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# Worth of equity option on balance sheet before its execution

#### greattallwang

##### New Member
Subscriber
This question comes from below sentence:

"The equity call option with an underlying value of $100 and a delta of 50 percent, is equivalent to having bought$50 worth of the S&P 500 index shown up on (the buyer"s) balance sheet."

The buyer pays a small premium for getting this equity option. Before this buyer executes the option, the buyer has not owned the underlying asset yet, so why this option can still be marked as $50 on the balance sheet? The buyer may post some IA for this trade, but this amount of IA should not be bigger or equal to$50. Thanks.

#### David Harper CFA FRM

##### David Harper CFA FRM
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Hi @greattallwang

You make a good point. That's from Malz and he is not referring to the recorded book (accounting) balance sheet, he is referring to an economic balance sheet. When he credits liabilities by $50 broker loan, that is not an actual$50 broker loan. Further, it is the nature of derivative contracts (especially FMV options which have zero intrinsic value at issuance) that there is no single best approach to "restoring" them from off-balance-sheet status to on-the-economic-balance-sheet. An economic balance sheet is fraught with judgement calls (but so is the accounting balance sheet). I can say this with confidence about stock options because years ago, I researched/wrote about them extensively (e.g., investopedia tutorial on accounting for stock options http://www.investopedia.com/features/eso/eso1.asp). In short, you have a good point, IMO.

But here is Malz introducing the economic balance sheet which is the context for his approach, and his approach of multiplying by delta has two advantages (that i can think of): one, it's dead simple (my estimate of true economic value would be more complicated), and two, it's very conservative (by overstating the increase in assets and liabilities) :

Malz Section 6.2.1 (emphasis mine): "We start with an economic balance sheet for the firm [Assets = Equity + Debt] ... While this looks familiar, it differs from the accounting balance sheet in that asset values are not entered at book or accounting values, but at market values, or at some other value, such as an option delta equivalent, that is more closely related to the market and credit risk generated by the asset. In Chapter 12, we use this concept to create more accurate measures of the firm’s indebtedness. Here, we will use an economic balance sheet to more accurately value the firm’s equity, that is, the part of the value of the assets belonging to the owners of the firm once the debt has been deducted."

... Malz Section 12.3.4 (emphasis mine): "Derivative securities are a means to gain an economic exposure to some asset or risk factor without buying or selling it outright. One motivation for market participants to use derivatives is as a means of increasing leverage. Leveraged ETFs are an example of an investment product that uses derivatives in order to create the economics of leveraged investment in, say, an equity index. Although derivatives are generally off-balance-sheet items in standard accounting practice, they belong on the economic balance sheet, since they may have a large impact on returns. Each side of a derivatives contract is synthetically long or short an asset or risk factor. But the market values of derivative securities are not equal to the value of the underlying asset, or the riskiness of the positions. Therefore, their market values or NPVs are generally not the best values to represent them. Rather, for purposes of measuring economic leverage, we wish to find, for each type of derivative, a cash-equivalent market value. As with most issues around the measurement and interpretation of leverage, as much judgment as science is involved."

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#### Gdb

##### Member
Hi @David Harper CFA FRM ,

I do get the idea of the economic exposure not being equal to the accounting book value exposure.
But I think I am joining the initial question in the thread of how we go from the mentioned call option on 100m S&P, to a broker loan of 50m as a liability (what is a broker loan? is this margin lending?), and a 50m exposure on assets?

The option value changes 50% of the change in S&P notional (this is how to interpret delta; correct?) I suppose this is where the 50m assets comes from? But not sure how to interpret the broker loan..
Also, I'm not sure where to put the paid premium in this reasoning?

Could you (or someone else reading this) help me in the right direction?

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Gdb I think the two key ideas (at least from my perspective) are (i) the delta-equivalent and (ii) the unfunded nature of a derivative such as this option. In this example (Malz), the $100.00 notional option has a delta of 0.50 and the idea of entering the option position (on a value of$100.00) is that $50.00 is the share-equivalent in delta terms, which is an instantaneous (temporary) first approximation. This simply means that, let's say, if the stock actually has a price of$100.00 and its option actually has a delta of 0.50, then both of the following have the same short-term reaction to an $X gain or loss in the stock price: • If the stock drops by$5.00, where (N) is the number of options, the option position value drops by approximately N* $5.00 * Δ = N*$5.00 * 0.50 * N = N * $2.50, and: • If the stock drops by$5.00, the delta-equivalent share position drops by approximately (N*Δ)* $5.00 = N * 0.50 *$5.00 = N * $2.50. In this way, the delta-equivalent share position (i.e.,$50 long S&P position) exists simply to mimic (reverse synthesize) the reaction to an option position on $100 notional value. Or more simply, how many outright shares would we own if we want the share position to respond to small price changes in the same way as our actual option position? (it's doubly approximate however: 1. only linear and 2. as soon as the shares change, so does the delta). The entry of the broker loan liability is because the option position is unfunded: if cash were used, then the corresponding entry to the debit (+$50 long S&P 500 position) would be credit (reduce) cash asset. An option is a synthetic equivalent to delta-shares, but the full replication includes borrowing to pay for the purchased delta shares because the option is not funded (In Hull's dynamic delta hedge--Hull Chapter 19), the idea is the written options are hedged by borrowing cash to buy delta-equivalent shares. There is not an actual loan here, rather the liability is entered to match the asset and represent the position on the (economic) balance sheet, My interpretation of Malz is that the option premium is simply not included here (perhaps he contemplates it as an expense, or not at all). Only because I did have several years of consulting to companies on issues related closely to this, I can confidently say that what Malz is doing here is clearly not the only way to represent options on the balance sheet. I was involved in several projects that put options on the economic balance sheet where the team had a hard time agreeing on a methodology. But to me it does have coherent logic, it's just based on a really rough approximation. I hope that's helpful, thanks!

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#### Gdb

##### Member

To be sure though; I think the option value should not be able to go below zero, should it?
Whereas a 50m long equity position funded by 50m common debt could, correct?

Thanks again!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber

Append: I just noticed the sentence immediately after the assumptions: "We assume that the nonoption positions are initiated at market-adjusted prices and spreads, and therefore have zero NPV." ... which doesn't help me (!) because I am not sure what it suggests for the option position. Unfortunately, he doesn't illustrate the BS in a subsequent interval when values (eg) drop, which would betray I think the exact assumption here w.r.t. notional/market/premium.

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#### Rohit

##### Member
@David Harper CFA FRM really struggling to grasp the concepts of derivatives and leverage wrt economic balance sheet. Do you think this has high testability for FRM P2? Any view on what type of questions/concepts can be tested. Many Thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Rohit I think it has low (to medium, but not certainly not high) testability. The thing about the economic balance sheet cluster: it's been consistently (i.e., every year) assigned since 2013, so it cannot be dismissed. Re type of questions, I think our associated PQ try to represent what could be answered (sorry i just don't have current time to pick thru the feedback threads to see what might have been similarly asked in the forum). I hope that's helpful,

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