Delta Hedging a European Call on a Stochastic Short Rate

Summary

The document outlines the usual discrete-time Black–Scholes delta hedge for a European call on a stock. It sells the option, offsets its delta with a stock position, and places the residual value in a risk-free account. At each step, the stock price is simulated, the hedge value is updated, and the stock and cash positions are rebalanced. The proposed terminal measure is the hedge portfolio value minus the call payoff.

The author asks whether the same approach can hedge a call paying the positive part of the difference between a short rate at expiry and a strike. The central complication is that a short rate is not a tradable stock, and the hedge portfolio’s funding and discounting depend on the evolving rate environment. The document poses the question but provides no model, hedge instruments, worked calculation, or evidence. Applying the stock recipe directly would therefore be incomplete; a rate model and suitable traded instruments would be needed to define and evaluate a hedge.

Key ideas

• The standard example hedges a stock call by rebalancing stock delta and a cash account at discrete intervals.
• The proposed short-rate call pays the positive part of the expiry short rate minus its strike.
• A short rate is not automatically a tradable underlying that can be held in a delta hedge.
• Rate-dependent funding and discounting complicate direct use of the Black–Scholes stock hedge recipe.
• The document raises the modeling question but does not provide a hedge construction or results.

Tags

# Delta hedge call option on short rate


# Delta hedge call option on short rate












Usually delta hedging an european call option in the black-scholes model is constructed of three assets; a call option, the underlying stock and the risk-free asset often assumed to have constant interest rate. The algorithm usually runs as follows:

At time $t=0$: Sell the european call option that expires at time T, and has time-0 price V, and take a position of delta in the underlying stock, $\Delta S_0$. The difference will be deposit in the risk-free asset as $b = V - \Delta S_0$.

For every discrete time step, M, of size $\Delta t = \frac{T}{M}$ until the option expires, $t_1, \dots, T_M$ we proceed as follows: Simulate a step in the underlying stock price; $$S_{t_j} = S_{t_{j-1}} \cdot \exp\{(r- \frac{\sigma^2}{2}) \Delta t + \sigma W^Q_{\Delta t}\}.$$

Update the value of the hedge position; $$V_{t_j}^h = \Delta_{t_{j-1}} S_{t_j} + b_{t_{j-1}} \exp\{r \cdot \Delta t\}.$$

Update delta kept in the underlying and finance/invest the residual using the risk-free asset; $$b_{t_j} = V_{t_j}^h - \Delta_{t_j} S_{t_j}.$$

Finally one can evaluate the PnL as $V^h_{T} - (S_T - K)^+$. The difference should be somewhat similar to the one on the attached image.

My question is whether it is possible to conduct a similar delta hedge experiment on a european call option on the short rate instead? And how such would be done explained in similar steps as above? Meaning the underlying stock is then replaced by some short rate process, r(t), where the payoff is; $$(r(T) - K)^+.$$ My uncertainty occurs as there is risk from the underlying, but also the discounting.

Thanks

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.