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kangsooman
Topic Author
Posts: 16
Joined: November 30th, 2006, 3:27 pm

### delta from bs equation

the initial delta value from bs equation is not associated with return of underline asset, my initial delta value form bs equation isd = (Log(30 / 30) + (0.05 + 0.3 ^ 2 / 2) * ((3000 - 0) / 3000)) / (0.3 * Sqr((3000 - 0) / 3000))delta = WorksheetFunction.NormSDist(d)=0.62but when you do simulation, the initial delta value depends on return of underline asset.For j = 1 To 300s(0, j) = 30For i = 1 To 3000s(i, j) = s(i - 1, j) + s(i - 1, j) * (r / 3000 + 0.3 *(Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd - 6) * Sqr(1 / 3000))Next iIf s(3000, j) < 30 Thendelta(j, 1) = 0Else: delta(j, 1) = 1End IfNext jresult of this simulationfor exampler=0.1 54%, r=0.11 60%, r=0.12 64% (probability of european call option exercised)my initial delta value form bs equation is 0.62 and is it wise way holding 0.62 stock per 1 call option at the begining of the option contract?or is it better taking delta from simulation depends on return of underline asset

quantmeh
Posts: 5974
Joined: April 6th, 2007, 1:39 pm

### delta from bs equation

the way you setup your simulation is not measuring delta. delta is a change in option price per 1$change in underlying asset. in your simulation case you look at the option price after one period, at the end of which the underlying price has changed approximately by return*S0. therefore, for different returns, you should expect different changes in option price, hence the different percentage of excercised options. if you really want to simulate deltra, then you have to factor in the difference in changes of underlying asset price. also, having rate 0.1 and period equal to 1, the approx change in asset price is$30*0.1~3\$. this may not really be a small enough change to ignore convexity.cheers

Aaron
Posts: 6433
Joined: July 23rd, 2001, 3:46 pm

### delta from bs equation

The risk-neutral probability that a vanilla European call will be exercised is d2, not d1. So you should subtract the 0.3^2/2 term, not add it. Also, you have used 0.05 for the interest rate in the formula, but run the simulation at 0.1, 0.11 and 0.12.These corrections will get you closer to the correct answer, but you simulation is not working correctly. It's too sensitive to the interest rate. You should get 0.57, 0.59 and 0.60 for r = 0.10, 0.11 and 0.12. Your code looks right, but I suspect an error in the implementation.

quantmeh
Posts: 5974
Joined: April 6th, 2007, 1:39 pm

### delta from bs equation

QuoteOriginally posted by: AaronAlso, you have used 0.05 for the interest rate in the formula, but run the simulation at 0.1, 0.11 and 0.12.i think that he's trying to simulate asset returns, not the risk-free rate. so, that part is Ok (though using sum of 12 Rnds to model Gaussian is not very nice).

kangsooman
Topic Author
Posts: 16
Joined: November 30th, 2006, 3:27 pm

### delta from bs equation

sorry for late my responds.I think there is little misunderstanding.My european call is exercised after one period, so I estimated the delta at expiry that way.I was practicing delta hedge simulation using delta from bs equation and natually, I used delta at expiry from simulationT:expiryIf s(T) < k Thendelta = 0Else: delta = 1not from bs equation.In my simulation, option exercise at expiry determined by underline asset price at option expiry.

quantmeh
Posts: 5974
Joined: April 6th, 2007, 1:39 pm

### delta from bs equation

QuoteOriginally posted by: kangsoomansorry for late my responds.I think there is little misunderstanding.My european call is exercised after one period, so I estimated the delta at expiry that way.i was trying to say that your delta hedge wont work because your period is too long, imhoi. delta and gamma is a partial derivative. if you compute it numerically then you have to choose a small increment of S. in your case it's ~10% increase, too big, imho

kangsooman
Topic Author
Posts: 16
Joined: November 30th, 2006, 3:27 pm

### delta from bs equation

I dont get it...s increment is small.r=10% and divided 3000times, so s increment is 10%/3000 as you see below. s(i - 1, j) * (r/3000 + 0.3 *(Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd - 6) * Sqr(1 / 3000))My final balance close to the initial option value from bs equation. expiry: after 1year, time steps:1/3000, s return:10%, vol:30%, option value from bs equation:4.27hedge frequency:30 and repeat this hedge simulation 1000timesaverage of My final balance is 4.45my hedge simulation is very simple one factor simulation.
Last edited by kangsooman on April 12th, 2007, 10:00 pm, edited 1 time in total.