Category Archives: Helpful Hints

Delta hedging, replicating portfolio, and risk neutral valuation approaches to pricing options

The delta hedging, replicating portfolio and risk neutral valuation perspectives for pricing options are important in that they enable us to think carefully as well as deeply concerning the economics of option pricing. For example, the delta hedging approach illustrates that an appropriately hedged portfolio consisting of either long-short call-share positions or long-long put-share positions is riskless and consequently must produce a riskless rate of return. Similarly, the replicating portfolio approach reminds us that a call option represents a “synthetic” margined stock investment, whereas a put option represents a “synthetic” short sale of the share combined with lending money. As we saw in our study earlier this semester of forward contracts, if there is any difference between the value of a derivative and its replicating portfolio, then one can earn profits with zero net investment and no exposure to risk. Thus, the “arbitrage-free” price for the derivative (option or forward) corresponds to the value of its replicating portfolio. The arbitrage-free pricing principle further implies that a risk-neutral valuation relationship exists between the derivative and its underlying asset, which in turn enables us to calculate risk neutral probabilities. Once we know what the risk-neutral probabilities are for up and down price movements, we can price options by discounting the risk-neutral expected value of the option payoff (i.e., its certainty-equivalent) at the riskless rate of interest.

During our next class meeting tomorrow, we will spend a bit more time on the risk neutral valuation approach, as well as extend our analysis from a single timestep to a multiple timestep setting. To do this, we need to introduce an important concept called backward induction. Backward induction involves beginning at the very end of the binomial tree and working our way back to the beginning. We’ll continue with the numerical example introduced during class today and illustrate the backward induction method for the delta hedging, replicating portfolio and risk neutral valuation approaches. We’ll discover that multiple timesteps imply that the delta hedging and replicating portfolio methods imply “dynamic” trading strategies which require portfolio rebalancing as the price of the underlying asset changes over time.

Problem Set 2 helpful hints

Problem Set 2 is now available from the course website at; its due date is Thursday, January 25.

Problem Set 2 consists of two problems. The first problem requires calculating expected value, standard deviation, and correlation, and using this information as inputs into determining expected return and standard deviation for 2-asset portfolios. We covered these concepts during last Thursday’s statistics tutorial; also see pp. 10-20 of the lecture note. The second problem involves using the standard normal probability distribution to calculate probabilities of earning various levels of return by investing in risky securities and portfolios. We will devote next Tuesday’s class meeting to this and related topics.

Problem Set 1 hint…

Problem Set 1 is due at the beginning of class on Tuesday, January 16. Here is a hint for solving the 4th question on problem set 1.

The objective is to determine how big a hospital must be so that the cost per patient-day is minimized. We are not interested in minimizing total cost; if this were the case, there would be no hospital because marginal costs are positive, which implies that total cost is positively related to the number of patient-days.

The cost equation C = 4,700,000 + 0.00013X2 tells you the total cost as a function of the number of patient-days. This is why you are asked in part “a” of the 4th question to derive a formula for the relationship between cost per patient-day and the number of patient days. Once you have that equation, then that is what you minimize, and you’ll be able to answer the question concerning optimal hospital size.