Mapping options

[ajsa]

Hi David,

I am not sure I understand what you described

"1. Long position in the stock equal to delta
2. Short position in the underlying asset financed by a loan equal to [(delta Shares) – value of call]"

I assume in 2 the underlying asset is the stock, so 1 and 2 will just offset each other resulting a long position equal to value of call?

Does financing by the loan mean there will be the 3rd component, short position of treasury bill? or you mean this is what 2 means actually?

thanks.

 

[David Harper CFA FRM]

Hi asja,

I admittedly gave that short shrift; it's from Jorion Ch 11.
(I should have an example in there...)

His example is a call option worth $4.20, with delta = 0.536, on a stock with price of $100
Such that $4.20 = (0.536)*100 - $49.40
i.e., call option = long partial share worth $53.60 funded by loan of $49.40 (so "out of pocket the $4.20" which equals the premium")
...then at expiration, repay loan and net position = gain on option

Re: "Does financing by the loan mean there will be the 3rd component, short position of treasury bill?"
Yes, exactly, so this merely reflects a "delta mapping" (an option has other risk factors) and in this case, the mapping is indeed to a short treasury bill (or short the rate). Just like short a bond will increase in value if rates increase, so a call option increases in value if riskless rate input increases.

David

 

[ajsa]

Hi David,

so can I simplify it to be the following?
1. long stock with the value of the call premium
2. short position of treasury bill equal to [(delta Shares) – value of call]

"2. Short position in the underlying asset financed by a loan equal to [(delta Shares) – value of call]”"
but how can one start a short position by borrowing a loan?

Also what about put?

Could you also explain the reason behind? put-call parity?

thanks.

 

[David Harper CFA FRM]

Hi asja,

Not quite, rather:
1. long delta*Share price; e.g., as above, delta of 0.536 multiplied by share price of $100 = long partial share with value of $53.60
2. short the present value of the strike price; e.g., as above, borrowing $49.40

Re: short position by borrowing a loan: instead of investing in a bond as a long position (i.e., spend our cash today which must be repaid at future date), we are short the bond, so we are are borrowing (i.e., borrow $49.40 today, so receive cash today, to be repaid in future)

the put is:
short underlying shares (delta of put * share price), and
long the bond (invest in cash)
i.e., lend cash and short the stock.

the reason is the same logic used to build the Black-Scholes, it is based on a dynamic hedge, but the above can be almost seen inside the Black-Scholes:

call = S*N(d1) - Strike*EXP[-rT]*N(d2)
put = discounted Strike - Stock*N(-d1)

note the call mapping is:
S*N(d1):
(+) long the amount of share price * delta ([N(d1)] plus
(-) short the cash in the amount of PV = discounted strike price, such that at exercise, cash is received equal to amount needed to exercise the option!

and put mapping is:
(+) long cash and (-) short put delta shares

David

 

[ajsa]

Hi David,

So for call, why N(d2) is not considered when shorting the cash?

Thanks.

 

[David Harper CFA FRM]

Hi asja,

Well, that's why I said it can "almost be seen" (I taught BSM to clients for about five years, it's the one FRM area where i actually have some bona fide depth!) ... trying to intuit the BSM directly without deep dive into the calculus is challenging (my opinion) and occassionally we are just dealing with metaphors

really, if we are talking about the call option value
(and for those wanting the deep dive, the best reference i know of is Neil Chriss' Black Scholes and Beyond)

the call option =
1. long a binary stock-or-nothing option with strike of K (i.e., pays asset or nothing) with value = N(d1)*Stock, plus
2. short a binary option with strike of K and payout of K (i.e., pays strike or nothing) with value = N(d2)*discounted Strike

where N(d1) = option delta and N(d2) = probability the option will be exercised in the *risk-neutral* world

...so the best i've been able to do (reflected in the videos) in regard to a direct intuition is:
minimum value of option = S - Discounted strike; i.e., value if vol = 0
then wrap in the N(d1) and N(d2) to adjust for the "probability-effect" introduced by volatility

hope that helps...David

 

[Sorend]

Hi, I'm new here

I trying to create a formula or a model in excel for calculation the probability (1% -> 99%) of a short index option being in-the-money given a spot price, timerange, volitility and strike price How can this be done?

Soren