SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives

SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October 31, 2015

Outline Terminology 1 Terminology 2 3 4 5 6

Portfolio Terminology A market consists of a safe asset usually referred to as a bond, together with one or more uncertain assets usually referred to as stocks. A portfolio is a set of holdings, consisting of the bond and the stocks. If there are n stocks, we think of the portfolio as an (n + 1)-dimensional vector (a 0, a 1,..., a n ), where each a i is a real number. a 0 is the quantity of the bond we hold and a i is the quantity of stock i that we hold.

Portfolio (Cont d) Terminology We can think of a portfolio as a vector in R n+1. Each a i can be positive or negative. Negative holdings correspond to borrowing money or shorting stocks. Note: In a multi-period investment problem, a i is actually a i (t), for i = 0,..., T 1, where T is the duration of the planning period. So we keep adjusting our portfolio from one time instant to the next. Note: We take our last investment decision at time T 1, whose outcome will become known at time T.

Self-Financing Terminology Our investment strategy has to be self-financing. In other words, we cannot introduce fresh money into the system. The logic is that any extra money we had at the start would be reflected in the holding a 0 (0) of the safe asset. This introduces a constraint on the portfolios at successive times, as shown next.

Self-Financing (Cont d) The sequence of events is as follows: At time 0, we have a certain amount of money available to us. The prices of the safe asset S 0 (0) and of the risky assets S 1 (0),..., S n (0) are known to us when we make the initial investment decision. Suppose the amount of money we have available to us is V (0). So the initial portfolio vector a(0) R n+1 must satisfy n a i (0)S i (0) = V (0). i=0

Self-Financing (Cont d) At time t = 0, the risky assets S 1,..., S n are to be viewed as random variables. But at time t = 1, they are no longer random, but have known values S 1 (1),..., S n (1). So at time t = 1, the value of our portfolio is V (1) = n a i (0)S i (1). i=1 At time t = 1 we are free to reallocate the money as we wish. But we must stay within the available funds. This can be expressed as n a i (1)S i (1) = V (1) = i=1 n a i (0)S i (1). i=1 This is called the self-financing constraint.

Options Terminology A call option is an instrument that gives the buyer the right, but not the obligation, to buy a stock a prespecified price called the strike price K. (A put option gives the right to sell at a strike price.) A European option can be exercised only at a specified time T. An American option can be exercised at any time prior to a specified time T. If S t is the price of the stock as a function of time, T is the maturity date, and K is the strike price, then the value of the European option is {S T K} +, the nonnegative part of S T K. The value of the American option at time t T is {S t K} +.

European vs. American Options The European option is worthless even though S t > K for some intermediate times. The American option has positive value at intermediate times but is worthless at time t = T.

Derivatives or Contingent Claims An option (call or put) is a special case of a contingent claim whereby the value of the instrument is contingent upon the value of an underlying asset at (or before) a specified time. The terminology derivative is also used in the place of contingent claim because value of the instrument is derived from that of an underlying asset.

Typical Questions Terminology Suppose there is only one risky asset (the stock) and one safe asset (the bond). Suppose the period T and the strike price K for an option have been mutually agreed upon by the buyer and seller of the option. What is the minimum price that the seller of an option should be willing to accept? What is the maximum price that the buyer of an option should be willing to pay? How can the seller (or buyer) of an option hedge (minimimze or even eliminate) his risk after having sold (or bought) the option?

Outline Terminology 1 Terminology 2 3 4 5 6

Many key ideas can be illustrated via one-period binomial model. We have a choice of investing in a safe bond or an uncertain stock. B(0) = Price of the bond at time T = 0. It increases to B(1) = (1 + r)b(0) at time T = 1, where the number r (the guaranteed return) is known at time T = 0. S(0) = Price of the stock at time T = 0. { S(0)u with probability p, S(1) = S(0)d with probability 1 p.

Simplification Terminology Assumption: d < 1 + r < u; otherwise problem is meaningless! Rewrite as d < 1 < u, where d = d/(1 + r), u = u/(1 + r). Convert everything (bond and stock price) to constant currency, and drop the superscript prime. So: Value of bond at time T = 1 is the same as the value of the bond at time T = 0, or B(1) = B(0). Stock price at time T = 1 is { S(0)u with probability p, S(1) = S(0)d with probability 1 p, where d < 1 < u.

Explanation of Nomenclature Why is this called a one-period binomial model? Because We are following the stock for just one time period, and The stock price has only two possible values at time T = 1.

Options Terminology An option gives the buyer the right, but not the obligation, to buy the stock at time T = 1 at a predetermined strike price K. Again, assume S(0)d < K < S(0)u. Otherwise the problem makes no sense. If stock goes up, the value of the option at time T = 1 is S(1) K = S(0)u K. If the stock goes down, the value of the option at time T = 1 is zero, because the option is worthless. So if X is the value of the option at time T = 1, then { {S(1) K}+ if S(1) = S(0)u, X = 0 if S(1) = S(0)d.

Contingent Claims Terminology More generally, a contingent claim is a random variable X such that { Xu if S(1) = S(0)u, X = if S(1) = S(0)d. X d

Option Example Terminology Suppose B(0) = S(0) = 1, u = 1.3, d = 0.9, p = 0.8, 1 p = 0.2. So the stock price S(1) has values { 1.3 with probability 0.8, S(1) = 0.9 with probability 0.2. So the expected value of the stock price at time T = 1 is E[S(1)] = 1.3 0.8 + 0.9 0.2 = 1.22. So on average the stock offers a positive return.

Option Example (Cont d) Suppose the strike price is K = 1.1. Then the value of the option X satisfies { 0.2 with probability 0.8, {S(1) K} + = 0 with probability 0.2. The expected value of the option is given by E[{S(1) K} + ] = 0.16.

Option Example (Cont d) A more general contingent claim is specified by { Xu = 4 if S(1) = S(0)u, X d = 1 if S(1) = S(0)d. The expected value of this contingent claim is E(X) = 4 0.8 1 0.2 = 3.

An Incorrect Intuition Question: How much should the seller of such a claim charge for the claim at time T = 0? Is it the expected value of the claim, namely E[X, p] = px u + (1 p)x d? NO! The seller of the claim can hedge against future fluctuations of stock price by using a part of the proceeds to buy the stock himself.

An Incorrect Intuition (Cont d) Consider the option with strike price K = 1.1, whose expected value is 0.16. Suppose the buyer of the option pays 0.16 to the option seller. If the seller of the option just sits on the stock, hoping that it won t go up, then on average he will break even. If the stock goes up, he has to pay 0.2. But he already has 0.16; so he has to pay only 0.04, with probability 0.8. If the stock goes down, he owes nothing, and gets to keep the 0.16 with probability 0.2. Net profit is zero (break-even).

An Incorrect Intuition (Cont d) But the seller can invest the 0.16 in the same stock against which he has sold the option. If the stock goes up, he loses 0.2 on the option, but his stock is now worth 0.16 1.3 = 0.208, so he makes a net profit of 0.08, with probability of 0.8. If the stock goes down, then he loses nothing on the option, but his stock is now worth 0.16 0.9 = 0.144, with probability 0.2.

An Incorrect Intuition (Cont d) So the expected return by selling the option and hedging by investing the proceeds in the stock is 0.08 0.8 + 0.144 0.2 = 0.0928 > 0. More to the point, this profit is risk-free, because the seller of the option makes money for every outcome!

Arbitrage-Free Price The previous calculation shows that the price of 0.16 for the option is too high, because the seller can make a risk-free profit he makes a profit whether the stock goes up or goes down! This is known as arbitrage. So what is the fair or arbitrage-free price for this option? The answer is: 0.05. The correct strategy for the seller of the option is to take the 0.05 from selling the option, borrow another 0.45, and buy 0.5 of the stock.

Outline Terminology 1 Terminology 2 3 4 5 6

: General Idea Build a portfolio at time T = 0 such that its value exactly matches that of the claim at time T = 1 irrespective of stock price movement. Choose real numbers ξ and b (quantity of stocks and bonds respectively) such that (remember that B(1) = B(0)) ξs(0)u + bb(0) = X u, ξs(0)d + bb(0) = X d,

: Algebraic Details In vector-matrix notation [ S(0)u S(0)d [ξ b] B(0) B(0) ] = [X u X d ]. This equation has a unique solution for a, b if u d. It is called a replicating portfolio.

Numerical Example: Option Suppose B(0) = S(0) = 1, and u = 1.3, d = 0.9. The value of the option with a strike price K = 1.1 is X u = 0.2 if the stock goes up, and X d = 0 is the stock goes down. Then the unique replicating portfolio is given by [ S(0)u S(0)d [ξ b] = [X u X d ] B(0) B(0) [ ] 1 1.3 0.9 = [0.2 0] 1 1 = [0.5 0.45]. ] 1 So the option seller borrows 0.45 and buys 0.5 units of the stock.

Numerical Example (Cont d) Check: If stock goes up, then value of portfolio at time T = 1 is 0.5 1.3 0.45 = 0.2. If the stock goes down, then at time T = 1, 0.5 0.9 0.45 = 0. The cost of implementing this strategy is 0.5 0.45 = 0.05, which is the cost of the option.

Numerical Example: Contingent Option Suppose B(0) = S(0) = 1, and u = 1.3, d = 0.9. The contingent claim is X u = 4, X d = 1. Then the unique replicating portfolio is given by [ S(0)u S(0)d [ξ b] = [X u X d ] B(0) B(0) [ ] 1 1.3 0.9 = [4 1] 1 1 = [12.5 12.25]. ] 1 So option seller shorts 12.25 units of bonds (i.e. borrow $ 12.25) and buys 12.5 units of stock.

Numerical Example (Cont d) Check: If stock goes up, then value of portfolio at time T = 1 is 12.5 1.3 12.25 = 4. If the stock goes down, then at time T = 1, 12.5 0.9 12.25 = 1. The cost of implementing this strategy is 12.5 12.25 = 0.25, which is the cost of the contingent claim.

Interpretation: Risk-Neutral Measure The unique solution for a, b is [ u d [ξ b] = [X u X d ] 1 1 ] 1 [ 1/S(0) 0 0 1/B(0) Amount of money needed at time T = 0 to implement the replicating strategy is [ ] [ ] S(0) qu c = [ξ b] = [X B(0) u X d ], q d ]. where q := [ qu q d ] = [ u d 1 1 ] 1 [ 1 1 ] = [ 1 d u d u 1 u d ]

Interpretation: Risk-Neutral Measure (Cont d) Note that q u, q d > 0 and q u + q d = 1. So q := (q u, q d ) is a probability distribution on S(1). Moreover it is the unique distribution such that E[S(1), q] = S(0)u 1 d u d + S(0)du 1 u d = S(0), i.e. such that the stock also becomes risk-neutral like the bond. Important point: q depends only on the returns u, d, and not on the associated real world probabilities p, 1 p.

Interpretation of Cost of Replicating Portfolio Thus the initial cost of the replicating portfolio [ ] qu c = [X u X d ] = E[X, q] is the discounted expected value of the contingent claim X under the unique risk-neutral distribution q. q d

Numerical Example Revisited We had u = 1.3, d = 0.9. So the unique risk-neutral distribution q such that E[S(1), q] = S(0) is given by q = [0.25 0.75]. For the option, the cost of the replicating portfolio is 0.2 0.25 + 0 0.75 = 0.05. If [X u X d ] = [4 1], then the cost of the replicating portfolio is E[X, q] = 4 0.25 + ( 1) 0.75 = 0.25.

Arbitrage-Free Price for a Contingent Claim Theorem: The quantity c = [X u X d ] [ qu q d ] = E[X, q] is the unique arbitrage-free price for the contingent claim. Suppose someone is ready to pay c > c for the claim. Then the seller collects c, invests c c in a risk-free bond, uses c to implement replicating strategy and settle claim at time T = 1, and pockets a risk-free profit of c c. This is called an arbitrage opportunity.

Example of Arbitrage Recall the option with strike price K = 1.1. Suppose someone is willing to pay, say, 0.07 for the option. The seller can us 0.05 to implement the replicating strategy, which guarantees that he comes out even at period 1, no matter what happens. The excess of 0.02 is his risk-free profit. Hence this is an arbitrage opportunity. On the other side, suppose you are the buyer of the option, and someone is foolish enough to sell you the option for 0.04. You can then sell your own option for 0.05, use the replicating strategy to break even at period 1, and the balance of 0.05 0.04 = 0.01 is your risk-free profit. Conclusion: The quantity E[X, q] is the unique price at which neither the buyer nor seller has an arbitrage opportunity.

Optimal Hedging Strategy It is not enough to set the right price for the option or contingent claim (price discovery); it is also necessary to use an optimal hedging strategy. Recall that the optimal hedging strategy is to charge 0.05 for the option, borrow 0.45, and buy 0.50 of the stock. If the stock goes up, the seller loses 0.2 on the option, but the value of his holding is 0.50 1.3 0.45 = 0.20. If the stock goes down, the seller loses nothing on the option, but the value of his holding is 0.50 0.9 0.45 = 0. This is why it is a replicating portfolio.

Suboptimal Hedging Strategy What happens if the seller doesn t follow this strategy? Suppose the seller borrows only 0.25 and buy 0.30 of the stock. If the stock goes up, the seller loses 0.2 on the option, and his portfolio is worth 0.30 1.3 0.25 = 0.14 < 0.20. So he loses 0.06. If the stock goes down, the seller loses nothing on the option, and his portfolio is worth 0.30 0.9 0.25 = 0.02. So he makes a profit of 0.02. So the expected return is 0.06 0.8 + 0.02 0.2 = 0.044. Similar argument if he borrows too much instead of too little.

Alternate Formulation of Arbitrage-Free Price V (0) = x d q d +x u q u = x d(u 1) + x u (1 d) u d = x d + x u x d u d (1 d). This can be given a pictorial interpretation, as in the next slide.

Pictorial Interpretation of Arbitrage-Free Price x x x u V (0) x d x d V (0) x u d 1 u α d 1 u α (a) (b)

Summary Terminology Fact 1. There is a unique synthetic distribution q on S(1) such that E[S(1), q] = S(0), so that the stock also becomes risk-neutral. This distribution q depends only on the two possible outcomes, but not on the associated real world probabilities. Fact 2. The unique arbitrage-free price of a contingent claim (X u, X d ) is the discounted expected value of the claim under the risk-neutral distribution q. Fact 3. There is a unique replicating strategy that allows the seller of the derivative to hedge his risk completely irrespectiveof outcome.

Outline Terminology 1 Terminology 2 3 4 5 6

Bond price is deterministic: B t+1 = (1 + r t )B t, t = 0,..., T 1, where the interest rates r 0,..., r T 1 are known beforehand. Stock price can go up or down: S t+1 = S t u t or S t d t. Note: No reason to assume that either the r t or the u t, d t are constant they can vary with time.

Normalized Multiple Period Binomial Model Since return on bond at each time is known beforehand, express everything in constant currency: B t = B 0 for all t, and S t+1 = S t u t or S t d t where the returns u t, d t have been normalized with respect to the bond returns. So d t < 1 < u t for all t. Since stock can go up or down at each time instant, there are 2 T possible sample paths for the stock. Define {u, d} T to be the set of strings of length T where each symbol is either u or d. Then each string h {u, d} T defines one possible evolution of the stock price.

Illustration: Three Periods S 2 u 2 S 0 u 0 S 1 u 1 S 2 d 2 S 2 u 2 S 0 S 1 d 1 S 2 d 2 S 2 u 2 S 0 d 0 S 1 u 1 S 1 d 1 S 2 d 2 S 2 u 2 S 2 d 2

For each sample path h {u, d} T, at time T there is an associated payout X h. Note: In other words, final payout may depend not only on the final stock price S T, but also the path to the final price. Claim becomes due only at the end of the time period (European contingent claim). In an American option, the buyer chooses the time of exercising the option. Questions What is the arbitrage-free price for this claim? How does the seller of the claim hedge against variations in stock price?

Replicating Strategy for T Periods We already know to replicate over one period. We can extend the argument to T periods. Suppose j {u, d} T 1 is the set of stock price transitions up to time T 1. Let S j denote the stock price at time T 1 corresponding to this set of stock price movements. So now there are only two possibilities for the final sample path: ju and jd. The stock price at time T can therefore go up by u T or down by d T. As before, compute the risk-neutral probability distribution [ ] 1 dt 1 u T 1 1 q T 1 = =: [q u,t q d,t ]. u T 1 d T 1 u T 1 d T 1

Replicating Strategy for T Periods (Cont d) Only two possible payouts: X ju and X jd. Denote these by c ju and c jd respectively. At time T 1, compute a cost c j and a replicating portfolio [a j b j ] to replicate this claim, namely: c j = (c ju q u,n + c jd q d,n ). [ Sj S [a j b j ] = [c ju c jd ] j B 0 B 0 ] 1.

Replicating Strategy for T Periods (Cont d) Do this for each j {u, d} T 1. So if we are able to replicate each of the 2 T 1 payouts c j at time T 1, then we know how to replicate each of the 2 T payouts at time T. Repeat backwards until we reach time T = 0. Number of payouts decreases by a factor of two at each time step.

Two-Period Example Bond price B t = 1 for all t, and S 0 = 1. S 1 can go up to 1.3 or down to 0.9 times S 0. S 2 can go up to 1.2 or down to 0.9 times S 1. Option at final time with a strike price of K = 1.05. Next slide shows possible paths and corresponding payouts.

Two-Period Example (Cont d) 0.51 1.56 1.3 0.12 1.17 1 0.03 1.08 0.9 0.00 0.81

Two-Period Example (Cont d) Risk-neutral distribution at period t = 1 is given, as before, by [ ] [ ] 1 [ ] ] [ qu,1 u1 d q 1 := = 1 1 1/3 = = 1 1 1 2/3 q d,1 q d,0 [ 1 d1 u 1 d 1 u 1 1 u 1 d 1 Similarly risk-neutral distribution at period t = 0 is given, as before, by [ ] [ ] 1 [ ] ] qu,0 u0 d q 0 := = 0 1 = = 1 1 1 [ 1 d0 u 0 d 0 u 0 1 u 0 d 0 [ 1/4 3/4 ]. ].

Two-Period Example (Cont d) 0.07 1 0.25 1.3 0.01 0.9 0.51 1.56 0.12 1.17 0.03 1.08 0.00 0.81

Two-Period Example (Cont d) At t = 1, top node price equals (1/3) 0.51 + (2/3) 0.12 = 0.25, while at the bottom top node price equals At t = 0, node price equals (1/3) 0.03 + (2/3) 0.00 = 0.01. (1/4) 0.25 + (3/4) 0.01 = 0.07. This is the arbitrage-free price for the option. If we had more complex payouts, the arbitrage-free price can be computed entirely similarly.

Replicating Strategy in Multiple Periods The unique arbitrage-free price is c 0 := X h q h. h {u,d} T c 0 is the expected value of the claim X h under the synthetic distribution {q h } that makes the discounted stock price risk-neutral. Moreover, c 0 is the unique arbitrage-free price for the claim.

Implementation of Replicating Strategy Seller of claim receives an amount c 0 at time t = 0 and invests a 0 in stocks and b 0 in bonds, where [ S0 u [ξ 0 b 0 ] = [c u c d ] 0 S 0 d 0 B 0 B 0 ] 1. Due to replication, at time t = 1, the portfolio is worth c u if the stock goes up, and is worth c d if the stock goes down. At time t = 1, adjust the portfolio according to [ S1 u [ξ 1 b 1 ] = [c i0 u c 1 S 1 d 1 i0 d] B 1 B 1 ] 1, where i 0 = u or d as the case may be. Then repeat.

Two-Period Example (Cont d) Arbitrage-free price at time t = 0 is 0.07. The optimal hedging strategy at time t = 0 is given by [ 1.3 0.9 [ξ 0 b 0 ] = [0.25 0.01] 1 1 ] 1 = [0.60 0.53]. So at time t = 0, seller of the option takes 0.07 from the buyer, borrows 0.53, and buys 0.60 units of stock.

Two-Period Example (Cont d) Suppose that stock goes up at time t = 1. Then value of portfolio is 0.25 (that is how it was chosen). Optimal hedging strategy at time t = 1 is given by [ 1.2 0.9 [ξ 1 b 1 ] = [0.51 0.12] 1 1 ] 1 = [1.30 1.05]. So the option seller borrows 1.05, adds to the 0.25 on hand, and buys 1.3 units of stock.

Two-Period Example (Cont d) Suppose that stock goes down at time t = 1. Then value of portfolio is 0.01 (that is how it was chosen). Optimal hedging strategy at time t = 1 is given by [ 1.2 0.9 [ξ 1 b 1 ] = [0.03 0.00] 1 1 ] 1 = [0.10 0.09]. So the option seller borrows 0.09, adds to the 0.01 on hand, and buys 0.10 units of stock. This is known as portfolio rebalancing at each time step.

Self-Financing Terminology It is true that ξ 0 S 1 + b 0 B 1 = ξ 1 S 1 + b 1 B 1 whether S 1 = S 0 u 0 or S 1 = S 0 d 0 (i.e. whether the stock goes up or down at time t = 1). So no fresh infusion of funds is required after time n = 0. This property has no analog in the one-period case. It is also replicating from that time onwards. Observe: Implementation of replicating strategy requires reallocation of resources T times, once at each time instant. Therefore this theory assumes no transaction costs.

Complete Markets Terminology The multi-period binomial model is an example of a complete market, because for every sample path, there is a hedging strategy that can ensure that the final payout is zero. If this is not the case, then we get an incomplete market.

Outline Terminology 1 Terminology 2 3 4 5 6

(Not Discussed) Multinomial model: At each time instant, the stock can take more than two values. Stochastic interest rates: The guaranteed returns on bonds r n are themselves random variables. Transactions: Moving from one type of asset to another entails costs. Multiple stocks and/or multiple bonds. Each of these situations can be handled, but not with elementary algebra. Finally, if make time continuous instead of discrete, we get the famous Black-Scholes theory of option pricing.

Outline Terminology 1 Terminology 2 3 4 5 6

Continuous-Time Processes Take limit at time interval goes to zero and T ; binomial asset price movement becomes geometric Brownian motion: S t = S 0 exp [(µ 12 ) ] σ2 t + σw t, t [0, T ], where W t is a standard Brownian motion process. µ is the drift of the Brownian motion and σ is the volatility. Bond price is deterministic: B t = B 0 e rt. Claim is European and a simple option: X T = {S T K} +. What we can do: Make µ, σ, r functions of t and not constants. What we cannot do: Make σ, r stochastic! (Stochastic µ is OK.)

Black-Scholes Formula Theorem (Black-Scholes 1973): The unique arbitrage-free option price is ( log(s0 /K ) C 0 = S 0 Φ σ + 1 ) ( T 2 σ T K log(s0 /K ) Φ σ 1 ) T 2 σ T, where Φ(c) = 1 2π c e u2 /2 du is the Gaussian distribution function, and K = e rt K is the discounted strike price.

Black-Scholes PDE Terminology Consider a general payout function e rt ψ(e rt x) to the buyer if S T = x (various exponentials discount future payouts to T = 0). Then the unique arbitrage-free price is given by C 0 = f(0, S 0 ), where f is the solution of the PDE f t + 1 2 σ2 x 2 2 f = 0, (t, x) (0, T ) (0, ), x2 with the boundary condition f(t, x) = ψ(x). No closed-form solution in general (but available if ψ(x) = (x K ) + ).

Replicating Strategy in Continuous-Time Define C t = C 0 + t 0 f x (s, S s )ds s, t (0, T ), where the integral is a stochastic integral, and define α t = f x (t, S t ), β t = C t α t S t. Then hold α t of the stock and β t of the bond at time t. Observe: Implementation of self-financing fully replicating strategy requires continuous trading and no transaction costs.

Practical Considerations Black-Scholes theory gives closed-form formulas when the derivative is a simple option. For more complicated derivatives, a good approach is to divide the total time [0, T ] into N equal intervals, create a binomial model, and then let N. The Financial Instruments Toolbox of Matlab contains several tools for approximating continuous-time processes by binomial trees, and for price determination. The Cox-Ross-Rubinstein (CRR) tree converges very slowly, and the Leisen-Reimer tree is to be preferred. Other approaches are to solve the Black-Scholes PDE using numerical techniques.

Extensions to Multiple Assets Binomial model extends readily to multiple assets. Black-Scholes theory extends to the case of multiple assets of the form [( S (i) t = S (i) 0 exp µ (i) 1 ) ] 2 [σ(i) ] 2 t + σw (i) t, t [0, T ], where W (i) t, i = 1,..., d are (possibly correlated) Brownian motions. Analog of Black-Scholes PDE: C 0 = f(0, S (1) 0,..., S(d) 0 ) where f satisfies a PDE. But no closed-form solution for f in general.

American Options Terminology An American option can be exercised at any time up to and including time T. So we need a super-replicating strategy: The value of our portfolio must equal or exceed the value of the claim at all times. If X t = {S t K} +, then both price and hedging strategy are same as for European claims. Very little known about pricing American options in general. Theory of optimal time to exercise option is very deep and difficult.

Sensitivities and the Greeks Recall C 0 = f(0, S 0 ) is the correct price for the option under Black-Scholes theory. (We need not assume the claim to be a simple option!) = C 0, Γ = = 2 C 0, S 0 S 0 S 2 0 Vega = ν = C 0 σ, θ = f(t, X t), ρ = f(t, X t). t r Delta-hedging : A strategy such that = 0 return is insensitive to initial stock price (to first-order approximation). Delta-gamma-hedging : A strategy such that = 0, Γ = 0 return is insensitive to initial stock price (to second-order approximation).

Outline Terminology 1 Terminology 2 3 4 5 6

Can Geometric Brownian Motion Model be Changed? Black-Scholes theory is predicated on the assumption that asssets follow a geometric Brownian motion model, or asset returns have a Gaussian distribution. We have seen that, even with respectable stocks such as the Dow Jones 30 stocks, this is simply not true. A stable distribution with α around 1.8 or 1.7 provides better approximation. So what happens to option pricing theory in this situation? Short Answer: It pretty much falls apart.

Incomplete Markets Terminology The binomial model had the very useful feature that there exists a replicating portfolio. In other words, ignoring transaction costs and permitting continuous trading, it is possible to rebalance one s portfolio constantly so as to ensure that the current value of the portfolio exactly equals the value of the derivative. Such a situation is called a complete market. Unfortunately, geometric Brownian assets are the only model that leads to complete markets. All others lead to incomplete markets.

Incomplete Markets (Cont d) This means that there is no unique arbitrage-free price. Instead, there is a range of prices at which neither the buyer nor the seller can gain arbitrage (i.e., risk-free profit. However, there are various possible hedging strategies, and it is not clear which one(s) is (are) best. This a subject of current research.