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Black-Scholes from Random Walk Derivation - Issue

July 24th, 2019, 7:32 pm

Hi All,

I am trying to understand the derivation bnehind the BS from a lognormal random walk, however, in the notes i have it just seems to suddenly replace (dX)^2 with dt.

Following another textbook, I see the same derivation, but no explanation? I know it is probably me being incompoetent, but why is it true that we can do this?

ImageDerivation shown at the link above.
 
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bearish
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Re: Black-Scholes from Random Walk Derivation - Issue

July 25th, 2019, 11:17 am

You are starting in the wrong place. First, try to thoroughly understand the one-step binomial model. That could take weeks to do, but it contains virtually all of the finance involved in standard option pricing theory, and the math is elementary. Next, advance to the two-step. If you get a handle on that, then going to a binomial model with n steps, each of length T/n with log stock price moves proportional to sqrt(T/n), shouldn’t be too hard. Now, let n go to infinity. Along the way, you should notice that the option price converges to the value given by the Black-Scholes formula.
 
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Alan
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Re: Black-Scholes from Random Walk Derivation - Issue

July 26th, 2019, 5:56 pm

Or, the cookbook recipe to working with stochastic differential equations is the following multiplication table:

[$]dt \times dt = dt \times dW_t = dW_t \times dt = 0 [$]
[$]dW_t \times dW_t = dt[$]
(All higher powers are zero).

After all, to drive a car, you don't have to be a mechanic.  
Of course, I understand you are asking for an explanation, but first, just memorize the rules. Formally, the mysterious [$]dW_t[$] behaves like [$]\sqrt{dt}[$] and any power higher than [$]dt[$] can be dropped. Sort of like, in grade school, you first memorize the rote rules for computations; later you take algebra. Personally, I would suggest first extensive playing around with just the rules -- then, later turn to understanding them. 

There are many good textbook treatments of SDEs and Brownian motion. Bearish's example is also extremely useful for understanding the BS "no-arbitrage" hedging argument. But, I'm not so sure it's easy to see why the binomial price will converge to the BS formula price -- even though it's true. You might want to read Black & Scholes J. Political Econ. article (1973).  
 
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katastrofa
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Re: Black-Scholes from Random Walk Derivation - Issue

July 26th, 2019, 10:45 pm

dX ~ N(0,dt) i.i.d. => dX^2 = dt almost surely
 
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bearish
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Re: Black-Scholes from Random Walk Derivation - Issue

July 27th, 2019, 12:51 am

That is the point, or at least a big part of it, but delivered in a very physicist way. I'm perfectly happy to engage in a non-standard stochastic analysis perspective, where something like dt may be given a solid foundation, but that does come with its own body of overhead.
 
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katastrofa
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Re: Black-Scholes from Random Walk Derivation - Issue

July 27th, 2019, 5:59 pm

dt is the split second that Achilles needs to catch the bloody turtle ;-)
I understand that this is mathematical finance formalism, but for a physicist (*) the differential form of the BS equation is undefined simply because Ito integral doesn't follow the fundamental theorem of calculus. I am capable of understanding the integral formulation though!

(*) I'm not a physicist, I'm a socialist (I do social science).
Image
 
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bearish
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Joined: February 3rd, 2011, 2:19 pm

Re: Black-Scholes from Random Walk Derivation - Issue

July 27th, 2019, 7:21 pm

Or, the cookbook recipe to working with stochastic differential equations is the following multiplication table:

[$]dt \times dt = dt \times dW_t = dW_t \times dt = 0 [$]
[$]dW_t \times dW_t = dt[$]
(All higher powers are zero).

After all, to drive a car, you don't have to be a mechanic.  
Of course, I understand you are asking for an explanation, but first, just memorize the rules. Formally, the mysterious [$]dW_t[$] behaves like [$]\sqrt{dt}[$] and any power higher than [$]dt[$] can be dropped. Sort of like, in grade school, you first memorize the rote rules for computations; later you take algebra. Personally, I would suggest first extensive playing around with just the rules -- then, later turn to understanding them. 

There are many good textbook treatments of SDEs and Brownian motion. Bearish's example is also extremely useful for understanding the BS "no-arbitrage" hedging argument. But, I'm not so sure it's easy to see why the binomial price will converge to the BS formula price -- even though it's true. You might want to read Black & Scholes J. Political Econ. article (1973).  
The convergence argument isn't actually that bad. Consider an n-period binomial model of a call option with strike K on a non-dividend paying stock with U>R>D>0. With [$] q=\frac{R-D}{U-D} \quad \overline{q}=\frac{UR-UD}{UR-RD} \quad j^*  =  \frac{ln(K/S) - n ln D}{ln(U/D)} [$]  and B(k,n,p) as the binomial distribution function giving the probability of at most k successes in n independent trials with single success probability p, we can elicit from the binomial tree (lattice, really)  that the initial value of the call is given by  [$]  C = S ( 1- B(j^*-1,n,\overline{q}) ) - R^{-n} K (1- B(j^*-1,n,q )) [$]. Now, let R, U and D depend on n: [$] R = e^{r \frac{T}{n}} \quad U = R e^{\sigma \sqrt{ \frac{T}{n}}} \quad D = R e^{-\sigma \sqrt{ \frac{T}{n}}}[$], substitute, simplify and go to a standard probability text book for the convergence of the binomial distribution to the normal distibution. And simplify a little bit more. 
 
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ISayMoo
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Re: Black-Scholes from Random Walk Derivation - Issue

July 27th, 2019, 7:43 pm

As an aside, the convergence of one distribution to another is sometimes abused. For example, the confidence interval for the binomial proportion based on a normal approximation is pretty bad even for large sample size. Because the convergence is not uniform w/r to the true probability of X = 1.
 
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bearish
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Re: Black-Scholes from Random Walk Derivation - Issue

July 27th, 2019, 8:09 pm

Yeah. I guess the most relevant convergence concept here is that of the expected value of sufficiently well behaved functionals of the path, i.e. some form of weak convergence.
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