One-Period Binomial Hedging of a Long Call Position
Summary
This worked example asks how a bank that owns a one-period European call can invest in stock and cash so its total position earns the stated risk-free rate across both possible stock outcomes. The call costs 1.20, pays in the up state, and has no payoff in the down state. The key accounting point is that the option is already owned: its purchase price should not be subtracted again from the cash available for the hedge. Solving for a stock position and money-market investment gives a short position of one-half share, with the proceeds invested as cash. At expiry, the stock and cash components combine with the option payoff to produce the target amount in either state.
The example illustrates one-period replication and the importance of defining portfolio wealth consistently when an asset is already held. It is a small, frictionless binomial exercise rather than a general hedging recipe; it assumes the specified prices and rate and omits transaction costs, financing constraints, and rebalancing before expiry.
Key ideas
- A hedge should account for the call as an existing asset rather than subtracting its purchase price again from cash.
- The example solves for a short position of one-half share and invests the short-sale proceeds in the money market.
- The stock, cash, and option payoff together reach the target in either terminal state.
- The calculation assumes a single period and ignores trading frictions.
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# Hedging a long position-one period from Steven Shreve Stochastic Calculus for Finance
# Hedging a long position-one period from Steven Shreve Stochastic Calculus for Finance
The following question is taken from Steven Shreve Volume 1, Chapter 1, Exercise $1.6$ (Hedging a long position-one period)
> Consider a one period binomial stock model with $S_0=4$, $S_1(H)=8$ and $S_1(T)=2$. The interest rate is $25$%. Consider a bank that has a long position in the European call written on the stock price. The call expires at time one and has strike price $K=5$. It is determined that the time-zero price of this call to be $V_0=1.20$. At time zero, the bank owns this option, which ties up capital $V_0=1.20.$ The bank wants to earn the interest rate $25\%$ on this capital until time one (i.e. without investing any more money, and regardless of how the coin tossing turns outs, the bank wants to have $$\frac{5}{4}\cdot 1.20 = 1.50$$ at time one, after collecting the payoff from the option (if any) at time one). Specify how the bank's trader should invest in the stock and money markets to accomplish this.
My attempt:
Let $X_0, X_1$ be portfolio values at time zero and one respectively. From the question, we have $$X_1(T)= X_1(H) = 1.50.$$ From the wealth equation, we have $$X_1(H) = \Delta_0S_1(H) + 3 +(1+r)(X_0-\Delta_0S_0 - 1.20),$$ $$X_1(T) = \Delta_0S_1(T) + 0+ (1+r)(X_0-\Delta_0S_0- 1.20).$$ Substituting appropriate values, we have $$1.50 = 8\Delta_0 + 3 +\frac{5}{4}(X_0-4\Delta_0- 1.20),$$ $$1.50 = 2\Delta_0 + 0 +\frac{5}{4}(X_0-4\Delta_0- 1.20).$$ Solving the 2 equations above for $\Delta_0$ and $X_0$ leads to $$\Delta_0 = -\frac{1}{2}, X_0= 1.20.$$ I do not know how to proceed from here. Is my attempt above correct?
## Answer by XIIIX (score 1)
https://quant.stackexchange.com/a/57788
In both of your wealth equations, there is no need to subtract 1.2 from your investment in money market because you already own the option.
Solve it you will get $\Delta=-\frac{1}{2}$ and $X_0=0$. Now, it means you need to short sell $-\frac{1}{2}$ shares, get $2\\\$$, and put $2\\\$$ into your investment account to guarantee that at $t=1$, you will get $1.5 \\\$ $ (you only get $1.5 \\\$ $ from that account because part of it is needed to cancel with the option payoff).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.