Now, if you recall something about density functions, then you will understand why we're letting Q, the risk neutral probability that S_T is in this interval is equal to the risk neutral density times the width of the interval. So we're saying, the risk neutral probability that S_T is between K minus Delta K and K plus Delta K, that is approximately equal to the risk-neutral density evaluated at K times the width of the interval to Delta K. That just follows from the property of PDF's. We actually explained this in one of the additional modules on probability that we also recorded. They're also available on the course platform. So remember, if you've got a PDF in general, so this is our PDF F of x, and suppose we want to compute the interval that the random variable x is inside x_0 to x_0 plus Delta x. Well, the density satisfies that the probability that the random variable x is in x_0 plus Delta x where Delta x small, that's approximately equal to F of x_0 times the width of the integral which is Delta x. This is a standard property of probability density functions and that's all we're using here. So we're saying the probability that S_T is inside this interval here is equal to the density which is f_T, times the width of the interval which is two Delta k. So therefore we're going to get B_0 is approximately equal to e to the minus rT, times f_T of K times Delta K squared. We have a two here but that cancels with the two there and we got a Delta K times Delta K which is Delta K squared. So what we've done now is we've come up with two expressions for the value of the butterfly strategy. We have this expression here in equation six which is exact, and we have this expression here which is an approximation. But as Delta K goes to zero, this approximation also becomes exact. So what we're going to do is we're going to equate equation seven with equation six, and then solve for f_T of K. Or in other words bring f_T of K over to the left-hand side. So we will see f_T of K is approximately equal to e to the rT times this expression on the right-hand side of six here, divided by Delta K squared. If we now let Delta K go to zero in eight, well, if you recall your calculus, you'll see that all you're doing when you do this, is actually computing the second partial derivative of the call option price with respect to the strike. So what we're seeing is that by constructing a butterfly strategy where Delta K goes to zero, we're actually able to come up with the risk-neutral probability density function for S_T. F_t is equal to e to the rT Delta 2C Delta K squared. So the volatility surface gives us the marginal risk-neutral distribution of the stock price S_T for any fixed time T. So this is a really interesting observation. We see option prices for finite number of strikes and maturities, we compute those implied volatilities, we then actually fit the volatility surface to those finite number of points. I mentioned earlier that we need to fit the volatility surface very carefully. The reason is, if we want to be able to compute something like f_T of K, then we're computing partial derivatives and second partial derivatives. So we need to make sure we do things, we fit things very smoothly. This means that given the implied volatility surface, Sigma K, T, we can compute the price P_0 of any derivative security whose payoff f only depends on the underlying stock price at a single and fixed time capital T. Now maybe I've chosen f unfortunately here because I used f subscript t to denote the risk neutral PDF of S-T. Here f, this f here has got nothing to do with this. This is the risk neutral PDF of S_T, f here is just some arbitrary function representing the payoff of some derivative security. So just to be clear, the f that I'm using here has nothing to do with the f subscript t on the previous slide which was the risk-neutral probability density function of S_T. Why is it we can compute this quantity here? Well, if I know the risk-neutral density of S_T, I can just perform an integration against that risk-neutral density to compute this quantity here. So therefore, I can compute the price of any derivative security if the payoff of that derivative security only depends on the stock price at a single and fixed time T. That's because I will know F subscript T, the risk-neutral density for that stock price. Therefore I can evaluate this expectation on the right-hand side. However, knowing the volatility surface tells us nothing about the joint distribution of the stock price at multiple times T_1 up to T_n. This is not surprising since the volatility surface is constructed from European option prices, and European option prices only depend on the marginal distributions of S_T. Just to be clear, the joint distribution of the stock price at multiple times T_1, T_n, what I'm referring to there is the following. It will be this distribution T_1 up to T_n of S_T_1 up to S_T_n. So this is the risk-neutral joint distribution of the stock price at time T_1 up to T_n. What we're saying here is that we don't know anything about this joint distribution. The only thing that we can learn from the volatility surface is the marginal distribution for each individual time T_1 up to T_n. Here's an example. Suppose we wish to compute the price of a knockout put option with time T payoff given to us here. So it's this piece here is like a regular put option. So this is like a regular European put option, the maximum of K minus S_T and zero. However, we only get that payoff if the minimum stock price over the interval of zero to capital T is greater than or equal to B. So B here represents a barrier. If the stock price ever falls below B, then this indicator function here becomes zero, and so we get nothing. Remember, the indicator function, this indicator function can take on two possible values. It takes on the value one or zero. It takes on the value one if the minimum of S_T is greater than or equal to B and that's the minimum over zero less than or equal to little t, less than equal to capital T, and it takes on the values here or otherwise. So this is an example of a knockout put option. The point I'm trying to make here is that we cannot compute the price of this option just using the implied volatility surface. The implied volatility surface will only give us the marginal risk-neutral distribution of the stock price. It doesn't give us the joint distribution. In order to evaluate this, I would need to compute the following. So if the value of the security is P_0, it will be equal to the expected value using risk-neutral probabilities, e to the minus rT times the maximum of K minus S_T and zero, times the indicator function of S_T being greater than or equal to B. The point I'm trying to make is, in order to compute the expectation, I would need to know the joint risk neutral distribution for the stock price at all times between zero and capital T. But I don't know that. I cannot compute that risk-neutral distribution from the implied volatility surface. So in practice and we'll come back to this soon, we would need to use some sort of model, some arbitrage free model to estimate the price of this quantity.