Stochastic Calculus, FAQ

Question 1

Define a convex function

Answer 1

A function f is convex if for every x and y in the interval, f(lambdax + (1-lambda)y) <= lambdaf(x) + (1-lambda)f(y), for any lambda in [0, 1]

Question 2

State Jensen’s Inequality

Answer 2

If x is a convex function and x is a r.v. then E[f(x)] >= f(E[x])

Question 3

State two insights from Jensen’s Inequality

Answer 3

1. Explains why non-linear instruments have inherent value

2. If a contract has convexity in a r.v., an allowance must be made in pricing

Question 4

State four complications with delta hedging

Answer 4

1. Need to estimate delta from the model, but the model may be incorrect
2. Continuous rebalancing
3. Transaction costs
4. Underlying must move consistently with the assumptions of the model (e.g. stock GBM)

Question 5

State three explanations of risk-neutral pricing

Answer 5

1. Hedging correctly in a Black-Scholes framework eliminates all risk
2. Change measure from the real world to risk neutral framework (arbitrage invariant under measures)
3. Price non-vanilla options using vanilla options via a replicating portfolio

Question 6

Define equivalent measures

Answer 6

Two measures are equivalent if they have the same sample space and possibilities (but same probabilities not required)

Question 7

State four implications of Girsanov’s Theorem

Answer 7

1. Every equivalent measure is given by a drift change
2. There is only one equivalent risk-neutral measure
3. Not needed for classic Black-Scholes but useful for more complicated analysis
4. Important for fixed-income with different maturities

Question 8

State two definitions of complete markets

Answer 8

1. Derivative products can be artificially reconstructed from more basic instruments
2. There exist the same number of linearly independent securities as there are states of the world in the future

Question 9

State three reasons why markets are incomplete in reality

Answer 9

1. Lognormal woth random volatility has too many states in the world
2. Jump diffusion implies more states than underlying securities
3. Some variables cannot be hedged

Question 10

State two methods of pricing in incomplete markets

Answer 10

1. Actuarial Pricing - consider long run average with the CLT
2. Consistent Pricing - replocate securities and assume no arbitrage for pricing

Question 11

State three ways to price with real world probabilities

Answer 11

1. Modern portfolio theory and utility functions
2. Certainty equivalent under a real random walk
3. Price = Real Expected Value - Multiple of Standard Deviation

Question 12

State two real world pricing concerns

Answer 12

1. Need to be able to measure real world probabilities and parameters
2. Need to measure risk aversion

Question 13

List requirements needed to apply risk-neutrality

Answer 13

1. Complete Market
2. Enough traded quantities to hedge risk
3. Continuous hedging
4. No transaction costs
5. Accurate parameters
6. No jumps

Question 14

State four common confusions in risk-neutral pricing

Answer 14

1. Forward price is the expected future value (it is not)
2. Forward curve is the market expected value of the spot rate (also has premium for risk aversion)
3. Using risk neutral pricing it is possible to replace mu with r (only under certain assumptions)
4. Delta is the probability that an option ends up ITM (that probability depends on RW not RN)