On the role of replicating portfolios in the pricing of financial derivatives in general

Replicating portfolios play a central role in terms of pricing financial derivatives. Here is what we have learned so far about replicating portfolios in Finance 4366:

  1. Buying forward is equivalent to buying the underlying on margin, and selling forward is equivalent to shorting the underlying and lending money. Like options, forwards and futures are priced by pricing the replicating portfolio and invoking the “no-arbitrage” condition. If the forward/futures price it too low, then one can earn positive returns with zero risk and zero net investment by buying forward, shorting the underlying and lending money. Similarly, if the forward futures price is too high, one can earn positive returns with zero risk and zero net investment by selling forward and buying the underlying with borrowed money. This is commonly referred to as “riskless arbitrage”; it’s riskless because you’re perfectly hedged, and it’s arbitrage because you are buying low and selling high.
  2. The replicating portfolio for a call option is a margined investment in the underlying. For example, in my teaching note entitled “A Simple Model of a Financial Market”, I provide a numerical example where the interest rate is zero, there are two states of the world, a bond which pays off $1 in both states is worth $1 today, and a stock that pays off $2 in one state and $.50 in the other state is also worth one dollar. In that example, the replicating portfolio for a European call option with an exercise price of $1 consists of 2/3 of 1 share of stock (costing $0.66) and a margin balance consisting of a short position in 1/3 of a bond (which is worth -$0.33). Thus, the value of the call option is $0.66 – $0.33 = $0.33.
  3. Since the replicating portfolio for a call option is a margined investment in the underlying, it should come as no surprise that the replicating portfolio for a put option consists of a short position in the underlying combined with lending. Thus, in order to price the put, you need to determine and price the components of the replicating portfolio; we will begin class tomorrow by determining the the relative weightings (delta and beta) for the put’s replicating portfolio.
  4. If you know the value of a call, the underlying, and the present value of the exercise price, then you can use the put-call parity equation to figure out the price for the put option; i.e., {C_0} + PV(K) = {P_0} + {S_0} \Rightarrow {P_0} = {C_0} + PV(K) - {S_0}. Since we know the price of the call ($0.33), the present value of the exercise price ($1), and the stock price ($1), then it follows from the put-call parity equation that the value of the put is also 33 cents. More generally, if you know the values of three of the four securities that are included in the put-call parity equation, then you can infer the “no-arbitrage” value of the fourth security.

Replicating portfolios for Call and Put options

As we discussed during today’s class meeting, the replicating portfolios for long calls and long puts resemble the replicating portfolios for long forwards and short forwards. Specifically, the replicating portfolio for a long call comprises a margined investment in the underlying, whereas the replicating portfolio for the long put involves shorting the underlying and lending money. The primary difference between the replicating portfolios for forward contracts vis-a-vis replicating portfolios for options is that replicating portfolios for options involve fractional long/short positions in the underlying combined with fractional short/long positions in a riskless bond, whereas replicating portfolios for forward contracts require unitary positions (i.e., 100% long/short position in the underlying combined with 100% short/long position in a riskless bond).

My teaching note entitled “A Simple Model of a Financial Market” lays out the logic behind these claims concerning replicating portfolios for calls and puts. Suppose two states of the economy occur 1 period from now: good (g) and bad (b). In state g, a share of stock is worth $2 whereas in state b, a share of stock is worth $0.50; today’s price for a share of stock is {S_0} = \$1. Also suppose that today’s and next period’s riskless bond price is {B_0} = {B_1} = \$1. Furthermore, a call option with an exercise price of K = $1 is worth {C_{1,g}} = \$1 and {C_{1,b}} = \$0. Let {\Delta} correspond to the number of shares of stock in the replicating portfolio, and {\beta} correspond to the number of bonds. It follows that the value of the replicating portfolio in state g, {V_{1,g}} = \Delta({S_{1,g}}) + {\beta}({B_1}) = \Delta(\$2) + {\beta}(\$1) = \$1, and the value of the replicating portfolio in state b, {V_{1,b}} = \Delta({S_{1,b}}) + {\beta}({B_1}) = \Delta(\$.5) + {\beta}(\$1) = 0. Thus we have two equations in two unknowns:

{V_{1,g}} = \Delta (\$2) + \beta(\$1) = \$1, and {V_{1,b}} = \Delta (\$0.50) + \beta(\$1) = \$0.

By subtracting the equation for {V_{1,b}} from the equation for {V_{1,g}}, we obtain \Delta = 2/3, and {\beta} = -1/3, which implies that {V_0} = \Delta ({S_0}) + \beta ({B_0}) = 2/3(1) - 1/3(1) = \$1/3. Since the payoff on the replicating portfolio is identical to the payoff on the long call position, it follows that {V_0} = {C_0} = \$1/3 represents the arbitrage-free price for the call option.

Next, consider the replicating portfolio for the put option with an exercise price of K = $1. Since {S_{1,g}} = \$2 and {S_{1,b}} = \$0.50, this implies that {P_{1,g}} = \$0 and {P_{1,b}} = \$0.50. In state g, the value of the replicating portfolio {V_{1,g}} = \Delta({S_{1,g}}) + {\beta}({B_1}) = \Delta(\$2) + {\beta}(\$1) = \$0, and the value of the replicating portfolio in state b, {V_{1,b}} = \Delta({S_{1,b}}) + {\beta}({B_1}) = \Delta(\$.5) + {\beta}(\$1) = \$0.50 Thus we have two equations in two unknowns:

{V_{1,g}} = \Delta (\$2) + \beta(\$1) = \$0, and {V_{1,b}} = \Delta (\$0.50) + \beta(\$1) = \$0.50.

By subtracting the equation for {V_{1,b}} from the equation for {V_{1,g}}, we obtain \Delta = -1/3, and {\beta} = 2/3, which implies that {V_0} = \Delta ({S_0}) + \beta ({B_0}) = -1/3(1) + 2/3(1) = \$1/3. Since the payoff on the replicating portfolio is identical to the payoff on the long put position, it follows that {V_0} = {P_0} = \$1/3 represents the arbitrage-free price for the put option.

2019 CFA exam scholarship opportunity (Deadline: January 30, 2019)

In case if any of you are planning on sitting for CFA (Chartered Financial Analyst) exams this coming June and December, you might consider taking advantage of this scholarship opportunity:

Student/Awareness Scholarships

June 2019 and December 2019 CFA Exams

What is it?

The CFA Institute grants “Affiliated Schools” such as Baylor several scholarships per year in accordance with Official Scholarships Rules. Scholarship awards reduce the CFA Program enrollment and exam registration fees to $350, which includes the eBook curriculum.

How to apply?

If you are not enrolled in the CFA Program, you must create a CFA Institute account in order to receive your login information and access the scholarship application. The online application can be found here: https://www.cfainstitute.org/en/programs/cfa/scholarships/student

When is the application deadline?

Candidates should (1) submit the application form on the CFA Institute website and (2) email Brandon_Troegle@baylor.edu prior to January 30, 2019, though earlier applications are encouraged. In the body of the email, please include:

  • A summary statement on why you should be considered for the scholarship. This statement should include what you hope to achieve by pursuing the CFA charter, your career goals, and you can discuss academic achievements/performance including GPA information in you wish.
  • Expected graduation date
  • Major(s)
  • Did you apply for the Access Scholarship (the other scholarship)? If not, why not?
  • Any other information you believe will aid in the scholarship decision

What is the evaluation process and criteria?

Awards will be made based on a combination of factors including interest in and rationale for pursuing the CFA charter, academic accomplishments, and other personal characteristics that indicate the applicant is a strong scholarship candidate.

The Birthday Paradox: an interesting probability problem involving “statistically independent” events

This past Thursday, we discussed the concept of statistical independence and focused attention on some important implications of statistical independence for probability distributions such as the binomial and normal distributions.

Here, I’d like to call everyone’s attention to an interesting (non-finance) probability problem related to statistical independence. Specifically, consider the so-called “Birthday Paradox”. The Birthday Paradox pertains to the probability that in a set of randomly chosen people, some pair of them will have the same birthday. Counter-intuitively, in a group of 23 randomly chosen people, there is slightly more than a 50% probability that some pair of them will both have been born on the same day.

To compute the probability that two people in a group of n people have the same birthday, we disregard variations in the distribution, such as leap years, twins, seasonal or weekday variations, and assume that the 365 possible birthdays are equally likely.[1] Thus, we assume that birth dates are statistically independent events. Consequently, the probability of two randomly chosen people not sharing the same birthday is 364/365. According to the combinatorial equation, the number of unique pairs in a group of n people is n!/2!(n-2)! = n(n-1)/2. Assuming a uniform distribution (i.e., that all dates are equally probable), this means that the probability that no pair in a group of n people shares the same birthday is equal to p(n) = (364/365)^[n(n-1)/2]. The event of at least two of the n persons having the same birthday is complementary to all n birthdays being different. Therefore, its probability is p’(n) = 1 – (364/365)^[n(n-1)/2].

Given these assumptions, suppose that we are interested in determining how many randomly chosen people are needed in order for there to be a 50% probability that at least two persons share the same birthday. In other words, we are interested in finding the value of n which causes p(n) to equal 0.50. Therefore, 0.50 = (364/365)^[n(n-1)/2]; taking natural logs of both sides and rearranging, we obtain (ln 0.50)/(ln 364/365) = n(n-1)/2. Solving for n, we obtain 505.304 = n(n -1); therefore, n is approximately equal to 23.[2]

The following graph illustrates how the probability that a pair of people share the same birthday varies as the number of people in the sample increases:

New Picture (1)

[1] It is worthwhile noting that real-life birthday distributions are not uniform since not all dates are equally likely. For example, in the northern hemisphere, many children are born in the summer, especially during the months of August and September. In the United States, many children are conceived around the holidays of Christmas and New Year’s Day. Also, because hospitals rarely schedule C-sections and induced labor on the weekend, more Americans are born on Mondays and Tuesdays than on weekends; where many of the people share a birth year (e.g., a class in a school), this creates a tendency toward particular dates. Both of these factors tend to increase the chance of identical birth dates, since a denser subset has more possible pairs (in the extreme case when everyone was born on three days of the week, there would obviously be many identical birthdays!).

[2]Note that since 26 students are enrolled in Finance 4366 this semester, this implies that the probability that two Finance 4366 students share the same birthday is roughly p’(26) = 1 – (364/365)^[26(25)/2] = 59%.

Finance 4366