All posts by jgarven

Option Pricing Class Problem (part 3)

The complete solutions for the Option Pricing Class Problem are available in the Option Pricing Class Problem Solutions document.  Parts A and B feature solutions for put and call prices using the replicating portfolio, delta hedging, and risk neutral valuation methods.  Parts C and D feature solutions for longer dated (2 timesteps instead of just 1) puts and calls.  There, I rely upon the risk neutral valuation approach along with put-call parity.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty. Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden. For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

Option Pricing Class Problem (part 2)

This is the 2nd blog posting about Option Pricing Class Problem that I passed out in class on Thursday, November 7. Here’s the (replicating portfolio) solution to part A, which requires the calculation of the price of a one-year European put option on Ripple, Inc. stock, where = $56, = $60, r = 4%, \delta t = 1, u = 1.3, and d = .9:

According to the Replicating Portfolio Approach for pricing this put option, \Delta = \displaystyle\frac{{{P_u} - {P_d}}}{{uS - dS}} = \displaystyle\frac{{0-9.60}}{{72.80 - 50.40}} = -.4286\; and B = \displaystyle\frac{{u{P_d} - d{P_u}}}{{{e^{r\delta t}}(u - d)}} = \frac{{1.3(9.6) - .9(0)}}{{1.0408(.4)}} = 29.98. Then {V_{RP}} = C = \Delta S + B = -.4286(56) + 29.98 = \$5.98.

As we discussed during yesterday’s Finance 4335 class meeting, if you already know the arbitrage-free price of a call option, then the arbitrage-free price of an otherwise identical (European, same underlying (non-dividend paying) asset, same exercise price, and same time to expiration) put option can also be determined by applying the put-call parity equation:

C + K{e^{ - r\delta t}} = P + S \Rightarrow P = C + K{e^{ - r\delta t}} - S = \$4.33 + \$ 60{e^{ - .04}}  - \$ 56 = \$5.98.

As shown above, the put-call parity equation implies that one can create a “synthetic” put option by purchasing an otherwise identical call worth $C and bond worth K{e^{ - r\delta t}}, while also shorting one unit of the underlying asset (which generates proceeds worth $S). Now suppose that P \ne \$ 5.98; specifically, suppose that actual put is worth more (less) than the synthetic put.  If this were to happen, then one could earn riskless arbitrage profit by selling the actual (synthetic) put and buying the synthetic put (actual put).  Thus, we have reconfirmed that $5.98 is indeed the arbitrage-free price for the put. 

Similarly, “synthetic” versions of the call, the bond, and the underlying asset can be created using the following combinations of the other instruments:

  1. Synthetic Call: C = P + S - K{e^{ - r\delta t}} = \$ 5.98 + \$ 56 - \$ 60{e^{ - .04}} = \$ 4.33;
  2. Synthetic Bond: K{e^{ - r\delta t}} = P + S - C = \$ 5.98 + \$ 56 - 4.33 = \$ 60{e^{ - .04}} = \$ 57.65; and
  3. Synthetic Underlying Asset: S = C + K{e^{ - r\delta t}} - P = \$ 4.33 + \$ 57.65 - \$ 5.98 = \$ 56.

During tomorrow’s Finance 4335 class meeting, we will 1) discuss the Delta Hedging and Risk Neutral Valuation approaches to pricing calls and put (see pp. 25-34 of the Derivatives Theory, part 1 lecture note),  2) move on to a discussion of multi-period option pricing formulas (which are presented in the Derivatives Theory, part 2 lecture note), and 3) complete Parts C and D of the Option Pricing Class Problem.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty. Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden. For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

Ed Thorp extra credit opportunity

Here is yet another extra credit opportunity for Finance 4335. You may earn extra credit by listening to and reporting on the 1 hour, 40 minute podcast hosted by Barry Ritholtz’s Masters of Business podcast (link provided below) entitled “Ed Thorp, the Man Who Beat the Dealer and the Market”.  Among other accomplishments, Dr. Thorp  discovered an options pricing formula well before the Black-Scholes formula was ever published.

In order to receive extra credit for this assignment, you must submit (via email sent to fin4335@garven.com) a 1-2 page executive summary of what you learned from this podcast; it is due by no later than 5 p.m. on Friday, November 22. This extra credit assignment will replace your lowest quiz grade in Finance 4335 (assuming the extra credit grade is higher).

Ed Thorp, the Man Who Beat the Dealer and the Market
https://www.stitcher.com/podcast/bloomberg/masters-in-business/e/50784420?autoplay=true

Bloomberg View columnist Barry Ritholtz interviews Ed Thorp, one of the most storied people in finance. A math professor at MIT and UC Ivine, Thorp figured out how to beat Las Vegas at blackjack and baccarat, created statistical arbitrage, and ran a hedge fund that not only beat the market by a wide margin, but never had a losing quarter. He is the author of several books, including “Beat the Dealer” and “Beat the Market”; his latest book is “A Man for All Markets.” Thorp tells Ritholtz that the secret to beating the market is having an edge that’s specific, definable and mathematical. If you don’t, you should be in index funds instead.

Baylor Conversation Series – Yet Another Finance 4335 extra credit opportunity!

I have decided to offer the following extra credit opportunity for Finance 4335. You can earn extra credit by attending and reporting on the upcoming “conversation on civil discourse” featuring Robert P. George and Cornel West. This event is scheduled for Friday, November 15, beginning at 1:30 pm in Waco Hall.

If you decide to take advantage of this opportunity, I will use the grade you earn on your report to replace your lowest quiz grade in Finance 4335 (assuming that your grade on the extra credit is higher than your lowest quiz grade). The report should be in the form of a 1-2 page executive summary in which you provide a critical analysis of the George/West conversation. In order to receive credit, the report must be submitted via email to fin4335@gmail.com in either Word or PDF format by no later than Monday, November 18 at 5 p.m.

How Do Energy Companies Measure the Temperature? Not in Fahrenheit or Celsius

Instead of Fahrenheit or Celsius, a metric called “degree days” is used to capture variability in temperature. The risk management lesson here is that this metric makes it possible to create risk indices which companies can rely upon for pricing and hedging weather-related risks with weather derivatives.

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

Replicating portfolios play a central role in terms of pricing financial derivatives. Here is a succinct summary from today’s class meeting:

  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. During today’s class meeting, we priced a one timestep call option where the price of the underlying asset is $100, the exercise price is also $100, u = 1.05, d = .95, the interest rate r = 5%, and the timestep \delta t = 1/12. Given these parameters, the payoff on the call is $5 at the up (u) node and $0 at the down (d) node. The replicating value consists of half a share that is financed by a margin balance of $47.30; thus the “arbitrage-free” price of the call option is (.5(100) – 47.30) = $2.70.
  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, we need to determine and price the components of the replicating portfolio. We will begin class next Tuesday by reviewing our analysis of the replicating portfolio approach to pricing calls and puts, and move on to other pricing methods such as delta hedging and risk neutral valuation.

Option Pricing Class Problem

Here’s the Option Pricing Class Problem that I passed out in class today.  We will be working more on this class problem during our next class meeting when we examine 1) the replicating portfolio approach to pricing a put option, 2) the “delta hedging” and “risk neutral valuation” approaches to pricing calls and puts, and 3) extending these models from a single period to multiple periods.

Today, we focused our attention on replicating portfolio approaches to pricing forward contracts and (single time-step) European call options.  In the class problem, the latter concept appears in part B.  Here’s the solution to part B:

According to the Replicating Portfolio Approach, \Delta = \displaystyle\frac{{{C_u} - {C_d}}}{{uS - dS}} = \displaystyle\frac{{12.80 - 0}}{{72.80 - 50.40}} = .5714\; and B = \displaystyle\frac{{u{C_d} - d{C_u}}}{{{e^{r\delta t}}(u - d)}} = \frac{{1.3(0) - .9(12.80)}}{{1.0408(.4)}} = - 27.67. Then {V_{RP}} = C = \Delta S + B = .5714(56) - 27.67 = \$ 4.33.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty. Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden. For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

Midterm Exam 2 Booklet and Solutions

The Midterm Exam 2 Booklet and Solutions are now available for downloading from the course website.

Restrictions on Use: Section III.C.16 of Baylor’s Honor Code Policy and Procedures stipulates that using, uploading, or downloading any online resource derived from material pertaining to a Baylor course without the written permission of the professor constitutes an act of academic dishonesty. Since Professor Garven gives no such permission for Finance 4335 course content, this means that the use and redistribution of Finance 4335-related documents involving any source other than Professor Garven are expressly forbidden. For more information on the use restrictions of Finance 4335 course content, see http://bit.ly/4335honorcode.

Next up in Finance 4335…

This Thursday, we will introduce the topic of (financial) derivatives (specifically, options and futures) in Finance 4335. Besides their use in managing price-related risks (for more on this, see “A Beginner’s Guide to Hedging“), the basic principles behind these types of financial instruments also yield insights into how firm-specific risks affect corporate value, which is a concept that we will explore in some detail during upcoming Finance 4335 class meetings.

The assigned readings for class on Thursday include:

1. Derivatives and Options (Doherty, Chapter 6)
2. Teaching the Economics and Convergence of the Binomial and Black-Scholes Option Pricing Formulas, by James R. Garven and James I. Hilliard

I will close by pointing out that besides my lecture notes on Futures and Forwards  (see pp. 5-13 of the Derivatives Theory, part 1 lecture note), it’s hard to beat the following video tutorial on this topic: