Scholes (Black and Scholes 1973), and Robert C. Merton (Merton 1973), which originally focused on valuing options through dynamic trading of a bond and a stock. Ross greatly expanded this by showing how to value derivatives using any collection of instruments, referred to as risky streams.
We place cash flows on equal footing with prices and assume every trading strategy involves only a finite number of transactions based on available information.
Perhaps at the time, over half a century ago, people were not prepared for such an audatious expansion of the B-S/M theory. Ross conflated cash flows with instantaneous changes in the price of the underlying. We put the cash flows associated with owning financial instruments on equal footing with prices. B-S/M and Ross assumed continuous time trading is possible. This is a mathematical artifact of using Ito processes to model trading strategies that leads to untenable results1. Every trading strategy involves only a finite number of transactions based on available information. The Simple Unified Model does not solve the crucial problem of when to hedge, how much to buy, or how good the hedge is; it only provides a rigorous mathematical framework for future researchers to address limitations of existing theory.
The Simple Unified Model does not involve so-called real-world probability measures that are immediately replaced by risk-neutral probability measures. It does not involve stochastic processes, the Ito formula, partial differential equations, the Hahn-Banach theorem or, much less, weak-* topologies. It does not even involve probability. As Ross showed, the lack of arbitrage places geometric constraints on price dynamics. Positive measures having mass one show up, but they are not the probability of anything.
Ross’s valuation operator corresponds to a deflator {(D_t)_{t\in T}}: positive measures on an algebras of sets ({\mathcal{A}}_t)_{t\in T} representing available information. If repurchase agreements are available then the canonical deflator is the stochastic discount: the reciprocal of the return from rolling over one unit invested at prevailing repo rates. Market instruments have prices (X_t) and associated cash flows (C_t). Trading involves buying and selling instruments (\Gamma_j) at a finite number of increasing stopping times (\tau_j). 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 and is zero otherwise. Prices, cash flows, and trades determine the value, or mark-to-market, {V_t = (\Delta_t + \Gamma_t)\cdot X_t} of the strategy and the amounts {A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t} showing up in the trading account.
Arbitrage exists if there is a trading strategy (\tau_j, \Gamma_j) with A_{\tau_0} > 0, A_t\ge0 for t > \tau_0, and \sum_j \Gamma_j = 0. There is no arbitrage if there exist positive measures D_t on {\mathcal{A}}_t with X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|_{{\mathcal{A}}_t}. A direct consequence using the definitions of value and amount is V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{{\mathcal{A}}_t}. If a derivative instrument pays \hat{A}_k at \hat{\tau}_k and there exists a trading strategy (\tau_j, \Gamma_j) with A_{\tau_0} > 0, A_t\ge0 for t > \tau_0, and \sum_j \Gamma_j = 0. There is no arbitrage if there exist positive measures D_t on {\mathcal{A}}_t with X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|_{{\mathcal{A}}_t}. A direct consequence using the definitions of value and amount is V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{{\mathcal{A}}_t}. If a derivative instrument pays \hat{A}_k at \hat{\tau}_k and there exists a trading strategy (\tau_j, \Gamma_j) with A_t = \hat{A}_k when t = \hat{\tau}_k, A_t = 0 otherwise, and \sum_j \Gamma_j = 0 then the value of the hedge at time t satisfies V_t D_t = (\sum_{\hat{\tau}_k > t} \hat{A}_k D_{\hat{\tau}_k})|_{{\mathcal{A}}_t}. This formula show how to value hedgeable derivative instruments based on their payoffs, deflator, and available information.
The Black-Scholes/Merton model for the price of a bond and a stock is {X_t = (r\exp(\rho t), s\exp(\rho t + \sigma B_t - \sigma^2t/2))P}, where B_t is Brownian motion and the cash flows are zero, C_t = (0, 0). The deflator is D_t = \exp(-\rho t)P where P is Wiener measure. For a call option paying \hat{A}_T = \max\{S_T - k, 0\} at time \hat{\tau} = T the above formula at t = 0 yields V_0 = E[\max\{S_T - k, 0\}\exp(-\rho T)].
The Simple Unified Model is an extension of Stephen Ross’s (Ross 1978) “A Simple Approach to the Valuation of Risky Streams”. According to Ross
If there are no arbitrage opportunities in the capital markets, then there exists a (not generally unique) valuation operator, L.
He used the Hahn-Banach theorem to show the existence of L and observed it is not unique if the market is not complete. Ross generalized the B-S/M model of a bond, stock, and option to any collection of instruments (risky-streams). He identified cash flows as a derivative of a price process. We put cash flows on the first class footing they deserve. He assumed, as did B-S/M, that continuous time trading was possible. Every trading strategy in the real world involves only a finite number of trades.
We assume you are familiar with measure theory and stochastic processes, but are not necessarily an expert. See the Appendix for details. Let \Omega be the set of all possible market outcomes.
In the Black-Scholes/Merton theory \Omega = C[0,\infty) is the set of all continuous functions from [0,\infty) to the real numbers. They model possible stock prices as S_t(\omega) = s\exp(\mu t + \sigma \omega(t) - \omega(t)^2/2) for \omega\in\Omega. This vastly underestimates set of all possible market outcomes.
Partial information is modeled by a partition of \Omega. Elements of the partition are atoms. The model specifies partitions {\mathcal{A}}_t representing available information at each trading time t\in T.
Let \bar{{\mathcal{A}}} denote the smallest algebra of sets generated by a finite partition {\mathcal{A}} of \Omega. Every function D\colon{\mathcal{A}}\to\boldsymbol{R} can be extended to a measure on \bar{{\mathcal{A}}} by {\hat{D}(B) = \sum\{D(A)\mid A\subseteq B, A\in{\mathcal{A}}\}}, B\in\bar{{\mathcal{A}}}. A function X\colon\Omega\to\boldsymbol{R} is \hat{{\mathcal{A}}} measurable if and only if X is constant on each atom of {\mathcal{A}}. This allows us to write X\colon{\mathcal{A}}\to\boldsymbol{R} as a function on atoms and relegate the words ‘algebra’ and ‘measurable’ to the (Appendix)[#appendix].
Let I be the collection of all market instruments. Instruments can be bought or sold at price X_t\colon{\mathcal{A}}_t\to\boldsymbol{R}^I at any trading time t\in T. We assume, as is customary, that market prices are perfectly liquid and divisible.
Instruments have cash flows C_t\colon{\mathcal{A}}_t\to\boldsymbol{R}^I associated with their ownership. Stocks have dividends, bonds have periodic coupons and principal at maturity, futures have daily margin adjustments. The price of a futures is always zero in an arbitrage-free model. Money market accounts have zero cash flows.
Traders buy and sell instruments over time based on information available at the time of the trade. A trading strategy is a finite sequence \tau_0 < \cdots < \tau_n of increasing stopping times and trades \Gamma_j\colon{\mathcal{A}}_{\tau_j}\to\boldsymbol{R}^I. Trades accumulate to the position \Delta_t = \sum_{\tau_j < t}\Gamma_j at time t. We also write \Delta_t = \sum_{s < t} \Gamma_s where \Gamma_s = \Gamma_j when s = \tau_j. Note the position does not include the trades \Gamma_t at time t. This corresponds to the fact that it takes time after a trade is executed for it to settle.
The value, or mark-to-market, of a trading strategy is V_t = (\Delta_t + \Gamma_t)\cdot X_t and represents how much liquidating the current position and trades yet to settle at current market prices would yield.
Trading strategies incur cash flows in the trading account A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t: you receive cash flows proportional to existing positions and pay the current market price 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. Eventually the position must close out, otherwise you could simply borrow a dollar every day.
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 there are a finite number of cash flows 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 measure then there is no arbitrage.
Proof: Replace X_u D_u by M_u - \sum_{s\le u} C_s D_s in equation (1).
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}
Proof. Use \Delta_t + \Gamma_t = \Delta_{t + \epsilon} if \epsilon > 0 is sufficiently small.
Note how the value V_t corresponds to price X_t and account A_t
Trading strategies create synthetic market instruments.
Equations (1) and (2) are the skeleton keys to pricing derivative securities.
Suppose a cash settled derivative security specifies amounts \hat{A}_j be paid at times \hat{\tau}_j. If there is a trading strategy (\tau_j, \Gamma_j) with A_{\hat{\tau}_j} = \hat{A}_j for all j and A_t = 0 otherwise (aka self-financing) then a “perfect hedge” exists2.
Equation (2) yields \tag{3} V_t D_t = (\sum_{\tau_j > t} \hat{A}_j D_{\hat{\tau}_j})|_{{\mathcal{A}}_t} which can be computed from the derivative contract term sheet and the deflators (D_t). Since V_t = (\Delta_t + \Gamma_t)\cdot X_t, the Fréchet derivative D_{X_t}V_t of option value with respect to X_t is \Delta_t + \Gamma_t so \Gamma_t = D_{X_t}V_t - \Delta_t. Since \Delta_t is known at time t, this gives a recipe for computing a trading strategy where \Delta is delta and \Gamma is gamma in the customary sense used by traders. In general this hedge will not exactly replicate the derivative contract obligation.
The Simple Unified Model does not prescribe when the hedge should be executed. The untenable Black-Scholes/Merton solution is to hedge “continuously”. Every trading strategy executed in the real world involves only a finite number of trades.
Various authors have considered replication error.
(Derman 1999)
(Derman and Taleb 2005)
One simply writes down a martingale measure indexed by market instruments and deflator, then use equation (3) to value derivative securities. There is still work to be done on when and how much to hedge, and measure how well a hedge performs.
A European option has a single payoff \hat{A}_T at expiration T. In this case, equation (3) for the value of the option is {V_t D_t = (\hat{A}_TD_T)|_{{\mathcal{A}}_t}}.
A forward on an instrument with price process (S_t) is determined by a strike K and _expiration T. A forward has a single cash flow C_T = S_T - K.
V_t D_t = ((S_T - K)D_T)|_{{\mathcal{A}}_t}
The Black-Scholes/Merton model uses the martingale measure {M_t = (r, p(\sigma B_t - \sigma^2t/2))P}, where B_t is Brownian motion and P is Wiener measure. The deflator is {D_t = \exp(-\rho t)P}, where \rho is the constant interest rate. From equation (3) the model of prices is X_t = (R_t, S_t) where the bond price is {R_t = r\exp(\rho t)} and the stock price is {S_t = s\exp(\rho t + \sigma B_t - \sigma^2t/2) - \sum_{s\le t} C_s\exp(\rho(t - s))} where C_t is the stock dividend at time t. The current value of a European call with payoff \nu(S_T) at time T is V_0 D_0|_{{\mathcal{A}}_0} = (\nu(S_T) D_T)|_{{\mathcal{A}}_0} so V_0 = E[\nu(S_T) D_t)].
If there are fixed dividends d_j at t_j then C_{t_j} = d_j. If there are proportional dividends p_j at t_j then C_{t_j} = p_j S_{t_j}. Since divideds are usually announced less than a year in advance a more realistic model might specify a function x(t) that is 1 over the announcement period that tends to zero and assume dividends are mixture {C_t = x(t_j)d_j + (1 - x(t_j)p_j S_{t_j}.
More generally, if B_t is a Levy process where B_1 has mean zero and variance one then {M_t = (r, \sigma\exp(s B_t - t\kappa(\sigma))P} is an arbitrage-free model, where \kappa(\sigma) = \log E[\exp(\sigma X_1)] and \log E[e^{B_t}] = t \log E[e^{B_1}].
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Merton’s jump diffusion model is a special case. (ref???)
Every positive random variable with finite mean and log variance can be written {F = f\exp(sX - \kappa(s))} where {f = E[F]}, {s^2 = \operatorname{Var}(\log F)}, X has mean zero and variance 1, and {\kappa(s) = \log E[\exp(sX)]}. The vol s is related to \sigma by s = \sigma\sqrt{t}.
We have \begin{aligned} E[\max\{k - F, 0\}] &= E[(k - F)1(F \le k)] \\ &= kP(F\le k) - E[F 1(F\le k)] \\ &= kP(F\le k) - E[F] E[F/E[F] 1(F\le k)] \\ &= kP(F\le k) - fP^*(F\le k) \\ \end{aligned} where dP^*/dP = F/E[F]. Since F > 0 and E[F/E[F]] = 1, P^* is a positive measure with mass 1.
Exercise. If X is standard normal show {E[\max\{k - F, 0\}] = kN(x) - fN(x - s)} where N is the standard normal cumulative distribution and {x = (\log(k/f) + s^2/2)/s}.
Hint: Use E[\exp(N)] = \exp(E[N] + \operatorname{Var}(N)/2) if N is normal and E[\exp(N) g(M)] = E[\exp(N)] E[g(M + \operatorname{Cov}(M,N))] if N and M are jointly normal.
Exercise. Show if F > 0 that E[{\max\{k - F, 0\}] = kP(X\le x) - fP^*(X\le x)} where {x = (\log(k/f) + \kappa(s))/s} and dP^*/dP = \exp(sX - \kappa(s)).
Delta is the derivative of value v = E[\max\{k - F, 0\}] with respect to initial price f. \begin{aligned} \partial v/\partial f &= E[-1(F \le k)\partial F/\partial f] \\ &= -E[1(F \le k)\exp(sX - \kappa(s))] \\ &= -P^*(F \le k) \\ &= -P^*(X \le x) \\ \end{aligned}
Gamma is the second derivative of value v = E[\max\{k - F, 0\}] with respect to initial price f. \begin{aligned} \partial_f^2 v &= -\partial_f P^*(F \le k) \\ &= -\partial_f P^*(X \le x) \\ &= -\partial_x P^*(X \le x)\partial_f x \\ &= -\partial_x P^*(X \le x)/\partial_x f \\ &= \partial_x P^*(X \le x)/f s\\ \end{aligned}
Vega is the derivative of option value with respect to volatility. The vega of a put option is \begin{aligned} \partial_s v &= \partial_s E[\max\{k - F, 0\}] \\ &= E[-1(F \le k)\partial_s F] \\ &= -E[1(F \le k)F(X - κ'(s))] \\ &= -\partial_s P_s(X \le x)f \\ \end{aligned}
Exercise. Show \partial_s P_s(X\le x) = E[1(X \le x)F(X - κ'(s))]/f.
Exercise. If X is standard normal show P^*(F \le k) = N(-d_2) where {d_2 = (\log f/k - s^2/2)/s}.
Fixed income instruments pay cash flows c_j at times t_j. From equation (1), the value of a fixed income instrument at time t is determined by V_t D_t = (\sum_{t_j > t} c_j D_{t_j})|_{{\mathcal{A}}_t}.
A repurchase agreement R(f,t,\Delta t) has price X_t^{R(f,t,\Delta t)} = 1 at time t and a single cash flow C_{t+\Delta t} = \exp(f\Delta t) at time t + \Delta t. In practice, the forward rate r is quoted using a day count basis and the cash flow is 1 + r\delta where \delta is the day count fraction. Typically, the day count fraction is the number of days from t to t + \Delta t divided by 360 or 365. These are the Actual/360 and Actual/365 day count basis respectively.
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(\Delta t_j))} at time t_{j+1} where \Delta t_j = t_{j+1} - t_j. 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} = \exp(-f_j\Delta t_j)D_j} and {D_n = \exp(-\sum_{j < n} f_j\Delta t_j) D_{t_0}} is the canonical deflator at time t_n.
The continuous time analog is D_t = \exp(-\int_0^t f(s)\,ds)D_0 where f is the continuously compounded instantaneous forward rate.
A zero coupon bond maturing at time u has a single unit cash flow at u, C_u^{D(u)} = 1 and C_t^{D(u)} = 0 for t\not=u. In an arbitrage free model its price at time t\le u, X_t^{D(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. 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 \delta3 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)}\left(\frac{D_t(u)}{D_t(v)} - 1\right).
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.
Some swaps involve an exchange of notional amounts. If a swap has cash flows -1 at t_0, f\delta(t_{i-1},t_i) at t_i for 1 < i < n, and 1 + f\delta(t_{n-1},t_n) at t_n then the value of f making the price zero at time t is equal to the swap par coupon.
Exercise. Show f = F_t^\delta(t_0,\dots,t_n) makes the price zero.
Exposure…
Companies can default and may pay only a fraction of the notional owed on bonds they issued. A simple model4 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}. Note if T = \infty or R = 1 then {D_t^{R,T}(u) = D_t(u)}.
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} 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))}. In the above case \lambda_t^{T,R}(u) = -\frac{\log(P(T > u) + R P(t < T \le u))}{u - t}.
A limit order is a contract to buy or sell an instrument at a given level. We assume a money market instrument with price R_t exists and the price of the underlying instrument is S_t. A limit buy order has price 0 and and each contract has a single cash flow of C_\tau = (-L, 1) in the money market and instrument where \tau is the first time the price of the instrument hits level L. This effectively purchases one share of stock for price L at time \tau.
This note provides a replacement for the Black, Scholes (Black and Scholes 1973) and Merton (Merton 1973) theory of option valuation. Scholes and Merton won a Nobel Prize in Economics for showing how to value a derivative instrument using dynamic hedging: the value is the cost of setting up the initial hedge. The Achilles heel of their model is their assumption of continuous time trading. This leads to untenable results5, something Zeno pointed out 2,500 years ago.
Continuous time trading is a mathematical artifact of the theory of Ito processes and duped many academics into believing complete markets and perfect hedges exist. Every trader and risk manager knows this does not correspond to reality after a few days on the job. Trades occur at discrete times based on available information. The primary unsolved problem in mathematical finance is how to advise traders when and how much to hedge. There is still work to be done on how well hedges work once you accept the fact markets are not complete.
One benefit of working as a quant during the heyday of derivatives was having a front row seat to how the software implementations of the theory performed. When the hedges provided by your implementation of a model started losing money, or even gaining money, you would get a visit from someone paying your salary expecting an explanation. I quickly learned the importance of gamma profiles. We can do better than the trader aphorism, “Hedge when you can, not when you have to.”
When a theory in physics does not agree with observation, it is time to come up with a new theory. The tenor of the time at the end of the 19th century was that Newton’s theory of gravitation and Maxwell’s theory of electromagnetism had been successfully carried out. Lord Kelvin proclaimed “It is just a matter of adding decimal points to physical constants.”
The Achilles’s heel of classical theory at the time was its prediction of black body radiation. Experiment did not agree with observation.
Für die Richtung des dabeieinzuschlagenden Gedankenganges giebt der Hinblick auf dieUnhaltbarkeit der früher gemachten Voraussetzung einen Finger-zeig. – Max Planck
The untenability of the assumption made earlier provides an indication of the direction of the line of thought to be taken.
Max Planck came up with a simple solution to fit the data: assume photons were emitted in integer multiples of of the Planck constant. Instrument prices are integer multiples of a minimum trading unit, not real numbers. This fact is relevant for accurate valuation of 0-day options.
The untenable assumption of the Black-Scholes/Merton theory is that continuous time trading is possible. The indicated direction is to provide a theory reflecting the fact traders decide at discrete times based on available information what to trade.
Let \Omega be a set of possible outcomes. Partial information is modeled by a partition of \Omega. A collection of subsets of \Omega, \{A_j\}, is a partition if they are pairwise disjoint and their union is \Omega. Full information is knowing \omega\in\Omega. Partial information is knowing only to which atom \omega belongs. No information is modeled by the singleton partition \{\Omega\}.
An algebras of sets is a collection of sets closed under union and complement.
Exercise. Show algebras of sets are closed under intersection.
Hint: Let A' = \Omega\setminus A = \{\omega\in\Omega\mid \omega\not\in A\} be the set complement of A in \Omega. Use De Morgan’s Law (A\cap B)' = A'\cup B'.
We can identify a set A with its indicator function 1_A\colon\Omega\to\boldsymbol{R} defined by 1_A(\omega) = 1 if \omega\in A and 1_A(\omega) = 0 if \omega\not\in A. Note 1_{A\cup B} = 1_A + 1_B - 1_{A\cap B}, 1_{A'} = 1 - 1_A, and 1_{A\cap B} = 1_A 1_B,
Exercise. Prove De Morgan’s Law.
Hint: Start from 1_{A'\cup B'} = 1_{A'} + 1_{B'} - 1_{A'\cap B'}.
Calculations on algebras of sets is algebra.
If {\mathcal{A}} is a finite algebra of sets on \Omega then [\omega] = \cap\{A\in{\mathcal{A}}\mid\omega\in 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.
This shows we can identify a finite algebra of sets with its atoms.
A function X\colon\Omega\to\boldsymbol{R} is {\mathcal{A}}-measurable for an algebra of sets {\mathcal{A}} if \{\omega\in\Omega\mid X(\omega) \le x\} belongs to {\mathcal{A}} for all x\in\boldsymbol{R}.
Exercise. If {\mathcal{A}} is finite, show X is {\mathcal{A}}-measurable if and if and only if it is constant on atoms of {\mathcal{A}}.
This shows X\colon[{\mathcal{A}}]\to\boldsymbol{R} is a function. We jettison the word ‘measurable’ and say X is known given {\mathcal{A}}.
Integration is a linear functional: it assigns a function to a number where a constant times a function is assigned to the constant times the integral and the integral of the sum of two functions is the sum of the integrals. Integration involves measures. A finitely-additive measure is a set function \lambda\colon{\mathcal{A}}\to\boldsymbol{R} satisfying \lambda(A\cup B) = \lambda(A) + \lambda(B) - \lambda(A\cap B) and \lambda(\emptyset) = 0. Measures don’t count things twice and the measure of the empty set is 0.
If S is a set and f\colon S\to\boldsymbol{R} is a function on S define its norm \|f\| = \sup_{s\in S} |f(s)|. Let B(S) = \{f\colon S\to\boldsymbol{R}\mid \|f\| < \infty\} be the normed linear space of bounded functions on S. The dual of B(S), B(S)^*, is the set of all bounded linear functionals L\colon B(S)\to\boldsymbol{R}. A linear functional is bounded if there exists a constant M\in\boldsymbol{R} with |Lf| \le M\|f\| for all f\in B(S). The least such constant is the norm of L, \|L\|.
Every bounded linear functional gives rise to a finitely-additive measure \lambda on S by \lambda(A) = L1_A. Let ba(S) denote all finitely-additive measures on S. We now show how to identify B(S)^* with ba(S).
Exercise: Show \lambda is a measure.
Every finitely-additive measure gives rise to a linear functional. We say f is simple if it is a finite linear combination of indicator functions f = \sum_j a_j 1_{A_j}. Given a measure \lambda define Lf = \sum_j a_j \lambda(A_j).
Exercise. If \{A_j\} are pairwise disjoint show Lf = 0 implies f = 0.
Exercise. Show for any collection \{B_i\} we have \sum_i b_i 1_{B_i} = \sum_j a_j 1_{A_j} where \{A_j\} are pairwise disjoint.
Hint: Use 1_{E\cup F} = 1_{E\setminus F} + 1_{F\setminus E} - 1_{E\cap F} and induction.
This shows L is well-defined for simple functions.
Exercise. Given any bounded function g and \epsilon > 0 there exists an simple function f with \|g - f\| < \epsilon.
Hint: Let a_n = f(n\epsilon) and A_n = f^{-1}([n\epsilon, (n + 1)\epsilon)).
This shows the set of simple functions is dense in B(S). We can extend the definition from simple functions to all of B(S) since L is bounded
Exercise: If f\in B(S) and \lim_n f_n = f then \lim_n Lf_n = Lf.
Hint Use L is bounded.
This defines the integral Lf = \int_S f\,d\lambda.
We can define a norm on ba(S) by \|\lambda\| = \sup_{\{A_j\}} |\lambda(A_j)| where the supremum is over all pairwise disjoint subsets of S.
Exercise. Show \|\lambda\| = \|L\|.
If S is finite then B(S) can be identified with \boldsymbol{R}^S = \{f\colon S\to\boldsymbol{R}\} where s\mapsto f(s). Similarly, ba(S) can be identified with \boldsymbol{R}^S = \{\lambda\colon S\to\boldsymbol{R}\} where \{s\}\mapsto \lambda(\{s\}). This is good news when it comes to computer implementation, everything is just a finite vector of numbers.
The Simple Unified Model does not involve probability, however as an aid to those schooled in the classical theory we will reconnoiter some elementary facts.
A probability measure is a positive measure with mass 1. If P is a probability measure on \Omega then any function X\colon\Omega\to\boldsymbol{R} is a random variable. The expected value of X is E[X] = \int_\Omega X\,dP.
The conditional probability of B given A is P(B\mid A) = P(B\cap A)/P(A) for B,A\subseteq\Omega.
Exercise. Show B\mapsto P(B\mid A) is a probability measure on A.
This can be generalized to the conditional expectation of a random variable given an algebra of sets. We say Y = E[X\mid{\mathcal{A}}] if Y is known given {\mathcal{A}} and \int_A Y\,dP = \int_A X\,dP for all A\in{\mathcal{A}}.
If X is a random variable and P is a measure we can define the measure XP by (XP)(A) = \int_A X\,dP.
Exercise. Show XP is a measure.
Exercise Show if X\ge 0 and E[X] = 1 then XP is a probability measure.
Hint: Show XP is positive and (XP)(\Omega) = 1.
Exercise. Show Y = E[X|{\mathcal{A}}] if and only if Y(P|_{\mathcal{A}}) = (XP)|_{\mathcal{A}}.
Hint: If P is a measure on \Omega then P|_{\mathcal{A}} is the restriction of the measure to {\mathcal{A}}.
Exercise. Show if A is an atom of {\mathcal{A}} then E[X\mid{\mathcal{A}}](A) = \int_A X\,dP/P(A).
Conditional expectation is the average over each atom.
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} the algebras of sets on \Omega indicating the information available at each trading time.
Merton provided a closed form solution for valuing barrier options based on the reflection principal of Brownian motion in Section 9 of (Merton 1973). The classical theory implies the value of a barrier option that knocks in or out the second time the barrier is hit has the same value. It also implies the value of a barrier option that knocks in or out the millionth time it hits the barrier has exactly the same value.↩︎
Perfect hedges never exist.↩︎
The day count fraction \delta(u, v) is approximately v - u in years.↩︎
This is a very simplified model.↩︎
Merton provided a closed form solution for valuing barrier options based on the reflection principal of Brownian motion in Section 9 of (Merton 1973). The classical theory implies the value of a barrier option that knocks in or out the second time the barrier is hit has the same value. It also implies the value of a barrier option that knocks in or out the millionth time it hits the barrier has exactly the same value.↩︎