Introduction to Risk-Neutral Valuation- Ashwin Rao

Aug 25, 2010

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 ?

BINARY OPTION PAYOFF AND UNDERLYING DISTRIBUTION

30 40 50 60 70 80 90 100 110 120 130 140 1500

0.2

0.4

0.6

0.8

1

1.2

0

0.005

0.01

0.015

0.02

0.025

Binary Option Payoff

Underlying Distribution

Dis

trib

utio

n

Underlying

Pay

off

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 ?

LAW OF DIMINISHING MARGINAL UTILITY

1 2 3 4 5 6 7 8 9 10

-10

0

10

20

30

40

50

Marginal Satisfaction

Total Satisfaction

Number of chocolate bars eaten

Sat

isfa

ctio

n

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]

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

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

PAYOFF OF A FORWARD CONTRACT (AT TIME T)

0 10 20 30 40 50 60 70 80 90 100

-50

-40

-30

-20

-10

0

10

20

30

40

50

Forward price = 50

Asset price at time t

Pay

off

of f

orw

ard

at ti

me

t

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)

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

markets

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

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

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) ?

PAYOFF OF CALL AND PUT OPTIONS (AT TIME T)

Asset price at time t

Pay

off

at t

ime

t

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

40

45

50

Call Payoff

Put payoff

Strike price K = 50

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

PRICING BY REPLICATION WITH ASSET AND BOND

p

p

1-p

S

SOLVING, WE GET THE PRICE FORMULA

Note that the price formula is independent of p and

REARRANGING, WE GET AN INTERESTING FORMULA

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

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)

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

calculate the prices of derivatives as expected discounted payoffs