( 0) ,...,S N ,S 2 ( 0)... S N S 2. N and a portfolio is created that way, the value of the portfolio at time 0 is: (0) N S N ( 1, ) +...

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1 No-Arbitrage Pricing Theory Single-Period odel There are N securities denoted ( S,S,...,S N ), they can be stocks, bonds, or any securities, we assume they are all traded, and have prices available. Ω is the set of possible states of the world at the end of the time period (this is a one-period model) with Ω = {,..., }. The price of security j at time 0 is S j ( 0) and its price at time k at state of the world is S j ( k, ). So the time 0 prices are represented by the row vector: S( 0) = [ S ( 0) S ( 0)... S N ( 0) ], and the prices at time are given by the matrix: S (, ) S (, )... S N (, ) S S(,Ω ) = (, ) S (, )... S N (, ). Ṃ S (, ) S (, )... S N (, ) If we buy units of security j at time 0, with j Θ =, N and a portfolio is created that way, the value of the portfolio at time 0 is: S( 0)Θ = S (0) + S (0) N S N (0). At time, if state occurs, the value of the portfolio is: S k (, ) + k S (, ) k N S N (, ). k The general random value at time is: S (, ) + S (, ) N S N (, ) S(, Ω )Θ = S (, ) + S (, ) N S N (, ). S (, ) + S (, ) N S N (, ) Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski - 3 -

2 ( ) 0, i.e., investor s risk is limited to the amount We assume that S j k, originally invested. The amounts invested may be negative, when that j happens it means that security S j has been sold short. All of the amounts invested s can be written as a column vector and referred to as Θ. Such a j vector is called a trading strategy. Θ =, then the investor has purchased units of security S, sold short one unit of security S, and purchased units of security S 3, and there are only three securities traded in the market. Security S will be assumed to be the risk-free. Its value at times will be therefore S (, ) = + i regardless of. An arbitrage opportunity is a trading strategy such that S( 0) 0 and S(,Ω) > 0 The second inequality is a vector inequality, and it says that the vector on the left-hand side has all components no less than the vector on the right hand side, while at least one component is actually larger. In other words, for an arbitrage opportunity, the cost is zero or less, but the payoff at time is never negative, and positive in some states. Arbitrage opportunity means the investor holding it has an assured profit, without spending any money to earn it. The key insight of this theory is that absence of arbitrage means that prices are derived in a specific way, very much to actuaries liking. The simplest example is a market with two securities: a riskless one (bank account) and a risky one (stock). Both sell for an initial price of, and at time, the stock either goes up to u or down to d, while the bank account is always worth + i. You can prove that an arbitrage opportunity exists if, and only if, + i > u. Thus, unless the bank account s return is certain to be greater than the stock s return, the market is arbitrage-free. To go over this [ ] T be a trading strategy, and let us see when it is an proof, let Θ = arbitrage opportunity. We must have S( 0)Θ 0, or + 0. Then S(, Ω )Θ > 0, or ( + i) + u 0 and ( + i) + d 0. If 0, then since + i > > d, we have 0 and d ( + i) d d. This implies that = = 0. There is no arbitrage opportunity. Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski - 4 -

3 > 0 and ( + i ) u u. Therefore However, if < 0, then + i u. Since u > d, the third inequality is strict, and Θ is an arbitrage opportunity. Note that if we start with the assumption that + i u and choose = and = + i opportunity. A state price vector Ψ = L all positive such that S( 0) = ΨS(, Ω), i.e., ( ) = k S j 0 k = u then Θ is an arbitrage [ ] is a vector whose entries are ( ) for j =,,...,N. S j, k The above expression may not be unique if there are fewer than assets with linearly independent payoffs at time. Note that a security which pays in a given state of nature, and 0 in all other states, is called an Arrow- Debreu security. Let { e,...,e } be the standard basis for R. These are the payoffs of the Arrow-Debreu securities. Fundamental Theorem of Asset Pricing (first, simple version): The singleperiod securities market model is arbitrage free if and only if there exists a state price vector. In other words, the value of security j at time 0 can be obtained by taking the value in each possible state at time and multiplying it be the corresponding entry of the state price vector. The state price vector is similar to n E x in life contingencies = a combined discount factor that takes probability and interest into account. Proof (this is only for fun it is not going to be on the test). Suppose that a price vector exists. If for some trading strategy S(, Ω )Θ > 0 then S( 0)Θ = ΨS(, Ω)Θ > 0, because the state price vector is strictly positive. This implies that the market is arbitrage free. Now suppose that the market is arbitrage-free. We now only consider a proof in the case when there are assets with linearly independent payoffs at time. Then S(, Ω ) has full rank and N. Assume that the first assets are linearly independent. Write S( 0) = [ L( 0) R( 0) ] where L( 0) = [ S ( 0) S ( 0) L S ( 0) ] and R( 0) = [ S + ( 0) S + ( 0) L S N ( 0) ]. The same way, write S,Ω [ ] where ( ) = L( ) R( ) Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski - 5 -

4 and L( ) = R( ) = ( ) S (, ) L S (, ) ( ) S (, ) L S (, ) S, S, O S (, ) S, ( ) L S (, ) ( ) S + (, ) L S N (, ) ( ) S + (, ) L S N (, ) S +, S +, O S + (, ) S +, ( ) L S N (, ). ( )( L( ) ). For an Arrow- The matrix L() is invertible. Define Ψ = L 0 Debreu security with payoff e m consider the trading strategy Θ = ( ( L( ) e ) T 0 K 0. Then S (,Ω )Θ = L( ) ( ( L( ) )e ) m m = e m. [ ] Since this is a positive payoff, and the market does not allow arbitrage, it follows that S( 0)Θ = L( 0) (( L( ) ) )e m = Ψe m is strictly positive. This implies that the m-th component of the vector Ψ is strictly positive, and as m was arbitrary, this applies to all components. Now consider [ ] = ΨS(, Ω ) = L ( 0) L ( ) L ( ) R( ) = L( 0) L( 0)L( ) R( ) [ ]. If N =, the second part of this matrix does not exist and we found a state price vector. If N > then since the columns of L() form a basis for R. There is an ( N ) matrix K such that R( ) = L( )K. Because the model is arbitrage free, the prices of the N redundant securities must equal the cost of the unique portfolio of the linearly independent assets that produce the same payoff at time, i.e., R( 0) = L( 0)K = ΨL( )K = ΨR( ). Therefore, ΨS,Ω [ ] = [ ΨL( ) ΨR( ) ] = [ ] = S( 0). ( ) = Ψ L( ) R( ) = L( 0) R( 0) This ends the proof of this limited case. Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski - 6 -

5 Recall the simplest example: two securities market, bank account and a stock. The stock either goes up to u or down to d after one period, and the same notation as before. Then using the state price vector concept we get the system of equations: = ( + i) + ( + i) as the bank account is always worth + i, = u + d for the stock. Solving this set of equations we obtain: = ( + i ) d and ( + i) ( u d) = u + ( i) ( + i) ( u d) Notice that if + i > u then there is arbitrage, since this would mean that is negative. To make this example more specific, suppose that a stock with price.00 now can increase to.0 or decrease to 0.80 after one period. If the one-period risk-free interest rate is 0%, then S( 0) = [.00.00] and.0.0 S(, Ω ) = The state price vector is: =.( ).0.0 = ( ) 44 If there is a state price vector, then define: Q( k ) = ( + i). Then k Q( )S S j ( 0) = j (, ). The values Q( ) define the risk-neutral Ω + i probability measure on Ω. The risk-neutral probability measures are strictly positive and add up to, so this is a probability measure. Come back to the basic example with two securities (and two states, up and down for the stock): bank account and a stock. Using previously calculated state-price vector, we get Q( up) = q = + ( i ) d and u d ( ) = q = u ( + i) Q down u d. Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski - 7 -

6 The generalized version of the Fundamental Theorem of Asset Pricing: The following are equivalent: - The single-period model is arbitrage-free, - There exists a state price vector, - There exists a risk-neutral probability measure. Let X( ) be the payment made by the security X at time if state occurs, and assume that we do not have the market price for this security, but we can replicate it with securities that have market prices. To find its price, we try to create a portfolio such that, j S j ( ) j = X ( ) for all Ω. This portfolio so created is called the replicating portfolio, and if X can be replicated by some trading strategy, we say that X is attainable. The Q( )X( ) time-0 value of X, is ΨX =. So the price is the actuarial Ω + i present value of future cash flows. If there are two states, ( ) X ΨX = q + i ( ) X + q + i A market is called complete if all securities can be replicated. Theorem. A single-period, arbitrage-free model is complete if and only if there is a unique state price vector A single-period model in which security is a riskless is arbitrage-free and complete if and only if the risk-neutral probability measure is unique. Consider the following market: 0 0 S( 0) = [ ] S(, Ω) =. 4 3 What does this mean? There are two assets: a riskless one and a risky one. Each costs initially. The riskless asset grows to $ at the end of the period, regardless of the state, and the second security s value at the end of the period is 0 in states and, 4 in state 3, and 3 in state 4. Let us find the state price vectors:. Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski - 8 -

7 = = There are two equations ad four unknowns; and we end up with a parametric solution. Set = s and = t. Then: 4 Solving for 3 = + s t or s t = 3 = t or 3t = 4 3 = s t 3t 4 ( ) = 4s t 4 ( ). State price vectors must be positive, so we must impose some conditions on s and t to ensure that. Since = s and = t we have s > 0 and t > 0. 4 Also, = 3t 3 4 ( ) so t < 3. Finally, = ( 4s t) so 4s +t <. 4 The market is not complete, as you can see (the state-price vector is not unique!). Not being complete means that some securities do not have market prices. We will try to complete it. Consider the security with time payoffs C = [ 0 0 ]. Let us see what its price should be, while still maintaining the absence of arbitrage for this model. By the definition of the state-price vector, we must have: ΨC = = = 4s t 4 ( ) +( t) = 4 s t. This is a linear expression in s and t, and we can find it range of values on the area where s and t are allowed to be. This is a simple linear programming problem. The maximum value of is obtained as t approaches 3 and s approaches 0. The minimum price of 0 is obtained as t approaches 0 and s approaches 4. Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski - 9 -

8 In the second example we will add two more assets while preserving the arbitrage-free nature of the model. The existing market is: S( 0) = [ ] S( ) = In order to do this, each time an asset is added, the range for asset price must be sold to preserve the arbitrage-free nature of the model. We do it like this: - The minimum price of the new asset is calculated by first forming a portfolio from the existing assets that is always dominated by the new asset; then the new asset must sell for at least as much as the maximum price of this portfolio. - The maximum price of the new asset is calculated by first forming a portfolio from the existing assets that always dominates new asset; then the new asset must sell for no more than the minimum price of this portfolio. The first security to be added to the model is this Arrow-Debreu security e = [ 0 0 0]. Arrow-Debreu securities are generally the simplest ones to add in such a situation. To find the minimum price we must solve the following problem: aximize + subject to: The optimum in linear programming is always achieved at a corner point of the constraint region. So we need to find all corner points and values of the objective function at them. Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski

9 For the problem above: Corner point Valid? Intersect. and - No-Violates 3 Intersect. and 3 None None No Intersect. and 4 -/3 Violates Intersect. and Yes Intersect. and Yes Intersect. 3 and Yes Now determine + at all valid points, which in this case is only the point (0,0). Now, = 0, so the minimum price of the new security is zero. To find the maximum price we must solve the following problem: inimize + subject to: Intersection of Valid? and - No-Violates 4 and 3 None None No and 4 -/3 Yes and No-Violates and No-Violates 3 and No-Violates Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski - 3 -

10 So the solution is found at =, = and the maximum price is = 3. Thus the new asset may be added at any price in the interval 0, 3 You can continue to add securities one at a time, continuing in this fashion, until the number of securities equals the number of possible states at time. However, it becomes very difficult to solve for the price range manually after the third security, since there will be more than two variables in your constraints; this will generally require programming or optimizing software. Once the number of securities equals the number of possible states at time, you can solve for a unique state price vector that will serve as a state price vector for the original model as well. ultiperiod odel There are states of nature Ω =,,..., { } and T + time points k = { } and S j ( k) is the value of 0,, T. There are N assets S = S,S,...,S N security j at time k. The one-period interest rate for the riskless asset (if such an asset is available) is assumed to be i k = S ( k +). In the next section S ( k) we will discuss the binomial form of this model. Assumptions: - Investors observe all prices, and remember all the past and present. - The investors start off at time 0 with full knowledge about Ω, P, and the ( ) for all j,k and. The price S j ( 0, ) is constant values of S j k, with respect to ; for the investors at time 0, any is possible. - There is a unique sample path for each Ω, so at time T the investors know the true state of nature. - Finally, at intermediate points in time, k < T, no additional information is given to the investors that would help them learn the true state of nature. The information known by investors at a point in time can be used to partition the possible states into ones that are still possible, and ones that can Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski - 3 -

11 no longer be possible; these subsets are called histories. They can then use this information to modify/construct trading strategies. The value of portfolio under a specific trading strategy at time k, immediately before any transactions are made, is V ( k, ) = S( k, ) ( k, ), i.e., the value of each security at time k times the number of shares of each security held at the prior time period). The value of the portfolio immediately after any transactions are made is: V ( k, ) = S( k, ) ( k, ), i.e., the value of each security at time k times the number of shares of each security held at the current time period. The net cash flow out of the portfolio, c ( k, ) is: ( ) if k = 0 ( ) V ( k, ) if k T ( ) if k = T V k, V k, V k, A trading strategy is self-financing if c ( k,w) = 0 for k T. This means that no money has come in to or out of the portfolio except at the beginning and the end (times 0 and T). We say that a multiperiod securities market admits arbitrage if there is a selffinancing trading strategy, called an arbitrage opportunity, such that: V ( 0) = 0) ( 0) 0 V ( T) = S( T) ( T ) 0 V ( T) = S( T) ( T ) 0 The above state the following: The initial cost of the portfolio is zero or less, The strategy is self-financing, The payoff at time T for this strategy is nonnegative in all states and strictly positive in at least one state. The idea is again that without any investment one earns a riskless profit. Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski

12 There will be now two important concepts used in the definition of the riskneutral probability measure. Normalized prices of assets follow a discrete stochastic process. That process if called adapted if investors who know the past and present prices of securities, know the current prices of a given normalized asset we are considering. That process is called a martingale if it is adapted, and the expected value of future normalized price is the current normalized price. A risk-neutral probability measure on Ω (which now consists of paths, not just single events) is a probability measure assigning positive probability to each Ω and such that the normalized price of each security is a martingale. A stochastic process Ψ = - ( 0, ) =, Ω - It is adapted, - It is strictly positive, - ( k, ) S j k, H { ( k):k = 0,,...,T } is the state price process if: ( ) = ( k +, ) H histories H, all k, and all j. S j k +, ( ) for all time k The ultiperiod Fundamental Theorem of Asset Pricing: For a multiperiod model of a securities market that satisfies the assumptions of this section, the following are equivalent: - The model is arbitrage-free, - There exists a state price process, - There exists a risk-neutral probability measure. Valuation of European options and other cash flows A European option or other set of cash flows is valued by creating a replicating portfolio and then find the price of the replicating portfolio, assuming the option or other set of cash flows is attainable. Completeness in the ultiperiod odel An arbitrage-free multiperiod model is said to be complete if every adapted cash flow stream c is financed by some trading strategy. For an arbitrage-free multiperiod model, the following statements are equivalent: - The model is complete, - The state price process Ψ is unique, - The risk-neutral probability measure Q is unique. Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski

13 Exercise: Society of Actuaries Course 6 Examination, ay 000, Problem No. 8 The market model consists of securities and a bank account. The bank account pays interest of 0% per year and is risk-free. The price of each security today is 00. There are three possible scenarios for the prices of the securities in year: Security X Security Y Scenario 0 0 Scenario 55 0 Scenario Calculate the state price vector for this securities market model, if one exists. If there is no state price vector, explain why. Solution = = = 00 = 4 55 = = 5 The state price vector is: = Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski

14 Exercise. You are given securities market described by: S( 0) = [ ],S( ) = 3. Is this market arbitrage-free? Is it complete? Solution. The risk-free security doubles, while the risky security appreciates by 50% or depreciates. Thus the risk-free security always beats the risky security, the market is not arbitrage-free. Because it is not arbitrage-free, it is not complete. You can also solve this by solving the system of equations for the state price vector, and you will see that the solution contains negative entries. Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski

15 Exercise 3. You are given securities market described by: S( 0) = [ ],S ( )= Find a state price vector. If a state price vector does not exist, explain why. Solution. Let [ ] be the state price vector. Then we must have: + = = and 0.60 = 0.50, = 5 6, = 6. [ ] = Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski

16 Exercise 4. You are given securities market described by: S( 0) = [ ],S ( )= Calculate the arbitrage-free price of a European put on the second asset with the strike (exercise) price of.0. Solution. By the same approach as in the previous problem you can find the following state price vector: [ ] = 3 3. The put described in this problem has cash flows price equals = 3 3 = 5. 0 at time, and its 0.0 Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski

17 Exercise 5. You are given securities market described by:.0.0 S( 0) = [ ],S ( )= Find the risk-neutral probabilities. Solution. We can see that the risk-free rate is 0%. Risk-neutral probabilities are: q = + i d u d = = 3,q = u ( +i) u d = = 3. Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski

18 Exercise 6. You are given securities market described by: 0 S( 0) = [ ],S ( )= 0. This market is not complete. Find an Arrow-Debreu security, and its price, such that adding it to this market makes the market complete, while keeping it arbitrage-free. Solution. 0 Note that adding this vector 0 as a third column makes the matrix S() invertible. Therefore it would be best to price this e 3 Arrow-Debreu security. State price vector must satisfy: + + =, = 3 so that + 3 =, = and therefore we must have 0 < <, =,0< 3 =. 0 The price of 0 must equal. In fact, we can pick it to be any number 3 between 0 and, and then let =. We can, for example pick 3 =, and the market becomes 3 4 S( 0) = 0 0 4,S( ) = 0, clearly complete and arbitrage-free. 0 Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski

19 Exercise 7. A multi-period securities market model follows the binomial stock price model. You are given that u =., d = 0.9, i = 0.04, and the initial stock price is $0. Compute the arbitrage-free price of a European call option on the stock that expires in two period and has a strike price of $. Solution. One period risk-neutral probabilities are: q u = + i d u d = = 3 7,q = u ( + i) d u d Over two periods, the stock can go: = = 4 7. Up-up to. $0 = $8.80, with call payoff of $7.80, and probability Up-down or down-up to. 0.9 $0 = $.08, with call payoff of $.08, and probability = Down-down to 0.9 $0 = $6.93, with call payoff or $0.00, and probability Expected present value, and the arbitrage-free price of the call, equals $ $ $0.00 = = ( $.43+$0.53 +$0.00) = $ Spring 003 Notes for SoA Course 6 exam, Copyright 003 by Krzysztof Ostaszewski - 4 -