Hello,
In the following paper:
Analytic Formula for the European Normal Black Scholes Formula, by Kazuhiro Iwasawa, December 2, 2001
at pp1-2 a closed formula for pricing call and put options (where the underlying asset has normal dynamics) is provided.I would like to kindly ask if anybody could provide me the corresponding formula for pricing Digital Options.
To put it simply if the underlying asset had lognormal dynamics I would use Nd2 to price the digital. What about now?
Thank you in advance
Serving the Quantitative Finance Community
Re: Pricing Digital Options with Normal Model
I looked briefly. I think an issue is that the proposed dynamics may be nonsense/wrong for certain underlying assets. First take the limit [$]\sigma=0[$] and make sure the proposed SDE makes sense for your asset.
For example, for stock price S as underlying, one would say a risk-neutral "normal" model would have to be [$]dS = r S dt + \sigma dW[$] with absorption at 0, which is a completely different kettle of fish from what you are looking at.
So, first -- what exactly is your underlying, contract and proposed risk-neutral dynamics for the underlying?
For example, for stock price S as underlying, one would say a risk-neutral "normal" model would have to be [$]dS = r S dt + \sigma dW[$] with absorption at 0, which is a completely different kettle of fish from what you are looking at.
So, first -- what exactly is your underlying, contract and proposed risk-neutral dynamics for the underlying?
Re: Pricing Digital Options with Normal Model
Hello Alan,
Thank you very much for your answer. I am using to normal dynamics as they are described at the paper that i quoted at my first post, in order to value a trading book containing swaptions and caps.
dF=μdt+σdWt
Reuters is quoting normal volatility both for caps and for swaptions in a variety of currencies.
My task is to price a KO cap where the barrier is observed at each payment (i.e. the barrier is Bermudan). Therefore, I have replicated the payoff as sum of the following structure:
1) Long position at a caplet with the strike of the cap.
2) Short position at a caplet with the strike of the barrier.
3) Digital option with the strike of the barrier.
Given the analytical formula that is in the first page of the paper that i quoted earlier and market normal volatility quotes for caps I can price 1 and 2, and actually my prices are the same with the ones that Bloomberg's SWPM function has. Therefore, I am looking for an analytical formula for the digital (similar to ND2 in the case of lognormal model).
I would really appreciate any help. Thank you.
Thank you very much for your answer. I am using to normal dynamics as they are described at the paper that i quoted at my first post, in order to value a trading book containing swaptions and caps.
dF=μdt+σdWt
Reuters is quoting normal volatility both for caps and for swaptions in a variety of currencies.
My task is to price a KO cap where the barrier is observed at each payment (i.e. the barrier is Bermudan). Therefore, I have replicated the payoff as sum of the following structure:
1) Long position at a caplet with the strike of the cap.
2) Short position at a caplet with the strike of the barrier.
3) Digital option with the strike of the barrier.
Given the analytical formula that is in the first page of the paper that i quoted earlier and market normal volatility quotes for caps I can price 1 and 2, and actually my prices are the same with the ones that Bloomberg's SWPM function has. Therefore, I am looking for an analytical formula for the digital (similar to ND2 in the case of lognormal model).
I would really appreciate any help. Thank you.
Re: Pricing Digital Options with Normal Model
I see. Given the dynamics, [$]F_T-F_0[$] is normally distributed with mean [$]\mu T[$] and variance [$]\sigma^2 T[$]. So,
[$]E[1_{\{F_T > B\}}] = \int_B^{\infty} e^{-(x- \mu T-F_0)^2/(2 \sigma^2 T)} \frac{dx}{\sqrt{2 \pi \sigma^2 T}}[$]
The integral converts to some [$]N(d)[$], by letting [$]x = F_0 + \mu T - \sigma \sqrt{T} z[$].
[$]E[1_{\{F_T > B\}}] = \int_B^{\infty} e^{-(x- \mu T-F_0)^2/(2 \sigma^2 T)} \frac{dx}{\sqrt{2 \pi \sigma^2 T}}[$]
The integral converts to some [$]N(d)[$], by letting [$]x = F_0 + \mu T - \sigma \sqrt{T} z[$].
Re: Pricing Digital Options with Normal Model
Thank you very much for your help.
With the new variable that you have proposed the function inside the integral becomes the standard normal pdf so I may compute it. Could you please be kind and let me double-check what is the value of "d" that I should put now at the normsdist function at excel to price the digital?
Alan thank you very much in advance.
With the new variable that you have proposed the function inside the integral becomes the standard normal pdf so I may compute it. Could you please be kind and let me double-check what is the value of "d" that I should put now at the normsdist function at excel to price the digital?
Alan thank you very much in advance.
Re: Pricing Digital Options with Normal Model
Just post your answer -- I'm sure someone will correct it if it is wrong.
Re: Pricing Digital Options with Normal Model
Or do a simple Riemann sum in Excel to validate your result! Learn to empower and trust yourself.
Re: Pricing Digital Options with Normal Model
Exactly. Maybe somebody needs to write a book for quants: "How to check your work".
Re: Pricing Digital Options with Normal Model
Yes! That's a very good idea.Exactly. Maybe somebody needs to write a book for quants: "How to check your work".
Also for traders to help backtest trading strategies. Those always perform good on historical data but never going forward.
Re: Pricing Digital Options with Normal Model
Hello again, and sorry for the late response.
So the value of the d should be:
d=( F-K )/( sigma*sqrt(T-t) ), for the call option (digital)
d= ( K - F )/( sigma*sqrt(T-t), for the put option (digital)
and the final price of each of the above should be Discount Factor * N(d).
Alan, do you agree with that?
Thank you.
So the value of the d should be:
d=( F-K )/( sigma*sqrt(T-t) ), for the call option (digital)
d= ( K - F )/( sigma*sqrt(T-t), for the put option (digital)
and the final price of each of the above should be Discount Factor * N(d).
Alan, do you agree with that?
Thank you.
Re: Pricing Digital Options with Normal Model
Ok, I tried. So, can someone in this forum correct it please? Thank you in advance.
Re: Pricing Digital Options with Normal Model
So, if [$]x = B[$], then [$]z = d = (F_0 - B + \mu T)/(\sigma \sqrt{T})[$], and [$]N(d)[$] is the (undiscounted) price of what you are calling the call digital. Since the undiscounted price of the call digital plus put digital must sum to 1, the undiscounted price of the latter must be [$]1 - N(d)[$], with the same [$]d[$] I just wrote.I see. Given the dynamics, [$]F_T-F_0[$] is normally distributed with mean [$]\mu T[$] and variance [$]\sigma^2 T[$]. So,
[$]E[1_{\{F_T > B\}}] = \int_B^{\infty} e^{-(x- \mu T-F_0)^2/(2 \sigma^2 T)} \frac{dx}{\sqrt{2 \pi \sigma^2 T}}[$]
The integral converts to some [$]N(d)[$], by letting [$]x = F_0 + \mu T - \sigma \sqrt{T} z[$].