premia to compensate investors for bearing different types of stock market risk. Keywords: risk neutral volatility; implied volatility; option pricing; risk aversion risk neutral value under the Q measure, and will rarely equal the real world value stock at a pre-specified price (futures price) at a pre-specified time (maturity, call it TF e-rtXt is a martingale under the risk-neutral measure. Why is this true? contract payout assuming the stock follows the risk-neutral (rather than real world ) of a random variable under the new measure as an expectation under the Under the risk neutral measure Q, the stock price process becomes as follows. dS. S. = (µ − q)dt + σdZP (replacing dZP with dZQ − (ต−

## 7 Apr 2012 What is the Risk-Neutral Probability Measure? That is, under Q, the expected value of the return on a asset X from t to T is the risk-free rate r Assume a stock S in a single step framework, where the initial price is S0 .

Under the measure Pe the discounted stock price is a martingale, which amounts to the simple relation eE 1 r +1 S(1) = S(0) , and means that in the Ee-average the stock behaves like the riskless bond growing from S(0) to (1+ r)S(0) in unit time. The risk neutral measure is the set of probabilities for which the given market prices of a collection of trades would be equal to the expectations of the winnings or losses of each trade. Remark: It is risk-neutral because in this alternative reality the price paid by player A for the game contains no risk premium — the price is exactly equal to the value of the expected winnings of the game. Under a risk-neutral measure, the option price is the expected discounted value of the option’s payoff. The risk-neutral measure exists if and only if the market is arbitrage free. 6 Basically, the price process of an option under a risk-neutral measure is given by In mathematical finance, a risk-neutral measure, (also called an equilibrium measure, or equivalent martingale measure), is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure.

### (PVFP) under the risk neutral valuation and the real world valuation with the o Simulate N possible ending stock price using the P-measure process for the

Compare with futures prices, which are martingales under the risk neutral measure. Note that when interest rates are deterministic, this implies that forward prices and futures prices are the same. For example, the discounted stock price is a martingale under the risk-neutral measure: () = ∗ [() | ()]. The forward price is given by (,) = (,). Thus, we have (,) = () back out the set of risk-neutral probabilities that these prices imply, calculate the expectation of the derivative trade’s payoff under these risk-neutral proabilities, that is the price of the derivative. The risk-neutral measure is in some sense the flip-side of the concept of risk premium. 4 4. RISK NEUTRAL MEASURES isamartingale. Thus,themartingalerepresentationtheoremcanbeusedtoexpress thisasanItôintegral(withrespecttoW˜ ). Withalittlealgebraicmanipulation onecanshowthatD(t)−1Y(t) isthewealthofaselfﬁnancingportfolio. Sincethe terminalwealthisclearlyV(T),thismustbeareplicatingportfolio. Remark 2.7. Risk neutral investors are not concerned with the risk of an investment. However, risk-averse investors have a greater fear of losing money, The term risk-neutral can sometimes be misleading because some people may assume it means that the investors are neutral, unconcerned or unaware of risk, risk-neutral valuation:the case of a stock price Stocks are expected to provide a higher return than the risk-free rate, the risk premium being equal to the (3 of the stock times the differential between the equity index return and the risk free return.