Questions tagged [numerairechange]

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10
votes
4answers
2k views

Understanding $N(d_1)$ and how to use the stock itself as the numeraire?

Assume the stock price follows a geometric Brownian motion Then in Black-Scholes pricing model, $N(d_2)$ is the risk-neutral probability that the option expires in-the-money. However, it is said that $...
8
votes
1answer
7k views

Baye's rule for conditional expectations (Proof review)

The Baye's rule for conditional expectations states $$ E^Q[X|\mathcal{F}]E^P[f|\mathcal{F}]=E^P[Xf|\mathcal{F}] $$ With $f=dQ/dP$ - thus being the Radon-Nikodyn derivative and $X$ being ...
8
votes
2answers
3k views

How to use the stock as a numeraire to price a derivative with payoff of the form $(S_T f(S_T))^+$?

I have $\frac{dS_t}{S_t} = rdt + \sigma dW_t$ as usual under the money-market numéraire and I need to price options with payoffs $$(S_T f(S_T))^+$$ How do I express the stock dynamics using the ...
5
votes
1answer
176 views

Change of measure's impact on parameter value

This is a follow-up question on Price of a prepayment-based claim. Consider a zero-coupon bond of maturity $T$ with price $P_0$ for which the borrower can reimburse the principal $N$ at any time $\...
4
votes
1answer
1k views

Numéraire — couldn't understand the wiki explanation

I'm trying to understand Numéraire concept so am reading the wiki page: I couldn't understand the last formula's 2nd equation: $$ E_{Q}\left[\left.\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}\frac{S(T)}{N(T)}\...
8
votes
1answer
693 views

How to use a change of numeraire to price this option?

I recently asked this question regarding how to price an option with payoff: $$\text{Payoff}_T = (A_TR_T - A_T \lambda)^+ $$ Let's assume for generality that $A_t$ and $R_t$ are GMB's: $$dA_t = \...
3
votes
1answer
272 views

Deriving Black Scholes PDE under stock as a numeraire

There are many ways to derive the Black Scholes PDE. The Martingale way would be to demand the option price is driftless according to particular measures. Below I derive the correct PDE using the bank ...
3
votes
4answers
309 views

Using a Constant as a Numeraire

Please provide steps to justify the below. 1) Can we use a constant as a numeraire? Related Question: Scaling Stock Price and Strike etc. by a Constant The rest of standard Geometric Brownian ...
4
votes
2answers
760 views

Is the money market account (MMA) numeraire and the forward measure equivalent?

Suppose we have a risk-neutral measure $\tilde{\mathbb{P}}$. The money market account is given as $M(t) = e^{\int^t_0 R(s) ds}$, while the price of the zero-coupon bond at time $t$ that matures at $T$ ...