Lecture 17/18/19 Options II

Size: px
Start display at page:

Download "Lecture 17/18/19 Options II"

Transcription

1 1 Lecture 17/18/19 Options II Alexander K. Koch Department of Economics, Royal Holloway, University of London February 25, February 29, and March In addition to learning the material covered in the reading and the lecture, students should be able to work out a number of price bounds for options (depending on their type, the time left to expiry, and their strike price); understand the basic notion of binomial option pricing and the idea of arbitrage with a tracking portfolio; be able to price a plain-vanilla option in a two-period binomial framework; understand what the hedge ratio ( Delta ) means. Required reading: Bodie, Kane, and Marcus (2008) (Chapter 21) Supplementary reading: Elton and Gruber (1995) (Chapter 8) M.S. Scholes (1998), Derivatives in a Dynamic Environment, American Economic Review, 88, (*)

2 2 1 Price bounds for options 1.1 Assumptions and notation We will make use of the following notation: strike (exercise) price of an option: X; maturity date: T ; price of the underlying asset at date t: S t ; dividend payments: D; (constant) risk-free rate of return (p.a.): r f ; price of a European call/put at date T: c t /p t ; price of an American call/put option at date T: C t /P t. 1.2 Bounds on the value of a European call The most obvious lower bound on the value of an option is that an option can never be worth less than zero - the holder can always simply let it expire. To derive another lower bound, consider a stock that pays dividend D just before maturity. Compare the payoff from the following two portfolios: A ( option position ): long one European call option on one share of stock; B ( leveraged equity position ): long one share of stock and short a zero bond with face value X + D and maturity T. Table 1 gives the position values at date t and at maturity date T. As we can see, with certainty, portfolio A is at least as valuable as portfolio B at maturity. Thus, by no arbitrage, the value of portfolio A at any date t < T must also be at least that of portfolio B. This provides us with another lower bound on the value of a European call option: the price of the underlying minus the present value of strike price and dividends. Combined with the restriction of nonnegative value, we have: c t max[0, S t P V t (X) P V t (D)]. The maximum payoff from an option at maturity is max[s T X, 0] S T. Therefore, no arbitrage implies that at any date t no one would be willing to pay more than the current price of the underlying asset, S t to purchase the option to buy it at date T for X > 0.

3 3 Position value value at T if at t S T X S T > X A c t 0 S T X B S t S T + D S T + D X+D (X + D) (X + D) (1+r f ) (T t)/365 S t X+D (1+r f ) (T t)/365 S T X S T X value A value B value A = value B Table 1: Payoffs of option position versus leveraged equity position. Result 1 The value of a European call option lies within the following bounds: S t c t max[0, S t P V t (X) P V t (D)]. (1) 1.3 Bounds on the value of an American call An American call option is at least as valuable as a European one with the same characteristics, since it provides the additional option to exercise early (if no early exercise occurs, the payoff is the same as that for the European call). Thus, S t C t c t. Clearly, the upper bound for the European call option also applies for an American call option: the maximum payoff from exercising the option at any date t = 0,..., T max[s t X, c t ] S t. As we will see, absent dividends, such an option will never be exercised early. No dividends on underlying asset We know from our previous discussion that an American option guarantees at least the payoff from that on a European option, by choosing not to exercise it before maturity: C t c t max[0, S t P V t (X)]. (2) if the American option is exercised at some date t < T, it yields a payoff of: C t (if exercised) = S t X. (3) Now since P V t (X) < X, we conclude that the payoff from holding on to the option [equation (2)] is greater than from exercising it at date t < T [equation (3)]. Therefore, the option is worth more alive than dead, i.e., the option to exercise early has no value.

4 4 Result 2 An American call option on a non-dividend-paying stock will never be exercised before the maturity date. From this we conclude, that for non-dividend-paying underlying assets, the values of a European and an American call option are identical: C t = c t, t = 0,..., T. Dividends on underlying asset Suppose now that the underlying stock makes a dividend payment D on date T. As we will see, with dividends the lower price bound in (2) can be made tighter, i.e. we can find an even higher value for the lower bound. Consider an American call option which has not been exercised prior to date t. We know that, if it is not exercised on t, it will be worth as much as a European call option: c t max[0, S t P V t (X) P V t (D)]. If the option is immediately exercised (which of course only pays if S t > X), it yields S t X. Suppose that the option is in the money, i.e., S t > X, but that P V t (X + D) > S t. Then, the value of the option if exercised might exceed the value of holding on to it for another day. Result 3 An American call option on stock that pays dividends during its time-to-maturity may be exercised before the maturity date. Thus, with dividend-paying stocks, the right to exercise early potentially has a value. 1.4 Bounds on the value of a European put Since the payoff at maturity of a European put is max[0, X S T ] X, the maximum value that a put can have clearly is X. As with call options, a put option always has nonnegative value. To derive another lower bound for a European put option on non-dividend paying stock, consider the payoff from the following two portfolios: A ( option-stock position ): long one share of stock and one European put option on one share of stock; B ( fixed income position ): long one zero bond with face value X and maturity T. Table 2 gives the position values at date t and at maturity date T. As we can see, with certainty, portfolio A is at least as valuable as portfolio B at maturity. Thus, by no arbitrage, the value of portfolio A at any date t < T must also be at least that of portfolio B:

5 5 Position value value at T if at t S T X S T > X A S t S T S T p t X S T 0 S t + p t X S T B X (1+r f ) (T t)/365 X X value A = value B value A > value B Table 2: Payoffs of option-stock position versus fixed-income position. Result 4 The value of a European put option lies within the following bounds: X p t max[0, P V t (X) S t ]. (4) 1.5 Bounds on the value of an American put Due to the early exercise feature, we know that the American put option is at least as valuable as its European counterpart: P t p t max[0, P V t (X) S t ]. In the case that the put is exercised early at date t < T (which of course only pays if S t < X), the payoff is X S t. Since P V t (X) < X, early exercise might be profitable. Thus, an American put is more valuable than its European counterpart. Result 5 The value of a American put option lies within the following bounds: X P t max[0, X S t ]. (5) It is straightforward to repeat the above exercise including dividends. What emerges is that, in contrast to the case of American call options, dividend payments make early exercise less likely for put options. The intuition is straightforward, dividends increase the lower bound on the European put options, which is the value of an American option not exercised early. From results 2 and 5, it follows immediately that the put-call parity derived in the last lecture for European options does not hold for American options.

6 2 The binomial option pricing model Most option pricing models rely on advanced mathematical techniques, which are beyond the scope of this course. The simple binomial option pricing model, developed by Cox, Ross, and Rubinstein (1979) and Rendleman and Bartter (1979), illustrates the logic of pricing by no-arbitrage that underlies these more complex models. Moreover, it provides an easily implementable option pricing tool. The model imposes the following assumptions: frictionless capital markets (no transaction costs, no taxes); price-taking behavior; all market participants have the same information; no short-sales constraints; the risk-free rate of return is r f per period; the stock price at any date t, S t can only move up to S t+1 = u S t or down to S t+1 = d S t, where u > > 1 > d > 0, P rob( up ) = q, and P rob( down ) = 1 q. We will only look at plain vanilla options, i.e. options with standard exercise terms and no special clauses. 2.1 The one-period version Let the risk-free rate of return for one period be r f = Suppose that a stock currently sells at S 0 = 100 and can either increase (by a factor of u = 2) to 200 or decrease (by a factor of d = 0.5) to 50 at year-end. Let the probability of an upward movement be q = 0.5. Consider a call option with strike price X = 125 which expires at the end of the year. The payoff of the option depends on the realized end-of-period stock price S 1 (see figure 1): 75 = S 1 X if S 1 = u S 0 = 200 value of call at end of period = 0 if S 1 = d S 0 = 50 The first step in valuing this call option is to construct a riskless hedge portfolio. Consider a portfolio long H shares of stock and short one call option written on the stock. Figure 2 illustrates how this portfolio s value evolves. It is worth H S 0 c 0 at the beginning of the period. The end-of-period value depends on whether the stock price moves up (value: 6

7 7 Beginning of period End of period Beginning of period End of period q S 1 =us 0 [ 200] q c u =max[0,us 0 -X] [ 75] S 0 [ 100] c 0 1-q 1-q S 1 =ds 0 [ 50] c d =max[0,ds 0 -X] [ 0] Stock price Value of a call option with strike price X Example: S 0 = 100, u=2, d=0.5, X= 125 Figure 1: Payoffs in the one-period binomial model Beginningof- period value Endof-period value q HuS 0 -c u risk-free portfolio if HuS 0 -c u = HdS 0 -c d HS 0 -c 0 1-q HdS 0 -c d hedge ratio: c H = S 0 c u d ( u d). Hedge portfolio Figure 2: Payoffs in the one-period binomial model

8 8 Position value value at end of period if at t = 0 S 1 = 50 S 1 = 200 long one stock S 0 = 100 S 1 = 50 S 1 = 200 short (written) 2 calls 2 c sum c Table 3: Hedging with underlying asset. H u S 0 c u ) or down (value: H d S 0 c d ). To obtain a riskless portfolio, we must choose H so that the end-of-period portfolio value is constant across states: H d S 0 c d = H u S 0 c u c u c d range of option values at end of period H = = S 0 (u d) range of stock values at end of period (hedge ratio). H is called the hedge ratio, it gives the number of shares that need to be purchased to offset the risk from one call written (i.e., short position) on the stock. Or, put differently, H is the number of shares you need to sell for every long call position held. Returning to our initial example values, we can compute the hedge ratio to be H = 1/2. That is, two written call options can be perfectly hedged using one share of stock (see Table 3). The second step exploits the assumption that there should be no arbitrage opportunities in a well-functioning capital market. Since the return on the hedge portfolio is risk-free, it must be equal to the risk-free rate of return: H u S 0 c u H S 0 c 0 = c 0 = H S 0 [( ) u] + c u. (6) Substituting in for the hedge ratio H in (6) yields [ ( ) ( )] (1 + rf ) d u (1 + rf ) c 0 = c u + c d u d u d Note that the terms in brackets sum up to unity. That is, we can denote by 1. (7) p = ( ) d u d and 1 p = u ( ) u d to obtain c 0 = [p c u + (1 p) c d ] 1. (8) Inspecting (8), we can see that the price of a call option is equal to the expected value of the call option at the end of the period (using the probabilities p and 1 p) discounted

9 9 with the risk-free rate. In other words, the call is priced as if the probability of an upward movement in the stock price was p and all investors were risk-neutral. Therefore, p and 1 p are called risk-neutral probabilities. Note that nothing in our argument required making any assumptions on investors actual risk preferences. Moreover, the objective probability of an upward movement in the stock, q, has no impact on the value we derived for the call option. The option is priced relative to the given parameters S 0, u, d, X, and r f. If investors disagreed on the probabilities for movements in the stock price, this would only indirectly affect the option s value through the market views being reflected in the stock price S 0. In our example, we obtain a risk-neutral probability of p Note that this is different from the objective probability q = 0.5 that we assumed to hold. The call option is priced at c 0 = Looking again at table 3, we see that indeed the return on the hedge portfolio is equal to the risk-free rate: = r f = The concept used to price the option is simple: from securities for which we know the price construct a risk-free hedge portfolio involving the option to be priced. No arbitrage then implies that the return on this portfolio must be equal to the risk-free rate. The same approach underlies more complex option pricing models as well (they just use more realistic assumptions on the price distribution for the underlying asset). 2.2 The pricing of European call options Using the insights from the previous section, it is easy to use the binomial model to price plain vanilla European call options on underlying assets which involve no payments until the maturity date (e.g. a non-dividend paying stock). We start off, by pricing a two-period option. The two-period model It is simple to generalize the one-period model in the case of European call options. Consider the extension to two periods in figure 3. Working backward through the tree, we can treat each second-period branch as a one-period model, and thus compute the input values for the next lower branch to be worked on. We will illustrate this using the values of our initial example for an option with strike price X = 75 now (see also figure 3). Starting at the upper end branch (2a in figure 3), we see that the option value at date 2 is either c uu = max[0, u u S 0 X] = 325 or c ud = max[0, u d S 0 X] = 25. Thus, we can compute the value of the option at date 1, ensuing an initial increase in the stock price, c u = p c uu + (1 p) c du

10 10 Strike price: X= 75 1 q c u 2a S 1 =us 0 [ 200] pcuu + ( 1 p) c = 1+ rf [ ] ud q 1-q S 2 =uus 0 [ 400] c uu =max[0,uus 0 -X] [ 325] S 2 =uds 0 [ 100] c ud =max[0,uds 0 -X] [ 25] S 0 [ 100] 1-q 2b q S 2 =dus 0 [ 100] c du =max[0,dus 0 -X] [ 25] ( 1 ) pcu + p cd c0 = 1+ r [ 51.83] f S 1 =ds 0 [ 50] pcdu + ( 1 p) c cd = 1+ rf [ 8.95] dd 1-q S 2 =dds 0 [ 25] c dd =max[0,dds 0 -X] [ 0] Figure 3: Payoffs in the two-period binomial model Working back the lower end branch (2b in figure 3), we see that c du = c ud and c dd = max[0, d d S 0 X] = 0. Thus, c d = p c du + (1 p) c dd Using these results as inputs in the starting branch (1 in figure 3), c 0 = p c u + (1 p) c d The T-period model The model can be extended to T periods. Note that c 0 = p c u + (1 p) c d = p2 c uu + p (1 p) c ud + (1 p) p c du + (1 p) 2 c dd ( ) 2. Since c ud = c du, all that counts are the number of upward and downward movements and not their order. Thus, we can use the binomial distribution to compute the probabilities of these events. After T -periods with n upward movements and T n downward movements, the option value is: max[0, u n d T n S 0 X]. (9)

11 11 The probability of ending up in this node of the branch is given by the binomial distribution: 1 T n p n (1 p) T n. Hence, T c 0 = T n n=0 p n (1 p) T n max[0, u n d T n S 0 X] 1 ( ) T. (10) Note that the payoff of a call option is positive only if it finishes off in-the-money, i.e. S T > X. Therefore, the option payoff given in (9) is zero for all n for which S T = u n d T n S 0 X. This helps simplify the formula in (10): T c 0 = S 0 T p n (1 p) T n n n n un d T n ( ) T X ( ) T T T p n (1 p) T n (11) n n n } {{ } = P rob(s T > X) under riskneutral probabilities where n is the smallest integer for which the option finishes in the money, i.e. n = min{n N : u n d T n S 0 > X}. 2.3 The pricing of European put options To price a plain vanilla European put option on a non-dividend paying stock, we can simply invoke the put-call parity relation (see last lecture) and use the price of the European call option as an input: p 0 = c 0 + X ( ) T/365 S The pricing of American call options We can invoke result 2, which tells us that an American call option on a non-dividend-paying stock will never be exercised early. Thus, the American call option has the same value as its European counterpart, for which we know how to price it: 1 Recall that for a binomial T n 0 C 0 = c 0. 1 A = T! n! (T n)!.

12 The pricing of American put options The only problematic case is the pricing of an American put option. Recall that the put-call parity relation does not hold for American options because it may be optimal to exercise early an American put on a non-dividend-paying stock (in contrast to an American call). The advantage of the binomial model is that we can easily modify it, to account for the possibility of early exercise. 2 We will illustrate the approach in a two-period example. The share price on a stock that pays no dividend over the next two periods currently is S 0 = 50. In each period, the stock price can either increase (by a factor u = 1.2) or decrease (by a factor d = 0.6). The yield curve is flat and the risk-free rate of return r f = 0.1 per period. What is the fair price of an American put option, P 0, with exercise price X = 50? Again, we can start with a one-period version of the model and form a hedge portfolio. Recall that we defined the hedge ratio as the number of shares that you need to sell for a long position in options on that stock. As before, the hedge ratio H is chosen so that the portfolio is riskless, i.e. it has a constant end-of-period value: H u S 0 + P u = H d S 0 + P d H = P u P d range of option values at end of period = S 0 (u d) range of stock values at end of period (hedge ratio). Note that since P u < P d, the hedge ratio will be negative, in contrast to the call option case. That is, we can offset the risk of a long put option by going long in shares of the stock. By no-arbitrage, the one-period portfolio return must be equal to the risk-free rate. Thus, given the end-of-period payoffs P u and P d we can price the option at the beginning of the period: ( ) (H S 0 P 0 ) = H u S 0 P u P 0 = u ( ) u d } {{ } =p P d + ( ) d u d } {{ } =1 p P u 1 (substituting in H). To analyze the two-period model, of course, we need to know the option s value ensuing an up or down move in the stock price at each period. As above, we start from the last period: at maturity the payoff of the American put is equal to that of the European counterpart (using 2 All option pricing models for American put options do not have a closed-form solutions and therefore need to rely on numerical methods for deriving the option price.

13 13 value of European put (no early exercise) early exercise q P uu =p uu =max[0,x-uus 0 ] [ 0] P u =max[p u,x-us 0 ] q [ 2.12=max[ 2.12, - 10] ] [no early exercise] 1-q P 0 1-q value of European put (no early exercise) early exercise q P ud =p ud =max[0,x-uds 0 ] [ 14] P du =p du =max[0,x-dus 0 ] [ 14] [P 0 = 4.64] [p 0 = 3.95] P d =max[p d,x-ds 0 ] [ 20=max[ 15.45, 20] ] [early exercise] 1-q P dd =p dd =max[0,x-dds 0 ] [ 32] Figure 4: Valuing a two-period American put option our example s values). P uu = p uu = max[0, X u 2 S 0 ] = 0 P du = P ud = p ud = max[0, X u d S 0 ] = 14 P dd = p dd = max[0, X d 2 S 0 ] = 32 Using these interim results, we can price the European put option (which cannot be exercised before maturity) at date 1 as follows in our example: 1 p u = [p p ud + (1 p) p uu ] 2.12 pd = [p p dd + (1 p) p du ] To determine the value of the American put, we need to compare this payoff from not exercising early to that from early exercise at date 1: stock price early exercise no exercise value of American put u S 0 = 60 X u S 0 = 10 p u = 2.12 P u = max[x u S 0, p u ] = 2.12 d S 0 = 30 X d S 0 = 20 p d = P d = max[x d S 0, p d ] = 20 As we can see, the American option will be exercised early if the stock moves down in the first period. Hence, we plug in P u = 2.12 and P d = 20 into (12) to obtain: P In

14 contrast, the European put would be valued at p Thus, the early exercise option has a premium of The procedure is summarized in figure 4. 3 The Black-Scholes option pricing model The binomial model had the advantage of being tractable and extremely flexible (e.g., it is easy to price American put options). Clearly, the assumption that the stock price can only move up or down by a fixed amount in each time period is restrictive. However, one can add realism by simply splitting a given unit of time (e.g. a month) into more and more time periods. Cox, Ross, and Rubinstein (1979) prove that the limit of the binomial call pricing formula is the Black and Scholes (1973) continuous time pricing formula. The Black and Scholes (1973) formula for pricing call options on non-dividend paying stocks is widely used in practice. In recognition of the significant impact it had on finance theory and practice, Myron Scholes and Robert C. Merton, who also contributed to developing option pricing, were awarded the 1997 Nobel prize (Fisher Black died in 1995). The Black-Scholes option pricing formula for a non-dividend paying call option is: c 0 = S 0 N(d 1 ) X e r f T N(d 2 ) (12) d 1 = ln(s 0/X) + (r f + σ 2 /2) T σ T d 2 = d 1 σ T, (13) N( ) is the cdf of the standard normal distribution. That is, the current option value c 0 is a function of the current stock price S 0, the strike price X, the risk-free rate r f (annualized rate with continuous compounding), the instantaneous variance of the annualized stock return, σ 2, and time to maturity T (in years). Note: the price of the option does not depend on the expected rate of return on the stock. The formula is easy to use given tabulated values for the standard normal cdf or using a computer program. 3 3 An example is the Excel spreadsheet available from the website: sites/dl/free/ /116763/ch21_bs_stu.xls. 14

15 15 Interpretation of the formula To interpret the Black-Scholes formula, it is useful at this point to compare it with the T-period binomial model. We restate here the pricing formula for a T-period European call option (11): T c 0 = S 0 T p n (1 p) T n n n n un d T n ( ) T X ( ) T T T p n (1 p) T n n n n } {{ } = P rob(s T > X) under riskneutral probabilities where n is the smallest integer for which the option finishes in the money, i.e. n = min{n N : u n d T n S 0 > X}. The pricing equation from the Binomial model and that from the Black- Scholes model (12) have a similar structure: the stock price S 0 is multiplied with a factor to compute the expected value of the stock (using the risk neutral probabilities), conditional on finishing in the money; from this we subtract the discounted value of the bond, conditional on finishing in the money. So one can view the second N(d) term in the Black-Scholes formula (12) as the probability that the option will finish in the money and the first N(d) term has an analogous interpretation to the first term in the pricing equation above from the Binomial model. Thus, if the option is almost certain to be in the money at maturity, the price is approximately S 0 X e r f T, which is the discounted value of the option payoff (often called the intrinsic value of an option): the claim on the stock is worth S 0 today and the exercise price is discounted. In contrast, if the probability of finishing in the money is almost zero, the option value is roughly zero, as we would expect. For intermediate cases, the N(d) terms adjust the price of the option to reflect the probability of profitable exercise at maturity. Option Delta and other Greeks Another interesting feature of this formula, is that the derivative of the option value with respect to the underlying stock s price is simply: 4 delta = c 0 S 0 = N(d 1 ). This partial derivative is called the delta of the option. It measures the first-order effect of a change in the underlying asset s price on the call option s price. Thus, one can approximately insure a portfolio long one call option against (small) movements in the underlying asset s 4 Note that when taking the derivative one needs to take into account that N(d 1) and N(d 2) also depend on S 0. The effects through them happen to cancel out.

16 16 price by establishing a short position of H = delta stocks. Then, following a small change in S 0 the change in portfolio value is approximately c 0 H S 0 S 0 S }{{} 0 =1 = 0 if H = c 0 S 0. Such a portfolio would be called delta neutral. We can interpret delta as the hedge ratio at which we must continuously keep our hedge portfolio to maintain a riskless position (if the other factors affecting option value remain constant). Note, that the concept of delta hedging relies on a first-order approximation. The option value is a convex function in the underlying asset s price (see panel (a) in figure 5). Therefore, the approximation is not exact. To correct for this, one can use a second-order term as well (the so-called option gamma ). Other such first-order approximations are commonly used for the impact of changes in the other factors that affect the option value. Because these derivatives are (almost all) denoted by Greek letters, they are often called the Greeks : rho measures interest rate sensitivity, theta measures the sensitivity to changes in time to maturity, vega measures the sensitivity to changes in volatility. As one can see from figure 5, the price of a call option increases with longer time to maturity (panel (b)), with higher interest rate (panel (c)), and with higher variance (panel (d)). The price is roughly linear in these factors so that a first-order approximation can be reasonable. The impact of time to maturity is intuitive: more time left means more chances of the option to move very far into the money. More volatility in the underlying asset has the same impact on the value of the option: since the downside of the call value is limited by the zero floor but the upside is unlimited more variance increases the value of the option. If you are interested in exploring these relations further, use the Excel application referred to in footnote 3 or have a look at the website: aspx? References Black, Fisher, and Myron Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy 81, Bodie, Zvi, Alex Kane, and Alan J. Marcus, 2008, Investments (Irwin McGraw-Hill: Chicago). Cox, J., S. Ross, and M. Rubinstein, 1979, Option pricing: A simplified approach, Journal of Financial Economics 7,

17 17 Elton, Edwin J., and Martin J. Gruber, 1995, Modern Portfolio Theory and Investment Analysis (Wiley: New York). Rendleman, Richard J., and Brit J. Bartter, 1979, Two-state option pricing, Journal of Finance 34,

18 18 call price call price S-PV(X) ('intrinsic value') price of underlying asset S 0 time to maturity T (a) Price of call option as a function of underlying stock price (b) Price of call option as a function of time to maturity call price call price risk-free rate r f return variance of underlying asset (c) Price of call option as a function of risk-free rate (d) Price of call option as a function of underlying asset s Figure 5: Black-Scholes prices for a non-dividend paying call option return variance

Chapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.

Chapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options. Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of risk-adjusted discount rate. Part D Introduction to derivatives. Forwards

More information

Options: Valuation and (No) Arbitrage

Options: Valuation and (No) Arbitrage Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial

More information

Two-State Option Pricing

Two-State Option Pricing Rendleman and Bartter [1] present a simple two-state model of option pricing. The states of the world evolve like the branches of a tree. Given the current state, there are two possible states next period.

More information

2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold

2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold Chapter 21 : Options-1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. Black-Scholes

More information

Caput Derivatives: October 30, 2003

Caput Derivatives: October 30, 2003 Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor

More information

The Binomial Option Pricing Model André Farber

The Binomial Option Pricing Model André Farber 1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a non-dividend paying stock whose price is initially S 0. Divide time into small

More information

Example 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).

Example 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r). Chapter 4 Put-Call Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.

More information

Model-Free Boundaries of Option Time Value and Early Exercise Premium

Model-Free Boundaries of Option Time Value and Early Exercise Premium Model-Free Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 33124-6552 Phone: 305-284-1885 Fax: 305-284-4800

More information

Option Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values

Option Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X

More information

Introduction to Binomial Trees

Introduction to Binomial Trees 11 C H A P T E R Introduction to Binomial Trees A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram that represents di erent possible paths

More information

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.

More information

Overview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies

Overview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies Options and Derivatives Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Overview Option basics and option strategies No-arbitrage bounds on option prices Binomial option pricing Black-Scholes-Merton

More information

Option Valuation. Chapter 21

Option Valuation. Chapter 21 Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price

More information

One Period Binomial Model

One Period Binomial Model FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing

More information

Two-State Model of Option Pricing

Two-State Model of Option Pricing Rendleman and Bartter [1] put forward a simple two-state model of option pricing. As in the Black-Scholes model, to buy the stock and to sell the call in the hedge ratio obtains a risk-free portfolio.

More information

Option pricing. Vinod Kothari

Option pricing. Vinod Kothari Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate

More information

Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration

Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put:

More information

Lecture 5: Put - Call Parity

Lecture 5: Put - Call Parity Lecture 5: Put - Call Parity Reading: J.C.Hull, Chapter 9 Reminder: basic assumptions 1. There are no arbitrage opportunities, i.e. no party can get a riskless profit. 2. Borrowing and lending are possible

More information

BINOMIAL OPTION PRICING

BINOMIAL OPTION PRICING Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex option-pricing

More information

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald) Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared

More information

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the

More information

7: The CRR Market Model

7: The CRR Market Model Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The Cox-Ross-Rubinstein

More information

Lecture 21 Options Pricing

Lecture 21 Options Pricing Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Put-call

More information

Options/1. Prof. Ian Giddy

Options/1. Prof. Ian Giddy Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Options2

More information

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12. Week 9 Binomial Trees : Hull, Ch. 12. 1 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral

More information

Lecture 3: Put Options and Distribution-Free Results

Lecture 3: Put Options and Distribution-Free Results OPTIONS and FUTURES Lecture 3: Put Options and Distribution-Free Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distribution-free results? option

More information

Consider a European call option maturing at time T

Consider a European call option maturing at time T Lecture 10: Multi-period Model Options Black-Scholes-Merton model Prof. Markus K. Brunnermeier 1 Binomial Option Pricing Consider a European call option maturing at time T with ihstrike K: C T =max(s T

More information

Options Pricing. This is sometimes referred to as the intrinsic value of the option.

Options Pricing. This is sometimes referred to as the intrinsic value of the option. Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the Put-Call Parity Relationship. I. Preliminary Material Recall the payoff

More information

Options. + Concepts and Buzzwords. Readings. Put-Call Parity Volatility Effects

Options. + Concepts and Buzzwords. Readings. Put-Call Parity Volatility Effects + Options + Concepts and Buzzwords Put-Call Parity Volatility Effects Call, put, European, American, underlying asset, strike price, expiration date Readings Tuckman, Chapter 19 Veronesi, Chapter 6 Options

More information

CHAPTER 21: OPTION VALUATION

CHAPTER 21: OPTION VALUATION CHAPTER 21: OPTION VALUATION PROBLEM SETS 1. The value of a put option also increases with the volatility of the stock. We see this from the put-call parity theorem as follows: P = C S + PV(X) + PV(Dividends)

More information

Introduction to Options. Derivatives

Introduction to Options. Derivatives Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived

More information

Lecture 4: The Black-Scholes model

Lecture 4: The Black-Scholes model OPTIONS and FUTURES Lecture 4: The Black-Scholes model Philip H. Dybvig Washington University in Saint Louis Black-Scholes option pricing model Lognormal price process Call price Put price Using Black-Scholes

More information

Options Markets: Introduction

Options Markets: Introduction Options Markets: Introduction Chapter 20 Option Contracts call option = contract that gives the holder the right to purchase an asset at a specified price, on or before a certain date put option = contract

More information

Exam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D

Exam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS

More information

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

More information

American and European. Put Option

American and European. Put Option American and European Put Option Analytical Finance I Kinda Sumlaji 1 Table of Contents: 1. Introduction... 3 2. Option Style... 4 3. Put Option 4 3.1 Definition 4 3.2 Payoff at Maturity... 4 3.3 Example

More information

Part V: Option Pricing Basics

Part V: Option Pricing Basics erivatives & Risk Management First Week: Part A: Option Fundamentals payoffs market microstructure Next 2 Weeks: Part B: Option Pricing fundamentals: intrinsic vs. time value, put-call parity introduction

More information

OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options

OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (risk-neutral)

More information

Option Basics. c 2012 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 153

Option Basics. c 2012 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 153 Option Basics c 2012 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 153 The shift toward options as the center of gravity of finance [... ] Merton H. Miller (1923 2000) c 2012 Prof. Yuh-Dauh Lyuu,

More information

Martingale Pricing Applied to Options, Forwards and Futures

Martingale Pricing Applied to Options, Forwards and Futures IEOR E4706: Financial Engineering: Discrete-Time Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the

More information

Binomial lattice model for stock prices

Binomial lattice model for stock prices Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }

More information

Finance 436 Futures and Options Review Notes for Final Exam. Chapter 9

Finance 436 Futures and Options Review Notes for Final Exam. Chapter 9 Finance 436 Futures and Options Review Notes for Final Exam Chapter 9 1. Options: call options vs. put options, American options vs. European options 2. Characteristics: option premium, option type, underlying

More information

The Black-Scholes Formula

The Black-Scholes Formula FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

More information

Financial Options: Pricing and Hedging

Financial Options: Pricing and Hedging Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equity-linked securities requires an understanding of financial

More information

CHAPTER 7: PROPERTIES OF STOCK OPTION PRICES

CHAPTER 7: PROPERTIES OF STOCK OPTION PRICES CHAPER 7: PROPERIES OF SOCK OPION PRICES 7.1 Factors Affecting Option Prices able 7.1 Summary of the Effect on the Price of a Stock Option of Increasing One Variable While Keeping All Other Fixed Variable

More information

FINANCIAL OPTION ANALYSIS HANDOUTS

FINANCIAL 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 information

EC3070 FINANCIAL DERIVATIVES

EC3070 FINANCIAL DERIVATIVES BINOMIAL OPTION PRICING MODEL A One-Step Binomial Model The Binomial Option Pricing Model is a simple device that is used for determining the price c τ 0 that should be attributed initially to a call option

More information

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008. Options

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008. Options FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options the so-called plain vanilla options. We consider the payoffs to these

More information

S 1 S 2. Options and Other Derivatives

S 1 S 2. Options and Other Derivatives Options and Other Derivatives The One-Period Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating

More information

Convenient Conventions

Convenient Conventions C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff

More information

Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options

Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder

More information

Lectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week

Lectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week Lectures Sergei Fedotov 20912 - Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics

More information

Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008

Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 : A Stern School of Business New York University Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 Outline 1 2 3 4 5 6 se notes review the principles underlying option pricing and some of

More information

1 The Black-Scholes model: extensions and hedging

1 The Black-Scholes model: extensions and hedging 1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes

More information

Call Price as a Function of the Stock Price

Call Price as a Function of the Stock Price Call Price as a Function of the Stock Price Intuitively, the call price should be an increasing function of the stock price. This relationship allows one to develop a theory of option pricing, derived

More information

Finance 400 A. Penati - G. Pennacchi. Option Pricing

Finance 400 A. Penati - G. Pennacchi. Option Pricing Finance 400 A. Penati - G. Pennacchi Option Pricing Earlier we derived general pricing relationships for contingent claims in terms of an equilibrium stochastic discount factor or in terms of elementary

More information

Chapter 1: Financial Markets and Financial Derivatives

Chapter 1: Financial Markets and Financial Derivatives Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange

More information

Session X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London

Session X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London Session X: Options: Hedging, Insurance and Trading Strategies Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Option

More information

Session IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics

Session IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock

More information

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t. LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

More information

Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS 8-1 a. An option is a contract which gives its holder the right to buy or sell an asset at some predetermined

More information

Lecture 9. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 8

Lecture 9. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 8 Lecture 9 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 8 Lecture 9 1 Risk-Neutral Valuation 2 Risk-Neutral World 3 Two-Steps Binomial

More information

Numerical Methods for Option Pricing

Numerical Methods for Option Pricing Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly

More information

Lecture 12. Options Strategies

Lecture 12. Options Strategies Lecture 12. Options Strategies Introduction to Options Strategies Options, Futures, Derivatives 10/15/07 back to start 1 Solutions Problem 6:23: Assume that a bank can borrow or lend money at the same

More information

Hedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging

Hedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in

More information

Likewise, the payoff of the better-of-two note may be decomposed as follows: Price of gold (US$/oz) 375 400 425 450 475 500 525 550 575 600 Oil price

Likewise, the payoff of the better-of-two note may be decomposed as follows: Price of gold (US$/oz) 375 400 425 450 475 500 525 550 575 600 Oil price Exchange Options Consider the Double Index Bull (DIB) note, which is suited to investors who believe that two indices will rally over a given term. The note typically pays no coupons and has a redemption

More information

Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model

More information

Betting on Volatility: A Delta Hedging Approach. Liang Zhong

Betting on Volatility: A Delta Hedging Approach. Liang Zhong Betting on Volatility: A Delta Hedging Approach Liang Zhong Department of Mathematics, KTH, Stockholm, Sweden April, 211 Abstract In the financial market, investors prefer to estimate the stock price

More information

CHAPTER 22 Options and Corporate Finance

CHAPTER 22 Options and Corporate Finance CHAPTER 22 Options and Corporate Finance Multiple Choice Questions: I. DEFINITIONS OPTIONS a 1. A financial contract that gives its owner the right, but not the obligation, to buy or sell a specified asset

More information

FINANCIAL ECONOMICS OPTION PRICING

FINANCIAL ECONOMICS OPTION PRICING OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.

More information

Option Premium = Intrinsic. Speculative Value. Value

Option Premium = Intrinsic. Speculative Value. Value Chapters 4/ Part Options: Basic Concepts Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options An Option-Pricing Formula Investment in

More information

Lecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model

Lecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model Brunel University Msc., EC5504, Financial Engineering Prof Menelaos Karanasos Lecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model Recall that the price of an option is equal to

More information

CHAPTER 21: OPTION VALUATION

CHAPTER 21: OPTION VALUATION CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given

More information

GAMMA.0279 THETA 8.9173 VEGA 9.9144 RHO 3.5985

GAMMA.0279 THETA 8.9173 VEGA 9.9144 RHO 3.5985 14 Option Sensitivities and Option Hedging Answers to Questions and Problems 1. Consider Call A, with: X $70; r 0.06; T t 90 days; 0.4; and S $60. Compute the price, DELTA, GAMMA, THETA, VEGA, and RHO

More information

ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs)

ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs) Problem ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 0,, 2, 8. October 2, 200 (Thurs) (i) The current exchange rate is 0.0$/. (ii) A four-year dollar-denominated European put option

More information

CHAPTER 15. Option Valuation

CHAPTER 15. Option Valuation CHAPTER 15 Option Valuation Just what is an option worth? Actually, this is one of the more difficult questions in finance. Option valuation is an esoteric area of finance since it often involves complex

More information

Lecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the

More information

CHAPTER 5 OPTION PRICING THEORY AND MODELS

CHAPTER 5 OPTION PRICING THEORY AND MODELS 1 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. In this section, we will consider an exception to that rule

More information

EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals

EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals R. E. Bailey Department of Economics University of Essex Outline Contents 1 Call options and put options 1 2 Payoffs on options

More information

Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13

Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13 Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The Black-Scholes-Merton Random Walk Assumption

More information

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model 1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American

More information

1 The Black-Scholes Formula

1 The Black-Scholes Formula 1 The Black-Scholes Formula In 1973 Fischer Black and Myron Scholes published a formula - the Black-Scholes formula - for computing the theoretical price of a European call option on a stock. Their paper,

More information

TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13

TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:

More information

Chapter 21 Valuing Options

Chapter 21 Valuing Options Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher

More information

BINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract

BINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple

More information

Lecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12

Lecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12 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 information

American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options

American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus

More information

The Greeks Vega. Outline: Explanation of the greeks. Using greeks for short term prediction. How to find vega. Factors influencing vega.

The Greeks Vega. Outline: Explanation of the greeks. Using greeks for short term prediction. How to find vega. Factors influencing vega. The Greeks Vega 1 1 The Greeks Vega Outline: Explanation of the greeks. Using greeks for short term prediction. How to find vega. Factors influencing vega. 2 Outline continued: Using greeks to shield your

More information

FAIR VALUATION OF THE SURRENDER OPTION EMBEDDED IN A GUARANTEED LIFE INSURANCE PARTICIPATING POLICY. Anna Rita Bacinello

FAIR VALUATION OF THE SURRENDER OPTION EMBEDDED IN A GUARANTEED LIFE INSURANCE PARTICIPATING POLICY. Anna Rita Bacinello FAIR VALUATION OF THE SURRENDER OPTION EMBEDDED IN A GUARANTEED LIFE INSURANCE PARTICIPATING POLICY Anna Rita Bacinello Dipartimento di Matematica Applicata alle Scienze Economiche, Statistiche ed Attuariali

More information

10 Binomial Trees. 10.1 One-step model. 1. Model structure. ECG590I Asset Pricing. Lecture 10: Binomial Trees 1

10 Binomial Trees. 10.1 One-step model. 1. Model structure. ECG590I Asset Pricing. Lecture 10: Binomial Trees 1 ECG590I Asset Pricing. Lecture 10: Binomial Trees 1 10 Binomial Trees 10.1 One-step model 1. Model structure ECG590I Asset Pricing. Lecture 10: Binomial Trees 2 There is only one time interval (t 0, t

More information

Other variables as arguments besides S. Want those other variables to be observables.

Other variables as arguments besides S. Want those other variables to be observables. Valuation of options before expiration Need to distinguish between American and European options. Consider European options with time t until expiration. Value now of receiving c T at expiration? (Value

More information

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial

More information

Reference Manual Equity Options

Reference Manual Equity Options Reference Manual Equity Options TMX Group Equities Toronto Stock Exchange TSX Venture Exchange Equicom Derivatives Montréal Exchange CDCC Montréal Climate Exchange Fixed Income Shorcan Energy NGX Data

More information

Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

More information

Week 13 Introduction to the Greeks and Portfolio Management:

Week 13 Introduction to the Greeks and Portfolio Management: Week 13 Introduction to the Greeks and Portfolio Management: Hull, Ch. 17; Poitras, Ch.9: I, IIA, IIB, III. 1 Introduction to the Greeks and Portfolio Management Objective: To explain how derivative portfolios

More information

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13.

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13. Week 11 The Black-Scholes Model: Hull, Ch. 13. 1 The Black-Scholes Model Objective: To show how the Black-Scholes formula is derived and how it can be used to value options. 2 The Black-Scholes Model 1.

More information

Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance. Brennan Schwartz (1976,1979) Brennan Schwartz

Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance. Brennan Schwartz (1976,1979) Brennan Schwartz Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance Brennan Schwartz (976,979) Brennan Schwartz 04 2005 6. Introduction Compared to traditional insurance products, one distinguishing

More information

Stock. Call. Put. Bond. Option Fundamentals

Stock. Call. Put. Bond. Option Fundamentals Option Fundamentals Payoff Diagrams hese are the basic building blocks of financial engineering. hey represent the payoffs or terminal values of various investment choices. We shall assume that the maturity

More information

Chapter 5 Financial Forwards and Futures

Chapter 5 Financial Forwards and Futures Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment

More information