Help with Black Scholes and Dynamic Hedging

[alcasa]

Hi

This is my first post, so not sure if this is the right forum. Please let me know if there is somewhere more appropriate.

I have to explain dynamic hedging for a put option and one point I want to make is:

i) When you dynamic hedge you are replicating an option.
ii) The option costs money so dynamic hedging will cost money.
iii) The cost comes in because you have to rebalance and have losses due to gamma.(Buy high/Sell low)
iv) I believe you can estimate your gamma costs by 1/2 sigma^2 x $gamma x t. (Or a formula very similar to this.)
v) So you can think of the cost of an option as the present value of all your future rebalancing costs.

So the question is....conceptually is theta the same thing as point iv) plus interest costs?


Any help would be appreciated.

 

[Martinghoul]

Hi

This is my first post, so not sure if this is the right forum. Please let me know if there is somewhere more appropriate.

I have to explain dynamic hedging for a put option and one point I want to make is:

i) When you dynamic hedge you are replicating an option.
ii) The option costs money so dynamic hedging will cost money.
iii) The cost comes in because you have to rebalance and have losses due to gamma.(Buy high/Sell low)
iv) I believe you can estimate your gamma costs by 1/2 sigma^2 x $gamma x t. (Or a formula very similar to this.)
v) So you can think of the cost of an option as the present value of all your future rebalancing costs.

So the question is....conceptually is theta the same thing as point iv) plus interest costs?


Any help would be appreciated.

Firstly, from what you're describing, you've sold the option...

Conceptually, I prefer to think of theta as just point iv). The discounting effects are separate.

Also, note that in point v), present value of all your expected future rebalancing costs comprises just the time value part of the option cost. The option may have intrinsic value in addition to the time value.

 

[alcasa]

Firstly, from what you're describing, you've sold the option...

Conceptually, I prefer to think of theta as just point iv). The discounting effects are separate.

Also, note that in point v), present value of all your expected future rebalancing costs comprises just the time value part of the option cost. The option may have intrinsic value in addition to the time value.

Thanks very much for the reply. You are correct I have sold the option.

Obne last question, using the view above. Is it fair to say that directional market moves just change the timing of when I incur my costs. For example when the market moves down, I will make a hedging gain at present but in future since my gamma is larger my expected future hedge costs will be larger.

Thanks Again!

 

[Martinghoul]

Thanks very much for the reply. You are correct I have sold the option.

Obne last question, using the view above. Is it fair to say that directional market moves just change the timing of when I incur my costs. For example when the market moves down, I will make a hedging gain at present but in future since my gamma is larger my expected future hedge costs will be larger.

Thanks Again!

I am not sure I understand what you're saying here...

Let's say you've sold an ATM put (strike K) at time T. In the next period, at time T+1, the mkt sells off and the underlying is at K-1, which means that the K put you're short is now ITM. That means that your delta is now more than 0.5 long and you have to sell some underlying at K-1 to hedge.

Given all this, what hedging gain are you referring to? Also, what larger gamma?

 

[alcasa]

I am not sure I understand what you're saying here...

Let's say you've sold an ATM put (strike K) at time T. In the next period, at time T+1, the mkt sells off and the underlying is at K-1, which means that the K put you're short is now ITM. That means that your delta is now more than 0.5 long and you have to sell some underlying at K-1 to hedge.

Given all this, what hedging gain are you referring to? Also, what larger gamma?

Thanks again for the reply.

I apologize for not explaining this very well, I don't have an investment background and learned a lot of this through books, so my way of thinking about and expressing this may be somewhat distorted compared to the industry.

If I have sold a put and then dynamic hedge the risk, when the market goes down my initial short position can be bought back cheaper so I have a market value gain. This is the gain I am referring to.

I believe gamma would be a very small amount larger as the option is more in the money.

I am trying to reconcile the view that hedge costs are the sum of future reblancing costs with the fact my delta position can create gains and loss along the way but the net of all of this subtracted from the payoff of the put option should be equal to the original cost of the put option.

 

[Martinghoul]

Thanks again for the reply.

I apologize for not explaining this very well, I don't have an investment background and learned a lot of this through books, so my way of thinking about and expressing this may be somewhat distorted compared to the industry.

If I have sold a put and then dynamic hedge the risk, when the market goes down my initial short position can be bought back cheaper so I have a market value gain. This is the gain I am referring to.

I believe gamma would be a very small amount larger as the option is more in the money.

I am trying to reconcile the view that hedge costs are the sum of future reblancing costs with the fact my delta position can create gains and loss along the way but the net of all of this subtracted from the payoff of the put option should be equal to the original cost of the put option.

Yes, I understand the context, but you're confusing yourself here, I think...

In order to proceed, let's define 'rebalancing' properly. Specifically, let's say that your portfolio has to be delta-neutral at every discrete time interval when you're allowed to transact.

Let's say that your initial trade is to sell 1 lot of ATM puts with delta. So at time T, when the underlying is at K, your portfolio is delta-neutral and consists of 1 short ATM put (strike K) and 0.5 short of the underlying. What happens when the mkt sells off at time T+1? With the underlying now at K-1, your short put is now ITM and its delta is long more than 50%. Thus, your portfolio is now LONG and to rebalance it, i.e. bring your net delta back to 0, you need to SELL, not buy, more of the underlying at a price K-1.

This is, in fact, what happens when you're short gamma. You have to sell low and buy high, as you have pointed out yourself. To make you happy to do that, you are compensated by getting paid the time value of the option.

 

[N Rothschild]

martinghoul, from this post it seems you know alot more about options than me

 

[alcasa]

Yes, I understand the context, but you're confusing yourself here, I think...

In order to proceed, let's define 'rebalancing' properly. Specifically, let's say that your portfolio has to be delta-neutral at every discrete time interval when you're allowed to transact.

Let's say that your initial trade is to sell 1 lot of ATM puts with delta. So at time T, when the underlying is at K, your portfolio is delta-neutral and consists of 1 short ATM put (strike K) and 0.5 short of the underlying. What happens when the mkt sells off at time T+1? With the underlying now at K-1, your short put is now ITM and its delta is long more than 50%. Thus, your portfolio is now LONG and to rebalance it, i.e. bring your net delta back to 0, you need to SELL, not buy, more of the underlying at a price K-1.

This is, in fact, what happens when you're short gamma. You have to sell low and buy high, as you have pointed out yourself. To make you happy to do that, you are compensated by getting paid the time value of the option.

Thanks for your help, I think you have answered my question. My comment was based on just looking at the dynamic hedge side of the portfolio as opposed to looking at the total portfolio (dynamic hedge plus put.).

Thanks again for your help.

 

[geotzi]

Hi!
I have to apply dynamic hedging in a put option. Could you please suggest any book with examples??I am very confused and I have to deal with Black Scholes and a specific insurance product and I really need some help.
Thank you

 

[Martinghoul]

Hi!
I have to apply dynamic hedging in a put option. Could you please suggest any book with examples??I am very confused and I have to deal with Black Scholes and a specific insurance product and I really need some help.
Thank you

Hull comes to mind, but there are many others, including stuff on the web...

 

[geotzi]

I am searching for people who could help me online too. Do you know any person who could give me guidance??
Thank you for the answer

 

[Martinghoul]

I am searching for people who could help me online too. Do you know any person who could give me guidance??
Thank you for the answer

What sort of guidance?

 

[geotzi]

I have to apply dynamic hedging on a put option.I have found an example on call option, but it confused me and i cannot implement it on put option .... Specifically i have to hedge put options (e.g. with different strikes and maturities) and find a perfect Delta-Hedge in case I can rebalance say every second