Risk-Neutral Valuation

Valuation staff use this approach when they calculate option prices with a model. Risk management staff read it to check that price calculations and risk measures such as VaR use different probabilities. Trading and development staff use it when they build calculation engines such as a Monte Carlo pricer.

What risk-neutral valuation represents

A price found by risk-neutral valuation matches the cost of replication. Consider a hypothetical example in which one period is modeled, as on the page Binomial Trees. The price of the underlying asset today is 100, and after one period it will be either 110 or 90. A call option with a strike price of 100 pays 10 at the end of the period if the underlying is 110, and 0 if it is 90. The one-period risk-free rate (a rate with no credit risk or similar risks)¹ is 2%, so you discount by dividing by 1.02.

On the page "Binomial Trees," buying 0.5 units of the underlying and borrowing 44.12 replicates the payoff of the call. The cost of this combination gives the price of the option (the premium) as 5.88.

If you take the probability of a rise to be 0.6 and discount the expected payoff at the interest rate, you get the same 5.88. The 0.6 is the value found on the page "Binomial Trees" from the two possible prices of the underlying and the interest rate. The expected payoff is 0.6 × 10 + 0.4 × 0 = 6, and 6 ÷ 1.02 = 5.88. A probability like this 0.6, which gives the same price as the cost of replication, is called a risk-neutral probability. Pricing a derivative by discounting the expected value under this probability is called risk-neutral valuation. A risk-neutral probability is separate from the real-world probability that investors forecast, and the two usually differ².

The pricing formula

In a one-period binomial model, let the value of the option at the end of the period be $V_u$ if the underlying rises and $V_d$ if it falls. With the one-period risk-free rate $r$ and the risk-neutral probability of a rise $p$, the value today $V_0$ is given by the following formula. The formula that finds $p$ from the up and down factors of the underlying and the interest rate is covered on the page "Binomial Trees."

$$V_0 = \frac{p V_u + (1 - p) V_d}{1 + r}$$

Only the movement of the underlying price and the interest rate enter $p$. The shape of the option's payoff does not². So you can value both a call and a put option on the same underlying with the same $p$. With the up factor $u$ and the down factor $d$ of the underlying, $p$ falls between 0 and 1 when $d < 1 + r < u$. This is the same as the condition for no-arbitrage (no arbitrage opportunities)³.

In a tree with many steps, you start from the nodes at expiry and work back to time 0 one step at a time, taking the expected value with this formula and discounting it. Written over the whole period to expiry, the price is the discounted expected value, under the risk-neutral probability, of the payoff at expiry $V_T$, written $E^{Q}[V_T]$⁴. From here on, $r$ is a continuously compounded annual rate. With a time to expiry of $T$ years, the formula is as follows.

$$V_0 = e^{-rT} E^{Q}[V_T]$$

Deep diveRisk-neutral valuation in a continuous-time model

The Black-Scholes model describes the underlying price $S$ with geometric Brownian motion. Let $\mu$ be the expected return under the real-world probability and $\sigma$ the volatility. Under the risk-neutral probability, $\mu$ is replaced by the risk-free rate $r$, and the change in the price is given by the following formula⁵.

$$dS = rS \, dt + \sigma S \, d\widetilde{W}$$

$\widetilde{W}$ is a Brownian motion under the risk-neutral probability.

Worked example with different real-world probabilities

If the underlying's current price, the up and down factors, and the interest rate stay the same, the call price stays at 5.88 whatever real-world probability of a rise you forecast. Consider a hypothetical example in which a staff member at Company C forecasts the probability of reaching 110 as 0.8, or as 0.3, under the same terms as above. The table compares these cases with the risk-neutral probability of 0.6.

The 7.84 found with the real-world probability of 0.8 is not a price. Suppose Company C could sell the call for 7.84. It would use 5.88 of that to set up the replicating combination: buying 0.5 units of the underlying and borrowing 44.12. At the end of the period, this combination pays the same amount that Company C owes on the call in either case, so the difference of 1.96 remains whichever way the market moves. Conversely, if Company C could buy the call for 2.94, it could short sell 0.5 units of the underlying and invest 44.12, keeping the difference of 2.94. The only price at which such arbitrage does not work is the cost of replication, 5.88⁶. Transaction costs and restrictions on short selling are ignored here.

The real-world probability has no direct effect on the price because the payoff of the replicating combination is fixed in both the up case and the down case, whatever the probabilities. However, when the real-world probability changes, one or more of the underlying's current price, the interest rate, and the size of its moves can change with it, and the option price then changes through them⁷.

The underlying's expected return and the forward price

Under the risk-neutral probability, the underlying's expected return equals the risk-free rate. In the example, the expected underlying price at the end of the period is 0.6 × 110 + 0.4 × 90 = 102. This is the same as today's price of 100 grown at the 2% rate, 100 × 1.02⁸. With Company C's real-world probability of 0.8, it is 0.8 × 110 + 0.2 × 90 = 106, an expected return of 6%.

In reality, the underlying's expected return is the risk-free rate plus an extra return for bearing risk (the risk premium). The risk-neutral probability corresponds to the probability that would apply if investors required no risk premium (if they were risk neutral), and this is where the name comes from. It is a computational probability for finding prices. It does not mean that the real expected return is the risk-free rate⁹.

In a model with a constant interest rate, like the one on this page, the price of a forward contract (the forward price) also equals the expected underlying price at expiry under the risk-neutral probability. For an underlying that pays no dividends, the forward price is today's price divided by the discount factor to expiry¹⁰. For the underlying in the example, it is 100 × 1.02 = 102, which matches the expected value of 102 above. Deriving the forward price from replication is covered on the page No-Arbitrage and Replication.

How the price changes with the interest rate and volatility

The risk-neutral probability is set by the risk-free rate and the size of the underlying's moves, so the option price changes with these two. In a three-step tree example in lecture notes, raising the risk-free rate increases the risk-neutral probability of a rise, and the call price goes up. The rate used for discounting also rises, but the higher probability of the states with larger payoffs more than offsets it¹¹. The simulator below, starting from its initial values, shows that a higher rate gives a higher price. The size of the moves is set by the volatility, and when the volatility rises, the option price rises¹². The real expected return, on the other hand, does not enter the pricing formula⁷.

The initial values of the simulator below are a call that is a European option, with an underlying price of 100, a strike price of 100, a time to expiry of 1 year, a risk-free rate of 5% (continuously compounded), a volatility of 20%, and no dividends. It is valued with a two-step Cox-Ross-Rubinstein (CRR) tree. Each step is 0.5 years, the underlying price rises by a factor of 1.1519 or falls by a factor of 0.8681 at each step, and the risk-neutral probability of a rise is 0.5539. The payoff at expiry is 32.69 at the node where the underlying price is 132.69, and 0 at the nodes at 100 and 75.36. The value at the upper node at step 1 is the expected value of the two nodes at the next step, discounted for half a year: $e^{-0.025} \times (0.5539 \times 32.69 + 0.4461 \times 0) = 17.66$. The same calculation back to time 0 gives 9.54.

In the same tree, the expected underlying price at expiry is 105.13. This is the same as today's price of 100 grown at 5% continuously compounded for one year, $100 \times e^{0.05}$. Setting the rate to 0% lowers the risk-neutral probability to 0.4647 and the price to 7.06, and setting it to 10% raises them to 0.6454 and 12.32. With a volatility of 10%, the two-step price is 6.32, and with 30%, it is 12.89. Increasing the number of steps to 100 gives a price of 10.43, approaching the price from the Black-Scholes model, 10.45.

Pricing and risk measurement use different probabilities

Price calculation and risk measurement such as value at risk (VaR) use different probabilities. The table below compares the two.

Valuation models in practice describe the movements of the underlying and other variables under risk-neutral probabilities from the start. The model's parameters are then set so that the model matches the prices of instruments actively traded in the market. This is called calibration, and the calibrated model is used to price instruments whose market prices cannot be observed¹⁵. Because the model's parameters are fitted to market prices, the model's risk-neutral probabilities are also set by market prices, not by investors' forecasts.

Pricing with the Monte Carlo method finds the expected value in risk-neutral valuation with random numbers. You generate many paths of the underlying price from today to expiry with random numbers, discount the payoff on each path, and take the average¹⁶. Because the price is an expected value under the risk-neutral probability, the paths are generated with the underlying's expected return (the drift) set to the risk-free rate. Scenarios that forecast future values⁵ cannot use this setting over a long period¹⁴.

Because the payoff is calculated path by path, the method also makes it easier to value options whose payoff depends on how the price moves along the way. Lattice methods (such as the binomial tree) and finite difference methods need sharply more computation as the number of underlying assets grows, and in practice they can handle only about three underlying assets. The computation for the Monte Carlo method does not grow as much as the number of underlying assets increases¹⁶.

The main uses of the Monte Carlo method in financial practice are the "expected value type," which finds the mean of a distribution (pricing derivatives), and the "percentile type," which finds a point in the tail of a distribution (calculating VaR)¹⁷.

Assumptions and limitations

Risk-neutral probabilities, and therefore prices, are unique only in a market where every payoff can be replicated with the underlying and risk-free investing and borrowing (a complete market)¹⁹. The binomial model and the Black-Scholes model are both complete-market models¹⁹. Replication also assumes that there are no transaction costs, and that you can sell short and trade fractions of a unit²⁰.

Consider a one-period model in which the underlying price at the end of the period can take three values. The underlying and risk-free investing and borrowing, just two instruments, cannot match the payoffs in all three states, so some payoffs cannot be replicated. In such a market (an incomplete market), infinitely many risk-neutral probabilities give prices with no arbitrage opportunities, and the price is not unique²¹. In practice, calibration picks one risk-neutral probability that matches market prices¹⁵.

Notes

1. Martin Haugh, "Forwards, Swaps, Futures and Options" (2016), Section 4.2 (page 12). 小田信之 (Nobuyuki Oda), "信用リスクを反映した金融商品のプライシング" (Pricing financial instruments that reflect credit risk), Institute for Monetary and Economic Studies, Bank of Japan (1999), page 71 (in Japanese). ↩

2. 重川一郎, "数理ファイナンス入門" (Introduction to mathematical finance), Kyoto University (intensive lectures at the Faculty of Science, Ehime University, 2009), "リスク中立測度" (Risk-neutral measure), page 9, equation (1.2) and the paragraph that follows (in Japanese). The source's $q$ and $\beta$ correspond to $p$ and $1 \div (1 + r)$ on this page. ↩ ↩²

3. Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging" (2016), Proposition 6 and equation (5) in Section 2.1, "Martingale Pricing in the Binomial Model" (page 7). The source's $q$ and $R$ correspond to $p$ and $1 + r$ on this page. ↩

4. Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging," equation (6) on page 7, and Example 2, "Pricing a Call Option," on pages 7–8. ↩

5. European Securities and Markets Authority, Luca Giordano and Giovanni Siciliano, "Real-world and risk-neutral probabilities in the regulation on the transparency of structured products" (ESMA Working Paper No. 1, 2015), page 4, equation [B.7] on page 15, and equation [B.13] on page 16. ↩ ↩² ↩³

6. 重川一郎, "数理ファイナンス入門," "単期間モデル: コールオプションの例" (Single-period model: an example of a call option), page 3 (the traditional view that takes the expected value $5p$ as the price, and that the price should be 1), and "ポートフォリオ" (Portfolio) and "単期間のポートフォリオ" (Single-period portfolio), pages 5–7 (the price of 1 found by replication, and the trades of the seller and the buyer when the price is above or below 1) (in Japanese). ↩

7. Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging," Remark 2 and Remark 3 (page 9). ↩ ↩²

8. 重川一郎, "数理ファイナンス入門," "リスク中立測度" (Risk-neutral measure), page 9 (in Japanese). Under the risk-neutral probability, the expected value of the discounted share price equals today's share price ($E_Q[\beta S_1] = S_0$). ↩

9. ESMA Working Paper No. 1, the Abstract on page 3, page 4, and page 16. ↩

10. Martin Haugh, "Forwards, Swaps, Futures and Options," equation (1) in Section 1.1, "Computing Forward Prices" (page 2). ↩

11. Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging," Example 3, "A Counter-Intuitive Result" (page 8). ↩

12. Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging," page 19 ("the price of an option increases as the volatility increases"). ↩

13. ESMA Working Paper No. 1, Section 2.2, "Why risk-neutral probabilities cannot be used to forecast the future value of derivatives" (pages 16–17). ↩

14. ESMA Working Paper No. 1, Box 3, "VaR calculation and real-world probabilities" (page 17). ↩ ↩² ↩³ ↩⁴

15. Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging," Section 1.4, "Derivatives Pricing in Practice" (page 6). ↩ ↩²

16. Tatsuya Ishikawa and Yoshihiko Uchida, "モンテカルロ法によるプライシングとリスク量の算出について" (Pricing and risk measurement with the Monte Carlo method), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (June 2002), pages 85–86 (in Japanese; path-dependent options, and a comparison with lattice methods and finite difference methods for options on many underlying assets). ↩ ↩²

17. Tatsuya Ishikawa and Yoshihiko Uchida, "モンテカルロ法によるプライシングとリスク量の算出について," Section 2 (1), "モンテカルロ法の定義と実務での適用対象" (Definition of the Monte Carlo method and its uses in practice), page 54 (in Japanese; the table, and the calculation steps shared by the expected value type and the percentile type). ↩ ↩²

18. Tatsuya Ishikawa and Yoshihiko Uchida, "モンテカルロ法によるプライシングとリスク量の算出について," "図表 1" (Figure 1), "試行回数と計算精度の関係" (Relationship between the number of trials and calculation accuracy), page 55 (in Japanese). ↩

19. Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging," Theorem 5 on page 5, and the end of Section 1.3 on pages 5–6. ↩ ↩²

20. 重川一郎, "数理ファイナンス入門," page 3 (in Japanese); Martin Haugh, "A Brief Review of Derivatives Pricing & Hedging," Section 2 (page 6); and ESMA Working Paper No. 1, the Abstract on page 3. ↩

21. 重川一郎, "数理ファイナンス入門," "単期間3 項モデル" (Single-period trinomial model), page 8 (in Japanese). In a single-period trinomial model, the equations for replication have no solution, and there are infinitely many risk-neutral measures. ↩