Using Put-Call Parity and Strike Spreads to Infer Option Prices

Summary

The example shows how put-call parity and relationships across option strikes can be used to infer some option prices from quoted market data. Ignoring discounting, the response applies parity at the 70 strike to the given futures level and straddle price, then uses the put spread and butterfly quotes to solve for puts at the 50, 60, and 70 strikes. Parity at 60 then gives the call price and the 60 straddle value.

The method does not determine prices at every strike. The response notes that the 80 call and 40 put can only be bounded by prices at nearer strikes from the supplied information. Exact values require an additional assumption about the underlying distribution or another pricing input. The calculations also rely on the stated simplification of ignoring discounting and do not develop how a volatility smile would alter the market-making approach.

Key ideas

• Put-call parity links call and put prices at the same strike to the underlying forward or futures value.
• A straddle quote combined with parity can separate the call and put prices.
• Put spreads and butterflies provide equations relating option prices across strikes.
• Prices at unquoted outer strikes cannot be uniquely inferred from the supplied quotes alone.
• A distributional or volatility assumption is needed to estimate exact values beyond the observed strikes.

Tags

# Relations between Call and Put


# Relations between Call and Put












I am trying to solve a question in finance but I am pretty much stuck and would need your help :)

Suppose you know the following information about a market:

Future is at 66 70 strike straddle is trading at 27 50-60 put spread is at 2.5 50-60-70 put fly is at .2 Assume volatility is constant across strikes Using the prices given and relationships between options of various strikes, what are the fair values for the 80 Call, 60 Straddle, and 40 Put? Assume we had a volatility smile among the curve, how would this make your markets different?

I started by the following equations: C(70) - P(70) = 66 C(70) + P(70) = 27 P(50) - P(60) = 2.5 P(50) - 2P(60) + P(70) = 0.2

Thanks!

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/25229

You haven't written down your equations correctly. Ignoring discounting, the equations should be: C(70)-P(70)= -4 (not 66), from put-call parity. Also, C(70) + P(70)= 27; from these two we get C(70)= 11.5 and P(70)=15.5

Also P(60)-P(50)= 2.5 and P(70)-2P(60)+P(50)=0.2 from which P(70)-P(60)=2.7, hence P(60)=12.8 and P(50)=10.3 so now we know all the option prices for 50, 60, and 70 strikes

So C(60)-P(60)= 6 from put-call parity, giving C(60) = 18.8, and therefore straddle (60)= 31.6

we cannot exactly know where the 80 call and 40 put are, without making a distributional assumption. all we can sa.y is that C(80)<=C(70)= 11.5 and P(40)<=P(50)=10.3

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.