I have posted Problem set 4 on the course website. This problem set is based upon the “Properties of Stock Options” reading, and it consists of four problems. It is due at the beginning of class on Tuesday, February 12.
The fourth problem in this problem set references an Excel spreadsheet template called “Derivagem” which you can download from http://fin4366.garven.com/spring2019/DG300.xls. Open DG300.xls up in Excel, and you’ll encounter a dialog box which looks something like this:
Click on “Enable Macros”, and once the spreadsheet is open, select the “Equity_FX_Index_Futures_Options” worksheet. The upper left corner of the spreadsheet is where you input the data for the problem. For this problem, “Underlying Type” is “Equity”, “Option Type” is “Black-Scholes – European”, a $.50 dividend is paid in 6 months, which corresponds to .5 year, the current stock price is $41, volatility is 35%, and the risk-free rate of interest is 6%, the option’s life is 1 year, and the strike or exercise price is $40:
Once you have input these data, select the call button for the call option price and the put button for the put option price. Depending on whether you select “Put” or “Call”, when you click on “Calculate”, the spreadsheet will report back put and call option prices based on the Black-Scholes-Merton (BSM) option pricing formula. For now, this formula may be somewhat of a black box for many (if not most) of you, which is a “problem” we will surely rectify in the not-too-distant future in Finance 4366.
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:
- 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.
- 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.
- 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.
- 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., 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.
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 . Also suppose that today’s and next period’s riskless bond price is . Furthermore, a call option with an exercise price of K = $1 is worth and . Let correspond to the number of shares of stock in the replicating portfolio, and correspond to the number of bonds. It follows that the value of the replicating portfolio in state g, and the value of the replicating portfolio in state b, Thus we have two equations in two unknowns:
By subtracting the equation for from the equation for , we obtain , and , which implies that Since the payoff on the replicating portfolio is identical to the payoff on the long call position, it follows that 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 and , this implies that and . In state g, the value of the replicating portfolio and the value of the replicating portfolio in state b, Thus we have two equations in two unknowns:
By subtracting the equation for from the equation for , we obtain , and , which implies that Since the payoff on the replicating portfolio is identical to the payoff on the long put position, it follows that represents the arbitrage-free price for the put option.
The notion of “arbitrage-free” pricing is important not only in Finance 4366, but also in your other finance studies. In Finance 4366, we take it as given that investors are risk averse. However, it turns out that we don’t need to Continue reading On the importance of “arbitrage-free” pricing in finance
For a non-technical introduction to forward and futures contracts, it’s hard to beat the following video tutorial on this topic:
This week’s Intelligent Investor column in the Wall Street Journal presents an homage to the memory of Jack Bogle, the founder of Vanguard Group. Mr. Bogle passed away this past Wednesday at the age of 89, and as the inventor of index investing, he is arguably one of the most important public figures in the practice of finance of the past 50 years. Burton Malkiel’s WSJ op-ed in today’s paper entitled “The Secrets of Jack Bogle’s Investment Success” is also a must read!