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09-Extensions7(f)

# 09-Extensions7(f) - Extending the Linear Functional Form...

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Primbs, MS&E 345 1 Extending the Linear Functional Form

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Primbs, MS&E 345 2 Change of numeraire Martingales and equivalent martingale measures Risk neutral worlds, rates of return, and market price of risk Where is the pde hiding now? Random interest rates Time to think...
Primbs, MS&E 345 3 Return to our linear pricing functional: dP B B B S T T T ϕ 0 = = dQ B S T T ] [ 1 0 0 T Q rT rT T S E e dQ e S S S - = = = Our old pricing formula! dP B B T ϕ = 0 Bond: dP S S T ϕ = 0 Stock: Integrates to 1 dP B B B B T ϕ = = 0 0 0 1 dP B B dQ T ϕ 0 = Act like it is a probability density: dP B S B S T ϕ = 0 0 0 Assume interest rates are constant and start with \$1 in the bond: = T T Q B S E B S 0 0

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Primbs, MS&E 345 4 Normalize by the stock: dP B B T ϕ = 0 Bond: dP S S T ϕ = 0 Stock: dP S S S S T ϕ = = 0 0 0 1 Integrates to 1 Act like it is a probability density: dP S S dQ T S ϕ 0 = dP S B S B T ϕ = 0 0 0 dP S S S B T T T ϕ 0 = = S T T dQ S B = T T Q S B E S B S 0 0 We have derived a different pricing formula under a different equivalent measure!
Primbs, MS&E 345 5 We saw this trick earlier in our solution of Margrabe’s exchange one asset for another. When we divide by the bond, we can think of this as measuring everything in units of the bond. Alternatively, we can measure in units of the stock. This is what happens when we divide by the stock. This is known as a change of numeraire. A numeraire asset is the asset that we measure everything else in terms of. Be careful, it must be positive. Dividing by zero doesn’t make sense!

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Primbs, MS&E 345 6 The general approach is as follows: Step 1: Select a numeraire asset: (say S) Step 2: Define the equivalent measure dP S S dQ T S ϕ 0 = S Q is often called the “forward risk neutral measure” with respect to S . Step 3: Then for any other (price, payoff) pair, (f 0 , f T ): = T T Q S f E S f S 0 0
Primbs, MS&E 345 7 Change of numeraire Martingales and equivalent martingale measures Risk neutral worlds, rates of return, and market price of risk Where is the pde hiding now? Random interest rates Time to think...

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Primbs, MS&E 345 8 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 What do we do with early cash flows: Reinvest in the numeraire! Assume B t is the numeraire: t S at time t T t t B B S at time T = = t t Q T t T t Q B S E B B B S E B S 0 0
Primbs, MS&E 345 9 t t B S This means that has zero rate of return. = t t T T Q t t B S B S E B S , This is not good enough if we have a path dependent derivative: The previous argument shows that for 0 < t < T: = t t Q B S E B S 0 0 Alternatively, we could consider purchasing S at time t and selling at time T. Then we might think of writing our pricing formula as:

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Primbs, MS&E 345 10 A path dependent derivative: C time 0 time T time t For a lookback option, for instance, if I buy at time t, then I need to know then maximum price that was achieved in [0,t] in order to be able to price it.
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09-Extensions7(f) - Extending the Linear Functional Form...

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