All posts by jgarven

Formula sheet for midterm 2

I have posted the formula sheet for midterm 2 at http://fin4335.garven.com/fall2018/formulas_part2.pdf; this formula sheet is also linked http://fin4335.garven.com/formula-sheets/ (see item 2).

During tomorrow’s Finance 4335 class meeting, we’ll have a review session for the 2nd midterm which will be given in class this coming Thursday, November 8. I highly recommend that everyone work on the 2nd midterm exam from the Spring 2018 semester; I will post solutions for this sample exam after tomorrow’s class meeting. Also, it would be a good idea to review Problem Set 6, Problem Set 7, and Problem Set 8 (solutions are available at http://risk.garven.com/category/problem-set-solutions/).

Cox-Ross-Rubinstein (CRR) option pricing spreadsheet

Here’s a spreadsheet which calculates call and put prices using the CRR framework for n = 1, 2, 3, and 4 timesteps.  For discussion of this method, see pp. 9-11 of http://fin4335.garven.com/fall2018/lecture15.pdf lecture note which we covered in class yesterday.

The key idea here involves determining option payoffs without having to recreate the entire stock binomial tree which is shown on page 9 of the above referenced lecture note.  The way this is done involves determining for each call option the minimum number of up moves required in order to determine the nodes at which the option will be in-the-money.  By rounding up to the nearest integer “a” the value of \ln (K/S{d^n})/\ln (u/d) for n = 1, 2, 3, and 4 (see rows 11 and 12 below), we determine that the 1 timestep call will only be in the money at the u node, the 2 timestep call will only be in the money at the uu node, the 3 timestep call will only be in the money at the uuu and uud nodes, and the 4 timestep call will only be in the money at the uuuu and uuud nodes.  This greatly simplifies the calculations for call prices because we know that the call payoffs will all be zero at all other nodes (see row 13 below).  Having determined the arbitrage-free call prices, then we find the arbitrage-free put prices by applying the put call parity equation P = C + K{e^{ - rn\delta t}} - S (you can bring up the spreadsheet by clicking on the screenshot below):

Black-Scholes-Merton option pricing spreadsheet

Here’s a copy of the spreadsheet that I created during the section 1 (11-12:15) meeting of Finance 4335 today (you can bring up the spreadsheet by clicking on the screenshot below):In order to calculate the arbitrage-free price of a call option, we need to solve the following equation:

C = SN({d_1}) - K{e^{ - rT}}N({d_2}),

where {d_1} = \displaystyle\frac{{\ln (S/K) + (r + .5{\sigma ^2})T}}{{\sigma \sqrt T }} and {d_2} = {d_1} - \sigma \sqrt T .. The arbitrage-free price for the put may be obtained by applying the put-call parity equation.  By doing so, we obtain the put pricing equation for an otherwise identical (same underlying, same exercise price, same time to expiration) put:

P = K{e^{ - rT}}N( - {d_2}) - SN( - {d_1}),

where N( - {d_1}) = 1 - N({d_1}) and N( - {d_2}) = 1 - N({d_2}).

For calls and puts, we need five parameter values: S (current underlying asset price), K (exercise price, \sigma (volatility of underlying asset return), T (time to expiration, measured in number of years), and r (the annualized riskless rate of interest). These parameters all show up in cells B1:B5. The first step to solving call and put prices requires determining the values for {d_1}, {d_2}, N({d_1}), N({d_2}),N( - {d_1}), and N( - {d_2}); these values are listed in cells B7:B12. Once we obtain this information, it’s simply a matter of coding the equations for the BSM call and put prices; these prices show up in cells E1:E2.

Finally, I also include the components of the replicating portfolios for the call and put options. Of course, the values of these portfolios must be the same as the call and put option values; otherwise, there would be riskless arbitrage opportunities. As we showed way back during class on Tuesday, October 23, one replicates a call option by buying delta units of the underlying asset on margin, whereas one replicates a put option by shorting delta units of the underlying asset and lending money. In the Black-Scholes-Merton pricing model, the call delta corresponds to N({d_1}), whereas the put delta corresponds to N({-d_1}).

Merton: Applications of Option-Pricing Theory (shameless self-promotion alert)…

During class on Thursday, we will show how the multi-timestep binomial option pricing formula which we studied during today’s class meeting converges into the famous (and Nobel-prize winning) Black-Scholes-Merton option pricing formula. Speaking of the Black-Scholes-Merton option pricing formula, allow me to shamelessly plug a journal article that I published early in my academic career which Professor Robert C. Merton cites in his Nobel Prize lecture (Merton shared the Nobel Prize in economics in 1997 with Myron Scholes “for a new method to determine the value of derivatives”).

Here’s the citation (and link) to Merton’s lecture:

Merton, Robert C., 1998, Applications of Option-Pricing Theory: Twenty-Five Years Later, The American Economic Review, Vol. 88, No. 3 (Jun. 1998), pp. 323-349.

See page 337, footnote 11 of Merton’s paper for the reference to Neil A. Doherty and James R. Garven (1986)… (Doherty and I “pioneered” the application of a somewhat modified version of the Black-Scholes-Merton model to the pricing of insurance; thus Merton’s reference to our Journal of Finance paper in his Nobel Prize lecture)…

Today’s class problem solutions for parts A-D applying risk neutral valuation

Today, we discussed the risk neutral valuation approach to pricing options, and today’s class problem assignment was to work parts A through D of the  Option Pricing Class Problem, relying solely upon the risk neutral valuation approach.

As I pointed out during today’s class meeting, the replicating portfolio and delta hedging approaches both imply that a risk neutral valuation exists between an option (both the call and put varieties) and its underlying asset.  This is analytically shown in sections 5 and 6 (located on pp. 8-9) of my Binomial Option Pricing Model (single-period) teaching note which was assigned for October 23.  A particularly useful advantage of the risk neutral valuation approach (compared with the replicating portfolio and delta hedging approaches) is that it is computationally simpler and particularly well suited for modeling multi-timestep option pricing problems.

Here are the solutions for parts A through D (click on the image for a full-size PDF version that you can print out):