N(d1)

[bass]

Hi all, can anyone tell me the following:

stock price is $10
1 hour before expiry.
In that hour it is expected that there are the following possibilities:
-10% chance that the stock moves 10%
-20% chance the stock moves with 20%
-30% chance the stock moves with 5%
-40% chance the stock stays where it is.

from this information, can I calculate N(d1) and N(d1) for strike 11?

 

[David Harper CFA FRM]

HI @bass what's the question's source (we sometimes like to know just so time isn't wasted, in case it's not a good question)?

 

[bass]

Hi David, it was an interview question. I was expected to calculate the price of the 11 call and I assumed the probabilities are linked to N(d1&d2). But the question may be wrong also, as you said.

HI @bass what's the question's source (we sometimes like to know just so time isn't wasted, in case it's not a good question)?

 

[David Harper CFA FRM]

Hi @bass really, an interview question? Fascinating. Never seen this pattern. My first reflex is to calculate the variance per E(X^2) - E(X) = 10% - 30%^2 = 1.0% such that σ=sqrt(2%) = 10%. (Minor issue: this assumes positive returns +10%, +20%, +5% which is not exactly what's assumed!). Then we have all the BSM params except for the riskfree rate, but for such a short maturity, the d1 is basically invariant to the RF assumption. We have S=10, K=10, σ=0.10, Rf = X, T = 1/8*1/250, q = 0 (?) ... but then we quickly realize such a d1 is quite negative: an OTM call is asymptotic to zero as maturity approaches; e.g., d1 = -42 such that N(d1) = 0. What's cool is that we can intuit the ~zero without any calculations.

What's also interesting is same question but ATM (S = 10, K =10) where ln(S/K) = 0 such that N(d1) ~ 0.50.

P.S. I never calculate intra-day delta, would love any feedback on the 1/8th in one hour term assumption, T = 1/8*1/250. Anyone know if that's right? Thanks,

 

[Torsleno]

Would this simplified way of thinking work? let's take a long call
N(d1) * e-qt = Delta
computing the different values of delta at expiry, assuming only moves up (with moves down all the deltas would be 0)

Move up 5%/stays the same: delta = 0
Up 10%: S=K: delta = 0.5
Move up 20%: ITM call, delta ~1 since the time to expiry is very small

weighted delta by probability = 0.25 -> N(d1) ~= 0.25

After all, N(d1) does represent an approximation of the probability of the stock expiring in the money.

If I were to price an option with this info, I would just take the payoffs with the probabilities, and assume only moves up, or assume a probability for a move up and a move down (similar to a binomial tree, except the probabilities are not risk neutral probabilities).

 

[David Harper CFA FRM]

@ArnaudD I like your thinking, but I agree the problem is these are not risk-neutral probabilities. Building from your idea, below is a plot of the four outcomes (without probabilities). That's a payoff graph, and delta is the slope of a line (aka, rise/run) ... but which line? In a straightforward binomial, the (slope's) line is straightforward: (c_u - c_d)/ (Su - Sd). Here, we have ($1.00 - 0))/($12 - Sd) where it's unclear to me how we anchor the Sd. If we collapse the three non-payoffs into one (40%*10 +30%*10.50 + 10%*11)/80% = $10.31 is weighted average Stock down, then Δ = 1/(12 - 10.31) = 0.59.

... at which point, I'd be tempted to remedy the fact that I'm treating the movements as ups only. If "20% chance the stock moves with 20%" implies that stock can go down to $10*80% = $8.00, notice that we could argue (sans probabilities) that Δ = (1 - 0)/(12 - 8) = 0.25. Same as yours But I have technical issues to varying degrees with all of these solutions ...

 

[bass]

It was an online application question David, before a possible interview. I suspect the answer is easier given the limited time I had for it or they take a shortcut which is not completely sound. I will try to find out what they were looking for. It may take some time. I appreciate the quick and extensive replies!

ps I am pretty sure the % moves are meant to be potentially both up and down. Otherwise I guess the stock wouldn’t be at 10.

 

[Torsleno]

In limited time, I would just have thought: what is the chance that this call expires in the money? I would have taken moves ups and down and assumed 50% chance for each.
I also correct my previous answer, I think the ATM shouldn't be included in this case (ATM our payoff is 0).

-10% chance that the stock moves 10% -> ATM
-20% chance the stock moves with 20% -> 10% chance ITM (50% * 20% * 1)
-30% chance the stock moves with 5% -> OTM
-40% chance the stock stays where it is -> OTM

So the delta should be 0.10 (on the long call)

Intuitively, I would also think:
- In terms of boundaries: max move down is -$3, max move up is +$1, with the spot at $10 these represent respectively -0.3 and +0.1 delta. I don't think the delta could move beyond that [-0.3;0.1] range. If you're short call you can get up to 0.3 times the current spot and if you're long you can get up to 0.1 times.
I would be curious to have some thoughts about this methodology, maybe @David Harper CFA FRM ?