胖胖 · 2023年10月14日

Don't understand the financing part of the equation

NO.PZ2023020101000023

问题如下：

Cummins states that long-/short-hedge fund managers seek to identify and exploit any mispricing that may exist between the price of an option and the price of its underlying stock, utilizing a replicating strategy. Cummins asks Spelding to assess the three scenarios outlined in Exhibit 2, based on the information in Exhibit 1 and assuming that the price of a one-year European-style call option is $19.25.

Exhibit 1: Binomial Model Variables and Values

Exhibit 2: Scenarios and Replicating Strategies

With respect to the replicating strategies, which scenario is most likely correct:

选项：

Scenario 1.

Scenario 2.

Scenario 3.

解释：

The $19.25 price of the call option exceeds its value of $15.44, as calculated based on both the no-arbitrage approach and the expectations approach. Accordingly, the replicating strategy per 100 shares is to (1) sell 1 option, (2) buy h shares, and (3) borrow h * (up/down factor price + up/down call payoff).

The call option calculations follow:

• No-arbitrage approach:

Hedge ratio

$h=\frac{c^+-c^-}{S^+-S^-}=\frac{35-0}{135-75}=\frac{35}{60}=0.5833$

Call Option value

$c=hS+PV(-hS^-+c^-=0.5833\ast100+\frac{{(-75\ast0}{.5833)}}{1.02}+0=\$15.44$

• Expectations approach:

Probability of an up move π=0.45

Call Option value

$c=\frac{35\ast0.45+0}{1+0.2}=\$15.44$

C=hS0- PV(bond)

If we are buying synethic call, isn't it buy h shares of stock and short bond which is lend money.

I am having a hard time understanding the the + and - sign of the PV term.

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pzqa35 · 2023年10月16日

嗨，从没放弃的小努力你好：

The no-arbitrage model refers to use short calls and h underlying to construct a risk-free investment portfolio, hedge against the risk of future underlying price increases.

So the position at time 0 is hS₀-c₀, and at time 1 it is hS₁⁺-c₁⁺, or hS₁^--c₁^-. In a no- arbitrage model, we set hS₁⁺-c₁⁺= hS₁^--c₁^-, so hS₀-c₀=PV(hS₁⁺-c₁⁺)=PV(hS₁^--c₁^-), and the inverse solution is c₀= hS₀- PV(hS₁⁺-c₁⁺)= hS₀-PV(hS₁^--c₁^-)

For a long bond, the cash flow is make payment at the beginning of the period and receive cash flow income in the future, which is equivalent to a lend money, and then receive interest and principal. So the corresponding short bond is to receive cash flow at the beginning of the period, and to pay interest and principal at the middle and end of the period, which is equivalent to borrowing money.

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胖胖 · 2023年10月14日

I am confused as to why short bond = borrow money

if you are selling a bond - you are lending money.

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Don't understand the financing part of the equation

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