1. Week 5 Options: Pricing

2. Pricing a call or a put (1/3) To price a call or a put, we will use a similar methodology as we used to price the portfolio of (C- P). We have to ask ourselves whether we can somehow replicate the payoff on a call (or a put) by trading the underlying asset, S, and a bond (borrowing or lending money). As an example, consider pricing a call on the $, priced in Yen, with a strike of 100 Yen/$. We will think of a very simple world where the price of the underlying asset in the future, S T, can be only one of two possibilities: u S or d S, where S is the current rate of 100 Yen/$.

3. Pricing a call or a put (2/3) Assume that the price at maturity is either (u S) or (d S). For example, if u=1.2 and d=0.8, then S T is either 120 or 80. In this world, how would you replicate the payoff on a call of strike 100? –Note that the call pays off 20 Yen if the spot is 120 Yen/$, and zero otherwise. Answer: Buy the PV of 0.5$ [i.e., 0.5/(1+r*n/360)], and borrow the present value of 40 Yen, this will replicate the payoff of the call. In other words, the position (0.5/(1 +r*T/360)) S - 40/(1+rT/360) replicates the payoff on the call. Thus, the value of the call is exactly the same as the value of the above portfolio.

4. Pricing a call or a put (3/3) Suppose this is a one year call with strike 100, the foreign interest rate, r*=5.5%, and the domestic interest rate is 0.50%. Then how much would you pay for this call? The value of the replicating portfolio is: 0.5S/(1 + 0.055) - 40/(1 + 0.005) = 7.59 Yen Because the payoff on both the replicating portfolio and the call are exactly the same, the price of the call should also be 7.59 Yen.

5. Hedging a Call How do you figure out the replicating portfolio? Rule: if a derivative security can be replicated it can be hedged perfectly (so that the payoff at maturity is not risky.) So if you want to figure out the replicating portfolio, you should figure out how to hedge the option. The direction of the hedge is as follows: if you are long (short) the call, you should be short (long) the underlying asset. If you are long (short) the put, then you should be long (short) the stock.

6. Creating the hedged portfolio Consider the call of the previous example: Up State (u): uS = 120, C u =20 Down State (d): dS = 80, C d = 0 Because the hedged portfolio is riskless it should have the same cash flow (B) every period. Suppose you are short one call, then to construct the hedged portfolio, you should go long some amount (call it “delta”) of the underlying asset. This portfolio should give you the same amount, $B in either state, u or d.

7. Delta-Hedged Portfolio (1/3) We can calculate the hedge ratio (amount of spot required to hedge a call), as the present value of the sensitivity of the call price to the price of the underlying asset at maturity, i.e., delta =(1/(1+ r*T/360)) x (C u - C d )/(uS-dS). For our example, if the foreign interest rate is 5.5%, the delta is (1/(1+0.055))*(20/40) = 0.4739 So to hedge one short call, you will go long 0.47 units of the underlying asset. For example, the call was an option on US$ in terms of Yen (as in our previous example), then you would go long 0.47 US$ for every call.

8. Delta-hedged portfolio (2/3) Now consider the following portfolio: short 1 call, and long “delta” of the underlying asset. -C + (delta) S. What is the value of this portfolio at maturity if S T is either 120 or 80? Answer: B= 40. To calculate this, note that you are long delta=0.4739 units of the foreign currency ($). Now as the $ earns interest at the rate r*=5.5%, at maturity you are long 0.4739(1.055)=0.5 US$. For the up state: -C T + (delta)S T = -20 + 0.5(120)=40. You can check for the down state (S=80) that you would get the same amount at maturity.

9. Delta-hedged portfolio (3/3) Also, work out the details for a portfolio that is long 1 call and short delta stock [C - (delta) S]. And you will see that in this case, B = -40. In either case, now it is easy to work out how much to pay for the call. As the hedged portfolio guarantees you a fixed amount, B, in the future, you can equate the value of the hedged portfolio to the present value of B. Thus: -C + (delta=0.4739)x(S=100) = [1/(1+0.005)]x(B=40). So: C=7.59 Yen This is the same answer we got earlier.

10. Pricing a Put Note that we can price a put by the same procedure. Alternatively, if we know the price of the call, we can figure out the price of the put, by put-call parity. Thus, the price of the 1 year put of strike 100 is: P = C – S/(1+r*n/360) + X/(1+rn/360) = 7.59 – 100/(1+0.055) + 100/(1+0.005)= 12.31

11. Summary: Factors that Affect Option Price 1. Current spot rate, S t. :If the foreign currency appreciates, the call (on the FC) increases in value, while the put decreases in value. 2. Strike: Increase in strike decreases price of call, but increases price of put. 3. Foreign interest rate: Increase in r* decreases price of call, increases price of put. 4. Domestic interest rate: Increase in r increases price of call, decreases price of put. 5. Volatility (u or d): Increase in volatility increases price of both call and put. 6. Maturity: Longer maturity increases price of call, but has ambiguous effect on put (but usually increases its price)

12. The Black-Scholes Formula In practice, of course, we make use of a more sophisticated model to price an option. Typically, we will use the Black-Scholes formula to price the option, as well as to figure out how to hedge the option.