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An interview question on Monte Carlo method

cycbill

C++ Certificate
Joined
6/4/22
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40
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I've just got a Monte Carlo question for a quant strategist interview from a bulge bracket bank.
Question: Let's say you use Monte Carlo method to price a European Option. Vol is constant 15%, rate and dividend are zero. You use 50k paths. And assume you do continuous delta hedging on each MC path. In the end, how will this strategy be value -- zero or what value, and why?

Can you share your opinion to this question?
 
cycbill said:
I've just got a Monte Carlo question for a quant strategist interview from a bulge bracket bank.
Question: Let's say you use Monte Carlo method to price a European Option. Vol is constant 15%, rate and dividend are zero. You use 50k paths. And assume you do continuous delta hedging on each MC path. In the end, how will this strategy be value -- zero or what value, and why?

Can you share your opinion to this question?
Click to expand...
I saw something like this in Shreve’s first book- dicrete time Modelling.
If we assume a constant volatility, with binomial trees and time steps =25k, we will get 50k prices @ t = N (a lot more paths though).

Then we start with some wealth to buy some stocks and hedge it @ t = n + 1 based on the information we have @ t = n. Whenever our wealth is less than what is required we borrowed from the bank.

Maybe something like this can be extended to MC paths (continous) and we delta-hedge for t=n+1based on the info at t=n, rather than speculating?
(I have only read the first book, maybe this is discussed in the second one)
 
cycbill said:
I've just got a Monte Carlo question for a quant strategist interview from a bulge bracket bank.
Question: Let's say you use Monte Carlo method to price a European Option. Vol is constant 15%, rate and dividend are zero. You use 50k paths. And assume you do continuous delta hedging on each MC path. In the end, how will this strategy be value -- zero or what value, and why?

Can you share your opinion to this question?
Click to expand...
My 2c. Suppose you have a long European call option position. To delta hedge, you must short delta. If you hedge continuously, the value of your portfolio would be zero at all times, no matter what the stock price movements.

If instead, you hedge discretely in a bullish market, you are selling delta at higher and higher prices. When the market falls, you must buy delta at lower prices. This sell high-buy low of the delta results in a profit.

But hey, I suggest you actually run the Monte Carlo simulation!
 
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