Hi everybody!Performing some MC simulations for Asset Paths and Asian Options pricing I noticed something pretty strange..according to Paul Wilmott Introduces QF, chap. 8.4 B&S equation In the Black & Sholes equation (risk neutral) applying Ito's lemma to asset paths and then to option price the drift term (mu) is cancelled out; so since the drift term is the only link with risk adversion, we can use, without getting it wrong, the risk neutral assumption even if we apply this model to a real world (not risk neutral); and of coure we use the risk-free rate r instead of muassumed this, if I simulate an asset path I must use mudS = mu S dt + sigma S dzand if I simulate an asset path, to price an option (path dependent , asian, etc...) I can use the risk-free rate "r"And I'll obtain 2 different asset path if I use them in 2 different ways ?????Excuse me but It's quite difficult to explain this in english, for me, and I hope anybody could at least understand what I'm saying....(be patient with all the mistakes... plz)edo

Only mu will give the actual asset distribution. However, for valuing an option you can use mu = r and get the same option value. Try running your simulation for different values of mu and see if you get the same values for the options.

of course you can use mu = r, and you'll get the same option price using the B&S formula...but things will be different princing a path-dependent option with Monte Carlo Simulation and using mu = r.the point is:if I'm just interestet in the asset path I must use muif I want an asset path to price an path dep. option I can use r.so wich is the right path ?remember that pricing with montecarlo is nothing but evaluating the asset path for 100K timesand then just discout at risk-free rate the option payoff....thanks.edo.

In theory you have a choice. In practice you should use r. Here are descriptions of the choices:1. Assume a drift rate of mu. Find the expectation of the option payoff under the real-world measure. Discount the expectation back at some rate that reflects your degree of risk aversion.>> Problems: (1) you don't know what mu will be, (2) you can estimate but not observe the real-world measure and (3) you probably don't have much idea what rate to discount back at in order to reflect your risk preferences.2. Assume a drift rate of r. Find the expectation of the option payoff under the risk-neutral measure. Discount the expectation back at the risk-free rate.>> You know what r is, you can use the prices of other derivatives to calibrate to the risk-neutral measure, you don't have to rely on knowing what your risk preferences are.>> However, you can only use this approach if the underlying shares are continuously frictionlessly traded. Also I can only show that this works for processes involving Brownian motion, i.e. where the asset prices follows a process of the form dX = mu(X,t)dt + sigma(X,t) dW(t) . I don't know if it works for other processes. EDIT: Just to confirm, this approach does work for path-dependent contracts.

I disagree that the risk-neutral assumption can be generally used for pricing path-dependent options.Consider an underlying security that sells for $1 today and will pay either $0 or $2 in a year. Interest rates are zero. A one-year call option at $1 is worth $0.50. I know that because it has the identical payout to half a share of the underlying. Using risk-neutral pricing I get the $1 by saying a risk-neutral investor will assume there is a 0.5 probability of $2, and so will value the option at 0.5*$1 = $0.50.The underlying can take one of four paths, $1 today to $2 in 6 months to $2 in one year, $1 to $0 to $0, $1 to $1 to $2 and $1 to $1 to $0. What is an option worth that pays $1 only for the first, path $1 to $2 to $2? It can be anything from $0 to $0.50.The trouble is that the risk neutral assumption only tells you that the probabilities of $0 and $2 are equal, both in six months and one year. It tells you nothing about the probability of $1 in 6 months. That number could be 0 or 1 or anything in between.

Your disagreement is based on the premise that we don't know the conditional probabilities, e.g. the probability of going from $1 in six months to $2 in one year. You have conjured up this premise by using an example that only allows for the six month and one year probabilities to be known, but that does not allow the conditional probabilities (1 yr | 6 mths) to be known. This trick is called misdirection.Putting it another way, you have introduced the possibility that there exists more than one martingale measure. This is equivalent to assuming incomplete markets. I agree that if markets are incomplete then another approach is needed. But I explicitly assumed complete markets in my previous post.To reiterate, under the assumption of complete markets there exists only one martingale measure. The fundamental theorem of asset pricing states that the price of a properly normalised security is equal to the expected value under this martingale measure. This is precisely what a montecarlo simulation is used for: calculating the expected value under the martingale measure and then normalising i.e. discounting back to value date at the risk-free rate.To be very clear on this: if markets are complete then all probabilities (including conditional probabilities) are uniquely defined. This means that even path-dependent contracts can be valued by MC simulation using the risk-free rate and the risk-neutral measure.

Sorry, I didn't mean to misdirect.What I should have said is that assuming complete markets for path dependent options is a much stricter assumption than assuming complete markets for options whose value depends only on the underlying price at expiration. As a practical matter, I think it's too strict to serve as a reliable basis for pricing. I prefer to price path dependent derivatives, except for some specials cases, using a realistic drift rate.

This will not add to what the wise men already said, though it may clarify:Use the risk neutral distribution for pricing, the objective distribution for hedging and other path dependent stuff. The price is an average. So different paths do not matter as long as they have the same average.

KrazyKThanks for trying to clarify. I disagree with everything you wrote."Use the risk neutral distribution for pricing, the objective distribution for hedging and other path dependent stuff."As I think Pat once remarked, pricing and hedging are two sides of the same coin. The price you charge for a derivative should depend on how much it will cost you to hedge it. Using different distributions to price and hedge makes no sense at all."The price is an average."The price is only an average under one particular probability measure, which is the risk-neutral measure. Even this is only true if you make stringent assumptions about market completeness."So different paths do not matter as long as they have the same average."Yes they do matter. If you own a 100/120 up-and-out call and the share price is 110 on expiry, then the option is worth 10 to you on paths which didn't reach 120 and zero on paths that did reach 120, even if the averages along those paths are the same.

"What I should have said is that assuming complete markets for path dependent options is a much stricter assumption than assuming complete markets for options whose value depends only on the underlying price at expiration. As a practical matter, I think it's too strict to serve as a reliable basis for pricing. I prefer to price path dependent derivatives, except for some specials cases, using a realistic drift rate."AaronI agree with a lot of this. However, I'd like to clarify two sets of practical points.1. Market completeness assumptionThe simplest assumption to make is that markets are complete. Edoardobarro initiated this thread by asking about the case in which an asset price follows GBM, dS = mu.S dt + sigma.S dW. More generally it's commonplace for derivatives desks to assume a complete volatility surface a la Dupire or Derman-Kani, in which the asset price is assumed to follow a process dS = mu(S,t) dt + sigma(S,t) dW. If markets are assumed complete, as in these examples, then the correct approach to MC simulation is to use risk free rates and the risk-neutral measure.However, as you wrote, assuming complete markets is a very strict assumption to make for path-dependent contracts. An example of this would be pricing a forward-start option using a volatility surface derived from vanillas. The information contained in this vol surface about the conditional probabilities is uncertain at best.In this case, we might go to the other extreme and assume that we know nothing about the price process of the underlying asset. We would look to see what hedging instruments are available and construct a static hedge. Using a static hedge will give us upper and lower bounds for the price of our derivative. In this case we will not know the price of our derivative with much accuracy but we will have a very clear idea of how wrong we might be. This contrasts with the case of the completeness assumption, where we *know* the price of our derivative with great accuracy but have no idea how wrong we might be.Pragmatically, the best approach is somewhere in between these two extremes. We should use the market prices of existing derivatives not to calibrate our models, but to form static hedges. We should then assume some fairly general process incorporating stochastic volatility and/or jumps to price the residual *left over* by the static hedges. This approach tells us the price of our derivative and also allows us to work out how wrong we might be about this price so that we can make a reasonable bid-ask spread and/or reserve appropriately. The downside to this approach is that we have introduced non-traded quantities, such as volatility, for which we need to estimate the market price of risk. This approach is similar in spirit to ITO33's HERO idea.So the answer to Eduardobarro's question "what drift rate should I use if I assume that the underlying asset follows GBM?" is that he should use the risk free rate. However, I very much agree with you that it is often not good enough to assume market completeness when dealing with path-dependent options and therefore that other approaches are needed.2. Questions"I prefer to price path dependent derivatives, except for some specials cases, using a realistic drift rate."a. Using a "realistic" i.e. real-world drift rate implies that you need to take expectations under the real-world measure. How do you estimate the real-world probabilities? How accurate do you believe this to be?b. Using the real-world drift rate and taking expectations under the real-world measure means that you need to discount back at a rate that takes risk preferences into account. How do you estimate this rate? Why?

and now some dubts arises...I'm estimating a simple portfolio VaR with one asset and one option on this asset, So I must simulate two different paths, one with mu for the asset and one with r for the option ?

This is an Aaron question if there ever was one.

QuoteOriginally posted by: Johnnya. Using a "realistic" i.e. real-world drift rate implies that you need to take expectations under the real-world measure. How do you estimate the real-world probabilities? How accurate do you believe this to be?b. Using the real-world drift rate and taking expectations under the real-world measure means that you need to discount back at a rate that takes risk preferences into account. How do you estimate this rate? Why?I think we agree pretty well, except that I think using a realistic drift rate is a more tractable approach than you do. I point to insurance companies, who do it all the time. They price your fire insurance not by computing a riskless hedge, but by trying to estimate a real world probability.In general, I think the historical frequency of some occurance is often a reliable estimate of its future probability. I prefer to use risk-neutral pricing if available, but I'm not afraid to model odds. If you ask me to price a binary call that pays $1 if the S&P500 is over 1,500 in three years; I use a risk-free hedge model. If you ask me to price a range note that pays $1 every day the closing S&P500 rounds to a prime number; I'm going to model the probabilities. I may be drifting into misdirection again by using such an extreme example, but there are many real world contracts for which I would rather know the historical performance of such contracts than the Black-Scholes value.The discount rate is not hard, I use the CAPM. I know the risk-free rate, r_f, and I have assumed some rate on the underlying, r_u (which may be dynamic). I compute a delta, d, and discount the derivative at d*r_u + (1-d)*r_f.I should also add that my main professional activity is not pricing or hedging, but risk management. Someone else has already priced the thing and we own it, and a hedge has been placed against it. I need to judge the residual risk, not offer opinions about the price or hedge.

For computing var, you should use real world quantities to determine the state of the world in which to price. Once you are in that state of the world, the price of the stock is known and the price of the option should be computed using r. MJ