VaR option / Bond

[mh2452]

Hi

May be I am getting confused but I am trying to reconcile the intuition behind the VaR for an equity option with that of a bond

If I am not mistaken,
1- VaR Stock = Stock * deviate * vol for lets say 1 day
then VaR Option = VaRstock * Delta - 0.5 VaRstock^2 * gamma

2- VaR(Yield) = deviate * vol yield
VaR bond = price * VaR Yield * duration - 0.5 VaR yield * price * convexity

I am having a hard time understand the presence of ^2 in first case, not in second case etc

Anyone would let me know?

thx

 

[David Harper CFA FRM]

Hi @mh2452 But the convexity term does square the yield shock. See https://forum.bionicturtle.com/threads/p1-t4-328-delta-gamma-value-at-risk-var.7203/#post-28378
ie..,

In mathy terms, Taylor truncated to only two terms says: df = δf/δS*δS + 0.5*δ^2f/δS^2*dS^2 + ....
In the option context, that is df = ΔS*delta + 0.5*gamma*ΔS^2

• ΔS*delta is represented by the blue line. Consequently, ΔS informs df (change in option value) obviously:
  • For a long put position, risk is +ΔS with linear impact: -df = +ΔS*(-delta)
  • For a short put position, risk is -ΔS with linear impact: +df = -ΔS*(-delta)
• But 0.5*gamma*ΔS^2 is always additive in the Taylor Series because (ΔS^2) is positive regardless of up/down stock price
  • For a long put position, that implies the positive gamma terms mitigates the negative delta term (-df); i.e., reduces risk
  • For a short put position, that implies the positive gamma terms exacerbates the postive delta term (+df); i.e., increases risk

In short, the same Taylor Series: f = δf/δS*δS + 0.5*δ^2f/δS^2*dS^2 ... is applied to both:

• Options: df = delta*ΔS + 0.5*gamma*ΔS^2 <-- risk factor is change in underlying stock price
• Bonds: dP/P = -D*Δy + 0.5*convexity*Δy^2 <-- risk factor is yield change

These are generic (instantaneous changes), to get value at risk (VaR), just replace ΔS with σ(S)*deviate(α) and Δy with σ(y)*deviate(α) as VaR just scale (multiplies) the volatility of the risk factor. I hope that helps!

 

[jshall]

I'm a little confused: is *MINUS* or *PLUS* appropriate?

VaR_opt = [VaR_stock x Delta] *MINUS* [0.5 x VaR_stock^2 x gamma]
(This was posted by mh2452 and I know that I have this equation in my notes somewhere)

or

VaR_opt = [VaR_stock x Delta] *PLUS* [0.5 x VaR_stock^2 x gamma]
(Which is how I am interpreting David Harper's post above)

 

[David Harper CFA FRM]

Hi @jshall I recommend intuition here, not rote memorization. In the Taylor approximation, you can see that the gamma term, 0.5*gamma*ΔS^2, always adds because ΔS is squared. This is just the curvature (convexity) of the call-versus-share plot. df is change in price and from the perspective of the option price, the convexity adds to the linear delta term. But you are showing VaR_option rather than df or ΔP. VaR depends on your perspective (ie, long or short). If you are long the option, risk is a drop in the stock price (left on the x axis) and the additive gamma term (the curvature/convexity) adds to price but reduces your VaR; if you are short the option, risk is an increase in price and the (still mathematically) additive gamma term increases your VaR. Thanks,

 

[jshall]

That makes a lot of sense. Thanks, David!