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Exotic options

ShaktiRathore

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Hi all,
here just starting a thread on exotic options. These are wonderful options in a sense that they are modifications of the original options and they can suit needs of variety of investors.
I would describe about these options in this thread only for coming days.
lets start about the types of them following are the types of exotic options, well this is financial Engineering for you:):
  • Package
  • Binary options
  • Non Standard American options
  • Lookback options
  • Forward Start options
  • Asian options
  • Compound options
  • Chooser options
  • Barrier options
  • Options to exchange one asset for another
  • Options involving various assets
  • Volatility ans Variance swaps
which one is your favorite :)
thanks
 

ShaktiRathore

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A Strangle Combination: Long call(Exercise price K2)+ long put(Exercise price K1) where K2>K1
P/Off=call P/Off+put P/Off
P/Off=max(S-K2,0)+max(K1-S,0)
for S<K1<K2 => P/Off=0+K1-S=K1-S
for K1<S<K2 => P/Off=0+0=0
for K1<K2<S => P/Off=S-K2+0=S-K2
From above its clear that strangle pays when either the stock goes low(good downward shift) or when stock goes up. If a trader expects that a stock is highly volatile and that its high volatility in coming days can cause good down or up moves then he shall try this one i am sure:) Stocks that are expected to remain stable might not be the choice.
Similarly is a straddle except that the exercise price is same for both put and call option. K2=K1=K
P/Off=max(S-K,0)+max(K-S,0)
for S<K => P/Off=K-S
for S>K => P/Off=S-K

thanks
 

ShaktiRathore

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Now lets know something about Non standard American options:
1. bermudan options are nothing but plain vanilla european options with additional rights to exercise at certain specific dates before expiration. The bermuda options gives more flexibility than european options but are more restrictive than American options.
2. options in which there is a lock out period during which the option cannot be exercised and the rest of the period when the option can be exercised before maturity.
3.Options in which strike prices changes over the life of the option e.g. Warrants, Convertibles
Warrants are nothing but call options that gives the holder the right to buy the equity at a specific price in future. Convertibles on the other hand gives the bond holders the right to convert the bond into a specific amount of equity at an agreed upon price.

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ShaktiRathore

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Forward Options are the options that start their life at future time T1. They are often given to employees in from of Employees Stock options(ESOP). Their strike price often assumes the value of the asset at the time T1 when option comes into existence. So if the asset price is S1 after time T1 then the strike price of option becomes X=S1 now after time say t the asset price equals S.
option value at time t+T1=max(S-X,o)=max(S-S1,0)
so the value of option today assuming risk free rate as r=max(S-S1,o)*exp(-r(T1+t))

Another type of options are compound options that is the options that give right to buy or sell an option. Some variants are call on call, put on call, put on put and call on put.

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ShaktiRathore

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Chooser options:
Chooser options are a combination of put and call options. The chooser option allows investors to choose whether the option they hold is a call or a put at time T1 with maturity T2>T1.
At time T1 value of option is max(c,p) ...1
from put call parity relation,
p+S*exp(-qT)=c+K*exp(-rT)...taking into consideration dividend q and T=T2-T1
or p=c+K*exp(-rT)-S*exp(-qT)....2
put 2 in 1=>
value of chooser option
=max(c, c+K*exp(-rT)-S*exp(-qT) )
=max(c, c+K*exp(-rT)-S*exp(-qT) )
=c+max(0,K*exp(-rT)-S*exp(-qT) )
=c+exp(-qT)*max(0,K*exp(-(r-q)T)-S)

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ShaktiRathore

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Barrier options:
The options are special where options comes into existense when stock price hits a particular price barrier are called the In options. Whereas when options dies when stock price hits a particular price barrier are called the Out options.
When stock price hits barrier from below then options are up options and when stock price hits barrier from above then options are down options.
For a call option the various combinations possible are : up and out, up and in, down and out, down and in(4)
For a put option the various combinations possible are : up and out, up and in, down and out, down and in(4)
In this way there are a total of 8 combinations of barrier options that are possible.
The parity relation that holds are:
c=c(up and in)+c(up and out)
c=c(down and in)+c(down and out)
p=p(up and in)+p(up and out)
p=p(up and in)+p(up and out)
One can imagine that if exercise price of call is K=30, now suppose the barrier price for call up and in and call up and out is 40 ,
Now various values of stock price the we examine the value of combination of call up and in and call up and out,
i) S<30: c(up and in)=0,c(up and out)=0 value of call option c=0
ii)30<S<40: c(up and in)=0,c(up and out)=S-30 value of call option c=S-30
iii)S>40: c(up and in)=S-30,c(up and out)=0 value of call option c=S-30
so from above conditions its clear that relation c=c(up and in)+c(up and out) holds for all values of stock price where we assumed the exercise price of barrier options and plain vanilla options are same. The above results can also be proves for other combinations. This closes our today's learning of exotic options

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ShaktiRathore

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Binary options are another class of exotic options:
1) Cash or Nothing
2) Asset or Nothing
Cash or Nothing option pays a fixed amount of cash Q if at time to maturity S(T)>K otherwise pays nothing.
The net pay today,
=PV of future payment at time T if S(T)>K
= PV of [(future payment)*(probability of S(T)>K)]
= PV of [Q*N(d2)]
= exp(-rT)*[Q*N(d2)]=Q*exp(-rT)*N(d2)
so Cash or Nothing value today= Q*exp(-rT)*N(d2)
Asset or Nothing pays S(T) if S(T)>K otherwise pays nothing.
The net pay =[S(T)*exp(-rT)]*N(d1)=S0*exp(-qT)*N(d1)
The call option is equivalent to a package of long Asset or Nothing and short Cash or Nothing(payoff Q=K)
call option value=Asset or Nothing pay-Cash or Nothing pay
call option value= S0*exp(-qT)*N(d1) -K*exp(-rT)*N(d2) which is nothing but call option value

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ShaktiRathore

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Lookback options are of following types:
1)Floating Lookback call options: The options pay (S(T)-S(min)) at time T where S(T) is the value of asset at time T and S(min) is the minimum value of the asset observed during the time period T.
2)Floating Lookback put options: The options pay (S(max)-S(T)) at time T where S(T) is the value of asset at time T and S(max) is the maximum value of the asset observed during the time period T.
3)Fixed Lookback call options: The options pay max(S(max)-K,0) at time T where K is the value of exercise price for the asset and S(max) is the maximum value of the asset observed during the time period T.
4)Fixed Lookback put options: The options pay max(K-S(min),0) at time T where K is the value of exercise price for the asset and S(min) is the minimum value of the asset observed during the time period T.

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ShaktiRathore

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Shout options: Buyer can shout once during the life of the option. So buyer of shout option gets additional option to shout once during option life time.
S(T): stock price at time T(maturity)
S(t): stock price at time t(time of shout)
So final payoff is either payoff at time T or payoff at time t
option payoff= max(S(T)-K,0)+max(S(t)-K,0)
i) S(T)>S(t)>K=> get payoff at time T=>option payoff=S(T)-K
ii) S(T)<S(t)<K=> =>option payoff=0
iii) S(t)>S(T)>K=> get payoff at time t=>option payoff=S(t)-K
iv) S(T)<K<S(t)=> get payoff at time t=>option payoff=S(t)-K
v) S(T)>K>S(t)=> get payoff at time T=>option payoff=S(T)-K
The final formula for option payoff is also,
max(S(T)-S(t),0)+S(t)-K

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ShaktiRathore

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Asian Options determines payoffs based on the average stock price. Average stock price over a period T is compared with the stock price at expiration and payoff is determined based on whether the average stock price is greater or less than the stock price at expiration.Similarly for average strike option the average strike price over a period T is compared with the stock price at expiration and payoff is determined based on whether the average strike price is greater or less than the stock price at expiration
Average price option pays: call: max(S(avg.)-K,0) and put: max(0,K-S(avg.))
Average strike option pays: call: max(S(T)-S(avg.),0) and put: max(0,S(avg.)-S(T))
there is no exact valuation for asian options but can be valued approximately by assuming that average stock price is lognormally distributed.

Exchange options:
Option to exchange one asset for another(remember Barter system!!). For e.g. its option to exchange one unit of U for one unit of V so the payoff is max(V(T)-U(T),0)

Basket options:
These are the options which gives the holder the right to sell or buy a portfolio of assets.

That's it for today
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ShaktiRathore

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Volatility Swaps is an agreement to exchange realized volatility between time 0 and T with a prespecified volatility set at time 0. Both volatility being multiplied with a pre-specified principal.
Variance Swaps is an agreement to exchange realized variance between time 0 and T with a prespecified variance set at time 0. Both variances being multiplied with a pre-specified principal.
Valuation of these swaps can be done analytically. First expected variance/volatility can be calculated from option/put price of different maturities and once these expected variances/volatilities are known over different maturities between time 0 and T than expected average realized volatility can be calculated from which the fixed volatility/variance can be compared to get the value of the volatility/variance swaps. So the values of such options can be calculated if enough options trade.

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#12
hi Shakti, regarding to the Strangle combination, i have a question that similar to yours, Long call(Exercise price K2)+ short put(Exercise price K1) where K2>K1.no dividends and same maturity.
how do i work out the lower bounds and upper bounds?
many thanks.
 

ShaktiRathore

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A Strangle Combination: Long call(Exercise price K2)+ short put(Exercise price K1) where K2>K1
P/Off=call P/Off+put P/Off
P/Off=max(S-K2,0)-max(K1-S,0)
for S<K1<K2 => P/Off=0-(K1-S)=-(K1-S)<0
for K1<S<K2 => P/Off=0-0=0
for K1<K2<S => P/Off=S-K2-0=S-K2
So upper bound is S-K2 and lower bound is -(K1-S).The values of these upper and lower bound depends on the value of stock S.
S-> infinity=> p/off ->infinity
S-> 0=> p/off->-K1

So lower bound is -K1 and upper bound is infinity.


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#15
Hi @ShaktiRathore , i read a lot of your posts and the way you handle the query precisely, really great. I have a query please help me to understand that is How to calculate Bond portfolio VaR using Historical Simulation and Monte carlo simulation. I know how to calculate by parametric method but i am not able to find he way around of other two ways, please help me out, thanks....
 

Rohit

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#17
Hi,

What type of questions in FRM test can we expect on this topic. This is such a vast area that memorizing everything is not possible. Any help will be much appreciated..Thanks !
 
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