I am working on valuing some options with monte carlo simulation and my first step was to generate normally distributed returns which is fairly easy to do.

This approach works fine when risk free rate is assumed to be zero, now most papers on the topic propose adjusting for drift using the risk free rate. They link this to black scholes and how black scholes having proved that the rf is the rate that matters and not the stocks expected growth rate. Now before I dive in to understand the derivation etc, can someone shed some light on the reasoning so that I can better understand the assumptions.

Fundamentals on the underlying are not factored into option pricing. If company A and company B are growing at different rates, their options will not be priced with more or less premium given that the implied volatility is the same. Options do, however, provide premium for the opportunity cost to the writer of the option for holding the capital. This is where the risk free rate come into play.

In monte carlo simulation a drift equal to the risk free rate gives a result that matches black scholes and the binomial, now my question is if we are trying to simulate where the stock price is going, why not use our expectations of the stocks returns.

One answer I found is that if the stock is expected to yeild more than RF someone can borrow and invest in the stock and if it is expected to yeild less than RF we can short it and invest in RF. But then with that logic why would anyone invest in the stock…

Part of the reason for this is that there is an assumed connection between the volatility and the rate of growth in a risk neutral pricing model. If the fundamentals say that one company should be growing faster than the other, then they should have different volatilities if the price of risk is constant for all things investible.

^^^ good point. I was assuming implied volatility was a purely technical/ probability derived value. You have inspired me to examine it further.

EDIT: I did “examine it further” and implied volatility is not derived from anything… well, it is a chicken and egg argument. Implied volatility is determined after the market supply/ demand pressures set the prices of the options. I orignally thought is was the implied volatlity that was derived first and then plugged into Black Scholes to get the option prices but actually it is the option prices that are plugged into Black Scholes to give the implied volatity. ( it is historical volatility that is based on pure technical behavior of the underlying). Anyway, anything option traders presume to be signifcant to the underlying (includng growth expectations) will drive the price of the options. Just thought I would add that perspective for anyone who was curious.

bro this explanation is complete BS. Not sure on what amaturish blog this answer was published, but it’s a waste of kilobytes.

If you want to simulate the stock price in order to observe the real-world probability distribution of the future stock prices, you need to use whatever the expected return for that stock is, which you find using the CAPM for instance. There are plenty of situations where one must use the expected return of the stock as the drift when they perform simulations in order to find the real-world probabilities. Usually those situations don’t involve discounting. For instance, I might be interested in knowing the real-world probability that the stock price will exceed some threshold, or the expected time until the stock crosses some barrier. In those situaitons, I won’t use risk-neutral simulation.

Risk-neutrality comes into the picture when you invite discount rates to the party for the purpose of calculating present values. For instance, if I need to price a call option and I simulate the real-world stock price using the stock price’s expected return, what is the discount rate for the call option payoff? It is NOT the same drift. The call option has a different risk profile than the underlying, and its discount rate is not constant but varies with its moneyness, time to maturity, etc. Turns out that if you perform a “change of measure” so that instead of the stock’s real-world expected return, you use the risk-free as the drift, then the risk-free rate is also the right discount rate for the call option (or any other derivative!), which becomes very convenient for pricing such derivatives. So we do risk-neutral simulation of the stock price path because that also “fixes” the right discout rate for the derivative payoff. For further detail about why this change of measure works the way it does, read up on Girsanov theorem.

The change of measure concept explains a lot of things and so does the distinction between real world/risk neutrality.

Is there anyway the change of measure can be explained in a way simple enough for someone without extensive math background to understand? Every reading I find is very mathmatical, I am not looking to learn how to derive the concept, just to understand what it is doing and move on.

Take your very first equation and make volatility 0. You know exactly what’s going to occur, i.e. it’s risk free.

Assume 5% for the Rf return. Investment is at 100 today, therefore you know with 100% certainty it will be at $105.13 in 1 year (continuous compounding). Screw continuous for purposes of this, let’s say it’ll be at $105. Therefore, it’s drifting upwards based on the risk free rate. Your $100 call option, expiring in 1 year, will have a future value of $5. Discount that back to today, the call will be ~$4.76.

Now, apply the same idea to a case where volatility is not 0, and you will still get an upward drift due to the discount rate.