quant question

I own an at the money call option on IBM with 1 year to maturity. What is my 1 year expected return?

1) Get a finance book and look up black-scholls model.


2) Look up the value in a newspaper.


3) If you believe that IBM accounting is a bunch of BS, the value of an at the money call of any duration is zero. I personally see more downside than up in IBM calls.

The difference of the price below the expected price - based on IBM's implied volatility - at expiration, would be something more than the risk-free rate, but something less than the volatility-adjusted rate, based purely on the massive SD of the call.


To take a wild guess, I think buyers would equal sellers... when...


It is so easy to hedge, meaning there should be pure interest-rate buyers, but it should also be easy to manufacture.


But you already own it, so its immediate utility is the bid. I would say that, for foregoing that bid - and hoping instead to assign - you will probably expect to make double the risk-free rate over hitting the bid and buying t-bills, and even more than if you rolled it into distressed CP, for instance.


Moreover, your expected return might be slightly lower if it is at-the-money at the cash price.


But I'll have to think about that...


Cool question!


eLROY

1) spell the name right

2) todays value has nothing to do with my expected value

3) This is completely wrong.


Do you know the definition of expected value?

The call option gives you leverage on the underlying. For those who are familiar with Black Scholes, you know that N(d2) the risk neutral probability of exercise. In the d2 formula, change the drift from the risk free rate to the expected drift and you get the probability that your call will expire in the money. For the payout, for all S>K, you have a probability density function. Multiplying P(S(x))*(S(x)-K) will give you the expected payout of your derivative.


What about the 6 month expected return?

Do you plan to hold it to expiration?

Javelin,


It seems you computed the expected payout, but not the immediate popular utility for that future payout. Meaning, you forgot to subtract the immediate bid, and compare the resulting difference to the immediate bid, to get the expected return that payout represents.


So the real question - if I am not mistaken - is what is the immediate bid? Ordinarily, such a volatile payout would command a large risk premium. But natural hedges exist by which you can filter out the brownian motion, and capture only the risk-free drift.


In other words, calls, purchased in combinations with short puts and short stock - with stock proceeds invested in an interest-bearing account - should yield something near the risk-free rate, right? Meaning, people will bid the call up to where they can short the stock and sell the put and do a hair better than T-bills.


So is the call too expensive for someone who doesn't hedge? Not if he has asymmetric beliefs 1) about the underlying, or 2) about the future implied-volatility beliefs of prospective bidders. But of course, since he doesn't hedge the Brownina motion, he is looking at a different payout distribution entirely, apart from just mutation in the carry curve.


Moreover, it seems there would be special risk-spread or utility curves for asymmetric or skewed volatility. The utility of the call as a leg of the synthetic T-bill is theoretically about the same as the utility of the call as leverage on the underlying. Otherwise, you need to jack up vega in the put and the call... Plus, you have ...


Am I out of my element on this one?


eLROY

You want the expected payout on a call option. Samuelson originally assumed the option is discounted at one constant rate and the stock is discounted at another. Black and Scholes then solved their equation by discounting everything at the riskless rate.


You can calculate the expected call payout with the Black-Scholes formula using the expected future stock price instead of the current stock price and setting the riskless rate equal to 0. If the expected continuously compounded stock return is "mu", volatility is "sigma", and current at-the-money stock price is "S", then the answer is


S*exp(mu)*N(mu/sigma+.5*sigma) - S*N(mu/sigma-.5*sigma),


which is approximately .4*sigma*S when mu=0.

This answer assumes that the option is held to expiration, and the distribution of the underlying is log normal, with known future volatility. The poser can decide to use these assumptions; but it's also possible to make other assumptions.

The immediate bid, by which i presume you mean to be the price of the call option, is given by the Black Scholes formula(under certain assumptions).


The important point to make is that the price of a call has nothing to do with either the expected payout or utility functions. Its simply a no arbitrage argument between the option and a replicating portfolio of the underlying and a risk free security.

Two things about Black Scholes.


The first is that everyone knows that its assumptions are incorrect.


The second is that everyone uses it.


What people do is simply fudge the volaility input. So you can a vol smile or skew across strikes, a vol term structure across time and for things like swaptions another axis across maturities.

I mostly agree..Black Scholes is only the dimmest of light bulbs in a dark room.

I may be misunderstanding, but it seems like javelin is talking about how to put a price on the call option. But the way I interpreted the original question was that "options trader" ("OT") already held the IBM call option and wanted to know what his expected return was. For now, let us assume that he is going to hold it to expiration (keep in mind that OT has not confirmed that this assumption is valid).


Then the way to determine the expected return is to:


1. Assign a present value P to the option -- this could be (a) the price he paid, (b) the last traded price, (c) the current offer, (d) the output of an option valuation formula, (e) etc.


2. Determine the probability distribution of the underlying at expiration (in this case, the price of IBM one year from now). Kim Lee provides one way of doing this, but there other ways to determine this distribution, and the result does not have to be the same for different individuals. For example, trader A may "know" that the variance of IBM is greater than that implied by the current option price, or trader B "knows" that IBM may issue much larger than (publicly) expected dividends during the next year, or trader C knows that IBM's current price correctly reflects public knowledge that there is a 25% chance that IBM will go under during the next year (e.g., because it has bet the farm on some make or break strategy).


3. Deduce the probability distribution of the value of the option at expiration (from the probability distribution of the underlying).


4. Compute E, the expected value of the option at expiration.


5. Compute Y = ln(E/P), the expected continuously compounded annual rate of return.


If we assume everything required to derive Black-Scholes (including an efficient market), then Kim Lee provides the way to compute Y. Note that this will be *higher* than R, the risk free (continuously compounded) rate of return. First of all, S, the expected return on the underlying, figures to exceed R by the current equity risk premium (the additional expected rate of return that the market requires in order to accept the stock's risk). Second, Y will exceed S because the option is implicitly leveraged with respect to the underlying.


For example, if the option is implicitly leveraged 3 to 1 with respect to the underlying, then we'd expect Y = R + 3 (S - R). If we further assume R = 1%, S = 5%, then Y = 13%. But just because the expected return on the option is 13%, there's no free lunch here because you have to take on correspondingly higher risk to acquire its higher rate of return.