I'm stuck on the Binomial Model (chapter 5 of the Paul Wilmott Introduces Quantitative Finance) which seems to lead to the counter-intuitive result that the price of a call option depends on the interest rate and not on the growth rate of the underlying asset.It seems to me that this result can be reached thanks to the no-arbitrage argument, which in turn can be applied thanks to the hypothesis that the asset price can change to only two possible values at any time step.To convince myself that the same result can be reached even in a more general situation, I tried to devise a different model, where at each time step, given a starting price, the next price is characterised by a gaussian probability distribution. Moving through the time steps, the variance of the gaussian distribution increases and its mean follows the asset price growth rate. At the time of the option expiry, it is possible to integrate the distribution using the option payoff as weight, therefore calculating the average value of the option at expiry. This average value can then be discounted by the interest rate and should give the option value at the start of the calculation.In the resulting formula the option value seems to be indeed dependent on the growth rate of the asset (as well as the interest rate).Have I made any unrealistic/invalid assumptions with this model?Or perhaps I made a mistake in the calculations. Has anyone else already tried a similar approach, finding results that are consistent with the binomial model?

You have to use the no-arbitrage argument in this model as well. Otherwise, you could borrow at the risk-free rate to finance you hedged position and make a risk-less profit.

QuoteOriginally posted by: FulvioI'm stuck on the Binomial Model (chapter 5 of the Paul Wilmott Introduces Quantitative Finance) which seems to lead to the counter-intuitive result that the price of a call option depends on the interest rate and not on the growth rate of the underlying asset.This is the fundamental breakthrough of the Black-Scholes-Merton world - that option pricing is independent of preferences. At the time of this discovery, it is urban legend that the result was so counter-intuitive that the paper was rejected out of hand by several journals, before it was eventually published (in a not especially prestigeous journal). Thus, to be baffled by such a phenomenon is probably not to be a source of embarrasment.On a not unrelated point, although I find your efforts commendable, I get the impression that you are spending a lot of time reinventing the wheel (of knowledge). Another approach would be to read further, in a possibly more superficial way, to see where one is heading, and then review. In a few chapters time there are genuine unresolved problems to chew over.

QuoteOriginally posted by: FulvioTo convince myself that the same result can be reached even in a more general situation, I tried to devise a different model, where at each time step, given a starting price, the next price is characterised by a gaussian probability distribution. Moving through the time steps, the variance of the gaussian distribution increases and its mean follows the asset price growth rate. At the time of the option expiry, it is possible to integrate the distribution using the option payoff as weight, therefore calculating the average value of the option at expiry. This average value can then be discounted by the interest rate and should give the option value at the start of the calculation.In the resulting formula the option value seems to be indeed dependent on the growth rate of the asset (as well as the interest rate).Have I made any unrealistic/invalid assumptions with this model?Or perhaps I made a mistake in the calculations. Has anyone else already tried a similar approach, finding results that are consistent with the binomial model?Fulvio --If I read you correctly, you are projecting to a future date using the asset price growth rate, and then discounting the expected value back to present at "the interest rate." What "interest rate" did you mean? Risk-free? If you use a risk-free interest rate that is not equal to the asset price growth rate (when they are equal, this is the hallowed risk-neutral assumption ...), you won't get the Black-Scholes price. I think it's a bad assumption, in any case, to use a single interest rate to discount back an expected value -- why would the value of $1 in the event that your underlying goes up to $100 in price be the same as the value of $1 in the event that your underlying goes only to $10 in price, which is what you are implicitly assuming if you discount pay-offs under both outcomes back to present using the same interest rate.

Thanks for the replies, yes I meant the risk-free interest rate and I can see that it's a bad idea to use it to discount the expected price of an option which is not risk-free.

Fulvio, not to beat the horse to death, but another suggestion.There are some really nice arguments explaining arbitrage pricing in Baxter and Rennie chapter II. It seems to me that your Gaussian model might a bit of overkill if I understand you correctly. The whole idea seems to be that arbitrage pricing places a much stronger condition on what the fair price of an option should be in the Cox-Ross-Rubinstein tree model, or the Black-Scholes continuous case. The way a lot of authors seem to explain it is by using arbitrage arguments and logical contradiction. The reasoning is usually as follows: 1) pick a price for the option that is anything but the arbitrage price. 2) Construct an arbitrage argument to show how you can lock in a risk free profit, or how someone will be able to profit off you.3) Do this exercise for prices above AND below the arbitrage-free price and you will then see that the only fair price is the arbitrage-free price.So within the models that we are studying right now (no dividends, geometric brownian motion asset prices, constant interest rates, etc.) instead of reinventing the wheel as one poster said, quote a price, ANY price that is not the arbitrage price, and work out who gets a free lunch and how. It's what I am struggling with right now as well...

It seems that no-one has yet mentioned the key assumption. The price of an option is only independent of the drift rate if markets are complete and the underlying asset can be traded continuously and without frictions. If this assumption is removed the whole risk-independent edifice collapses.

Hi Fluvio!I'll try my best at giving some insight behind the problem you encountered, even if I see that some well known Wilmott:ers have given some good advice.The method you are trying (with the gaussian increments) is basically the Monte Carlo approach, which is equivalent to the PDE-method originally used by Blacks & Scholes, through what is known as the Feynman-Kac theorem.If we thus first look at what Black & Scholes did, and find rationale behind their approach, we should be able to find rationale behind the Monte Carlo method as well.The rationale behind the Black & Scholes method (that we are particularly interested in, and which Graeme touched on) is that we assume that we can form a (temporarily) risk-neutral portfolio by using both the underlying asset as well as the option, as they both have a common source of risk. In the short run, this common source of risk should impact both the option and the underlying similarily, the only difference being a scaling-factor (i.e. what is now known as the options delta-value). If you have a risk-neutral position however, the return of this position should be the risk free rate of return, right? So over a short period of time, this portfolio should have a return equal to the risk free rate. Now, imagine a new portfolio, consisting of at least the underlying stock. By holding d underlyings (where d = the options delta value) we can replicate the option. However, the delta value is not constant, but you will need to rebalance the portfolio. You will thus need some kind of extra non-stochastic asset, in our case we use a bond or a cash-account.We know that the second portfolio will replicate the option (as it is this portfolios main purpose, and as it is (so far only) assumed that we can always find the "scaling-factor" i.e. the delta value), so the value of a third portfolio, consisting of 1 long option and 1 short portfolion #2 will have a constant value of zero. Using this portfolio, we can find the price of the option.Now, you may be well aware of all this, however it is essential to understand that this is not done just for fun, but for a reason. The reason is the without the replicating portfolio, it is impossible to price options theoretically correct. In the Monte Carlo method you describe, you do not have a replicating portfolio, and this is the reason why your method is not consistent with the Black & Scholes model. We can use the Monte Carlo method however we then need to adjust the drift of the stock to a risk-neutral drift and make assumptions of risk-neutral behaviour of investors. These adjustments are due to the Feynman-Kac theorem I mentioned previously.Another alternative is to use the binomial model. In this model you are creating the risk-neutral portfolio in each step. The Binomial model makes some unrealistic assumption, I agree. The main one is that you discretize the outcomes to only 2 states (up or down), and time is cast in discrete time intervalls. However as you make the time increaments smaller and smaller (and in the limit infinitely small), Binomial steps converge to the gaussian distribution. The Binomial model thus converges not only to the Black & Scholes model, but also to the Gaussian Monte Carlo model you are really looking for, i.e. gaussian increaments (over discrete time intervalls) but with risk-neutral portfolios generated in your strategy.

Thanks guys, your comments have clarified things a lot I wasn't convinced with the binomial model because I was only considering the case where in a time step the asset price can end up either above or below the option's strike price. In that case, with just two states to consider it is possible to set-up a risk free portfolio, but not with multiple states. Isn't there some risk when the asset price is crossing the option's strike price that the binomial model artificially eliminates? Though I guess that this risk would be negligible anyway even with a gaussian model, since for most of the time the asset price will be sufficiently clear of the option's strike price.

QuoteOriginally posted by: hazeriderYou have to use the no-arbitrage argument in this model as well. Otherwise, you could borrow at the risk-free rate to finance you hedged position and make a risk-less profit.In theory yes; in reality you can only guess what sigma (=cost of convexity) is, so this would not be riskless. The option is purchased for a fixed volatility, but the delta hedge is exposed to floating volatility, and a variance swap can connect the two. In many cases, your assumptions about vol are the most important factor in your option model...