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Gambler
Topic Author
Posts: 15
Joined: July 23rd, 2001, 7:22 pm

### Options on options shortcuts?

Is there any simple shortcuts for valuation of options on options?

Paul
Posts: 11437
Joined: July 20th, 2001, 3:28 pm

### Options on options shortcuts?

What do you mean by a 'shortcut'?

Collector
Posts: 4950
Joined: August 21st, 2001, 12:37 pm

### Options on options shortcuts?

I don’t know any tricks fore such options, so why not just develop a sexy approximation for atm-atm compound options. That is where the underlying option is at-the-money forward, and also the option on the option is at-the-money. For call-on-call and put-on-call S*0.4*v*Sqrt(t)*(0.5+0.2*v*Sqrt(T))Where S is the asset price, v is the volatility, t is time to maturity compound option, T is time to maturity on underlying option. Take a close look! The first part is simply the value of a plain vanilla option with time to maturity tS*0.4*v*Sqrt(t) The second part is the delta of a call option with time to maturity T 0.5+0.2*v*Sqrt(T)For a call-on-put and put-on-put we get plain vanilla option with time to maturity t multiplied by the delta of a put option with time to maturity TS*0.4*v*Sqrt(t)*(0.5-0.2*v*Sqrt(T))How accurate:Assume S=70 (Futures), v=0.2, t=0.5, T=1.0, risk free rate 0.1Underlying option Strike=70, Underlying option value = 5.5759Option on option strike=5.5759Using the Geske formula (involving cumulative bivariate distribution) I get 2.1022Using my approximation I get 2.1383 You can find more info and VBA code for the Geske formula in my book: "The Complete God to Option Pricing Formulas. "Sorry, God should be Guide I think...Regards,EGH (The Collector)

Collector
Posts: 4950
Joined: August 21st, 2001, 12:37 pm

### Options on options shortcuts?

The first one that can show me how I came up with this solution will get a free copy of my book. (You can choose between English and Chinese, or I can read it for you in Norwegian)

DB

### Options on options shortcuts?

Dear Espen,your book is indeed quite useful. Shockingly I got it from a Waterstones where it was marked "obsolete" for a fiver.

Collector
Posts: 4950
Joined: August 21st, 2001, 12:37 pm

### Options on options shortcuts?

So now I understand why only 25% of my sales comes from Europe. Looking at the bright side of life, Waterstones comment should soon make it a collectible item.

Julian
Posts: 86
Joined: August 23rd, 2001, 12:19 pm

### Options on options shortcuts?

Hi Collector,Here is how you have obtained your solution.the underlying call is at the money forward, that isS = Exp(-r T ) K where K is the strike of the underlying option and ris the interest rate.with this you obtain the value of the underlying call asS (N(+Sqrt(T)*v/2)-N(-Sqrt(T)*v/2))if you expand in power series for low T and keep the leading term you obtain its value asS Sqrt(T/(2 Pi))*vwhich is S*0.39894.. * Sqrt(T)*v that is approx. S*0.4*v*Sqrt(T)Now using Ito we have the call volatility isisv*N(+Sqrt(T)*v/2)/(N(+Sqrt(T)*v/2)-N(-Sqrt(T)*v/2))Since the call-on-call is atm forward also, we can use the first formula replacingS -> S Sqrt(T/(2 Pi))*vT -> tv -> v*N(+Sqrt(T)*v/2)/(N(+Sqrt(T)*v/2)-N(-Sqrt(T)*v/2))that is the price of the call-on-call isS * Sqrt(T/(2 Pi))*v*Sqrt(t/(2 Pi))* v*N(+Sqrt(T)*v/2)/(N(+Sqrt(T)*v/2)-N(-Sqrt(T)*v/2))orS * Sqrt(t/(2 Pi))* v * Sqrt(2/Pi) * Sqrt(T)*v/2 N(+Sqrt(T)*v/2)/(N(+Sqrt(T)*v/2)-N(-Sqrt(T)*v/2))now expanding for low T we obtainS * Sqrt(t/(2 Pi))* v * (1 + Sqrt(T/(2 Pi))*v)/2that is your formulathe put-on call is similarI look forward to your book!Julian

Paul
Posts: 11437
Joined: July 20th, 2001, 3:28 pm