title: One-Period Model author: Keith A. Lewis institution: KALX, LLC email: kal@kalx.net classoption: fleqn abstract: Simplest formal model of a financial market. …
c The One‑Period Model is the simplest framework for rigorously representing a financial market over a single time horizon. The model defines the initial prices of tradable instruments and their terminal cash flows contingent on the realized outcome. In the absence of arbitrage prices are subject to geometric constraints determined by the cash flows.
We make the usual unrealistic assumptions that prices are real numbers instead of integral multiples of each instrument’s minimum trading increment/tick size and there is no bid-ask spread depending on the amount being bought or sold, much less any consideration of credit or tax issues. We also ignore the fact instruments can only be purchased in integral multiples of their minimum share/lot size. The Appendix proposes a model that can incorporate more realistic assumptions.
Quants turn mathematical models into software used for trading. If a model is deployed without ensuring it is arbitrage-free then buy-side clients will exploit mispricings by going long on trades that are undervalued and short on overpriced ones. Eventually trading reality catches up and the sell-side firm loses money. Even worse, a “clever” trader might find an internal arbitrage that gives the illusion of making profits1.
The Fundamental Theorem of Asset Pricing characterizes arbitrage-free models and provide an arbitrage if they are not. As (Ross 1978) showed, this is a purely geometric result having nothing to do with probability. Positive measures having mass one make an appearance, but they are not the probability of anything.
Ross made the untenable assumption that continuous time trading is possible and the category error of defining a cash flow as a jump in price. Stock prices jump between market close and market open but there is no associated cash flow. A cash flow is a payment made by the instrument issuer to each instrument holder. Stocks pay dividends, bonds pay coupons, futures pay the change in end-of-day quotes (and always have price zero). The Fundamental Theorem of Asset Pricing shows models of cash flows entail geometric constraints on arbitrage-free prices.
The One-Period Model specifies a finite set of tradable instruments I. The set of possible outcomes \Omega represents what can happen over the period. The initial prices are given by a vector {x\in\boldsymbol{R}^I}2, indexed by the instruments. Terminal cash flows are defined by a vector-valued function {C\colon\Omega\to \boldsymbol{R}^I} where {C(\omega)\in\boldsymbol{R}^I} are the cash flows for each instrument corresponding to outcome {\omega\in\Omega}.
A position \xi\in\boldsymbol{R}^I is the number of shares purchased in each instrument at the beginning of the period. The cost of acquiring the position is {\xi\cdot x = \sum_{i\in I}\xi(i) x(i)} and results in {\xi\cdot C(\omega)} at the end of the period. The realized return of a position \xi\in\boldsymbol{R}^I is {R_\xi = \xi\cdot C/\xi\cdot x} provided \xi\cdot x\not=0.
Exercise. Show R_{t\xi} = R_{\xi} for any non-zero t\in\boldsymbol{R}.
This is actually a deleterious feature of the model. Going long (t > 0) or short (t < 0) typically affects the realized return. It also implies a portfolio strategy can be scaled to arbitrarily large positions. At some point you will run out of instruments to buy and sell.
We assume redundant instruments are removed from the model. If {\xi\cdot C = 0} then one instrument is a linear combination of the others and can be removed. This can be repeated until {\xi\cdot C\not=0} for any \xi\in\boldsymbol{R}^I. If so, the map \xi\in\boldsymbol{R}^I to \xi\cdot C\in\boldsymbol{R}^\Omega is one-to-one. A model is complete if this map is onto. This cannot be the case if the cardinality of I is less than the cardinality of \Omega. In general the number of instruments is much smaller than the number of possible outcomes.
Classical literature often specifies terminal prices as a function X\colon\Omega\to\boldsymbol{R}^I rather than cash flows C. From a rigorous standpoint, prices do not actually exist at the end of the period since there is no further economic activity available. The classical approach implicitly assumes the initial position is liquidated at the end of the period at prevailing prices yielding a payment of \xi\cdot X. In practice, cash flows are paid proportional to position whether or not any trading occurs.
For example, a zero coupon bond has an initial price/discount and pays a unit cash flow at termination. If you are uncomfortable using cash flows C instead of prices X when the instrument is a stock, C may be interpreted as the firm’s liquidation value or the proceeds from a stock buy back by the company at the end of the period.
The Multi-period model clarifies the relationship between prices and cash flows.
The Capital Asset Pricing Model is a one-period model where a probability measure on possible outcomes is specified.
Arbitrage exists in a one-period model if there is a position \xi\in\boldsymbol{R}^I with {\xi\cdot x < 0} and {\xi\cdot C(\omega)\ge0} for all {\omega\in\Omega}: you make money acquiring the initial position and never lose money at the end of the period.
Some authors define arbitrage as a portfolio satisfying {\xi\cdot x = 0} and {\xi\cdot C\ge0} is strictly positive on some set having positive probability. We haven’t specified a probability measure so we can’t use this definition. Moreover, no trader would consider this to be an arbitrage. Even though the position costs nothing other than agita to put on, the above definition has nothing definite to say about how much they will make nor how likely it is they will make it.
Our stronger probability-free definition is still not good enough for traders and risk managers. Even though {\xi\cdot x} is strictly negative they will slap absolute value signs around every number and compute {|\xi|\cdot|x|} as a proxy for how much capital will be tied up putting on the position. No business would approve using a million dollars from their funding account just to make a penny up front even though that technically satisfies our mathematical definition of arbitrage.
The assumption of no arbitrage places constraints on initial prices that are determined by cash flows. The constraints involve a cone.
Recall a cone K is a subset of a vector space closed under positive scalar multiplication and vector addition: if x\in K then tx\in K for t > 0 and if x,y\in K then {x + y\in K}.
Exercise. A cone is convex.
Hint: Show x,y\in K implies {tx + (1-t)y\in K} for {0 < t < 1}.
Exercise. The set of arbitrage positions is a cone.
The smallest cone containing the possible cash flows C is the set of finite linear combinations with positive coefficients {\{\sum_i C(\omega_i) d_i\mid \omega_i\in\Omega, d_i > 0\}}. If x = \sum_i C(\omega_i) d_i is in the cone and \xi\cdot C is non-negative on \Omega then {\xi\cdot x\ge 0} so no arbitrage exists.
Exercise. If x belongs to the smallest closed cone containing the range of C then there is no arbitrage.
The contrapositive is also true.
Theorem. Arbitrage exists in a one-period model if x does not belong to the smallest closed cone containing the range of C. If x^* is the closest point in the cone then \xi = x^* - x is an arbitrage.
In general the arbitrage is not unique. We will establish the theorem using the purely geometric
Lemma. If x\in\boldsymbol{R}^n and K is a closed cone in \boldsymbol{R}^n with x\not\in K then there exists {\xi\in\boldsymbol{R}^n} with {\xi\cdot x < 0} and {\xi\cdot y \ge0} for {y\in K}.
Proof. Let x^* be the point in K closest to x. It exists since K is closed and is unique since K is convex. Let \xi = x^* - x and note \xi\not=0.
We have ty + x^*\in K for any t > 0 and y\in K so \|x^* - x\| \le \|ty + x^* - x\|. Simplifying gives {t^2||y||^2 + 2ty\cdot\xi\ge0}. Dividing by t > 0 and letting t decrease to 0 shows {\xi\cdot y\ge0} for all y\in K.
We have (t + 1)x^*\in K for t + 1 > 0 so \|x^* - x\| \le \|tx^* + x^* - x\|. Simplifying gives {t^2||x^*||^2 + 2tx^*\cdot\xi^*\ge 0} for t > -1. Dividing by t < 0 and letting t increase to 0 shows {\xi\cdot x^*\le 0}.
Since {0 < ||\xi||^2 = \xi\cdot (x^* - x) \le -\xi\cdot x} we have {\xi\cdot x < 0}.
The lemma proves the FTAP and that \xi = x^* - x implements an arbitrage.
A risk-neutral pricing measure is any positive, finitely additive measure D on \Omega with x = \int_\Omega C\,dD. The FTAP shows no arbitrage implies this set is not empty. Every such measure corresponds to a positive linear functional on the vector space of bounded functions on \Omega. See (Dunford and Schwartz 1958). Risk-neutral pricing measures are not generally unique.
If D is a risk-neutral pricing measure then Q = D/D(\Omega) is a positive measure having mass 1 so it satisfies the definition of a probability measure. Every portfolio has the same expected realized return under a risk-neutral measure so perhaps this should be called a risk-blind measure.
Exercise. If D is a risk-neutral measure then the expected realized return R = E^Q[R_\xi] = 1/D(\Omega) is constant for any portfolio \xi\in\boldsymbol{R}^I with \xi\cdot x\not=0.
Hint: The expectation is with respect to the “probability” measure Q = D/D(\Omega).
This exercise is a wake-up call to the fact risk-neutral measures are useless for risk management. The variance of a realized return can be arbitrarily large but a risk-neutral measure cannot detect excess returns to compensate for this risk.
A zero coupon bond pays 1 unit at the end of the period on every outcome. A portfolio \zeta\in\boldsymbol{R}^I with {\zeta\cdot C(\omega) = 1} for all {\omega\in\Omega} is a zero coupon bond. The discount of a zero coupon bond is its price {\zeta\cdot x = \int_\Omega \zeta\cdot C(\omega)\,dD(\omega) = D(\Omega)}.
Exercise. If \zeta is a zero coupon bond with only one non-zero component then that component is equal to the discount.
Exercise. If x = \int_\Omega C\,dD show x = E^Q[C]D(\Omega).
This formula can be read “Prices are expected discounted cash flows.” It is a mathematically rigorous one-period example of the method used by (Graham and Dodd 1934) in Security Analysis for valuing equities. Except Graham and Dodd only considered real-world probabilities, long before risk-neutal probabilities were invented.
We now apply the FTAP to particular models.
A common misconception is that the price of a zero coupon bond must not be greater than its notional since this would imply negative interest rates. This actually occurred in Europe between 2014 and 2020 but did not give rise to arbitrage opportunities. As shown above, the arbitrage-free constraint is only that the price is positive.
A very simple and unrealistic one-period model consists of a bond with price 1 at the beginning of the period that has a cash flow 2 at the end and a stock with price 1 that has a cash flow of either 1 or 3. The model is x = (1, 1) and {C(\omega) = (2, \omega)} where {\omega\in\{1,3\} = \Omega}. This is arbitrage-free if and only if we can find {d_1,d_3 \ge0} with {x = C(1)d_1 + C(3)d_3}. The bond component implies {1 = 2d_1 + 2d_3} and the stock component implies {1 = 1d_1 + 3d_3} so {d_1 = d_3 = 1/4}.
This shows the risk-neutral measure is D(\{1\}) = D(\{3\}) = 1/4. The risk-neutral probability measure Q = D/D(\Omega) is Q(\{1\}) = Q(\{3\}) = 1/2.
Exercise. Show E^Q[C]D(\Omega) = \int_\Omega C\,dD = (1, 1) is the initial bond and stock price.
If we add a call option with strike 2 and price v then the model becomes {x = (1, 1, v)}, {C(\omega) = (2, \omega, \max\{\omega - 2,0\})} where v is the option value. Since the bond and stock components determine {d_1 = d_3 = 1/4} the option component is {v = \max\{1 - 2, 0\}(1/4) + \max\{3 - 2, 0\}(1/4) = 1/4}.
A similar argument shows any European option paying \nu(\omega) at expiration has value {v = (\nu(1) + \nu(3))/4}. Every option payoff is linear in a binomial model.
Note this argument does not depend on probability. If the real-world probability of the stock staying at 1 is 0.1 and the stock tripling to 3 is 0.9 then the expected payoff is (0(0.1) + 1(0.9))/2 = 0.45. As John Illuzi at Banc of America securities pointed out when I showed him this, “Do you mean I can buy the option for 0.25 and get 0.45 on average? I’d take that trade all day long!” He also identified the risk-blind nature of risk-neutral probability. “But not if I got shot in the head if the option ever finished out-of-the-money.”
We have already seen every risk-neutral measure has the same expected realized return. This example shows even if the measure is unique it implies infinite risk aversion.
The 1-2-3 model is a special case of the binomial model having instruments a bond and stock where {x = (r, s)}, {C(\omega) = (R, \omega)}, and {\Omega = \{L, H\}} with {L < H}. The bond has realized return R/r and the stock can go from price s to either L or H. This is arbitrage-free if and only we can find d_L,d_H \ge 0 with x = C(L)d_L + C(H)d_H. Considering the bond and stock components we have d_L = (Hr/R - s)/(H - L) and d_H = (s - Lr/R)/(H - L). The model is arbitrage-free if and only if these are both non-negative so L/R \le s/r \le H/R.
Adding an option with payoff \nu(\omega) we have its arbitrage-free value is v = \nu(L)d_L + \nu(H)d_H = \frac{(H\nu(L) - L\nu(H))r/R + (\nu(H) - \nu(L))s}{H - L}.
This can be simplified for call options.
Exercise. If \nu(\omega) = \max\{\omega - K, 0\} where L \le K \le H then v = (H - K)(Hr/R - s)/(H - L).
Exercise. Show this agrees with the 1-2-3 Model for a call option with strike 2.
Even though the option price is completely determined by the bond and stock in the binomial model we can use this to get bounds on the option price. Since v = (H - K)d_H we always have v \ge 0 and v\le ???.
A somewhat more realistic model is the binomial model with sample space {\Omega = [L, H]} where the final stock price can be any value in the interval. First we show the no arbitrage constraint is still L/R\le s/r\le H/R.
Exercise. The smallest cone containing C(L) and C(H) is the same as the smallest cone containing C(\omega), L\le\omega\le H.
Hint: Show if L\le\omega\le H then C(\omega) = (1 - t)C(L) + tC(H) for some t\in[0,1].
Since cones are convex, this shows the smallest cone containing C(L) and C(H) also contains C(\omega) for \omega\in[L,H].
We can make the one-period model more realistic. The price of an instrument must be an integral multiple of its minimal trading increment, or tick size. Let \epsilon(i)\in\boldsymbol{R} be the tick size of instrument i\in I. Initial prices x\in\boldsymbol{Z}^I correspond to actual prices x(i)\epsilon(i). Likewise for final prices {X\colon\Omega\to\boldsymbol{Z}^I}.
The amount must be an integral multiple of its minimum share size, or lot size. Let \delta(i) be the lot size of instrument i\in I. Shares \xi\in\boldsymbol{Z}^I correspond to amounts \xi(i)\delta(i).
The atoms of finance are holdings: an instrument, amount, and owner. Let {\eta = (a,i,o)} denote owner o holds amount a of instrument i. A position is a collection of holdings. A transaction at time t is a triple {(t,\eta,\eta')} where \eta and \eta' are holdings. If the buyer o holds \eta and the seller o' holds \eta' at time t then after the transaction settles the buyer holds (a',i',o) and the seller holds (a,i,o'). The price of the transaction is X = a/a'.
Sellers quote prices for potential transactions. Buyers decide whether or not to execute the transaction. For example, if a seller quotes Ford stock for 12 USD per share then the buyer can exchange \$24 for 2 shares of Ford stock. This corresponds to the transaction where {\eta = (24, \$, o)} and {\eta' = (2, F, o')}. The price is {X = a/a' = 24/2} and we can use 12 USD/F to indicate a price of \$12 per share of Ford stock. Replacing the virgule ‘/’ by ‘= 1’ gives the mnemonic 12 USD = 1 F. After the transaction settles the buyer holds {\eta = (2, F, o)} and the seller holds {\eta' = (24, \$, o')} in their respective positions.
After a transaction the price is a well-defined number: the amount the buyer exchanged with the seller divided by the amount the seller exchanged with the buyer. This number is recorded in the books and records of each counterparty.
Prior to a transaction the price quoted by the seller is not a well-defined number. The difference between a quoted price and the actual price is called slippage. Sellers quote a bid price and an ask price. When a buyer buys one share they pay the seller’s ask price and when the buyer sells one share they only get the sellers bid price. The difference between the ask and the bid is the bid-ask spread and is usually positive. This is how sellers get paid for providing liquidity to buyers in the market.
The quotes are only valid for transaction of a small number of shares. As the number of shares becomes larger sellers increase the bid-ask spread. On an exchange the quote only holds out to level 1 depth before switching to level 2. Sellers might not know why so many shares are being purchased, but know demand will move the price up and they will increase their ask. At some point the number of shares to purchase will bump into the total number of shares issued and price becomes meaningless. When a buyer sells shares they face even more restrictions and are also charged a borrow cost over the period of time they are short. Prices can also depend on the particular buyer and seller of the transaction due to credit issues or regulations, among other things.
A more realistic model for prices in a one-period model is to replace x\colon I\to\boldsymbol{R} by x\colon I\times A\to\boldsymbol{R} where A is the set of possible amounts. The cost of setting up the position \xi\in\boldsymbol{R}^I is {\sum_{i\in I} \xi(i) x(i, \xi(i))}. For a fixed transaction cost \tau\in\boldsymbol{R} per share we have {x(i, a) = x_0(i) + \tau a} where x_0 is the mid price. If a limit order book is available then a very good approximation to the execution price is the function of cumulative limit order amounts versus order levels.
title: One-Period Model author: Keith A. Lewis institution: KALX, LLC email: kal@kalx.net classoption: fleqn abstract: Simplest formal model of a financial market. …
The One‑Period Model is the simplest framework for rigorously representing a financial market over a single time horizon. The model defines the initial prices of tradable instruments and their terminal cash flows contingent on the realized outcome. In the absence of arbitrage prices are subject to geometric constraints determined by the cash flows.
We make the usual unrealistic assumptions that prices are real numbers instead of integral multiples of each instrument’s minimum trading increment/tick size and there is no bid-ask spread depending on the amount being bought or sold, much less any consideration of credit or tax issues. We also ignore the fact instruments can only be purchased in integral multiples of their minimum share/lot size. The Appendix proposes a model that can incorporate more realistic assumptions.
Quants turn mathematical models into software used for trading. If a model is deployed without ensuring it is arbitrage-free then buy-side clients will exploit mispricings by going long on trades that are undervalued and short on overpriced ones. Eventually trading reality catches up and the sell-side firm loses money. Even worse, a “clever” trader might find an internal arbitrage that gives the illusion of making profits3.
The Fundamental Theorem of Asset Pricing characterizes arbitrage-free models and provide an arbitrage if they are not. As (Ross 1978) showed, this is a purely geometric result having nothing to do with probability. Positive measures having mass one make an appearance, but they are not the probability of anything.
Ross made the untenable assumption that continuous time trading is possible and the category error of defining a cash flow as a jump in price. Stock prices jump between market close and market open but there is no associated cash flow. A cash flow is a payment made by the instrument issuer to each instrument holder. Stocks pay dividends, bonds pay coupons, futures pay the change in end-of-day quotes (and always have price zero). The Fundamental Theorem of Asset Pricing shows models of cash flows entail geometric constraints on arbitrage-free prices.
The One-Period Model specifies a finite set of tradable instruments I. The set of possible outcomes \Omega represents what can happen over the period. The initial prices are given by a vector {x\in\boldsymbol{R}^I}4, indexed by the instruments. Terminal cash flows are defined by a vector-valued function {C\colon\Omega\to \boldsymbol{R}^I} where {C(\omega)\in\boldsymbol{R}^I} are the cash flows for each instrument corresponding to outcome {\omega\in\Omega}.
A position \xi\in\boldsymbol{R}^I is the number of shares purchased in each instrument at the beginning of the period. The cost of acquiring the position is {\xi\cdot x = \sum_{i\in I}\xi(i) x(i)} and results in {\xi\cdot C(\omega)} at the end of the period. The realized return of a position \xi\in\boldsymbol{R}^I is {R_\xi = \xi\cdot C/\xi\cdot x} provided \xi\cdot x\not=0.
Exercise. Show R_{t\xi} = R_{\xi} for any non-zero t\in\boldsymbol{R}.
This is actually a deleterious feature of the model. Going long (t > 0) or short (t < 0) typically affects the realized return. It also implies a portfolio strategy can be scaled to arbitrarily large positions. At some point you will run out of instruments to buy and sell.
We assume redundant instruments are removed from the model. If {\xi\cdot C = 0} then one instrument is a linear combination of the others and can be removed. This can be repeated until {\xi\cdot C\not=0} for any \xi\in\boldsymbol{R}^I. If so, the map \xi\in\boldsymbol{R}^I to \xi\cdot C\in\boldsymbol{R}^\Omega is one-to-one. A model is complete if this map is onto. This cannot be the case if the cardinality of I is less than the cardinality of \Omega. In general the number of instruments is much smaller than the number of possible outcomes.
Classical literature often specifies terminal prices as a function X\colon\Omega\to\boldsymbol{R}^I rather than cash flows C. From a rigorous standpoint, prices do not actually exist at the end of the period since there is no further economic activity available. The classical approach implicitly assumes the initial position is liquidated at the end of the period at prevailing prices yielding a payment of \xi\cdot X. In practice, cash flows are paid proportional to position whether or not any trading occurs.
For example, a zero coupon bond has an initial price/discount and pays a unit cash flow at termination. If you are uncomfortable using cash flows C instead of prices X when the instrument is a stock, C may be interpreted as the firm’s liquidation value or the proceeds from a stock buy back by the company at the end of the period.
The Multi-period model clarifies the relationship between prices and cash flows.
The Capital Asset Pricing Model is a one-period model where a probability measure on possible outcomes is specified.
Arbitrage exists in a one-period model if there is a position \xi\in\boldsymbol{R}^I with {\xi\cdot x < 0} and {\xi\cdot C(\omega)\ge0} for all {\omega\in\Omega}: you make money acquiring the initial position and never lose money at the end of the period.
Some authors define arbitrage as a portfolio satisfying {\xi\cdot x = 0} and {\xi\cdot C\ge0} is strictly positive on some set having positive probability. We haven’t specified a probability measure so we can’t use this definition. Moreover, no trader would consider this to be an arbitrage anyway. Even though the position costs nothing other than agita to put on, the above definition has nothing definite to say about how much they will make nor how likely it is they will make it.
Our stronger probability-free definition is still not good enough for traders and risk managers. Even though {\xi\cdot x} is strictly negative they will slap absolute value signs around every number and compute {|\xi|\cdot|x|} as a proxy of how much capital will be tied up putting on the position. No business would approve using a million dollars from their funding account just to make a penny up front even though that technically satisfies our mathematical definition of arbitrage.
The assumption of no arbitrage places constraints on initial prices that are determined by cash flows. The constraints involve a cone.
Recall a cone K is a subset of a vector space closed under positive scalar multiplication and vector addition: if x\in K then tx\in K for t > 0 and if x,y\in K then {x + y\in K}.
Exercise. A cone is convex.
Hint: Show x,y\in K implies {tx + (1-t)y\in K} for {0 < t < 1}.
Exercise. The set of arbitrage positions is a cone.
The smallest cone containing the possible cash flows C is the set of finite linear combinations with positive coefficients {\{\sum_i C(\omega_i) d_i\mid \omega_i\in\Omega, d_i > 0\}}. If x = \sum_i C(\omega_i) d_i is in the cone and \xi\cdot C is non-negative on \Omega then {\xi\cdot x\ge 0} so no arbitrage exists.
Exercise. If x belongs to the smallest closed cone containing the range of C then there is no arbitrage.
The contrapositive is also true.
Theorem. Arbitrage exists in a one-period model if x does not belong to the smallest closed cone containing the range of C. If x^* is the closest point in the cone then \xi = x^* - x is an arbitrage.
In general the arbitrage is not unique. We will establish the theorem using the purely geometric
Lemma. If x\in\boldsymbol{R}^n and K is a closed cone in \boldsymbol{R}^n with x\not\in K then there exists {\xi\in\boldsymbol{R}^n} with {\xi\cdot x < 0} and {\xi\cdot y \ge0} for {y\in K}.
Proof. Let x^* be the point in K closest to x. It exists since K is closed and is unique since K is convex. Let \xi = x^* - x and note \xi\not=0.
We have ty + x^*\in K for any t > 0 and y\in K so \|x^* - x\| \le \|ty + x^* - x\|. Simplifying gives {t^2||y||^2 + 2ty\cdot\xi\ge0}. Dividing by t > 0 and letting t decrease to 0 shows {\xi\cdot y\ge0} for all y\in K.
We have (t + 1)x^*\in K for t + 1 > 0 so \|x^* - x\| \le \|tx^* + x^* - x\|. Simplifying gives {t^2||x^*||^2 + 2tx^*\cdot\xi^*\ge 0} for t > -1. Dividing by t < 0 and letting t increase to 0 shows {\xi\cdot x^*\le 0}.
Since {0 < ||\xi||^2 = \xi\cdot (x^* - x) \le -\xi\cdot x} we have {\xi\cdot x < 0}.
The lemma proves the FTAP and that \xi = x^* - x implements an arbitrage.
A risk-neutral pricing measure is any positive, finitely additive measure D on \Omega with x = \int_\Omega C\,dD. The FTAP shows no arbitrage implies this set is not empty. Every such measure corresponds to a positive linear functional on the vector space of bounded functions on \Omega. See (Dunford and Schwartz 1958). Risk-neutral pricing measures are not generally unique.
If D is a risk-neutral pricing measure then Q = D/D(\Omega) is a positive measure having mass 1 so it satisfies the definition of a probability measure. Every portfolio has the same expected realized return under a risk-neutral measure so perhaps this should be called a risk-blind measure.
Exercise. If D is a risk-neutral measure then the expected realized return R = E^Q[R_\xi] = 1/D(\Omega) is constant for any portfolio \xi\in\boldsymbol{R}^I with \xi\cdot x\not=0.
Hint: The expectation is with respect to the “probability” measure Q = D/D(\Omega).
This exercise is a wake-up call to the fact risk-neutral measures are useless for risk management. The variance of a realized return can be arbitrarily large but a risk-neutral measure cannot detect excess returns to compensate for this risk.
A zero coupon bond pays 1 unit at the end of the period on every outcome. A portfolio \zeta\in\boldsymbol{R}^I with {\zeta\cdot C(\omega) = 1} for all {\omega\in\Omega} is a zero coupon bond. The discount of a zero coupon bond is its price {\zeta\cdot x = \int_\Omega \zeta\cdot C(\omega)\,dD(\omega) = D(\Omega)}.
Exercise. If \zeta is a zero coupon bond with only one non-zero component then that component is equal to the discount.
Exercise. If x = \int_\Omega C\,dD show x = E^Q[C]D(\Omega).
This formula can be read “Prices are expected discounted cash flows.” It is a mathematically rigorous one-period example of the method used by (Graham and Dodd 1934) in Security Analysis for valuing equities.
We now apply the FTAP to particular models.
A common misconception is that the price of a zero coupon bond must not be greater than its notional since this would imply negative interest rates. Negative rates actually occured in Europe between 2014 and 2020 but did not give rise to arbitrage opportunities. As we have seen above, the only constraint is the price of the zero coupon bond must be positive.
A very simple and unrealistic one-period model consists of a bond with price 1 at the beginning of the period that has a cash flow 2 at the end and a stock with price 1 that has a cash flow of either 1 or 3. The model is x = (1, 1) and {C(\omega) = (2, \omega)} where {\omega\in\{1,3\} = \Omega}. This is arbitrage-free if and only if we can find {d_1,d_3 \ge0} with {x = C(1)d_1 + C(3)d_3}. The bond component implies {1 = 2d_1 + 2d_3} and the stock component implies {1 = 1d_1 + 3d_3} so {d_1 = d_3 = 1/4}.
This shows the risk-neutral measure is D(\{1\}) = D(\{3\}) = 1/4. The risk-neutral probability measure Q = D/D(\Omega) is Q(\{1\}) = Q(\{3\}) = 1/2.
Exercise. Show E^Q[C]D(\Omega) = \int_\Omega C\,dD = (1, 1) is the initial bond and stock price.
If we add a call option with strike 2 and price v then the model becomes {x = (1, 1, v)}, {C(\omega) = (2, \omega, \max\{\omega - 2,0\})} where v is the option value. Since the bond and stock components determine {d_1 = d_3 = 1/4} the option component is {v = \max\{1 - 2, 0\}(1/4) + \max\{3 - 2, 0\}(1/4) = 1/4}.
A similar argument shows any European option paying \nu(\omega) at expiration has value {v = (\nu(1) + \nu(3))/4}. Every option payoff is linear in a binomial model.
Note this argument does not depend on probability. If the real-world probability of the stock staying at 1 is 0.1 and the stock tripling to 3 is 0.9 then the expected payoff is (0(0.1) + 1(0.9))/2 = 0.45. As John Illuzi at Banc of America securities pointed out when I showed him this, “Do you mean I can buy the option for 0.25 and get 0.45 on average? I’d take that trade all day long!” He also identified the risk-blind nature of risk-neutral probability. “But not if I got shot in the head if the option ever finished out-of-the-money.”
We have already seen every risk-neutral measure has the same expected realized return. This example shows even if the measure is unique it implies infinite risk aversion.
The 1-2-3 model is a special case of the binomial model having instruments a bond and stock where {x = (1, s)}, {C(\omega) = (R, \omega)}, and {\Omega = \{L, H\}} with {L < H}. The bond has realized return R and the stock can go from price s to either L or H. This is arbitrage-free if and only we can find {d_L,d_H \ge 0} with {x = C(L)d_L + C(H)d_H}. Considering the bond and stock components we have {d_L = (Hr/R - s)/(H - L)} and {d_H = (s - Lr/R)/(H - L)}. The model is arbitrage-free if and only if these are both non-negative so {L/R \le s \le H/R}.
Exercise. Find an arbitrage if s < L/R or s > H/R.
Hint: If s < L/R then buy the stock and short the bond. If s > H/R then sell the stock and buy the bond.
Adding an option with payoff \nu(\omega) we have its arbitrage-free value is v = \nu(L)d_L + \nu(H)d_H = \frac{(H\nu(L) - L\nu(H))R + (\nu(H) - \nu(L))s}{H - L}.
This can be simplified for call options.
Exercise. If \nu(\omega) = \max\{\omega - K, 0\} where L \le K \le H then v = (H - K)(H/R - s)/(H - L).
Exercise. Show this agrees with the 1-2-3 Model for a call option with strike 2.
Even though the option price is completely determined by the bond and stock in the binomial model we can use this to get bounds on the option price. Since v = (H - K)d_H we always have v \ge 0.
A somewhat more realistic model is the binomial model with sample space {\Omega = [L, H]} where the final stock price can be any value in the interval. The arbitrage-free conditions are still the same.
Exercise. The smallest cone containing C(L) and C(H) is the same as the smallest cone containing C(\omega), L\le\omega\le H.
Hint: Show if L\le\omega\le H then C(\omega) = (1 - t)C(L) + tC(H) for some t\in[0,1].
Since cones are convex, this shows the smallest cone containing C(L) and C(H) also contains C(\omega) for \omega\in[L,H].
An interesting exerise is to generlize this to multiple call options with strikes between the low and high stock values, {H < K_1 < \cdots < K_n < H}. The above exercise shows we only need consider the cone generated by {C(L), C(K_1), \ldots, C(K_n), C(H)}.
We can make the one-period model more realistic. The price of an instrument must be an integral multiple of its minimal trading increment, or tick size. Let \epsilon(i)\in\boldsymbol{R} be the tick size of instrument i\in I. Initial prices x\in\boldsymbol{Z}^I correspond to actual prices x(i)\epsilon(i). Likewise for final prices {X\colon\Omega\to\boldsymbol{Z}^I}.
The amount must be an integral multiple of its minimum share size, or lot size. Let \delta(i) be the lot size of instrument i\in I. Shares \xi\in\boldsymbol{Z}^I correspond to amounts \xi(i)\delta(i).
The atoms of finance are holdings: an instrument, amount, and owner. Let {\eta = (a,i,o)} denote owner o holds amount a of instrument i. A position is a collection of holdings. A transaction at time t is a triple {(t,\eta,\eta')} where \eta and \eta' are holdings. If the buyer o holds \eta and the seller o' holds \eta' at time t then after the transaction settles the buyer holds (a',i',o) and the seller holds (a,i,o'). The price of the transaction is X = a/a'.
Sellers quote prices for potential transactions. Buyers decide whether or not to execute the transaction. For example, if a seller quotes Ford stock for 12 USD per share then the buyer can exchange \$24 for 2 shares of Ford stock. This corresponds to the transaction where {\eta = (24, \$, o)} and {\eta' = (2, F, o')}. The price is {X = a/a' = 24/2} and we can use 12 USD/F to indicate a price of \$12 per share of Ford stock. Replacing the virgule ‘/’ by ‘= 1’ gives the mnemonic 12 USD = 1 F. After the transaction settles the buyer holds {\eta = (2, F, o)} and the seller holds {\eta' = (24, \$, o')} in their respective positions.
After a transaction the price is a well-defined number: the amount the buyer exchanged with the seller divided by the amount the seller exchanged with the buyer. This number is recorded in the books and records of each counterparty.
Prior to a transaction the price quoted by the seller is not a well-defined number. The difference between a quoted price and the actual price is called slippage. Sellers quote a bid price and an ask price. When a buyer buys one share they pay the seller’s ask price and when the buyer sells one share they only get the sellers bid price. The difference between the ask and the bid is the bid-ask spread and is usually positive. This is how sellers get paid for providing liquidity to buyers in the market.
The quotes are only valid for transaction of a small number of shares. As the number of shares becomes larger sellers increase the bid-ask spread. On an exchange the quote only holds out to level 1 depth before switching to level 2. Sellers might not know why so many shares are being purchased, but know demand will move the price up and they will increase their ask. At some point the number of shares to purchase will bump into the total number of shares issued and price becomes meaningless. When a buyer sells shares they face even more restrictions and are also charged a borrow cost over the period of time they are short. Prices can also depend on the particular buyer and seller of the transaction due to credit issues or regulations, among other things.
A more realistic model for prices in a one-period model is to replace x\colon I\to\boldsymbol{R} by x\colon I\times A\to\boldsymbol{R} where A is the set of possible amounts. The cost of setting up the position \xi\in\boldsymbol{R}^I is {\sum_{i\in I} \xi(i) x(i, \xi(i))}. For a fixed transaction cost \tau\in\boldsymbol{R} per share we have {x(i, a) = x_0(i) + \tau a} where x_0 is the mid price. If a limit order book is available then a very good approximation to the execution price is the function of cumulative limit order amounts versus order levels.
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