Why is ZVS > OAS? I don’t understand this eqn: ZVS= OAS+option cost. Can anyone help?

I believe this has to do with embedded call options. The opposite (ZVS < OAS) if their were embedded put options.

OAS is used when a bond has embedded options. For a callable bond, you require more yield than for an option free bond.

Suppose that you have two bonds, identical in all respects save one: one has a call option, the other does not.

The callable bond will sell at a lower price than the option-free bond. (Ignoring, for simplicity, that the OAS isn’t added to the zero-volatility spot curve), the OAS will be higher than the Z-spread: you have to discount at higher rates to get a lower price. The difference (OAS – Z-spread) is the option cost (in basis points): it’s positive because the issuer has to pay for the call option.

Suppose that you have two bonds, identical in all respects save one: one has a put option, the other does not.

The putable bond will sell at a higher price than the option-free bond. (Ignoring, for simplicity, that the OAS isn’t added to the zero-volatility spot curve), the OAS will be lower than the Z-spread: you have to discount at lower rates to get a higher price. The difference (OAS – Z-spread) is the option cost (in basis points): it’s negative because the issuer gets paid for the put option.

Just repeating what s2000 said (in my words):

If I own a callable bond, then I am at loss and issuer has some extra rights. So, if issuer has some extra rights, I as a bond holder should get something extra (Higher yield or lower Bond price). So, If I am getting a higher yield it means I am paid (by issuer) an option premium (thus lower bond price). So, it is positive. So my extra spread will be:

ZVS = OAS + option cost (extra I am getting)

In case of put this will be opposite and option cost will be negative…

Hope this helps

@S2000Magician: Thank you again for the beautiful explanation. Everything seems to be in place for me except the foll excerpt:

“(Ignoring, for simplicity, that the OAS isn’t added to the zero-volatility spot curve)”. Likewise, “(Ignoring, for simplicity, that the OAS isn’t added to the zero-volatility spot curve)”.

@exotichedge: That’s a wonderful explanation. It makes this equation second nature to me. Can’t thank you enough for this trick.(I am calling it a trick because it sticks this whole thing into my head)

The OAS is computed by creating an interest-rate tree (e.g., a binomial tree) extending into the future: this tree starts with the spot curve, but varies the future interest rates up and down from the spot curve values. You add the same spread to each node of the tree – not to each node of the spot curve – then use those adjusted interest rates to discount the cash flows, determining at each node whether or not the option will be exercised. The amount of variation in the nodes of the tree is the volatility of the tree.

The Z-spread – the zero-volatility spread – uses the spot curve unadjusted; in essence, it is a tree with zero volatility (hence the name).

Unfortunately, when authors try to show a picture of how the Z-spread, OAS, and option cost relate to each other, they show the OAS being added to the (zero-volatility) spot curve. It’s a useful picture, but is also slightly misleading.

My Pleasure

I just noticed the explanation again. Wouldn’t Z-spread be higher than OAS in case of callable bond?

This brings me to my next question: When we say discount at a higher rate, OAS should be higher than ZVS, but the equation says ZVS=OAS+option price, which is positive in this case; hence, ZVS>OAS.

Same query.

Kindly help @S2000magician.

Great! That’s some insight Can you help me on one more thing? What are these future intr rates? Are they short term forward rates?

Today’s spot rates.

So, today’s spot curve might have the 1-year spot rate at 4.0%, the 2-year spot rate at 4.5%, and the 3-year spot rate at 5.0%. The binomial tree might have:

- the 1-year up (U) rate at 4.40% and the 1-year down (D) rate at 3.64%
- the 2-year UU rate at 5.45%, the 2-year UD rate at 4.50%, and the 2-year DD rate at 3.72%
- the 3-year UUU rate at 6.66%, the 3-year UUD rate at 5.50%, the 3-year UDD rate at 4.55%, and the 3-year DDD rate at 3.76%

The volatility of this tree is 10% (i.e., the up factor is 1.1, the down factor is 1/1.1).

blackjack21: S2000magician:The OAS is computed by creating an interest-rate tree (e.g., a binomial tree) extending into the future: this tree starts with the spot curve, but varies the future interest rates up and down from the spot curve values. You add the same spread to each node of the tree – not to each node of the spot curve – then use those adjusted interest rates to discount the cash flows, determining at each node whether or not the option will be exercised. The amount of variation in the nodes of the tree is the volatility of the tree.

The Z-spread – the zero-volatility spread – uses the spot curve unadjusted; in essence, it is a tree with zero volatility (hence the name).

Unfortunately, when authors try to show a picture of how the Z-spread, OAS, and option cost relate to each other, they show the OAS being added to the (zero-volatility) spot curve. It’s a useful picture, but is also slightly misleading.

Great! That’s some insight Can you help me on one more thing? What are these future intr rates? Are they short term forward rates?

Today’s spot rates.

So, today’s spot curve might have the 1-year spot rate at 4.0%, the 2-year spot rate at 4.5%, and the 3-year spot rate at 5.0%. The binomial tree might have:

- the 1-year up (U) rate at 4.40% and the 1-year down (D) rate at 3.64%
- the 2-year UU rate at 5.45%, the 2-year UD rate at 4.50%, and the 2-year DD rate at 3.72%
- the 3-year UUU rate at 6.66%, the 3-year UUD rate at 5.50%, the 3-year UDD rate at 4.55%, and the 3-year DDD rate at 3.76%
The volatility of this tree is 10% (i.e., the up factor is 1.1, the down factor is 1/1.1).

Thank You!

S2000magician: blackjack21: S2000magician:The OAS is computed by creating an interest-rate tree (e.g., a binomial tree) extending into the future: this tree starts with the spot curve, but varies the future interest rates up and down from the spot curve values. You add the same spread to each node of the tree – not to each node of the spot curve – then use those adjusted interest rates to discount the cash flows, determining at each node whether or not the option will be exercised. The amount of variation in the nodes of the tree is the volatility of the tree.

The Z-spread – the zero-volatility spread – uses the spot curve unadjusted; in essence, it is a tree with zero volatility (hence the name).

Unfortunately, when authors try to show a picture of how the Z-spread, OAS, and option cost relate to each other, they show the OAS being added to the (zero-volatility) spot curve. It’s a useful picture, but is also slightly misleading.

Great! That’s some insight Can you help me on one more thing? What are these future intr rates? Are they short term forward rates?

Today’s spot rates.

So, today’s spot curve might have the 1-year spot rate at 4.0%, the 2-year spot rate at 4.5%, and the 3-year spot rate at 5.0%. The binomial tree might have:

- the 1-year up (U) rate at 4.40% and the 1-year down (D) rate at 3.64%
- the 2-year UU rate at 5.45%, the 2-year UD rate at 4.50%, and the 2-year DD rate at 3.72%
- the 3-year UUU rate at 6.66%, the 3-year UUD rate at 5.50%, the 3-year UDD rate at 4.55%, and the 3-year DDD rate at 3.76%
The volatility of this tree is 10% (i.e., the up factor is 1.1, the down factor is 1/1.1).

Thank You!

You’re welcome.

The OAS is computed by creating an interest-rate tree (e.g., a binomial tree) extending into the future: this tree starts with the spot curve, but varies the future interest rates up and down from the spot curve values. You add the same spread to each node of the tree – not to each node of the spot curve – then use those adjusted interest rates to discount the cash flows, determining at each node whether or not the option will be exercised.

What is this “same spread”?

S2000magician:The OAS is computed by creating an interest-rate tree (e.g., a binomial tree) extending into the future: this tree starts with the spot curve, but varies the future interest rates up and down from the spot curve values. You add the same spread to each node of the tree – not to each node of the spot curve – then use those adjusted interest rates to discount the cash flows, determining at each node whether or not the option will be exercised.

What is this “same spread”?

Let’s use the binomial tree I proposed above:

- the 1-year up (U) rate at 4.40% and the 1-year down (D) rate at 3.64%
- the 2-year UU rate at 5.45%, the 2-year UD rate at 4.50%, and the 2-year DD rate at 3.72%

If we calculate the OAS for a given bond to be 85bp, then the discount rates we use in this tree (for this bond, to get today’s market price) are:

- the 1-year U rate is 5.25% (= 4.40% + _
**0.85%**_) and the 1-year D rate is 4.49% (= 3.64% + _**0.85%**_) - the 2-year UU rate is 6.30% (= 5.45% + _
**0.85%**_), the 2-year UD rate is 5.35% (= 4.50% + _**0.85%**_), and the 2-year DD rate is 4.57% (= 3.72% + _**0.85%**_) - the 3-year UUU rate is 7.51% (= 6.66% + _
**0.85%**_), the 3-year UUD rate is 6.35% (= 5.50% + _**0.85%**_), the 3-year UDD rate is 5.40% (= 4.55% + _**0.85%**_), and the 3-year DDD rate is 4.61% (= 3.76% + _**0.85%**_)

At every node we’re adding the same spread – _ **85bp** _, the OAS – to the spot rate at that node.

(In fact, to calculate the OAS, we guess at the spread – maybe we try adding 50bp to each node – and see what price we get. We get a price higher than today’s market price, so we know we need to increase our discount rates. Maybe we try 100bp next, but that gives us a price lower than today’s market price. So we try something in between, maybe 75bp. And we keep adjusting our guess until we get today’s price.)

blackjack21: S2000magician:The OAS is computed by creating an interest-rate tree (e.g., a binomial tree) extending into the future: this tree starts with the spot curve, but varies the future interest rates up and down from the spot curve values. You add the same spread to each node of the tree – not to each node of the spot curve – then use those adjusted interest rates to discount the cash flows, determining at each node whether or not the option will be exercised.

What is this “same spread”?

Let’s use the binomial tree I proposed above:

S2000magician:

- the 1-year up (U) rate at 4.40% and the 1-year down (D) rate at 3.64%
- the 2-year UU rate at 5.45%, the 2-year UD rate at 4.50%, and the 2-year DD rate at 3.72%
If we calculate the OAS for a given bond to be 85bp, then the discount rates we use in this tree (for this bond, to get today’s market price) are:

- the 1-year U rate is 5.25% (= 4.40% + _
0.85%_) and the 1-year D rate is 4.49% (= 3.64% + _0.85%_)- the 2-year UU rate is 6.30% (= 5.45% + _
0.85%_), the 2-year UD rate is 5.35% (= 4.50% + _0.85%_), and the 2-year DD rate is 4.57% (= 3.72% + _0.85%_)- the 3-year UUU rate is 7.51% (= 6.66% + _
0.85%_), the 3-year UUD rate is 6.35% (= 5.50% + _0.85%_), the 3-year UDD rate is 5.40% (= 4.55% + _0.85%_), and the 3-year DDD rate is 4.61% (= 3.76% + _0.85%_)At every node we’re adding the same spread – _

85bp_, the OAS – to the spot rate at that node.(In fact, to calculate the OAS, we guess at the spread – maybe we try adding 50bp to each node – and see what price we get. We get a price higher than today’s market price, so we know we need to increase our discount rates. Maybe we try 100bp next, but that gives us a price lower than today’s market price. So we try something in between, maybe 75bp. And we keep adjusting our guess until we get today’s price.)

Woah! This is amazing! So, it boils down to numerical methods and computer scientists.

How else do you think we mathematicians stay gainfully employed?

How else do you think we mathematicians stay gainfully employed?

–off the topic— Removed.

Don’t we have to add OAS to a Z spread tree?

The *z*-spread is added to the *spot* rates: no volatility (hence, _ **z** _-spread: _ **zero-volatility** _ spread).

The OAS is added to the _one-period **forward** _ rates in a binomial tree: positive volatility.

They’re never added together.