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jackgurae
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How does P&L of delta-hedged option position accumulate over time?

February 10th, 2017, 3:44 am

Assume that I long ATM Call option on a stock at 20% implied volatility. This Call option will expire in 1 month. If, over this coming month, the underlying moves with 30% volatility and I delta-hedge this option. I expect to get profit from Gamma scalping. My expected P&L would be positive.

How would my P&L accumulate, on average,

Does P&L grow linearly over time, or does P&L grow exponentially over time?

Does moneyness of option affect accumulated P&L shape?
 
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frolloos
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Re: How does P&L of delta-hedged option position accumulate over time?

February 10th, 2017, 6:40 am

It depends which free lunch you'd like today. (Google paper with similar title by Wilmott and Ahmad)
 
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gatarek
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Re: How does P&L of delta-hedged option position accumulate over time?

February 12th, 2017, 9:30 pm

Or Bergomi's book.
 
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DavidJN
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Re: How does P&L of delta-hedged option position accumulate over time?

February 13th, 2017, 8:34 pm

I found it really helpful to use a Monte Carlo simulation exercise to strengthen intuition about this kind of stuff. You can try different, moneyness, hedge rebalancing rules, transactions costs, etc. and see how the p&l outcome is affected. It is even a good way to convince oneself that an option is a redundant security in the mythical Black Scholes world (i.e. that you end up with zero time-adjusted profit for a delta-hedged option when realized vol = transaction implied vol). More than anything, you learn that delta hedging is really not for the faint of heart!
 
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jackgurae
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Re: How does P&L of delta-hedged option position accumulate over time?

February 16th, 2017, 2:11 pm

frolloos wrote:
It depends which free lunch you'd like today. (Google paper with similar title by Wilmott and Ahmad)


I've read this paper like 10 times already. Very useful!

I've run Monte-Carlo simulation. There are trials that P&L grows exponentially first, and remain steadty. There are trials that P&L remain flat then sprout before option expires. It literally depends on paths. It is random. But on average, P&L accumulate linearly.
 
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tw
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Re: How does P&L of delta-hedged option position accumulate over time?

February 19th, 2017, 8:46 am

jackgurae wrote:
Assume that I long ATM Call option on a stock at 20% implied volatility. This Call option will expire in 1 month. If, over this coming month, the underlying moves with 30% volatility and I delta-hedge this option. I expect to get profit from Gamma scalping. My expected P&L would be positive.

How would my P&L accumulate, on average,

Does P&L grow linearly over time, or does P&L grow exponentially over time?

Does moneyness of option affect accumulated P&L shape?


It might interest you to think about the following question, that was often used at interviews where I used to work to see if candidates had any intuition beyond the mathematics of option trading.
Consider a market that conforms exactly to the Black-Scholes assumptions (GBM, constant parameters, continuous frictionless trading etc.).
Implied vols are offered exactly 10 vols over historical, realised volatility at all times. You cannot buy options cheaper than this.
You are only allowed to buy options and delta hedge them precisely according to Black-Scholes delta hedging rules. No cheating!
How can you make money in this environment? 
 
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jackgurae
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Re: How does P&L of delta-hedged option position accumulate over time?

February 19th, 2017, 2:02 pm

tw wrote:
jackgurae wrote:
Assume that I long ATM Call option on a stock at 20% implied volatility. This Call option will expire in 1 month. If, over this coming month, the underlying moves with 30% volatility and I delta-hedge this option. I expect to get profit from Gamma scalping. My expected P&L would be positive.

How would my P&L accumulate, on average,

Does P&L grow linearly over time, or does P&L grow exponentially over time?

Does moneyness of option affect accumulated P&L shape?


It might interest you to think about the following question, that was often used at interviews where I used to work to see if candidates had any intuition beyond the mathematics of option trading.
Consider a market that conforms exactly to the Black-Scholes assumptions (GBM, constant parameters, continuous frictionless trading etc.).
Implied vols are offered exactly 10 vols over historical, realised volatility at all times. You cannot buy options cheaper than this.
You are only allowed to buy options and delta hedge them precisely according to Black-Scholes delta hedging rules. No cheating!
How can you make money in this environment? 

Zero.
 
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tw
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Re: How does P&L of delta-hedged option position accumulate over time?

February 19th, 2017, 6:19 pm

jackgurae wrote:

Zero.

You mean zero probability of making a profit?
 
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jackgurae
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Re: How does P&L of delta-hedged option position accumulate over time?

February 20th, 2017, 1:28 am

tw wrote:
jackgurae wrote:

Zero.

You mean zero probability of making a profit?

Given that implied volatility is equal to realised volatility, I would say p&l will stay strictly at zero every moment til expiry and every trials.
 
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tw
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Re: How does P&L of delta-hedged option position accumulate over time?

February 20th, 2017, 2:23 pm

jackgurae wrote:
tw wrote:
jackgurae wrote:

Zero.

You mean zero probability of making a profit?

Given that implied volatility is equal to realised volatility, I would say p&l will stay strictly at zero every moment til expiry and every trials.

Well I wrote that you could only buy options and implied vols were 10 points above realised at each point in time.
So is it possible to make money?
 
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jackgurae
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Re: How does P&L of delta-hedged option position accumulate over time?

February 21st, 2017, 2:39 am

tw wrote:
jackgurae wrote:
tw wrote:
You mean zero probability of making a profit?

Given that implied volatility is equal to realised volatility, I would say p&l will stay strictly at zero every moment til expiry and every trials.

Well I wrote that you could only buy options and implied vols were 10 points above realised at each point in time.
So is it possible to make money?

Oh, my bad!
I would say Delta hedging will accumulate loss every time.
I have no idea how to make money if I'm forced to buy expensive option and delta-hedge it. To my knowledge, it's is impossible to make money.
 
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Paul
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Re: How does P&L of delta-hedged option position accumulate over time?

February 21st, 2017, 8:10 am

When I ask students the question about high implied vol it is usually those with a mathematical understanding that get the answer, the traders are more easily tricked! My version of this requires a) not making assumptions and b) understanding the maths behind hedging.
 
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tw
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Re: How does P&L of delta-hedged option position accumulate over time?

February 21st, 2017, 11:57 am

jackgurae wrote:
tw wrote:
jackgurae wrote:
Given that implied volatility is equal to realised volatility, I would say p&l will stay strictly at zero every moment til expiry and every trials.

Well I wrote that you could only buy options and implied vols were 10 points above realised at each point in time.
So is it possible to make money?

Oh, my bad!
I would say Delta hedging will accumulate loss every time.
I have no idea how to make money if I'm forced to buy expensive option and delta-hedge it. To my knowledge, it's is impossible to make money.

To put some numbers on it.  Implied is always 10 %age points over realised.
Initially: realised = 10%, implied = 20%. Buy 20% (e.g. ATMs)
Subsequently vols (both) go up. Now, realised  = 25%    implied = 35%.
What is the p&l?
 
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emac
Posts: 89
Joined: July 7th, 2009, 7:15 pm

Re: How does P&L of delta-hedged option position accumulate over time?

February 21st, 2017, 6:19 pm

tw wrote:
jackgurae wrote:
tw wrote:
Well I wrote that you could only buy options and implied vols were 10 points above realised at each point in time.
So is it possible to make money?

Oh, my bad!
I would say Delta hedging will accumulate loss every time.
I have no idea how to make money if I'm forced to buy expensive option and delta-hedge it. To my knowledge, it's is impossible to make money.

To put some numbers on it.  Implied is always 10 %age points over realised.
Initially: realised = 10%, implied = 20%. Buy 20% (e.g. ATMs)
Subsequently vols (both) go up. Now, realised  = 25%    implied = 35%.
What is the p&l?

You said we were in BS world with constant vol... so how can vol change?

When you say we need to hedge using BS delta, which vol do we use to calculate the delta?
 
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frolloos
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Location: Netherlands

Re: How does P&L of delta-hedged option position accumulate over time?

February 21st, 2017, 8:59 pm

[/quote]
You said we were in BS world with constant vol... so how can vol change?
[/quote]

That's the surprise - always expect all assumptions to be wrong :)

The joking aside, I suspect what TW is hinting at is that you can still generate positive p/l through other Greeks, eg vega.
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