This document introduces the concepts of risk-neutral valuation and derivative pricing. It defines derivative securities as contracts contingent on the value of an underlying asset. The key questions in pricing derivatives are determining the expected payoff and the information required to calculate it. The document explains how replication arguments and preventing arbitrage opportunities can be used to derive fair prices for forwards, calls, puts and other derivatives under the risk-neutral measure, where the expected return on the underlying asset is equal to the risk-free rate. Relaxing simplifying assumptions, more advanced stochastic models can also be used for risk-neutral pricing.

1. Introduction to Risk-Neutral Valuation
- Ashwin Rao
Aug 25, 2010

2. WHAT ARE DERIVATIVE SECURITIES ?
• Fundamental Securities – eg: stocks and bonds
• Derivative Securities - Contract between two parties
• The contract specifies a contingent future financial claim
• Contingent on the value of a fundamental security
• The fundamental security is refered to as the underlying
asset
• A simple example – A binary option
• How much would you pay to own this binary option ?
• Key question: What is the expected payoff ?
• What information is reqd to figure out the expected payoff
?

3. BINARY OPTION PAYOFF AND
UNDERLYING DISTRIBUTION
1.2 0.025
1
0.02
Binary Option
0.8
Payoff
0.015 Underlying
Distribution
Payoff
Distribution
0.6
0.01
0.4
0.005
0.2
0 0
30 40 50 60 70 80 90 100 110 120 130 140 150
Underlying

4. A CASINO GAME
• The game operator tosses a fair coin
• You win Rs. 100 if it’s a HEAD and 0 is it’s a TAIL
• How much should you pay to play this game ?
• Would you rather play a game where you get Rs. 50 for both H & T ?
• What if you get Rs. 1 Crore for H and Rs. 0 for T ?
• Depends on your risk attitude
• Risk-averse or Risk-neutral or Risk-seeking
• What is the risk premium for a risk-averse individual ?

5. LAW OF DIMINISHING MARGINAL UTILITY
50
40
30
Satisfaction
Marginal Satisfaction
20
Total Satisfaction
10
0
1 2 3 4 5 6 7 8 9 10
Number of chocolate bars eaten
-10

6. LAW OF DIMINISHING MARGINAL UTILITY
• Note that the total utility function f(x) is concave
• Let x (qty of consumption) be uncertain (some probability distribution)
• Then, E[ f(x) ] < f( E[x] ) (Jensen’s inequality)
• Expected Utility is less than Utility at Expected Consumption
• Consumption y that gives you the expected utility: f(y) = E[ f(x) ]
• So, when faced with uncertain consumption x, we will pay y < E[x]

7. LAW OF DMU RISK-AVERSION
• E[x] is called the expected value
• y is called the “certainty equivalent”
• y – E[x] is called the “risk premium”
• y – E[x] depends on the utility concavity and distribution
variance
• More concavity means more risk premium and more risk-
aversion
• So to play a game with an uncertain payoff, people would
Generally pay less than the expected payoff

8. A SIMPLE DERIVATIVE – FORWARD CONTRACT
• Contract between two parties X and Y
• X promises to deliver an asset to Y at a future point in time t
• Y promises to pay X an amount of Rs. F at the same time t
• Contract made at time 0 and value of F also established at
time 0
• F is called the forward price of the asset
• What is the fair value of F ?
• Expectation-based pricing to arrive at the value of F is
wrong

9. PAYOFF OF A FORWARD CONTRACT (AT TIME T)
50
Payoff of forward at time t
40
30
20
10
0
0 10 20 30 40 50 60 70 80 90 100
-10
-20
Forward price = 50
-30
-40
-50
Asset price at time t

10. TIME VALUE OF MONEY
• The concept of risk-free interest rate is very important
• Deposit Rs. 1 and get back Rs. 1+r at time t (rate r for time
t)
• So, Rs. X today is worth Rs. X*(1+r) in time t
• If I’ll have Rs. Y at time t, it is worth Rs. Y/(1+r) today
• So you have to discount future wealth when valuing them
today
• With continuously compound interest, er instead of (1+r)

11. ARBITRAGE
• The concept of arbitrage is also very important
• Zero wealth today (at time 0)
• Positive wealth in at least one future state of the world (at
time t)
• Negative wealth in no future state of the world (at time t)
• So, starting with 0 wealth, you can guarantee positive
wealth
• Arbitrage = Riskless profit at zero cost
• Fundamental concept: Arbitrage cannot exist in financial

12. USE THESE CONCEPTS TO VALUE A FORWARD
• Contract: At time t, you have to deliver asset A and receive Rs. F
• Assume today’s (t = 0) price of asset A = Rs. S
• Step 1: At time 0, borrow Rs. S for time t
• Step 2: At time 0, buy one unit of asset A
• Step 3: At time t, deliver asset A as per contract
• Step 4: At time t, receive Rs. F as per contract
• Step 5: Use the Rs. F to return Rs. Sert of borrowed money
• If F > Sert , you have made riskless money out of nowhere
• Make similar arbitrage argument for your counterparty
• Arbitrage forces F to be equal to Sert

13. REPLICATING PORTFOLIO FOR A FORWARD
• A Forward can be replicated by fundamental securities
• Fundamental Securities are the Asset and Bonds
• “Long Forward”: At time t, Receive Asset & Pay Forward Price
F
• “Long Asset”: Owner of 1 unit of asset
• “Long Bond”: Lend money for time t (receiving back Rs.1 at t)
• “Short positions” are the other side (opposite) of “Long
positions”
• “Long Forward” equivalent to [“Long 1 Asset”, “Short F
Bonds”]
• Because they both have exactly the same payoff at time t
• This is called the Replicating Portfolio for a forward

14. DERIVATIVES: CALL AND PUT OPTIONS
• X writes and sells a call option contract to Y at time 0
• At time t, Y can buy the underlying asset at a “strike price” of K
• Y does not have the obligation to buy at time t (only an “option”)
• So if time t price of asset < K, Y can “just ignore the option”
• But if time t price > K, Y makes a profit at time t
• At time 0, Y pays X Rs. C (the price of the call option)
• With put option, Y can sell the asset at a strike price of K
• What is the fair value of C (call) and of P (put) ?

15. PAYOFF OF CALL AND PUT OPTIONS (AT TIME T)
50
45
40
Payoff at time t
35
30
25
Call Payoff
20
Put payoff
15
10
5
0
Strike price K = 50
0 10 20 30 40 50 60 70 80 90 100
Asset price at time t

16. PRICING OF OPTIONS
• Again, it is tempting to do expectation-based pricing
• This requires you to know the time t distribution of asset
price
• We know expectation-based pricing is not the right price
• Note that Call Payoff – Put Payoff = Fwd Payoff when K = F
•This is useful but doesn’t help us in figuring out prices C and
P
• Like forwards, use replication and arbitrage arguments
• However, replication is a bit more complicated here
• Consider two states of the world at time t

17. PRICING BY REPLICATION WITH ASSET AND BOND
p
p
S
1-p

18. SOLVING, WE GET THE PRICE FORMULA
Note that the price formula is independent of p and

19. REARRANGING, WE GET AN INTERESTING FORMULA

20. WHAT EXACTLY HAVE WE DONE HERE ?
• We have altered the time t asset price’s mean to F = Sert
• Arbitrage-pricing is equiv to expected payoff with altered mean
• Altered mean corresponds to asset price growth at rate r
• But bond price also grows at rate r
• All derivatives are replicated with underlying asset and a bond
• So, all derivatives (in this altered world) grow at rate r
• In reality, risky assets must grow at rate > r (Risk-Aversion)
• Only in an imaginary “risk-neutral” world, everything will grow at
rate r
• But magically, arbitrage-pricing is equivalent to:
Expectation-based pricing but with “risk-neutrality”
assumption

21. RELAXING SIMPLIFYING ASSUMPTIONS
• Model a stochastic process for underlying asset price
• For example, Black Scholes: dS = μS dt + σ S dW
• Use Girsanov’s Theorem to alter process to “risk-neutral
measure” Q
• Risk-neutral Black Scholes process: dS = r S dt + σ S dWQ
•Bond process is: dB = r B dt
• Two-state transition works only for infinitesimal time dt
• So, expand into a binary (or binomial) tree to extend to time t
• Use “backward induction” from time t back to time 0
• At every backward induction step, do discounted expectation
• But using risk-neutral probabilities (derived from repl.
portfolio)

22. CONTINUOUS-TIME THEORY: MARTINGALE PRICING
• One has to assume the replicating portfolio is “self-financing”
• Any profits/losses are reinvested into the next step’s repl.
portfolio
• Underlying asset and its derivatives have a drift rate of r (in Q)
• So, discounted (by e-rt ) derivatives processes have no drift
(no dt term)
• Driftless processes are martingales
• So, in the risk-neutral measure, the martingale property is
used to