Qs on general formula for pricing

[Jun Wu]

Dear All

Hope you are well.

I looked back to the general formulae in section 1.1,

c=P(0,T*)*[F0*Phi(d1)-X*Phi(d2)]

1 For call option, I don't intuitively get why F0 came into the equation, it is the forward price at time 0 for the asset with maturity T, this is the price agreed at time 0 to deliver the asset at time T? How is this impacting the value of call option?

2 On CH13, pg 12 for pricing interest rate caps:

If the rate KR is assumed to be lognormally distributed with volatility sK , then we can use the expression in Section 1.1 above, substituting: ·
RX for X
FK (the forward rate for the period between time kt and tk+1 ) for F0
tk+1 for T*
sigma^K for sigma

How does it make sense for FK to replace F0 here?

Thank you and Merry Christmas!

 

[Joe Hook]

Hi,

Merry christmas to you too!

Apologies. This isn't the easiest to explain via text but hopefully the below works for you. I've tried to keep it reasonable concise. If unclear, please feel free to come back to me.

The way I think about it if it helps is to think back to the call option formula for a share. The payoff on a call option, on a NON-DIVIDEND paying share, is the maximum of (ST-K,0). And the formula for this call option is:

c=St*Phi(d1)-K*exp(-r(T-t))*Phi(d2)

This formula takes possible payoffs ST - K (truncated for share prices that are indeed higher than the strike prices), multiplies them by the associated probabilities that the share price does indeed attain that value and exceeds the strike price and then discounts them back to time 0. The payoff ST is random and depends on volatility and time to expiry but is a function of the current share price St. Our expectation is that ST=St*exp(r(T-t)) ie the current share price grown at the risk-free rate. This is because we price the option in the risk-neutral world where we expect all risky assets to grow at the risk free rate.

For a call option on a bond we similarly need the current value of the bond but if this bond pays interest we also need to strip out the present value of any income that is incorporated in the bond price but will not be paid to the call option holder. Hence,

c=(B0-I)*Phi(d1)-K*exp(-r(T-t))*Phi(d2)

But since (B0-I)=F0*exp(-r(T-t)) we rewrite instead as:

c=exp(-r(T-t)*(F0*Phi(d1)-K*Phi(d2)) or more simply:

c=P(0,T)*(F0*Phi(d1)-K*Phi(d2))

So in both cases our first term, either F0 or St is the current value of the asset.

Similarly for the interest rate caplet, Fk is our current expectation for the interest rate that will apply between time tk and tk+1. The behaviour of this forward rate between now and when it will be used to calculate the payoff will then be wrapped up within the probability Phi(d1). However, because the payment is not being made at time tk but instead time tk+1 our P(0,t) is adjusted to be the discount factor from time tk+1.

Joe

 

[Jun Wu]

Hi,

Merry christmas to you too!

Apologies. This isn't the easiest to explain via text but hopefully the below works for you. I've tried to keep it reasonable concise. If unclear, please feel free to come back to me.

The way I think about it if it helps is to think back to the call option formula for a share. The payoff on a call option, on a NON-DIVIDEND paying share, is the maximum of (ST-K,0). And the formula for this call option is:

c=St*Phi(d1)-K*exp(-r(T-t))*Phi(d2)

This formula takes possible payoffs ST - K (truncated for share prices that are indeed higher than the strike prices), multiplies them by the associated probabilities that the share price does indeed attain that value and exceeds the strike price and then discounts them back to time 0. The payoff ST is random and depends on volatility and time to expiry but is a function of the current share price St. Our expectation is that ST=St*exp(r(T-t)) ie the current share price grown at the risk-free rate. This is because we price the option in the risk-neutral world where we expect all risky assets to grow at the risk free rate.

For a call option on a bond we similarly need the current value of the bond but if this bond pays interest we also need to strip out the present value of any income that is incorporated in the bond price but will not be paid to the call option holder. Hence,

c=(B0-I)*Phi(d1)-K*exp(-r(T-t))*Phi(d2)

But since (B0-I)=F0*exp(-r(T-t)) we rewrite instead as:

c=exp(-r(T-t)*(F0*Phi(d1)-K*Phi(d2)) or more simply:

c=P(0,T)*(F0*Phi(d1)-K*Phi(d2))

So in both cases our first term, either F0 or St is the current value of the asset.

Similarly for the interest rate caplet, Fk is our current expectation for the interest rate that will apply between time tk and tk+1. The behaviour of this forward rate between now and when it will be used to calculate the payoff will then be wrapped up within the probability Phi(d1). However, because the payment is not being made at time tk but instead time tk+1 our P(0,t) is adjusted to be the discount factor from time tk+1.

Joe

Thanks Joe! Yes that makes much more sense.

A few follow up queries please:
1 On (B0-I)=F0*exp(-r(T-t)) : Is this because F0 is the price we agreed/expected today at time t, to exchange the bond at time T? So if we discount back between T-t at risk free rate we get B0-I = the present value at time t of bond value stripped away income. I get confuse by the use of 0 and t, is current time t or 0?

2 Fk : Is this the rate expected at time 0, or t, being applicable between tk and tk+1?

3 Are the following correct?
Share: Phi(d1) = probability for share price does indeed attain ST (share price expected at time T if grown at risk free rate)
Bond Call : Phi(d1) = probability for bond price does indeed attain F0 (bond price expected at time t for exchange at T)
Rate caplet: Phi(d1) = probability for rate does indeed attain Fk (rate expected at time t being applicable between tk and tk+1)

Thank you and happy new year!

 

[Jun Wu]

Can i also check please, on page 17 of CH13, the paragraph in the end:

Thus, we have in effect replaced the unknown series of variable reference rates with the fixed swap rate, which is known with certainty at the strike date and which yields payments with the same (expected) present value (at the strike date) as the unknown reference rate payments.

On the bold texts - Why would the PV of fixed rate payments = PV of floating rate payments? I presume reference rate = (actual) float rate applicable for the swap starting T years later?

Thank you!

 

[CapitalActuary]

> Why would the PV of fixed rate payments = PV of floating rate payments?

Assuming this is talking about interest rate swaps: The fixed rate is set such that the PV of the fixed leg is equal to the PV of the floating leg.

 

[Joe Hook]

Thanks Joe! Yes that makes much more sense.

A few follow up queries please:
1 On (B0-I)=F0*exp(-r(T-t)) : Is this because F0 is the price we agreed/expected today at time t, to exchange the bond at time T? So if we discount back between T-t at risk free rate we get B0-I = the present value at time t of bond value stripped away income. I get confuse by the use of 0 and t, is current time t or 0?

2 Fk : Is this the rate expected at time 0, or t, being applicable between tk and tk+1?

3 Are the following correct?
Share: Phi(d1) = probability for share price does indeed attain ST (share price expected at time T if grown at risk free rate)
Bond Call : Phi(d1) = probability for bond price does indeed attain F0 (bond price expected at time t for exchange at T)
Rate caplet: Phi(d1) = probability for rate does indeed attain Fk (rate expected at time t being applicable between tk and tk+1)

Thank you and happy new year!

Apologies I have complicated matters by bringing in the T-t there. F0 is the value of the forward at time 0 and so the formula in the course is P(0,T) * (F0*PHI(d1)-X*PHI(d2)). This is valuing the call option at time 0. To value the call option at a later time t it would then, in effect, become P(t,T) * (Ft*PHI(d1)-X*PHI(d2)). Current time can be t or it can be 0, just depends on how we are looking at the problem. As long as the notation we then use is consistent we will be fine.

Fk is the rate expected at time t being applicable between tk and tk+1 but as above t could be 0. The formula in the course notes uses 0.

In all cases it's actually Phi(d2) that does this. For example for a call option on a share you can think of two things happening at maturity (if you exercise the option): you receive a share and you pay the strike price. Because the strike price is a fixed K, we just need to multiply this by a discount factor and the probability that we pay it, ie the probability that the share price exceeds the strike price Phi(d2). Phi(d1) also in essence contains that probability but it then also has to factor in all of the possible share prices at maturity ABOVE the strike price. For example, if a call option on a share is exercised with a strike price of £2, then the share price could be £2.01, £2.02, etc etc and we need to factor those prices, and the relevant probabilities in. So all in all Phi(d1) is a bit more complicated, but its derivation is found in CM2.

Joe