SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives
|
|
- Abner Garrett
- 5 years ago
- Views:
Transcription
1 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 October 31, 2015
2 Outline Terminology 1 Terminology
3 Outline Terminology 1 Terminology
4 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.
5 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.
6 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.
7 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
8 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.
9 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} +.
10 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.
11 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.
12 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?
13 Outline Terminology 1 Terminology
14 Outline Terminology 1 Terminology
15 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.
16 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.
17 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.
18 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.
19 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
20 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)] = = So on average the stock offers a positive return.
21 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.
22 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) = = 3.
23 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.
24 An Incorrect Intuition (Cont d) Consider the option with strike price K = 1.1, whose expected value is 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).
25 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 = 0.208, 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 = 0.144, with probability 0.2.
26 An Incorrect Intuition (Cont d) So the expected return by selling the option and hedging by investing the proceeds in the stock is = > 0. More to the point, this profit is risk-free, because the seller of the option makes money for every outcome!
27 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: 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.
28 Outline Terminology 1 Terminology
29 : 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,
30 : 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.
31 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) [ ] = [0.2 0] 1 1 = [ ]. ] 1 So the option seller borrows 0.45 and buys 0.5 units of the stock.
32 Numerical Example (Cont d) Check: If stock goes up, then value of portfolio at time T = 1 is = 0.2. If the stock goes down, then at time T = 1, = 0. The cost of implementing this strategy is = 0.05, which is the cost of the option.
33 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) [ ] = [4 1] 1 1 = [ ]. ] 1 So option seller shorts units of bonds (i.e. borrow $ 12.25) and buys 12.5 units of stock.
34 Numerical Example (Cont d) Check: If stock goes up, then value of portfolio at time T = 1 is = 4. If the stock goes down, then at time T = 1, = 1. The cost of implementing this strategy is = 0.25, which is the cost of the contingent claim.
35 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 ]
36 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.
37 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
38 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 = [ ]. For the option, the cost of the replicating portfolio is = If [X u X d ] = [4 1], then the cost of the replicating portfolio is E[X, q] = ( 1) 0.75 = 0.25.
39 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.
40 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 You can then sell your own option for 0.05, use the replicating strategy to break even at period 1, and the balance of = 0.01 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.
41 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 = If the stock goes down, the seller loses nothing on the option, but the value of his holding is = 0. This is why it is a replicating portfolio.
42 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 = 0.14 < So he loses If the stock goes down, the seller loses nothing on the option, and his portfolio is worth = So he makes a profit of So the expected return is = Similar argument if he borrows too much instead of too little.
43 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.
44 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)
45 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.
46 Outline Terminology 1 Terminology
47 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.
48 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.
49 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
50 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?
51 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
52 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.
53 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.
54 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 = Next slide shows possible paths and corresponding payouts.
55 Two-Period Example (Cont d)
56 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 = = /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 d0 u 0 d 0 u 0 1 u 0 d 0 [ 1/4 3/4 ]. ].
57 Two-Period Example (Cont d)
58 Two-Period Example (Cont d) At t = 1, top node price equals (1/3) (2/3) 0.12 = 0.25, while at the bottom top node price equals At t = 0, node price equals (1/3) (2/3) 0.00 = (1/4) (3/4) 0.01 = This is the arbitrage-free price for the option. If we had more complex payouts, the arbitrage-free price can be computed entirely similarly.
59 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.
60 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.
61 Two-Period Example (Cont d) Arbitrage-free price at time t = 0 is The optimal hedging strategy at time t = 0 is given by [ [ξ 0 b 0 ] = [ ] 1 1 ] 1 = [ ]. 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.
62 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 [ [ξ 1 b 1 ] = [ ] 1 1 ] 1 = [ ]. So the option seller borrows 1.05, adds to the 0.25 on hand, and buys 1.3 units of stock.
63 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 [ [ξ 1 b 1 ] = [ ] 1 1 ] 1 = [ ]. 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.
64 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.
65 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.
66 Outline Terminology 1 Terminology
67 (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.
68 Outline Terminology 1 Terminology
69 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.)
70 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.
71 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 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 ) + ).
72 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 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 ) + ).
73 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.
74 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.
75 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.
76 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.
77 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).
78 Outline Terminology 1 Terminology
79 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.
80 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.
81 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.
Martingale Pricing Theory in Discrete-Time and Discrete-Space Models
IEOR E4707: Foundations of Financial Engineering c 206 by Martin Haugh Martingale Pricing Theory in Discrete-Time and Discrete-Space Models These notes develop the theory of martingale pricing in a discrete-time,
More informationLECTURE 2: MULTIPERIOD MODELS AND TREES
LECTURE 2: MULTIPERIOD MODELS AND TREES 1. Introduction One-period models, which were the subject of Lecture 1, are of limited usefulness in the pricing and hedging of derivative securities. In real-world
More informationLecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.
Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous
More informationFINANCIAL OPTION ANALYSIS HANDOUTS
FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any
More informationCompleteness and Hedging. Tomas Björk
IV Completeness and Hedging Tomas Björk 1 Problems around Standard Black-Scholes We assumed that the derivative was traded. How do we price OTC products? Why is the option price independent of the expected
More informationThe Binomial Model. Chapter 3
Chapter 3 The Binomial Model In Chapter 1 the linear derivatives were considered. They were priced with static replication and payo tables. For the non-linear derivatives in Chapter 2 this will not work
More informationComputational Finance. Computational Finance p. 1
Computational Finance Computational Finance p. 1 Outline Binomial model: option pricing and optimal investment Monte Carlo techniques for pricing of options pricing of non-standard options improving accuracy
More informationFrom Discrete Time to Continuous Time Modeling
From Discrete Time to Continuous Time Modeling Prof. S. Jaimungal, Department of Statistics, University of Toronto 2004 Arrow-Debreu Securities 2004 Prof. S. Jaimungal 2 Consider a simple one-period economy
More informationForwards and Futures. Chapter Basics of forwards and futures Forwards
Chapter 7 Forwards and Futures Copyright c 2008 2011 Hyeong In Choi, All rights reserved. 7.1 Basics of forwards and futures The financial assets typically stocks we have been dealing with so far are the
More information******************************* The multi-period binomial model generalizes the single-period binomial model we considered in Section 2.
Derivative Securities Multiperiod Binomial Trees. We turn to the valuation of derivative securities in a time-dependent setting. We focus for now on multi-period binomial models, i.e. binomial trees. This
More informationMathematics of Finance Final Preparation December 19. To be thoroughly prepared for the final exam, you should
Mathematics of Finance Final Preparation December 19 To be thoroughly prepared for the final exam, you should 1. know how to do the homework problems. 2. be able to provide (correct and complete!) definitions
More informationEconomathematics. Problem Sheet 1. Zbigniew Palmowski. Ws 2 dw s = 1 t
Economathematics Problem Sheet 1 Zbigniew Palmowski 1. Calculate Ee X where X is a gaussian random variable with mean µ and volatility σ >.. Verify that where W is a Wiener process. Ws dw s = 1 3 W t 3
More informationAppendix: Basics of Options and Option Pricing Option Payoffs
Appendix: Basics of Options and Option Pricing An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise
More informationCorporate Finance, Module 21: Option Valuation. Practice Problems. (The attached PDF file has better formatting.) Updated: July 7, 2005
Corporate Finance, Module 21: Option Valuation Practice Problems (The attached PDF file has better formatting.) Updated: July 7, 2005 {This posting has more information than is needed for the corporate
More informationHomework Assignments
Homework Assignments Week 1 (p 57) #4.1, 4., 4.3 Week (pp 58-6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15-19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9-31) #.,.6,.9 Week 4 (pp 36-37)
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationBinomial Option Pricing
Binomial Option Pricing The wonderful Cox Ross Rubinstein model Nico van der Wijst 1 D. van der Wijst Finance for science and technology students 1 Introduction 2 3 4 2 D. van der Wijst Finance for science
More informationLecture Quantitative Finance Spring Term 2015
and Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 06: March 26, 2015 1 / 47 Remember and Previous chapters: introduction to the theory of options put-call parity fundamentals
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationMATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS
MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material.
More informationThe Multistep Binomial Model
Lecture 10 The Multistep Binomial Model Reminder: Mid Term Test Friday 9th March - 12pm Examples Sheet 1 4 (not qu 3 or qu 5 on sheet 4) Lectures 1-9 10.1 A Discrete Model for Stock Price Reminder: The
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
More informationHedging. MATH 472 Financial Mathematics. J. Robert Buchanan
Hedging MATH 472 Financial Mathematics J. Robert Buchanan 2018 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in market variables. There
More informationCalculating Implied Volatility
Statistical Laboratory University of Cambridge University of Cambridge Mathematics and Big Data Showcase 20 April 2016 How much is an option worth? A call option is the right, but not the obligation, to
More informationIntroduction to Binomial Trees. Chapter 12
Introduction to Binomial Trees Chapter 12 Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. Hull 2013 1 A Simple Binomial Model A stock price is currently $20. In three months
More informationChapter 15: Jump Processes and Incomplete Markets. 1 Jumps as One Explanation of Incomplete Markets
Chapter 5: Jump Processes and Incomplete Markets Jumps as One Explanation of Incomplete Markets It is easy to argue that Brownian motion paths cannot model actual stock price movements properly in reality,
More informationThe Pennsylvania State University. The Graduate School. Department of Industrial Engineering AMERICAN-ASIAN OPTION PRICING BASED ON MONTE CARLO
The Pennsylvania State University The Graduate School Department of Industrial Engineering AMERICAN-ASIAN OPTION PRICING BASED ON MONTE CARLO SIMULATION METHOD A Thesis in Industrial Engineering and Operations
More informationSYSM 6304 Risk and Decision Analysis Lecture 2: Fitting Distributions to Data
SYSM 6304 Risk and Decision Analysis Lecture 2: Fitting Distributions to Data M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 5, 2015
More informationIlliquidity, Credit risk and Merton s model
Illiquidity, Credit risk and Merton s model (joint work with J. Dong and L. Korobenko) A. Deniz Sezer University of Calgary April 28, 2016 Merton s model of corporate debt A corporate bond is a contingent
More informationMath 181 Lecture 15 Hedging and the Greeks (Chap. 14, Hull)
Math 181 Lecture 15 Hedging and the Greeks (Chap. 14, Hull) One use of derivation is for investors or investment banks to manage the risk of their investments. If an investor buys a stock for price S 0,
More informationHedging Credit Derivatives in Intensity Based Models
Hedging Credit Derivatives in Intensity Based Models PETER CARR Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU Stanford
More information1 Geometric Brownian motion
Copyright c 05 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is
More informationnon linear Payoffs Markus K. Brunnermeier
Institutional Finance Lecture 10: Dynamic Arbitrage to Replicate non linear Payoffs Markus K. Brunnermeier Preceptor: Dong Beom Choi Princeton University 1 BINOMIAL OPTION PRICING Consider a European call
More informationLecture 3: Review of mathematical finance and derivative pricing models
Lecture 3: Review of mathematical finance and derivative pricing models Xiaoguang Wang STAT 598W January 21th, 2014 (STAT 598W) Lecture 3 1 / 51 Outline 1 Some model independent definitions and principals
More informationMonte Carlo Simulations
Monte Carlo Simulations Lecture 1 December 7, 2014 Outline Monte Carlo Methods Monte Carlo methods simulate the random behavior underlying the financial models Remember: When pricing you must simulate
More information2 The binomial pricing model
2 The binomial pricing model 2. Options and other derivatives A derivative security is a financial contract whose value depends on some underlying asset like stock, commodity (gold, oil) or currency. The
More informationlast problem outlines how the Black Scholes PDE (and its derivation) may be modified to account for the payment of stock dividends.
224 10 Arbitrage and SDEs last problem outlines how the Black Scholes PDE (and its derivation) may be modified to account for the payment of stock dividends. 10.1 (Calculation of Delta First and Finest
More informationThe Black-Scholes Model
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh The Black-Scholes Model In these notes we will use Itô s Lemma and a replicating argument to derive the famous Black-Scholes formula
More informationMonte Carlo Methods in Structuring and Derivatives Pricing
Monte Carlo Methods in Structuring and Derivatives Pricing Prof. Manuela Pedio (guest) 20263 Advanced Tools for Risk Management and Pricing Spring 2017 Outline and objectives The basic Monte Carlo algorithm
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationChange of Measure (Cameron-Martin-Girsanov Theorem)
Change of Measure Cameron-Martin-Girsanov Theorem Radon-Nikodym derivative: Taking again our intuition from the discrete world, we know that, in the context of option pricing, we need to price the claim
More informationReplication strategies of derivatives under proportional transaction costs - An extension to the Boyle and Vorst model.
Replication strategies of derivatives under proportional transaction costs - An extension to the Boyle and Vorst model Henrik Brunlid September 16, 2005 Abstract When we introduce transaction costs
More informationMathematics in Finance
Mathematics in Finance Robert Almgren University of Chicago Program on Financial Mathematics MAA Short Course San Antonio, Texas January 11-12, 1999 1 Robert Almgren 1/99 Mathematics in Finance 2 1. Pricing
More information( ) since this is the benefit of buying the asset at the strike price rather
Review of some financial models for MAT 483 Parity and Other Option Relationships The basic parity relationship for European options with the same strike price and the same time to expiration is: C( KT
More informationOptions Markets: Introduction
17-2 Options Options Markets: Introduction Derivatives are securities that get their value from the price of other securities. Derivatives are contingent claims because their payoffs depend on the value
More informationDerivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester
Derivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester Our exam is Wednesday, December 19, at the normal class place and time. You may bring two sheets of notes (8.5
More informationOption Pricing. Chapter Discrete Time
Chapter 7 Option Pricing 7.1 Discrete Time In the next section we will discuss the Black Scholes formula. To prepare for that, we will consider the much simpler problem of pricing options when there are
More informationIntroduction Random Walk One-Period Option Pricing Binomial Option Pricing Nice Math. Binomial Models. Christopher Ting.
Binomial Models Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October 14, 2016 Christopher Ting QF 101 Week 9 October
More informationFIN FINANCIAL INSTRUMENTS SPRING 2008
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Greeks Introduction We have studied how to price an option using the Black-Scholes formula. Now we wish to consider how the option price changes, either
More informationStochastic Differential equations as applied to pricing of options
Stochastic Differential equations as applied to pricing of options By Yasin LUT Supevisor:Prof. Tuomo Kauranne December 2010 Introduction Pricing an European call option Conclusion INTRODUCTION A stochastic
More informationIn general, the value of any asset is the present value of the expected cash flows on
ch05_p087_110.qxp 11/30/11 2:00 PM Page 87 CHAPTER 5 Option Pricing Theory and Models In general, the value of any asset is the present value of the expected cash flows on that asset. This section will
More informationOutline One-step model Risk-neutral valuation Two-step model Delta u&d Girsanov s Theorem. Binomial Trees. Haipeng Xing
Haipeng Xing Department of Applied Mathematics and Statistics Outline 1 An one-step Bionomial model and a no-arbitrage argument 2 Risk-neutral valuation 3 Two-step Binomial trees 4 Delta 5 Matching volatility
More informationRisk Neutral Measures
CHPTER 4 Risk Neutral Measures Our aim in this section is to show how risk neutral measures can be used to price derivative securities. The key advantage is that under a risk neutral measure the discounted
More informationBinomial model: numerical algorithm
Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4
More informationAdvanced Numerical Methods
Advanced Numerical Methods Solution to Homework One Course instructor: Prof. Y.K. Kwok. When the asset pays continuous dividend yield at the rate q the expected rate of return of the asset is r q under
More informationLecture Notes for Chapter 6. 1 Prototype model: a one-step binomial tree
Lecture Notes for Chapter 6 This is the chapter that brings together the mathematical tools (Brownian motion, Itô calculus) and the financial justifications (no-arbitrage pricing) to produce the derivative
More informationAn Introduction to the Mathematics of Finance. Basu, Goodman, Stampfli
An Introduction to the Mathematics of Finance Basu, Goodman, Stampfli 1998 Click here to see Chapter One. Chapter 2 Binomial Trees, Replicating Portfolios, and Arbitrage 2.1 Pricing an Option A Special
More informationLattice (Binomial Trees) Version 1.2
Lattice (Binomial Trees) Version 1. 1 Introduction This plug-in implements different binomial trees approximations for pricing contingent claims and allows Fairmat to use some of the most popular binomial
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives November 5, 212 Option Analysis and Modeling The Binomial Tree Approach Where we are Last Week: Options (Chapter 9-1, OFOD) This Week: Option Analysis and Modeling:
More informationModule 10:Application of stochastic processes in areas like finance Lecture 36:Black-Scholes Model. Stochastic Differential Equation.
Stochastic Differential Equation Consider. Moreover partition the interval into and define, where. Now by Rieman Integral we know that, where. Moreover. Using the fundamentals mentioned above we can easily
More information2.1 Mathematical Basis: Risk-Neutral Pricing
Chapter Monte-Carlo Simulation.1 Mathematical Basis: Risk-Neutral Pricing Suppose that F T is the payoff at T for a European-type derivative f. Then the price at times t before T is given by f t = e r(t
More informationBluff Your Way Through Black-Scholes
Bluff our Way Through Black-Scholes Saurav Sen December 000 Contents What is Black-Scholes?.............................. 1 The Classical Black-Scholes Model....................... 1 Some Useful Background
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
More informationUtility Indifference Pricing and Dynamic Programming Algorithm
Chapter 8 Utility Indifference ricing and Dynamic rogramming Algorithm In the Black-Scholes framework, we can perfectly replicate an option s payoff. However, it may not be true beyond the Black-Scholes
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2017 14 Lecture 14 November 15, 2017 Derivation of the
More informationA No-Arbitrage Theorem for Uncertain Stock Model
Fuzzy Optim Decis Making manuscript No (will be inserted by the editor) A No-Arbitrage Theorem for Uncertain Stock Model Kai Yao Received: date / Accepted: date Abstract Stock model is used to describe
More informationBasic Arbitrage Theory KTH Tomas Björk
Basic Arbitrage Theory KTH 2010 Tomas Björk Tomas Björk, 2010 Contents 1. Mathematics recap. (Ch 10-12) 2. Recap of the martingale approach. (Ch 10-12) 3. Change of numeraire. (Ch 26) Björk,T. Arbitrage
More informationValuation of Options: Theory
Valuation of Options: Theory Valuation of Options:Theory Slide 1 of 49 Outline Payoffs from options Influences on value of options Value and volatility of asset ; time available Basic issues in valuation:
More informationNo-arbitrage theorem for multi-factor uncertain stock model with floating interest rate
Fuzzy Optim Decis Making 217 16:221 234 DOI 117/s17-16-9246-8 No-arbitrage theorem for multi-factor uncertain stock model with floating interest rate Xiaoyu Ji 1 Hua Ke 2 Published online: 17 May 216 Springer
More informationOutline One-step model Risk-neutral valuation Two-step model Delta u&d Girsanov s Theorem. Binomial Trees. Haipeng Xing
Haipeng Xing Department of Applied Mathematics and Statistics Outline 1 An one-step Bionomial model and a no-arbitrage argument 2 Risk-neutral valuation 3 Two-step Binomial trees 4 Delta 5 Matching volatility
More informationStochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models
Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete
More informationProbability in Options Pricing
Probability in Options Pricing Mark Cohen and Luke Skon Kenyon College cohenmj@kenyon.edu December 14, 2012 Mark Cohen and Luke Skon (Kenyon college) Probability Presentation December 14, 2012 1 / 16 What
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More informationThe Capital Asset Pricing Model as a corollary of the Black Scholes model
he Capital Asset Pricing Model as a corollary of the Black Scholes model Vladimir Vovk he Game-heoretic Probability and Finance Project Working Paper #39 September 6, 011 Project web site: http://www.probabilityandfinance.com
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationMartingale Measure TA
Martingale Measure TA Martingale Measure a) What is a martingale? b) Groundwork c) Definition of a martingale d) Super- and Submartingale e) Example of a martingale Table of Content Connection between
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationRichardson Extrapolation Techniques for the Pricing of American-style Options
Richardson Extrapolation Techniques for the Pricing of American-style Options June 1, 2005 Abstract Richardson Extrapolation Techniques for the Pricing of American-style Options In this paper we re-examine
More informationThe Binomial Lattice Model for Stocks: Introduction to Option Pricing
1/33 The Binomial Lattice Model for Stocks: Introduction to Option Pricing Professor Karl Sigman Columbia University Dept. IEOR New York City USA 2/33 Outline The Binomial Lattice Model (BLM) as a Model
More informationChapter 9 - Mechanics of Options Markets
Chapter 9 - Mechanics of Options Markets Types of options Option positions and profit/loss diagrams Underlying assets Specifications Trading options Margins Taxation Warrants, employee stock options, and
More informationPage 1. Real Options for Engineering Systems. Financial Options. Leverage. Session 4: Valuation of financial options
Real Options for Engineering Systems Session 4: Valuation of financial options Stefan Scholtes Judge Institute of Management, CU Slide 1 Financial Options Option: Right (but not obligation) to buy ( call
More informationLecture 16: Delta Hedging
Lecture 16: Delta Hedging We are now going to look at the construction of binomial trees as a first technique for pricing options in an approximative way. These techniques were first proposed in: J.C.
More informationTerm Structure Lattice Models
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh Term Structure Lattice Models These lecture notes introduce fixed income derivative securities and the modeling philosophy used to
More informationMulti-Asset Options. A Numerical Study VILHELM NIKLASSON FRIDA TIVEDAL. Master s thesis in Engineering Mathematics and Computational Science
Multi-Asset Options A Numerical Study Master s thesis in Engineering Mathematics and Computational Science VILHELM NIKLASSON FRIDA TIVEDAL Department of Mathematical Sciences Chalmers University of Technology
More informationMATH 425: BINOMIAL TREES
MATH 425: BINOMIAL TREES G. BERKOLAIKO Summary. These notes will discuss: 1-level binomial tree for a call, fair price and the hedging procedure 1-level binomial tree for a general derivative, fair price
More informationLecture 6: Option Pricing Using a One-step Binomial Tree. Thursday, September 12, 13
Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationReview of Derivatives I. Matti Suominen, Aalto
Review of Derivatives I Matti Suominen, Aalto 25 SOME STATISTICS: World Financial Markets (trillion USD) 2 15 1 5 Securitized loans Corporate bonds Financial institutions' bonds Public debt Equity market
More informationINTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS. Jakša Cvitanić and Fernando Zapatero
INTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS Jakša Cvitanić and Fernando Zapatero INTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS Table of Contents PREFACE...1
More informationFutures and Forward Markets
Futures and Forward Markets (Text reference: Chapters 19, 21.4) background hedging and speculation optimal hedge ratio forward and futures prices futures prices and expected spot prices stock index futures
More informationExtensions to the Black Scholes Model
Lecture 16 Extensions to the Black Scholes Model 16.1 Dividends Dividend is a sum of money paid regularly (typically annually) by a company to its shareholders out of its profits (or reserves). In this
More informationFinancial Markets & Risk
Financial Markets & Risk Dr Cesario MATEUS Senior Lecturer in Finance and Banking Room QA259 Department of Accounting and Finance c.mateus@greenwich.ac.uk www.cesariomateus.com Session 3 Derivatives Binomial
More informationOptimal trading strategies under arbitrage
Optimal trading strategies under arbitrage Johannes Ruf Columbia University, Department of Statistics The Third Western Conference in Mathematical Finance November 14, 2009 How should an investor trade
More informationIntroduction to Real Options
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh Introduction to Real Options We introduce real options and discuss some of the issues and solution methods that arise when tackling
More informationAdvanced Corporate Finance. 5. Options (a refresher)
Advanced Corporate Finance 5. Options (a refresher) Objectives of the session 1. Define options (calls and puts) 2. Analyze terminal payoff 3. Define basic strategies 4. Binomial option pricing model 5.
More informationLECTURE 4: BID AND ASK HEDGING
LECTURE 4: BID AND ASK HEDGING 1. Introduction One of the consequences of incompleteness is that the price of derivatives is no longer unique. Various strategies for dealing with this exist, but a useful
More informationDynamic Portfolio Choice II
Dynamic Portfolio Choice II Dynamic Programming Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Dynamic Portfolio Choice II 15.450, Fall 2010 1 / 35 Outline 1 Introduction to Dynamic
More information************* with µ, σ, and r all constant. We are also interested in more sophisticated models, such as:
Continuous Time Finance Notes, Spring 2004 Section 1. 1/21/04 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connection with the NYU course Continuous Time Finance. This
More informationOption Pricing Models. c 2013 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 205
Option Pricing Models c 2013 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 205 If the world of sense does not fit mathematics, so much the worse for the world of sense. Bertrand Russell (1872 1970)
More informationMSC FINANCIAL ENGINEERING PRICING I, AUTUMN LECTURE 6: EXTENSIONS OF BLACK AND SCHOLES RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK
MSC FINANCIAL ENGINEERING PRICING I, AUTUMN 2010-2011 LECTURE 6: EXTENSIONS OF BLACK AND SCHOLES RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK In this section we look at some easy extensions of the Black
More information