Slides of fin math lectures given at IIT-Delhi, IIT-Bombay and at Morgan Stanley

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