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# R25.P1.T4.ALLEN_Ch 2& 3:Topic:VAR_LINEAR_DERIVATIVES

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @gargi.adhikari Please note that we've since issued a greatly revised version of this study note (at https://www.bionicturtle.com/topic/study-notes-allen-chapters-2-3/ ) although unfortunately we didn't update this learning objective, which to your point, really needs some improvement (especially because it is so fundamental). The short answer to your good question is: we definitely have different constant values, the $250 is just coincident to the fact that it's the standard multiplier for an S&P futures contract (i.e., contract unit at http://www.cmegroup.com/trading/equity-index/us-index/sandp-500_contract_specifications.html ). To illustrate, (It occurs to me that, technically speaking, if the S&P index is denominated in dollars, which I believe it is, then the futures contract contract ought to be unitless 250 multiple rather than$250 ... hmmm ... wondering what i missing there )

This LO really does want a revision (which I will add this week) because the delta, Δ, is confusing (but the source Linda Allen is arguably confusing here). As L. Allen rightly points out, the most generic (and critical) formula here is for the linear approximation of a derivative (aka, truncated Taylor series with only the first term) is:
• VaR(derivative) = Δ*VaR(risk factor). In the case of an option on a stock: $VaR(c) = Δ*VaR($S); e.g., if call option delta 0.60 and S = $10.00 with daily σ = 1.0%, then 95% normal VaR(c) = 0.60*[$10.00*1.0%*1.645].
• In the case of the futures contract, the Δ is approximately one (please do know that futures delta is exactly exp(rT) per Hull, which is slightly per Hull but it's okay to round here) .... what I will add to the notes is the proper expression:
• VaR(S&P 500 futures contract) = Δ*VaR(risk factor) = (~1.0)*VaR(risk factor) = (~1.0)*[$250 * VaR(S&P Index). • And therefore technically it is ever better to say: VaR(S&P 500 futures contract) = exp[(r-q)*T]*[$250 * VaR(S&P Index].
• I don't think this is exactly in the source Allen, but hopefully you can see how this parses (i) the $250 is a multiplier that leverages the index value (and this is the leverage introduced by the notional value of the contract!) and so leverages the risk/upside opportunity versus the (ii) the delta of 1.0 where we are implicitly assuming the futures contract price moves 1:1 with the index price. I will post the updated note after I write it. In the meantime, great question and I hope this helps! Last edited: #### gargi.adhikari ##### Active Member @David Harper CFA FRM Thanks so much for the above explanation So is this the takeaway just to make sure I understood it right ...or if there are still some glitches in my understanding on this topic... 1) When the underlying is a Stock:- We use the Delta-Normal to calculate the VAR VAR = Delta * (Sigma * z) * Stock Price 2) a) When the underlying is a " FUTURES-Contract " Based on a " Commodity ":- VAR = Delta * Contract Size * Spot Price of Commodity Now DELTA for FUTURES-Contracts is =e^(r-q)T ...? So, VAR = e^(r-q)T * Contract Size * Spot Price of Commodity 2) b) When the underlying is a " FORWARD-Contract " Based on a " Commodity ":- VAR = Delta * Contract Size * Spot Price of Commodity Now DELTA for FORWARD-Contracts is=1 ...? So, VAR = 1 * Contract Size * Spot Price of Commodity 3)a)When the underlying is a " FUTURES-Contract" Based on a " Index ":- VAR = Delta * Contract Multiplier ($ 250 for S&P 50 Index ; $50 for Mini S & P Index etc) * Spot Price of Index Now Delta for FUTURES-Contracts =e^(r-q)T ...? So, VAR = e^(r-q)T * Contract Multiplier ($ 250 for S&P 50 Index ; $50 for Mini S & P Index etc) * Spot Price of Index 3)b)When the underlying is a " FORWARDS-Contract" Based on a " Index ":- VAR = Delta * Contract Multiplier ($ 250 for S&P 50 Index ; $50 for Mini S & P Index etc) * Spot Price of Index Now Delta for FORWARD-Contracts is =1 ...? So, VAR = 1 * Contract Multiplier ($ 250 for S&P 50 Index ; $50 for Mini S & P Index etc) * Spot Price of Index Also, Is the Delta of Forward-Contracts = 1 .....? and Delta of Futures-Contracts = e ^ (r- q) T ......? Much gratitude for confirming and correcting the takeaways above.  Last edited: #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @gargi.adhikari Sure thing, but please note there is a glitch when you replace VaR(index) with Spot Price of Index; or when you replace VaR(commodity price) with Spot Price of Commodity. If S(t) is the commodity spot price, the easiest approach is to use VaR(commodity price) = S(t)*σ[S(t)]*z(α), where z(α) is the normal deviate; e.g., z(95%) = 1.645. So here are some working assumptions: • Let a stock price, S(t) =$30.00 with volatility, σ[S(t)] = 20.0%
• Assume a call option on the stock has (percentage) delta of +0.60
• Assume the S&P Index = 2400 with volatility σ[S&P500] = 10.0%; i.e., deliberately lower
• Assume the price of corn is $4.00 per bushel with σ = 15%; and per the contract specifications (we don't choose these!) the corn's futures contract unit is 5,000 bushels Okay then running down the list, and making the dreaded normality assumption: • VaR(single share of stock; ie, underlying is a share) = S(t)*σ[S(t)]*z(α) =$30.00 * 20% * 1.645 = $9.87 • Linear approximation (ie., omitting gamma effects) of VaR(single call option on a share; ie, underlying is an option derivative) = Δ*VaR(risk factor) = Δ*S(t)*σ[S(t)]*z(α) = 0.6*$30.00*20%*1.645 = $5.92 • VaR(single S&P 500 futures contract) = Δ*Multiple*VaR(risk factor) = Δ*Multiple*S&P500(t)*σ[S&P500(t)]*z(α) = exp(rt) * 250 * 2400 * 10% * 1.645 = exp(rt) *$98,691
• VaR(single S&P 500 forward contract) = Δ*Multiple*VaR(risk factor) = Δ*Multiple*S&P500(t)*σ[S&P500(t)]*z(α) = 1.0 * Multiple * 2400 * 10% * 1.645; i.e., multiple can be anything (non standard)
• VaR[single corn futures contract] = Multiplier * Δ * VaR(corn price) = 5,000 * 1.0 * ($4.00 * 15% * 1.645) =$4,934
• VaR[single corn forward contract] = Multiplier * Δ * VaR(corn price) = Multiplier * exp(rT) * VaR(corn price); i.e., same but the multiplier can be anything (is non-standard)
Please note that all of these linear VaR approximations above fit into the general form VaR(derivative position) = multiple * Δ*VaR(risk factor). I hope that's helpful!

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##### Active Member
@David Harper CFA FRM Thank you so so much for laying out the above - super clear now  #### emilioalzamora1

##### Well-Known Member
hi @[email protected],

Forward delta is 1 (the change in the forward with respecto to a change in the underlying asset).

The difference why delta is NOT 1 for futures is based on the fact that interest rates are not constant. Forwards are settled at maturity date while futures are settled daily.

#### [email protected]

##### New Member
Hi , The forward contact price F(t,T) = s(t) - ke^(-r*(T-t)) . In that case df/ds ( delta)=1 . But mathematically howz that different for future contract price since dr/ds=0. So delta would be 1 mathematically. Am I missing something?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @[email protected] I've struggled with this in the past; e,g, here is a nine-year old thread where I am not sure that I make any sense https://www.bionicturtle.com/forum/threads/delta-of-forward-and-future.336/

But i wonder why i've never considered this simply, where Hull distinguished between forward value and futures price
• the value of a forward contract, f = [F(0) - K]*exp(-rt) where F(0) = S(0)*exp(rT) such that f = [S(0)*exp(rT) - K]*exp(-rt) = 1.0*S(0) - K*exp(-rT) and ∂f/∂S = 1.0; ie, K is constant
• the price of a futures contract per cost of carry, F(0) = S*exp(rT) where ∂f/∂S = exp(rT).
I think given time I could actually connect this to the fundamental narrative explanation for the difference (which is that the futures contract settles daily which creates cash in-flow/outflow at the margin--literally via the margin account), along the lines of: the forward contract future value is discounted back, which negates the risk free growth; but the futures contract price is immediately responsive to riskfree rate changes, or put another way, unlike the forward math, effectively it is not getting nullified by not being discounted back, which is economically similar to investing now at that risk free rate (and earning the gain). My phrasing could stand much improvement, hopefully this makes a bit of sense ... thanks,