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99 Risk Neutral Valuation

Risk Neutral Valuation

Suppose our economy includes stock \(S\), riskless money market account \(B\) with interest rate \(r\) and derivative claim \(f\)

Assuming there’s only 2 possible outcomes at time \(dt\)

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Naive Approach

Current price of a derivative claim is determined by current price of portfolio which exactly replicates the payoff of the derivative at maturity

Consider Forward contract with pays \(S-K\) at time \(dt\). One could think that its strike \(K\) should be defined by the “real world” transition probability \(p\) $$ p(S_1 - k) + (1-p) (S_2 - k) = p S_1 + (1-p) S_2 - k \ p = ½ \implies k_0 = (S_1 + S_2)/2 $$

  1. Borrow \(S_0\) to buy stock. Enter forward contract with strike \(k_0\)

  2. In time \(dt\) deliver stock in exchange for \(k_0\) and repay \(S_0 e^{r \ dt}\)

  3. If \(k_0 > S_0 e^{r \ dt}\), riskless profit

  4. If \(k_0 < S_0 e^{r \ dt}\), definite loss

Notes

  • Given current price of the stock and assumptions on the dynamics of stock price, there is no uncertainty about the price of a derivative.
  • Price is defined only by the price of the stock and not by the risk preferences of the market participants
  • Mathematical apparatus allows us to compute current price of a derivative and its risks, given certain assumptions about the market

General derivative claim

For a claim \(f\), find \(a\) and \(b\) such that $$ \begin{aligned} f_1 &= a S_1 + b B_0 e^{r dt} \ f_2 &= a S_2 + b B_0 e^{r dt} \ \implies f_0 &= a S_0 + b B_0 \end{aligned} $$

\[ \begin{aligned} a &= \dfrac{f_1 - f_2}{S_1 - S_2} \\ b &= \dfrac{S_1 f_2 - S_2 f_1}{(S_1 - S_2) B_0 e^{r \ dt}} \end{aligned} \]
\[ f_0 = e^{- r \ dt} \Big( f_1 q + f_2 (1-q) \Big) \\ q = (S_0 e^{r \ dt} - S_2)/(S_1 - S_2), & q \in (0, 1) \\ \implies q S_1 + (1-q) S_2 = e^{r \ dt} S_0 \]

Black-Scholes

Assumes that stock has log-normal dynamics $$ dS = \mu S dt + \sigma S dw $$ where \(W\) is a Brownian motion: \(dW\) is normally-distributed with mean 0 and standard deviation \(\sqrt{dt}\) 

We want t find a replicating portfolio such that $$ df = a dS + b dB $$

\[ (dS)^2 = \sigma^2 S^2 dt \]
\[ \dfrac{\partial f}{\partial t} + \dfrac{1}{2} \dfrac{\partial^2 f}{\partial S^2} \sigma^2 S^2 + \dfrac{\partial f}{\partial S} r S - rf = 0 \]
Last Updated: 2024-05-12 ; Contributors: AhmedThahir

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